	New content
Statistical Methods in Medical Research

Open access

Research article

First published online August 17, 2015

A review of instrumental variable estimators for Mendelian randomization

Stephen Burgess [email protected], Dylan S Small, and Simon G ThompsonView all authors and affiliations

https://doi.org/10.1177/0962280215597579

Abstract

Instrumental variable analysis is an approach for obtaining causal inferences on the effect of an exposure (risk factor) on an outcome from observational data. It has gained in popularity over the past decade with the use of genetic variants as instrumental variables, known as Mendelian randomization. An instrumental variable is associated with the exposure, but not associated with any confounder of the exposure–outcome association, nor is there any causal pathway from the instrumental variable to the outcome other than via the exposure. Under the assumption that a single instrumental variable or a set of instrumental variables for the exposure is available, the causal effect of the exposure on the outcome can be estimated. There are several methods available for instrumental variable estimation; we consider the ratio method, two-stage methods, likelihood-based methods, and semi-parametric methods. Techniques for obtaining statistical inferences and confidence intervals are presented. The statistical properties of estimates from these methods are compared, and practical advice is given about choosing a suitable analysis method. In particular, bias and coverage properties of estimators are considered, especially with weak instruments. Settings particularly relevant to Mendelian randomization are prioritized in the paper, notably the scenario of a continuous exposure and a continuous or binary outcome.

1 Introduction

Instrumental variable analysis is the exploitation of a natural experiment to obtain causal inferences in an observational setting. An instrumental variable is a factor that is correlated with the exposure (usually a putative causal risk factor), but is not associated with any confounder of the exposure–outcome association, nor is there any pathway by which the instrumental variable can influence the outcome other than via the exposure of interest.^1,2 Usually, an instrumental variable is in some way external to the relationship between the exposure and outcome and formally must be exogenous to the statistical model relating the exposure to the outcome (that is, not associated with the error term in the equation).³ A natural experiment is analogous to a randomized trial, except that the value of the exposure is influenced by nature rather than by an investigator on the basis of randomized allocation.⁴

A source of potential instrumental variables is genetic variants.⁵ A genetic variant is a section of genetic code that differs between individuals. Genetic variants are good candidate instrumental variables: the function of many genes is known and well characterized; genetic variants are fixed at conception, and so do not change due to environmental factors, thus avoiding reverse causation; and genetic variants are generally inherited independently, meaning that they tend to be specific in their associations. The use of genetic variants as instrumental variables in observational data has been termed ‘Mendelian randomization’.^6,7

Many reviews exist on the use of instrumental variables, in particular on the assumptions necessary to be an instrumental variable,^8,9 assessing the validity of instrumental variables,¹⁰ assumptions necessary for estimation of a causal effect parameter using instrumental variables,^11–13 methods for effect estimation with multiple instrumental variables,¹⁴ with binary outcomes,^15,16 methods for the estimation of odds ratios,¹⁷ and guidelines for the reporting of instrumental variable analysis.¹⁸ We seek to complement this literature by contributing a review to compare methods for instrumental variable analysis, with accompanying practical guidelines on their use. This review will focus on methodological issues that are particularly relevant to Mendelian randomization: the scenario of a continuous exposure and a continuous or binary outcome; statistical inference and the construction of confidence intervals; and the use of multiple instruments, in particular multiple weak instruments. A ‘weak instrument’ is an instrumental variable that explains a relatively small proportion of variance in the exposure.¹⁹ Although these are issues particularly germane to Mendelian randomization, they are also relevant to instrumental variable analyses in other contexts.

After framing the typical context of a Mendelian randomization investigation that we assume to hold (Section 2), we proceed to consider in turn the ratio of coefficients method (Section 3), two-stage methods (Section 4), likelihood-based methods (Section 5), and semi-parametric methods (Section 6). This order corresponds roughly to the complexity of the methods, with the simplest ones first. This is not an exhaustive list of methods that have been proposed for instrumental variable analysis, but it covers a wide range of commonly used methods. The methods are contrasted in terms of bias, coverage, efficiency, robustness to misspecification, and the existence of finite moments. We outline the problem of weak instruments and present a comparison of estimates from methods with weak instruments (Section 7). We also introduce a score-based (or ‘allele score’) approach to instrumental variable estimation, conceived for use with multiple weak instruments. We conclude by discussing methodological issues relating to the efficiency and validity of instrumental variable analyses, and to the use of instrumental variable methods in practice (Section 8).

2 Context of the problem

We assume that individual-level data are available from a single data source on an exposure X, an outcome Y, and an instrumental variable (IV) Z, or set of IVs

Z_{1}

Z_{2}

, …,

Z_{K}

. Usually the outcome is disease, although there is no methodological restriction as to what outcomes can be considered. The exposure is a putative causal risk factor for the outcome and is assumed to be continuous. A binary exposure is common for IV estimation in the context of a randomized trial to estimate the causal effect of treatment, where the IV is randomization.²⁰ In this case, randomization typically has a large effect on the treatment received for a subgroup of individuals known as the compliers.²¹ However, in Mendelian randomization, the exposure is almost always continuous, and genetic variants typically have small effects on the exposure that are thought to affect all individuals in the population. Although it is possible to consider a dichotomized risk factor as an exposure (such as presence of hypertension in the case of blood pressure), a valid IV for the continuous exposure is not generally a valid IV for the dichotomized exposure, as there is an alternative causal pathway from the IV to the outcome via the true value of the continuous exposure.¹⁰

2.1 Genetic variants used in Mendelian randomization

We here provide a simplistic introduction to genetics with the aim of explaining the format of genetic data used in applied Mendelian randomization investigations; more detailed information on genetics for Mendelian randomization can be found in other papers.^5,6,22

Genetic variants used as IVs in Mendelian randomization are usually single-nucleotide polymorphisms (SNPs). A SNP is a variation at a single position of the DNA sequence in some individuals in a population, where the DNA sequence has one letter replaced with another letter. The different possible letters that may appear at this position are known as alleles. Most SNPs are diallelic, having two alleles: one that is more common, known as the wild-type or major allele, and the other known as the variant or minor allele. For each individual, a SNP can be represented as a random variable by the number of variant alleles; this can take the values 0, 1, or 2, because each individual has two copies of the DNA sequence (two chromosomes). In some cases, the SNP may be regarded as a dichotomous variable, taking values 0 and 1: either for statistical reasons, if the variant allele is rare so that only a very small number of individuals in the sample have two copies, or for biological reasons, if individuals with a single copy of the variant allele have similar average values of the exposure as those with zero copies (recessive model) or as those with two copies (dominant model). Alternatively, the IV may be a copy number variant (a repeated section of the DNA code where the number of copies of the section varies between individuals),²³ or an allele score (Section 7.3); in these cases, the IV would be a discrete variable taking positive values or a continuous variable.

2.2 Causal estimand

It has been established that the core IV assumptions only identify bounds for the causal effect rather than a single causal estimate.^24,25 However, these bounds are extremely imprecise and even in large samples are rarely informative. We do not recommend their use in practical Mendelian randomization investigations. We here assume that the IV assumptions are satisfied; otherwise, even these bounds can be biased and highly misleading, particularly if the IVs are weak.²⁶

With a binary exposure and a single IV, under the assumption of monotonicity in the effect of the IV on the exposure, the causal estimand from an IV analysis is known as a local average treatment effect²⁷ or (in particular in the case of a binary IV) a complier-averaged causal effect.²⁸ It represents the average change in the outcome for a change in the exposure amongst those individuals whose value of the exposure is affected by their value of the IV.¹¹ Monotonicity means that the exposure for each individual would be increased (or alternatively for each individual would be decreased) or unchanged if that person had

Z = z_{1}

compared to if they had

Z = z_{0}

for all

z_{1} > z_{0}

. In the language of principal stratification for randomized trials, this is equivalent to the assumption that there are no defiers.²¹ For genetic variants that have proven biological links with the risk factor, the monotonicity assumption is plausible, as such biological effects seem likely to act in the same direction across all individuals. However, the monotonicity assumption is unlikely to hold if the IV is an allele score (Section 7.3), particularly if it is an unweighted or imprecisely weighted score. In any case, the monotonicity assumption is not testable from empirical data. If the monotonicity assumption is not satisfied, then the causal effect of the exposure on the outcome can be estimated consistently if it is constant for all individuals across the population.²⁹ A weaker version of this assumption, that of no modification of the causal effect across levels of the instrument within the treated, can be used to estimate the effect of treatment in the treated.³⁰

With a continuous exposure, to identify a causal parameter, it is necessary to make some parametric assumptions, for example homogeneity and linearity in the causal exposure–outcome association with no effect modification.^9,12 The causal estimand is then the average change in the outcome for a unit change in the exposure. If there are multiple IVs, it is convenient to assume that the causal effect of the exposure on the outcome is homogeneous, so that all the IVs identify the same causal quantity. However, this assumption is questionable, particularly if the genetic variants used as IVs affect different biological pathways and so affect the value of the exposure in different ways. This assumption is best addressed by a sensitivity analysis, comparing IV estimates based on different subsets of IVs, particularly if the subsets correspond to groups of genetic variants with distinct biological functions, and can be tested formally by an overidentification test (Section 8.3).

With a binary outcome, it is common in conventional observational studies to express the association of interest as a relative risk or an odds ratio, rather than as an average change in the outcome (which would represent an effect on the risk difference scale). This is problematic in the case of an odds ratio, as the odds ratio is a non-collapsibility measure of association, meaning that its value depends on the distribution of risk factors in the population that affect the outcome even if these variables are not confounders of the relationship between the exposure and outcome.³¹ It is therefore recommended that odds ratios should only be estimated in a case–control setting, as other measures of association are sensitive to case–control sampling. We return to the problem of defining and estimating a causal odds ratio in the context of the individual IV methods.

In this paper, we presume that a causal effect estimate is the desired outcome of a Mendelian randomization analysis. In many cases, testing for an association between an IV and the outcome, to assess the causal role of the exposure and the direction of the causal effect without estimating a causal parameter, is sufficient.³² Such a test can be performed by a variety of methods, but most commonly using linear regression with a continuous outcome and using linear or logistic regression with a binary outcome. Although a causal estimate is likely to differ from the effect of a clinical intervention on the same exposure in practice,⁷ an effect estimate may be desired, for instance, to give a quantitative indication of the importance of the exposure in relation to the outcome. We proceed to consider various methods for IV estimation in the context described above; however, deviations in many of these features (such as the availability of individual-level data, linearity of the causal effect, homogeneity of the IV estimate from different IVs) are addressed throughout the paper and in particular in the discussion.

3 Ratio of coefficients method

The ratio of coefficients method, or the Wald method,³³ is the simplest way of estimating the causal effect of the exposure (X) on the outcome (Y). The ratio method uses a single IV (Z); all other methods in this paper are able to include data on multiple IVs.

3.1 Continuous outcome

If the coefficient of the IV in the regression of the exposure on the IV is written as

{\hat{β}}_{X | Z}

and the coefficient of the IV in the regression of the outcome on the IV is written as

{\hat{β}}_{Y | Z}

, then the ratio estimate of the causal effect is

Ratiomethodestimate = \frac{{\hat{β}}_{Y | Z}}{{\hat{β}}_{X | Z}}

(1)

Intuitively, we can think of the ratio method as saying that the change in the outcome for a unit increase in the exposure is equal to the change in the exposure for a unit increase in the IV, scaled by the change in the exposure for a unit increase in the IV. The causal estimate is usually expressed as the change in the outcome resulting from a unit change in the exposure, although changes in the outcome corresponding to different magnitudes of change in the exposure could be quoted instead.

If the IV is dichotomous, taking values 0 and 1, then the ratio estimate simplifies to

Ratiomethodestimate (dichotomousIV) = \frac{{\bar{y}}_{1} - {\bar{y}}_{0}}{{\bar{x}}_{1} - {\bar{x}}_{0}}

(2)

where

{\bar{y}}_{1}

is the average value of the outcome amongst those with an IV value of 1, and so on. If the effect of the exposure on the outcome is assumed to be linear, then the ratio estimate is the causal effect on the outcome of an exposure of x+1 units versus an exposure of x units. If the effect is not linear, then theoretical and empirical considerations have demonstrated that the ratio estimate approximates an average causal effect of a population intervention in the exposure, for the population under analysis, with the assumption that the effect of the IV on the exposure is constant in the population³⁴ (Section 8.1).

We note that the ratio estimate can be calculated simply from the coefficients

{\hat{β}}_{X | Z}

and

{\hat{β}}_{Y | Z}

and as such only requires the availability of summarized data, not individual-level data, from the population. We return to consider IV estimation using summarized data in more detail in Section 8.6.

3.2 Binary outcome

Generally in epidemiological applications, disease is the outcome of interest. Disease outcomes are often dichotomous. We use the epidemiological terminology of referring to an individual with an outcome event as a case (Y=1), and an individual with no event as a control (Y=0).

With a binary outcome, the coefficient

{\hat{β}}_{Y | Z}

in the ratio estimate (equation (1)) is again taken from regression of the outcome on the IV. The regression model used could in principle be linear, so that the IV estimate represents the change in the probability of an event for a unit change in the exposure. However, with a dichotomous outcome, log-linear or logistic regression models are generally preferred by epidemiologists, where the IV estimate represents the log relative risk or log odds ratio, respectively, for a unit change in the exposure. In a case–control setting, measures based on risk differences depend on the choice of the ratio of cases to controls, whereas odds ratios are invariant to outcome-dependent sampling. More generally, relative risks and odds ratios (which approximate relative risks under the rare disease assumption³⁵) are based on multiplicative risk models, which may be more plausible than additive risk models, and are easier to communicate.³⁶

With a dichotomous IV, the ratio estimate simplifies as with a continuous outcome

Ratiomethodlogrelativerisk (orlogoddsratio) estimate (dichotomousIV) = \frac{{\bar{y}}_{1} - {\bar{y}}_{0}}{{\bar{x}}_{1} - {\bar{x}}_{0}}

(3)

where

{\bar{y}}_{j}

is the log of the probability of an event, or the log odds of an event, in IV subgroup j=0,1.

If the structural model relating the log of the probability of the outcome to the exposure is linear in the exposure with no effect modification, then the ratio estimate calculated using log-linear regression will be consistent for the causal log relative risk.¹² The analogous consistency property does not hold in the case of a logistic-linear relationship between the probability of the outcome and the exposure, as the odds ratio is not a collapsible measure of association (see Section 4.2).

3.3 Case–control data

When case–control data are available, it is usual to make inferences on the IV-exposure association using only the controls.³⁷ This makes the assumption that the association of the IV with the exposure in the controls is similar to that in the general population, which is true for a rare disease.³⁸ This is necessary for two reasons. The first is reverse causation, as measurements of the exposure in a retrospective setting may be distorted by a disease event. Second, over-recruitment of cases into the study means that the distribution of confounders in the case–control sample is different to that in the general population. An association may be induced between the IV and the confounders, leading to possible bias in the IV estimate.⁹ This affects not only the ratio method but all IV methods. The IV–outcome association estimate should not be affected provided that logistic regression is used, as logistic regression estimates are invariant to outcome-dependent sampling. It is also necessary to assume that the probability of recruitment into the study is not dependent on the genetic variants used as IVs. For example, a genetic variant associated with increased mortality after an outcome event would appear to be protective rather than deleterious if the individuals who died were excluded from the study.

If the outcome is common and its prevalence in the population from which the case–control sample was taken is known, such as in a nested case–control study, then inferences on the IV–exposure association can be obtained using both cases and controls, provided that measurements of the exposure in cases were taken prior to the outcome event. This analysis can be performed by weighting the sample so that the proportions of cases and controls in the reweighted sample match those in the underlying population.³⁸

3.4 Confidence intervals

Confidence intervals for the ratio estimate can be calculated in several ways.

3.4.1 Normal approximation

The simplest way is to use a normal approximation. With a continuous outcome, standard errors and confidence intervals can be taken from the two-stage least squares method (see Section 4). Alternatively, the following approximation can be used, based on the first terms of the delta method expansion for the variance of a ratio

Asymptotic standard error of ratio estimate = \sqrt{\frac{se ({\hat{β}}_{Y | Z}) 2}{{\hat{β}}_{X | Z} 2} + \frac{{\hat{β}}_{Y | Z} 2 se ({\hat{β}}_{X | Z}) 2}{{\hat{β}}_{X | Z} 4} - \frac{2 {\hat{β}}_{Y | Z} cov ({\hat{β}}_{X | Z}, {\hat{β}}_{Y | Z})}{{\hat{β}}_{X | Z} 3}}

(4)

If the numerator and denominator of the ratio estimator are uncorrelated, then the third term can be omitted; in any event, such correlation is unlikely to have a considerable impact on confidence intervals.³⁹

However, asymptotic normal approximations for the IV estimate may result in overly narrow confidence intervals, especially if the sample size is not large or the IV is weak (Section 7). This is because IV estimates are not normally distributed.¹⁹

3.4.2 Fieller’s theorem

Alternatively, if the regression coefficients in the ratio method

{\hat{β}}_{Y | Z}

and

{\hat{β}}_{X | Z}

are assumed to be normally distributed, critical values and confidence intervals for the ratio estimator may be calculated using Fieller’s theorem.^5,40 Fieller’s theorem gives confidence intervals that have one of three possible forms⁴¹:

The interval may be a closed interval [a,b],

ii.

The interval may be the complement of a closed interval (−∞,b]∪[a,∞), and

iii.

The interval may be unbounded.

where a and b are functions of the association estimates

{\hat{β}}_{Y | Z}

and

{\hat{β}}_{X | Z}

and their standard errors. Confidence intervals from Fieller’s theorem may be preferred to those from an asymptotic normal approximation when the IV is weak.⁴² A tool to calculate confidence intervals from Fieller’s theorem is available online (https://sb452.shinyapps.io/fieller).

3.4.3 Bootstrapping

As an alternative approach, also applicable to any of the following methods, confidence intervals can be calculated by bootstrapping.⁴³ The simplest way of constructing a bootstrapped confidence interval is by taking several random samples with replacement from the data of the same sample size. The empirical distribution of the IV estimator in the bootstrapped samples approximates the true distribution of the IV estimator.⁴⁴ However, there are some concerns about the behaviour of bootstrapped confidence intervals with weak instruments.⁴⁵

3.4.4 Other approaches

Alternative approaches for inference with weak instruments not discussed further here are confidence intervals based on inverting a test statistic, such as the Anderson–Rubin test statistic⁴⁶ or the conditional likelihood ratio test statistic.⁴⁷ A variance estimator for a generalized empirical likelihood approach with good coverage properties has also been developed.⁴⁸ These intervals give appropriate confidence levels under the null hypothesis with weak instruments but may be underpowered with stronger instruments. However, unlike Fieller’s theorem, Anderson–Rubin and conditional likelihood ratio test confidence intervals can be calculated based on multiple IVs. These methods have been discussed in detail elsewhere^49,50 and implemented in Stata⁵¹ and R.⁵² We recommend the Fieller’s theorem, Anderson–Rubin, or conditional likelihood ratio approaches for obtaining causal inferences and confidence intervals in practice with a weak instrument.

3.5 Absence of finite moments

One peculiar property of the ratio estimator is that its mean is not finite.^53,54 This implies that, if you generated data on the exposure and outcome from a model with a valid IV and calculated the ratio estimate a large number of times, the mean value of these ratio estimates would become arbitrarily high (or low). This is due to the fact that there is a finite probability that the denominator in the ratio estimate (

{\hat{β}}_{X | Z}

) is very close to zero, leading to a large IV estimate. In practice, this is unlikely to be a serious issue since, if

{\hat{β}}_{X | Z}

were close to zero, the IV would be considered invalid as the assumption that the IV is associated with the exposure would appear to be violated. Theoretically, the absence of a finite mean makes comparison of IV methods more difficult, as the (mean) bias of the ratio estimate, defined as the difference between the mean of the IV estimator over its distribution and the true value of the causal effect, cannot be calculated. We therefore additionally consider the median bias, the difference between the median of the IV estimator and the true causal effect, when comparing different methods for IV estimation.¹⁹

4 Two-stage methods

A two-stage method comprises two regression stages: the first-stage regression of the exposure on the IVs, and the second-stage regression of the outcome on the fitted values of the exposure from the first stage. Both two-stage and likelihood-based methods (Section 5) make parametric assumptions for the identification of causal effects; this is contrasted with the weaker semi-parametric associations of the methods introduced in Section 6.

4.1 Continuous outcome – Two-stage least squares

With continuous outcomes and a linear model, the two-stage method is known as two-stage least squares (2SLS). It can be used with multiple IVs. In the first-stage regression, the exposure (X) is regressed on the IV(s) (Z) to give fitted values of the exposure (

\hat{X} | Z

). In the second-stage regression, the outcome (Y) is regressed on the fitted values for the exposure (

\hat{X} | Z

) from the first-stage regression. The causal estimate is this second-stage regression coefficient for the change in the outcome caused by a unit change in the exposure.

With a single IV, the 2SLS estimate is the same as the ratio estimate. With multiple IVs, the 2SLS estimator may be viewed as a weighted average of the ratio estimates calculated using the instruments one at the time, where the weights are determined by the relative strengths of the instruments in the first-stage regression.⁵⁵

Suppose we have K instrumental variables available. With data on individuals indexed by i = 1, … , N who have exposure

x_{i}

, outcome

y_{i}

and assuming an additive linear model for the IVs

z_{ik}

indexed by k = 1, … , K, the first-stage regression model is

x_{i} = α_{0} + \sum_{k} α_{k} z_{ik} + ɛ_{Xi}

(5)

The fitted values

{\hat{x}}_{i} = {\hat{α}}_{0} + \sum_{k} {\hat{α}}_{k} z_{ik}

are then used in the second-stage regression model

y_{i} = β_{0} + β_{1} {\hat{x}}_{i} + ɛ_{Yi}

(6)

where

ɛ_{Xi}

and

ɛ_{Yi}

are independent error terms. The causal parameter of interest is

β_{1}

. If both models are estimated by standard least-squares regression, both the error terms are implicitly assumed to be homoscedastic and normally distributed.

Although estimation of the causal effect in two stages (a sequential regression method) gives the correct point estimate, the standard error from the second-stage regression (equation (6)) is not correct.^{56(p. 188)} This is because it does not take into account the uncertainty in the first-stage regression. Under homoscedasticity of the error term in the equation

y_{i} = β_{0} + β_{1} x_{i} + ɛ'_{Yi}

(7)

the asymptotic variance of the 2SLS estimator is

\hat{σ} 2 (X T Z (Z T Z) - 1 Z T X) - 1 = \hat{σ} 2 (\hat{X} T \hat{X}) - 1

(8)

where

\hat{σ} 2

is an estimate of the variance of the residuals from equation (7), and Z is a N by K + 1 matrix of the IVs (including a constant term) and X is a N by 2 matrix of the risk factor and a constant term. Robust standard errors are often used in practice, as estimates are sensitive to heteroscedasticity and misspecification of the equations in the second-stage model. However, in the continuous outcome case, consistent estimates are obtained even when the first-stage model is misspecified.⁵⁷

When all the associations are linear and the error terms normally distributed, the 2SLS estimator has a finite kth moment when there are at least (k+1) IVs.⁵³ Therefore, the mean of a 2SLS estimator is only defined when there are at least 2 IVs, and the variance is only defined when there are at least 3 IVs. Unlike in an observational analysis, in which estimates from conventional regression (ordinary least squares) are attenuated by measurement error in the exposure,⁵⁸ the bias of effect estimates from the 2SLS method (and consequently from the ratio method amongst others) is not affected by classical (non-differential) measurement error in the exposure.⁵⁹

4.2 Binary outcome

The analogue of 2SLS with binary outcomes is a two-stage estimator where the second-stage (exposure–outcome) regression uses a log-linear or logistic regression model. These methods are implemented in order to estimate a causal relative risk or odds ratio parameter. They can be implemented using a sequential regression method by performing the two regression stages in turn. Estimates from such an approach will be overly precise, as uncertainty in the first-stage regression is not accounted for; however, this over-precision may be slight if the standard error in the first-stage coefficients is low. This can be resolved by the use of a likelihood-based method (Section 5.3) or a bootstrap method.⁶⁰

As with the ratio IV estimator, in a case–control study, it is important to undertake the first-stage regression only in the controls, not the cases (Section 3.3). Fitted exposure values for the cases are obtained by substituting their IV values into the first-stage regression model.

Two-stage regression methods with non-linear second-stage regression models (such as logistic regression) have been criticized and called ‘forbidden regressions’.^{56(p. 190)} This is because the non-linear model does not guarantee that the residuals from the second-stage regression are uncorrelated with the instruments.⁶¹ There is current debate about the interpretation and validity of such estimates, especially when the measure of association is non-collapsible.

A measure of association, such as an odds ratio or relative risk, is collapsible if, when it is constant across the strata of the covariate, this constant value equals the value obtained from the overall (marginal) analysis. Non-collapsibility is the violation of this property.⁶² The relative risk and absolute risk difference are collapsible measures of association. Odds ratios are generally non-collapsible.⁶³ An odds ratio estimated in an observational study by conventional multivariable logistic regression is conditional on those covariates adjusted for in the analysis. Unless adjustment is made (Section 8.2), an odds ratio estimated in an IV analysis is marginal on these covariates. The odds ratio from a ratio or two-stage analysis method is conditional on the IV, but marginal in all other variables, including the exposure itself if it is continuous.⁶⁴ This approximates a population-averaged odds ratio from a randomized trial, where the intervention is to increase the exposure by one unit in all individuals across the population.⁴² A population-averaged odds ratio can be calculated as the average odds in the population for one value of the exposure distribution divided by the average odds for another value of the exposure distribution – for instance, the distribution of the exposure in the population increased uniformly by one unit, and the original distribution of the exposure in the population.

Despite the consequences of non-collapsibility, the two-stage estimator with a logistic second-stage model still provides a valid test of the null hypothesis.¹⁷

4.3 Adjusted two-stage method

An adjusted two-stage method has been proposed, where the residuals from the first-stage regression are included in the second-stage regression. This has been referred to as a control function approach,⁶⁵ or two-stage residual inclusion method.⁶⁶ If we have a first-stage regression of X on Z with fitted values

\hat{X} | Z

and residuals

\hat{R} | Z = X - \hat{X} | Z

, then the adjusted two-stage estimate comes from a second-stage regression additively on

\hat{X} | Z

and

\hat{R} | Z

(or equivalently on X and

\hat{R} | Z

). The residuals from the first-stage regression incorporate information on confounders.

If the outcome is continuous, then inclusion of these residuals in a second-stage linear regression model does not change the estimate, as the residuals are orthogonal to the fitted values. If the outcome is binary, inclusion of these residuals in a second-stage logistic regression model implies that the IV estimate is conditional on these residuals. Numerically, it brings the IV estimate closer to the subject-specific log odds ratio; the parameter

β_{1}

in a logistic-linear model⁶⁷

logitP (Y = 1 | X = x, U = u) = β_{0} + β_{1} x + β_{2} u

(9)

where U represents all other covariates (whether confounders or not) in the probability model. Some investigators have therefore recommended the adjusted two-stage method when the second-stage regression is logistic on the premise that it is less biased than the unadjusted two-stage method.⁶⁶

Under a particular choice of mathematical model in which the first-stage residual is equal (up to a multiplicative constant) to the term

β_{2} u

in equation (9), the adjusted two-stage estimate is consistent for the parameter

β_{1}

.⁶⁶ However, this mathematical model is unrealistic, and in general the adjusted two-stage estimate is biased for this parameter.^15,68 Further, when the covariates are unknown, as is usual in an IV analysis, it is not clear what variable is represented by the first-stage residuals, and so which covariates the adjusted two-stage estimate is conditional on and which it is marginal across. It is uncertain what odds ratio is being estimated by an adjusted two-stage approach, that is, to what question is the adjusted two-stage estimate the answer. We therefore do not recommend adjustment for the first-stage residuals in a two-stage method.

5 Likelihood-based methods

The above methods are not likelihood based and do not provide maximum likelihood estimates, which have the desirable properties of asymptotic unbiasedness, normality, and efficiency. So we next consider likelihood-based methods.

5.1 Limited information maximum likelihood

If each individual i = 1, … , N has exposure

x_{i}

, outcome

y_{i}

, and IVs

z_{ik}

indexed by k = 1, … , K, we can assume the following model

x_{i} = α_{0} + \sum_{k} α_{k} z_{ik} + ɛ_{Xi}

(10)

y_{i} = β_{0} + β_{1} x_{i} + ɛ_{Yi}

where the two error terms have a bivariate normal distribution.⁴⁶ (These error terms differ from those defined in equations (5) and (6).) The causal parameter of interest is

β_{1}

. Correlation between

ɛ_{X}

and

ɛ_{Y}

is due to confounding. We can calculate the maximum likelihood estimates of

β_{1}

and each of the other parameters in the model. This is known as limited information maximum likelihood (LIML; see Section 18.5 in Davidson and MacKinnon⁶⁹). Confidence intervals can be obtained by the assumption of asymptotic normality of the parameter estimates.

The term ‘limited information’ means that only limited information on the structure of the model is used. In particular, if there were multiple outcomes or risk factors, only parameters from a single equation are estimated. This is in contrast to full information maximum likelihood or three-stage least squares (3SLS),⁷⁰ in which the whole system of equations in the model is estimated simultaneously. Full-system methods have the potential advantage of efficiency, but the disadvantage of inconsistent estimation if one or more of these equations are misspecified. However, in the context of Mendelian randomization with a single outcome and a single risk factor, there is no distinction.

LIML has been called the ‘maximum likelihood counterpart of 2SLS’^{71(p. 227)} and gives the same causal estimate as the 2SLS and ratio methods with a single IV. As with 2SLS, estimates are sensitive to heteroscedasticity and misspecification of the equations in the model. Use of LIML has been strongly discouraged by some, as LIML estimates do not have defined moments for any number of instruments.⁵⁴ However, use has also been encouraged by others, especially with weak instruments, as the median of the distribution of the LIML estimator is close to unbiased even with weak instruments.^{56(p. 209)}

The LIML estimate can be intuitively understood as the effect

β_{1}

that minimizes the residual sum of squares from the regression of the component of the outcome not caused by the exposure,

(y_{i} - β_{1} x_{i})

, on the IV(s). Informally, the LIML estimator is the causal parameter for which the component of the outcome due to confounding is as badly predicted by the IVs as possible.

5.2 Bayesian methods

Inferences from a likelihood model can also be obtained in a Bayesian framework.⁷² Here, we consider a slightly different likelihood model from equation (10) that has been discussed previously in the literature.⁴² For each individual i, we model the observed exposure and outcome as coming from a bivariate normal distribution for

(X_{i}, Y_{i}) T

with mean

(ξ_{i}, η_{i}) T

and variance-covariance matrix ∑. The mean of the exposure distribution

ξ_{i}

is assumed to be a linear function of the instruments

z_{ik}, k = 1, \dots, K

, and the mean of the outcome distribution

η_{i}

is assumed to be a linear function of the mean exposure

ξ_{i}

⁷³

(X_{i} Y_{i}) \sim N_{2} ((ξ_{i} η_{i}), Σ) ξ_{i} = α_{0} + \sum_{k} α_{k} z_{ik} η_{i} = β_{0} + β_{1} ξ_{i}

(11)

This model is similar to that for the LIML method, except that the causal parameter

β_{1}

represents the causal effect between the true means

ξ_{i}

and

η_{i}

rather than the measured values of outcome and exposure. The model can be estimated in a Markov chain Monte Carlo framework, such as that implemented in WinBUGS.⁷⁴ The output is a posterior distribution, from which the posterior mean or median can be interpreted as a point estimate, and the 2.5th and 97.5th percentiles as a ‘95% confidence interval’. With vague priors, the posterior distribution is similar to the frequentist likelihood function.⁴²

An advantage of the Bayesian approach is that, under the modelling assumptions, valid inferences can be obtained that are not based on asymptotic results, but rather on the posterior distribution of the causal parameter. Simulations have shown that frequentist properties of inference (such as coverage levels) are more robust in the Bayesian approach with weak instruments.⁴²

5.3 Likelihood-based methods with binary outcomes

Maximum likelihood and Bayesian methods can be applied to binary outcomes. If we assume a linear model of association between the logit-transformed probability of an event (

π_{i}

) and the exposure (a logistic-linear model), and a Bernoulli distribution for the outcome event, as in the following model

x_{i} \sim N (ξ_{i}, σ_{X}^{2}) y_{i} \sim Bernoulli (π_{i}) ξ_{i} = α_{0} + \sum_{k = 1}^{K} α_{k} z_{ik} logit (π_{i}) = β_{0} + β_{1} ξ_{i} + β_{2} (x_{i} - ξ_{i})

(12)

Estimates can be obtained by maximization of the joint likelihood, or in a Bayesian framework, obtaining posterior distributions by Markov chain Monte Carlo methods. Inclusion of the coefficient

β_{2}

is similar to a control function approach (Section 4.3), in which the first-stage residual is adjusted for in the second-stage regression of a two-stage method. Alternatively, this term can be omitted, although formally this results in the likelihood model being misspecified.⁴² Simulations have suggested that both approaches give reasonable estimates, although they are affected by non-collapsibility.

Log-linear models can in principle be estimated in the same way, although care is needed to ensure that the probabilities

π_{i}

do not exceed 1 at any point in the estimation procedure.

5.4 Comparison of two-stage and likelihood-based methods

In the two-stage methods, the two stages are performed sequentially. The output from the first-stage regression is fed into the second-stage regression with no acknowledgement of uncertainty. In the likelihood-based methods, the two stages are performed simultaneously: the α and β parameters are estimated at the same time. Uncertainty in the first-stage parameters is acknowledged and feedback between the regression stages is possible. The uncertainty in the estimate of the causal parameter β₁ is therefore better represented in the likelihood-based approaches if there is non-negligible uncertainty in the first-stage regression model. This provides a heuristic justification why likelihood-based methods perform better than two-stage methods with weak instruments (see Section 7).

5.5 k-class estimators

The LIML and 2SLS estimators are part of a wider class of estimators known as k-class estimators.⁷⁵ The k-class estimator of the causal effect β₁ from equation (10) is

{\hat{β}}_{k} = [Y T (I - k P_{Z}) Y] - 1 [Y T (I - k P_{Z}) X]

(13)

where Y is a N by 1 matrix of the outcome, Z is a N by K+1 matrix of the IVs (including a constant term), and X is a N by 2 matrix of the risk factor and a constant term. The matrix

P_{Z} = I - Z (Z T Z) - 1 Z T

is the projection matrix of Z.⁷⁶ If k=1, then

{\hat{β}}_{k}

is the 2SLS estimator. If k is the smallest root of

det (Y T Y - k Y T P_{z} Y) = 0

, then

{\hat{β}}_{k}

is the LIML estimator. Several other choices are available, including the bias-corrected 2SLS estimator,⁵⁴ and Fuller’s k estimator.⁷⁷ These estimators are beyond the scope of this review.

6 Semi-parametric methods

A semi-parametric model has both parametric and non-parametric components. Typically semi-parametric estimators with IVs assume a parametric form for the model relating the outcome and exposure, but make no assumption on the distribution of the errors. Semi-parametric models are designed to be more robust to model misspecification than fully parametric models.⁷⁸

6.1 Generalized method of moments

The generalized method of moments (GMM) is a semi-parametric estimator designed as a more flexible form of 2SLS to deal with problems of heteroscedasticity of error distributions and non-linearity in the two-stage structural equations.^61,79 With a single instrument, the estimator is chosen to give orthogonality between the instrument and the residuals from the second-stage regression. Using bold face to represent vectors, if we have

E (Y | do (X = x)) = f (x; β)

(14)

then the GMM estimate is the value of β such that

\sum_{i} (y_{i} - f (y_{i}; β)) = 0 and \sum_{i} z_{i} (y_{i} - f (y_{i}; β)) = 0

(15)

where the summation is across i, which indexes study participants. Equation (14) is a structural model, where

E (Y | do (X = x))

is the expectation of the distribution that Y would take if we forced X to equal x for all individuals.⁸⁰

In the linear (or additive) case,

f (y_{i}; β) = β_{0} + β_{1} x_{i}

; in the log-linear (or multiplicative) case,

f (y_{i}; β) = expit (β_{0} + β_{1} x_{i})

; and in the logistic case,

f (y_{i}; β) = expit (β_{0} + β_{1} x_{i})

; where

β_{1}

is our causal parameter of interest and

expit (z) = (1 - exp (- z)) - 1

, the inverse of logit(z). We can solve the two equations in (15) numerically.¹⁶

GMM estimates are sensitive to the parameterization used. For example, estimates from the estimating equations (15) and from

\sum_{i} (y_{i} f (y_{i}; β) - 1 - 1) = 0 and \sum_{i} z_{i} (y_{i} f (y_{i}; β) - 1 - 1) = 0

(16)

may be different in finite samples, although they each assume the same structural model between the outcome and exposure. With a binary outcome, the second set of estimating equations is usually used, as this corresponds to a multiplicative error structure.¹⁵

When there is more than one IV,

z_{i}

becomes

z_{ik}

and we have a separate estimating equation for each instrument k=1, … , K. The orthogonality conditions for each instrument cannot generally be simultaneously satisfied. The estimate is usually taken as the minimizer of the objective function

(y - f (x; β)) T Z (Z T Ω Z) - 1 Z T (y - f (x; β))

(17)

where

Z = (1 z_{1} \dots z_{K})

is the N by K + 1 matrix of instruments, including a column of 1s. Although this gives consistent estimation for general matrix Ω, efficient estimation is achieved when

Ω_{ij} = cov (ɛ_{i}, ɛ_{j})

(i,j = 1, … , N), where ε_i is the residual

y_{i} - f (y_{i}; β)

.⁸¹

As the estimation of Ω requires knowledge of the unknown β, a two-step approach is suggested. We firstly estimate β* using

(Z T Ω Z) = I

, where I is the identity matrix, which gives consistent but not efficient estimation of β. We then use

e_{i} = y_{i} - f (y_{i}; β *)

to estimate

Z T Ω Z = \sum_{i} z_{i} z_{i} T ɛ_{i}^{2}

\sum_{i} z_{i} z_{i} T e_{i}^{2}

in a second-stage estimation.⁷⁹

6.2 Continuous updating estimator

An alternative approach is to express the weighting matrix Ω as a function of the causal parameter, and then to estimate the causal parameter and weighting matrix simultaneously by minimizing equation (17). This is known as the continuous updating estimator (CUE), as the weighting matrix updates during the estimation procedure.⁸² The CUE is similar to the LIML estimator in terms of confidence intervals and finite sample bias,⁴⁸ and the two estimators are equivalent if the residuals e_i defined above are homoscedastic.⁸³ However, the CUE does not make the homoscedasticity assumption. The CUE and two-step GMM estimators have the same asymptotic distribution.⁸³

6.3 G-estimation of structural mean models

G-estimation of a structural mean model (SMM) is another semi-parametric estimation method^84,85 designed in the context of randomized trials with incomplete compliance.^86,87 The potential outcome Y(x) is defined as the outcome which would have been observed if the exposure X were set to x. In particular, the exposure-free outcome Y(0) is the outcome which would have been observed if we had set X to 0 rather than it taking its observed value of x.⁷⁸

An explicit parametric form is assumed for the expected difference in potential outcomes between the outcome for the observed X = x and the potential outcome for X=0. In the continuous case, the linear or additive SMM is

E (Y (x) - Y (0) | X = x, Z = z) = β_{1} x

(18)

and β₁ is taken as the causal parameter of interest. In the context of non-compliance, this is referred to as the ‘effect of treatment on the treated’.⁸⁸ In this equation, there is no effect modification on the additive scale by Z.

As the expected exposure-free outcome E(Y(0)|X = x,Z = z) is statistically independent of Z, the causal effect is estimated as the value of β₁ which gives zero covariance between

E (Y (0) | X = x, Z = z) = E (Y (x) - β_{1} x | X = x, Z = z)

and Z. The estimating equations are

\sum_{i} (z_{ik} - {\bar{z}}_{k}) (y_{i} - β_{1} x_{i}) = 0 k = 1, \dots, K

(19)

where

{\bar{z}}_{k} = \frac{1}{N} \sum_{i} z_{ik}

and the summation is across i, which indexes study participants.

Where the model for the expected outcomes is non-linear, this is known as a generalized SMM. With a binary outcome, it is natural to use a log-linear (or multiplicative) SMM

log E (Y (x) | X = x, Z = z) - log E (Y (0) | X = x, Z = z) = β_{1} x

(20)

In this case, there is no effect modification by Z on the multiplicative scale.

The linear (additive) and log-linear (multiplicative) GMM and SMM approaches outlined give rise to the same estimating equations, and so software for GMM estimation can also be used for the estimation of a SMM.⁸⁹ In the case of a single IV, the linear GMM and SMM estimates coincide with the corresponding ratio estimate. With multiple IVs, the estimating equations are not guaranteed to have a unique solution, and so an objective function can be obtained and minimized in the same way as in the GMM approach. Efficient estimates can be obtained using the same two-step method outlined for the GMM approach in Section 6.1.⁹⁰

Unfortunately, a logistic SMM cannot be defined in the same way, as the moment conditions in the estimating equations depend on unmeasured covariates due to the non-collapsibility of the odds ratio.⁹¹ This problem also arises in the GMM case,¹⁵ although GMM is an estimation method and so does not require the functional form relating the exposure and outcome to be given the interpretation of a structural model. For a SMM, the problem can be resolved by estimating Y assuming an observational model⁹⁰

logit E (Y | X = x, Z = z) = β_{0 a} + β_{1 a} x

(21)

where the subscripts a indicate associational, and substituting estimates of Y into the structural model

logit E (Y (x) | X = x, Z = z) - logit E (Y (0) | X = x, Z = z) = β_{1 c} x

(22)

where the subscript c indicates causal. The associational parameters can be estimated by logistic regression, leading to estimating equations

\sum_{i} (z_{ik} - {\bar{z}}_{k}) expit (\hat{Y} (x) - β_{1 c} x_{i}) = 0 k = 1, \dots, K

(23)

where

\hat{Y} (x) = {\hat{β}}_{0 a} + {\hat{β}}_{1 a} x

6.4 Potential lack of unique estimate with binary outcomes

An issue with semi-parametric IV estimation in practice is that an estimate of the causal parameter may not be uniquely determined in finite samples. A parameter in a statistical model is identified if its value can be uniquely determined from the joint distribution of the observed variables in an infinite sample.⁹² However, for a semi-parametric instrumental variable analysis with a single IV and a binary outcome, the estimating equations may be almost flat in the vicinity of the solution to the equations (known as weak identification) or there may be multiple or no parameter values which satisfy the estimating equations.⁹³ This is especially likely if the IV is weak. Consequently, estimates and standard errors reported by automated commands for GMM or SMM estimation can be misleading.

It is recommended that applied investigators wanting to use a GMM or SMM approach should plot the objective function of the relevant estimating equations for a large range of values of the parameter of interest to check if there is a unique solution. If there is not, this should be reported as an indication that there is a lack of information on the parameter in the data. An alternative estimation technique can be used, such as a two-stage method, but identification will then rely on stronger assumptions.

A summary of features of the IV methods considered in this paper is given in Table 1. Having discussed various methods for IV estimation, we proceed to consider practical issues in the estimation of causal effects, in particular the use of weak instruments.

Table 1. Summary table of features of instrumental variable methods considered in this paper.

Ratio method	Can only be used with a single IV. Can be used with summarized data.
Two-stage methods	Efficient, but biased with weak instruments. Robust to misspecification in the first-stage model. Flexible to incorporate different exposure–outcome models.
Likelihood-based methods	Asymptotically efficient and less biased provided that the parametric assumptions hold. Requires coding by hand outside of the linear case.
Semi-parametric methods	Less efficient, less stringent assumptions. No guarantee of unique estimate in finite samples.

7 Weak instruments

A ‘weak instrument’ is defined as an IV for which the statistical evidence of association with the exposure is not strong. A weak IV differs from an invalid IV (one that does not satisfy the IV assumptions) in that the IV estimate using a weak IV is asymptotically unbiased; a weak IV can be made stronger by increasing the sample size. The F statistic in the regression of the exposure on the IV (also known as the Cragg–Donald F statistic⁹⁴) is usually quoted as a measure of the strength of an instrument.⁹⁵ IV estimates demonstrate systematic finite sample bias, typically in the direction of the observational (confounded) association between the exposure and outcome.⁹⁶ IVs with an F statistic less than 10 are often labelled as ‘weak instruments’.⁹⁷ The value was chosen as the bias of the 2SLS IV estimate with an IV having an expected F statistic value of 10 is limited to 10% of the bias of the observational association. Such a binary characterization of IVs as weak or strong is misleading, as the bias is a continuous phenomenon. Additionally, the expected magnitude of bias does not depend on the value of the F statistic for a particular dataset, but the expected value of the F statistic (also known as the concentration parameter⁹⁵). The F statistic is highly variable between datasets generated under the same data-generating mechanism, and so the F statistic in a given dataset cannot be relied on as an accurate predictor of the expected bias in that dataset.¹⁹

7.1 Problems of estimates using weak instruments

Aside from the bias suffered by IV estimates with weak instruments, the distribution of such IV estimates differs substantially from a normal distribution. Confidence intervals that rely on the asymptotic normality of IV estimates (such as those typically reported by statistical software packages for 2SLS or LIML methods) tend to underestimate the true uncertainty in the estimates, leading to inflated Type I error rates. With large numbers of IVs (10 or more), standard confidence intervals from the LIML method with weak instruments are too narrow and a correction is needed (known as Bekker standard errors).^98,83 Tables have been produced for the 2SLS method giving the expected value of the F statistic at which the maximum possible Type I error rate is less than 10%, 15%, 20%, and 25%.⁹⁹ (Although the maximum error rate is only achieved when the correlation between the exposure and outcome is one, so such extremes should not be expected in practice.) The power of a study with weak instruments may also be low, although this is not necessarily the case if there are many weak instruments. Correlation between IV estimates and standard errors can lead to bias being accentuated in a meta-analysis of IV estimates, as in the sampling distributions, estimates closer to the observational association have greater estimated precision.¹⁰⁰

These problems arise for IVs that are weak, but otherwise satisfy the IV assumptions. A more further problem with weak instruments is sensitivity to violations of the IV assumptions: while studies with stronger instruments can be insensitive to moderate violations of the IV assumptions, studies with weak instruments are extremely sensitive to tiny violations in the assumptions.¹⁰¹

7.2 Comparison of methods with single instrumental variable

As previously stated, if there is a single IV, the ratio, 2SLS, LIML, GMM and SMM IV estimates coincide. Although this IV estimate does not have a finite mean, simulations have shown that the median bias is close to zero even when the expected F statistic is around 5.⁴² An F statistic of 5 corresponds to a p-value for the association of the IV with the exposure of 0.03. It is unlikely that a variable would be used as an IV in practice unless the variable is expected to achieve a p-value of less than 0.03 in the sample under analysis. A causal effect and its direction can be assessed with a single IV by testing for an association between the IV and outcome; this association will be a valid test of the null hypothesis of no causal effect even with a weak instrument.⁹ Hence, unless bias is introduced via another mechanism (such as choosing between candidate IVs on the basis of their strength in the data under analysis¹⁰⁰), the problems of weak instruments should not arise with a single instrumental variable.

7.3 Multiple instrument variables and score-based approach

The bias of the IV estimate from the 2SLS method with a continuous outcome is approximately 1/E(F) of the bias of the observational association, where E(F) is the expected value of the F statistic from the first-stage regression. In contrast, the median bias of estimates from the LIML and CUE methods is close to zero, making them useful tools for sensitivity analysis. However, in finite samples with multiple IVs, the LIML method is generally less efficient than 2SLS.⁹⁸ A simulation analysis showed that confidence intervals from the LIML and CUE methods with large numbers of IVs were much wider than those from other IV methods, despite nominal coverage levels not being maintained⁸³; use of Bekker standard errors to maintain nominal coverage levels resulted in confidence intervals that were wider still.

A recent innovation in Mendelian randomization is to reduce data from multiple instrumental variables into a single univariate score, known as an allele score or genetic risk score.¹⁰² An unweighted score is constructed as the total number of exposure-increasing alleles present in the genotype of an individual. A weighted score can also be considered, in which each IV contributes a weight reflecting the effect of the corresponding genetic variant on the exposure. If an individual i has g_ik copies of the exposure-increasing allele for each variant k = 1, … , K, then their unweighted score is

z_{i} = \sum_{k = 1}^{K} g_{ik}

. If all genetic variants are diallelic SNPs, this score takes integer values between 0 and 2K. If the weight for variant k is w_k, then their weighted score is

z_{i} = \sum_{k = 1}^{K} w_{k} g_{ik} .

Either score can then be used in an IV analysis.

If the weights in a weighted score are derived naively from the data under analysis, then the score-based approach is effectively a 2SLS analysis, and there is no methodological benefit of the approach to a 2SLS analysis in terms of bias. However, if the weights are determined externally from prior knowledge or an independent data source, or from the data under analysis by a cross-validation or jackknife approach,¹⁰³ then effectively the IV analysis uses a single instrumental variable, reducing bias to negligible levels. Additionally, when the true effect sizes of the IVs do not vary greatly, score-based approaches have been shown to give greater precision than LIML and related methods, even when the weights for the score are misspecified or an unweighted score is used.^83,102 We would therefore recommend a score-based approach to IV analysis when there are a multiple potentially weak IVs.

A caveat is that the use of large numbers of IVs increases the possibility that one or more of the IVs is invalid, leading to potentially misleading estimates.¹⁰¹ It is therefore important not only to cite genetic associations with the outcome for the allele score but also for the constituent genetic variants, so that the consistency of estimates across the genetic variants can be assessed. Although allele scores can address a wide range of analysis questions in Mendelian randomization, more complex questions (such as analyses with multiple risk factors¹⁰⁴) cannot be addressed using an allele score method and require the use of alternative methods.

8 Discussion

In this final section, we discuss various practical topics relevant to IV estimation, as well as extensions to the scenario considered in this paper of individual-level data from a single source to estimate a linear causal effect.

8.1 Non-linear relationships

With a linear exposure–outcome relationship, the estimate from a linear IV analysis, such as using the ratio or 2SLS method, is the average change in the outcome for a uniform change (usually a 1 unit increase) in the distribution of the exposure across the population for all individuals whose exposure level is influenced by their value of the IV.²⁷ For a non-linear relationship, a linear IV estimate approximates the same population-averaged causal effect when the change in the distribution of the exposure associated with the IV is small, and when the linear IV estimate is scaled to represent the effect of a change in the exposure of similar magnitude to that associated with a change in the IV.³⁴

Although non-linear parametric two-stage IV methods have been proposed, inferences based on such methods have been shown to be highly sensitive to the choice of parameterization.^105,106 Equally, non-parametric two-stage IV methods have been explored,¹⁰⁷ but evidence on the shape of the causal relationship can only be inferred for the range of fitted values for the exposure (

\hat{X}

|Z). A recent development is the estimation of local IV estimates within strata of the exposure.³⁴ However, if the exposure is stratified on directly, misleading results may be obtained. This is because the exposure lies on the causal pathway between the IV and the outcome, and so conditioning on the exposure induces an association between the IV and confounders. This can be circumvented by initially subtracting the effect of the IVs on the exposure from the exposure measurement, to obtain the ‘IV-free exposure’. This quantity, representing the expected value of the exposure for an individual if their IVs took the value zero, can then be safely conditioned on.¹⁰⁸ Conditioning on this quantity is equivalent to conditioning on the residual from the first-stage regression,⁶⁵ known as the ‘control function approach’.¹⁰⁹ For this non-linear estimation approach to be valid, it is necessary for the association of the IV with the exposure in the population to remain constant at different levels of the exposure.¹¹⁰

8.2 Use of measured covariates

If we can find measured covariates which explain variation in the exposure or a continuous outcome, then we can incorporate such covariates into our analysis. Including covariates generally increases efficiency and hence the precision of the causal estimate. However, it may lead to bias in the causal estimate if a covariate is on the causal pathway between exposure and outcome or is a collider or causally downstream of a collider. A collider is a variable that is influenced both by the exposure or a causal consequent thereof and by the outcome or a causal consequent thereof.¹¹¹ There may also be bias if the analysis model including the covariate is misspecified. In a two-stage estimation, any covariate adjusted for in the first-stage regression should also be adjusted for in the second-stage regression¹¹²; failure to do so can cause associations between the IV and confounders leading to bias.^{56(p. 189)}

In some cases, adjustment for covariates is necessary to ensure validity of the IVs, as the IV assumptions only hold conditional on the covariates. An example in Mendelian randomization is the case of population stratification, in which the sample population consists of subpopulations that have different distributions of the IV and outcome.⁵ These subpopulations may correspond to ethnic subgroups in the population. An association between the IV and outcome may solely correspond to differences in ethnicity and not to any biological effect of the exposure. This can be addressed at least partially with genome-wide data, by adjusting for genetic principal components in the IV analysis,¹¹³ although there is no theoretical guarantee that adjustment for genetic principal components will resolve the problem of population stratification. In general, aside from these cases, we discourage adjustment for covariates in the primary analysis result for the assessment of a causal effect.

8.3 Overidentification tests

When more than one instrument is used, an overidentification test, such as the Sargan test,¹¹⁴ can be carried out to test whether the associations of the instruments with the outcome are statistically compatible with a single linear causal effect of the exposure. Overidentification means that the number of instruments used is greater than the number of exposures measured. (The latter is usually one, although analyses with multiple exposure variables have also been considered.^115,116) When there is more than one IV for a single exposure, separate causal estimates can be calculated using each IV in turn. The overidentification test assesses whether the parameters being estimated by each IV separately are the same, or equivalently whether the IVs have residual associations with the outcome once the main effect of the exposure has been removed.¹¹⁷ Failure of an overidentification test indicates heterogeneity in the effect estimates from each IV, which suggests that the IV assumptions may be violated for one or more genetic variants.

Overidentification tests are omnibus tests, where the alternative hypothesis includes failure of IV assumptions for one IV, failure for all IVs, a non-linear association between the exposure and outcome, and that different IVs identify different magnitudes of causal effect.¹¹⁸ However, in many practical cases they have low power,¹¹⁹ and so failure to reject an overidentification test should not be overinterpreted as positive evidence for the validity of the IV assumptions.

8.4 Endogeneity tests

Some applied Mendelian randomization analyses have reported on whether there is a difference between the observational and IV estimates as the primary outcome of interest¹²⁰; this can be formally tested using the Durbin–Wu–Hausman test.¹¹⁸ This is a test of equality of the observational and IV estimates, where a significant result indicates disagreement between the two estimates. It is called an endogeneity test.

While an informal comparison of the observational and causal estimates may be reasonable, for instance to judge whether a Mendelian randomization investigation has sufficient power for a null result to be informative, an endogeneity test should not be viewed as a primary analysis result. This is because the outcome of an endogeneity test neither implies the presence nor the absence of a causal effect. It is more appropriate to consider the presence or absence of a causal effect as the subject of investigation.³⁹ The confidence interval of the causal estimate gives the researcher a sense of the plausible size of any possible causal effect based on the data available.

8.5 Power of IV analyses

The confidence interval of an IV estimate is typically much wider compared with that of an observational estimate, for example from a regression analysis. The sample size required to obtain precise enough causal estimates to be clinically relevant can be very large.¹²¹ A rule of thumb for power is that the sample size for a conventional analysis should be divided by the coefficient of determination (R²) of the IV on the exposure.¹²² Formulae for power and sample size calculation have been published with a continuous outcome^123,124 and with a binary outcome,¹²⁵ and online tools for performing these calculations are available (http://glimmer.rstudio.com/kn3in/mRnd/ and http://sb452.shinyapps.io/power/).

8.6 Summarized data

In an applied context, it may not be practical or desirable to share individual-level data on the IV, exposure, and outcome. With a single IV, the ratio method can be implemented simply using summarized estimates on the associations of the IV with the outcome and with the exposure. Methods for obtaining IV estimates using summarized data on multiple IVs have been discussed elsewhere¹²⁶: these include a likelihood-based method for the summarized association estimates, and an inverse-variance weighted method combining the ratio estimates from each IV in a meta-analysis model.¹²⁷ A tool to implement these methods is available online (http://sb452.shinyapps.io/summarized/).

8.7 Subsample and two-sample Mendelian randomization

Especially when the outcome is binary, the sample size required in a Mendelian randomization experiment may be prohibitively large due to the expense of collecting data on the exposure. In this case, a subsample IV approach may be a cost-effective approach. Rather than collecting data on the exposure from the entire study sample, exposure data can be measured for a random subsample of (control) participants. As the association between the IV and the exposure is typically stronger than that between the IV and the outcome, the precision of the IV estimate may not be noticeably affected by reducing the sample size on which the exposure is measured. Simulations have shown that a subsample IV analysis with exposure data on only 10% of participants may retain 90% of the power of the full-sample IV analysis.¹²⁸ Estimates and confidence intervals with a single IV can be calculated using the ratio method and Fieller’s theorem (Section 3.4). With multiple IVs, a modified version of the two-stage least squares method can be used.¹²⁹

An alternative design is two-sample Mendelian randomization, in which the associations between the IV(s) and the exposure and between the IV(s) and the outcome are estimated from non-overlapping sets of individuals.¹³⁰ Although this may simply reflect the absence of a dataset with concomitant information on all three of the IV(s), exposure, and outcome, it is also a potentially efficient design strategy for Mendelian randomization, particularly in view of the increasing public availability of summarized data on genetic associations with risk factors and disease outcomes from large consortia. Several consortia with large numbers of participants, such as CARDIoGRAMplusC4D for coronary artery disease¹³¹ and DIAGRAM for type 2 diabetes,¹³² have published summarized data on the association of catalogues of genetic variants with either risk factors or disease status. These provide precise estimates of genetic associations which can be used to obtain causal estimates, provided the genetic variants included in the analysis are restricted to those for which the IV assumptions are valid. This makes two-sample Mendelian randomization using summarized data a worthwhile analysis strategy from the perspective of power, and one that should be encouraged in practice, particularly if there is available summarized data on genetic associations with the outcome.¹³³ The only caveat is that the two samples used should be as similar as possible (particularly with respect to nationality/ethnicity, but also to other population characteristics). This is because genetic variants that are valid IVs in one population may not be valid IVs in another population, and to ensure that the magnitudes of genetic associations with the risk factor in the first sample are relevant estimates of the genetic associations in the second sample.

8.8 Meta-analysis

The demands of power often demand synthesis of evidence from multiple, possibly heterogeneous studies. Data from multiple studies can be combined to provide a single causal estimate in a meta-analysis model. If it is possible to estimate the causal effect in each study, meta-analysis can be performed directly on these estimated causal effects, for example using inverse-variance weighting.¹³⁴ However, a study-level meta-analysis can accentuate weak instrument bias,¹⁰⁰ and so it is preferable to combine either individual-level or summarized data from each study. Additionally, it may be that some studies only provide data on one of the exposure or the outcome, and so a causal effect cannot be estimated from that study alone. A hierarchical model can be therefore proposed, in which the study-specific parameters are pooled in a between-study model. This could also be a more efficient use of data if the same genetic variants were used in separate studies. Such a model can be fitted using likelihood methods on either individual-level data or summarized data from each study.¹³⁵

8.9 Robust methods

Instrumental variable estimation methods that are robust to the violation of the IV assumptions for some or all of the ‘instruments’ have been the subject of recent methodological development. A full exposition of these methods is beyond the scope of this manuscript, but we provide a brief overview of some of these developments.

Given multiple candidate IVs, if over half of the ‘instruments’ satisfy the IV assumptions, then the median of the IV estimates using each of the ‘instruments’ individually is a consistent estimate of the causal effect.¹³⁶ A more efficient estimator than this simple median can be constructed by a weighted median method.¹³⁷ Here, rather than assuming over 50% of the ‘instruments’ are valid IVs, we assume that ‘instruments’ representing over 50% of the weight are valid IVs.

Alternatively, a consistent causal estimate can be obtained by allowing the ‘instruments’ to have direct effects on the outcome, but limiting the total magnitude of these direct effects, for example using L1-penalization.¹³⁸ Confidence intervals under the assumption that over 50% of the ‘instruments’ are valid IVs have also been developed.¹³⁹ Software for performing this penalization method is available in the R package sisVIVE.¹⁴⁰

Finally, direct effects of the IVs on the outcome can be accounted for by regressing the IV – outcome association estimates on the IV – exposure association estimates. If the associations of IVs with the outcome consist of a causal component, representing the effect of the exposure on the outcome, and a pleiotropic component, representing the direct effect of the IV on the outcome not via the risk factor, then the intercept from this regression model is an estimate of the average direct effect of each IV on the outcome.¹⁴¹ If the IV assumptions are satisfied, then the true direct effect of each IV on the outcome is zero. A test of whether the intercept estimate differs from zero is an assessment of the presence of directional (that is, unbalanced) pleiotropy. This is equivalent to Egger’s test, an assessment of small sample bias (including publication bias) in meta-analysis.^142,143

Under a weaker set of assumptions than the standard IV assumptions (namely that the direct effects of IVs on the outcome are distributed independently of instrument strength¹⁴⁴), the slope parameter from this regression with an intercept term is a consistent estimate of the causal effect. This method can be used to give a consistent estimate of the causal effect even if all the ‘instruments’ are invalid, but it requires the standard untestable assumption of no direct effects of the IVs on the outcome to be replaced with another untestable (albeit weaker) assumption.

8.10 Summary

Methods for IV analysis range from the very simple to the more complicated. The development of complex methods has been driven by the desire to produce efficient estimates, for example by integrating data on multiple IVs or multiple studies, or to provide robustness against weak instruments or misspecification of the modelling assumptions.

In conclusion, we emphasize that the main difficulty in an IV analysis is not the choice of analysis method but rather the choice of IVs and the assessment of the IV assumptions. If the IV assumptions do not hold, then inferences from any analysis method will be unreliable.²⁶ Although there are some contexts where a particular method is unsuitable (such as with large numbers of weak instruments), there is no single universal ‘best’ IV estimation method. Instead, the use of different IV methods provides sensitivity analyses to assess whether the estimate given by a particular choice of method is credible.

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: Stephen Burgess is supported by the Wellcome Trust (grant number 100114). Dylan Small was supported by a grant from the US National Science Foundation Measurement, Methodology and Statistics program. Simon G Thompson is supported by the British Heart Foundation (grant number CH/12/2/29428). No specific funding was received for the writing of this manuscript.

References

1. Greenland S. An introduction to instrumental variables for epidemiologists. Int J Epidemiol 2000; 29: 722–729.

Crossref

PubMed

ISI

Google Scholar

2. Martens EP, Pestman WR, de Boer A, et al. Instrumental variables: application and limitations. Epidemiology 2006; 17: 260–267.

Crossref

PubMed

ISI

Google Scholar

3. Zohoori N, Savitz DA. Econometric approaches to epidemiologic data: relating endogeneity and unobserved heterogeneity to confounding. Ann Epidemiol 1997; 7: 251–257.

Crossref

PubMed

ISI

Google Scholar

4. Rassen JA, Brookhart MA, Glynn RJ, et al. Instrumental variables I: instrumental variables exploit natural variation in nonexperimental data to estimate causal relationships. J Clin Epidemiol 2009; 62: 1226–1232.

Crossref

PubMed

ISI

Google Scholar

5. Lawlor D, Harbord R, Sterne J, et al. Mendelian randomization: using genes as instruments for making causal inferences in epidemiology. Stat Med 2008; 27: 1133–1163.

Crossref

PubMed

ISI

Google Scholar

6. Davey Smith G, Ebrahim S. ‘Mendelian randomization’: can genetic epidemiology contribute to understanding environmental determinants of disease? Int J Epidemiol 2003; 32: 1–22.

Crossref

PubMed

ISI

Google Scholar

7. Burgess S, Butterworth A, Malarstig A, et al. Use of Mendelian randomisation to assess potential benefit of clinical intervention. Br Med J 2012; 345: e7325.

Crossref

PubMed

Google Scholar

8. Hernán MA, Robins JM. Instruments for causal inference: an epidemiologist’s dream? Epidemiology 2006; 17: 360–372.

Crossref

PubMed

ISI

Google Scholar

9. Didelez V, Sheehan N. Mendelian randomization as an instrumental variable approach to causal inference. Stat Meth Med Res 2007; 16: 309–330.

Crossref

PubMed

ISI

Google Scholar

10. Glymour MM, Tchetgen Tchetgen EJ, Robins JM. Credible Mendelian randomization studies: approaches for evaluating the instrumental variable assumptions. Am J Epidemiol 2012; 175: 332–339.

Crossref

PubMed

ISI

Google Scholar

11. Angrist JD, Imbens GW, Rubin DB. Identification of causal effects using instrumental variables. J Am Stat Assoc 1996; 91: 444–455.

Crossref

ISI

Google Scholar

12. Didelez V, Meng S, Sheehan NA. Assumptions of IV methods for observational epidemiology. Stat Sci 2010; 25: 22–40.

Crossref

ISI

Google Scholar

13. Baiocchi M, Cheng J, Small DS. Instrumental variable methods for causal inference. Stat Med 2014; 33: 2297–2340.

Crossref

PubMed

ISI

Google Scholar

14. Palmer TM, Lawlor DA, Harbord RM, et al. Using multiple genetic variants as instrumental variables for modifiable risk factors. Stat Meth Med Res 2011; 21: 223–242.

Crossref

PubMed

ISI

Google Scholar

15. Clarke PS, Windmeijer F. Instrumental variable estimators for binary outcomes. J Am Stat Assoc 2012; 107: 1638–1652.

Crossref

ISI

Google Scholar

16. Palmer TM, Sterne JAC, Harbord RM, et al. Instrumental variable estimation of causal risk ratios and causal odds ratios in Mendelian randomization analyses. Am J Epidemiol 2011; 173: 1392–1403.

Crossref

PubMed

ISI

Google Scholar

17. Vansteelandt S, Bowden J, Babanezhad M, et al. On instrumental variables estimation of causal odds ratios. Stat Sci 2011; 26: 403–422.

Crossref

ISI

Google Scholar

18. Davies NM, Smith GD, Windmeijer F, et al. Issues in the reporting and conduct of instrumental variable studies: a systematic review. Epidemiology 2013; 24: 363–369.

Crossref

PubMed

ISI

Google Scholar

19. Burgess S, Thompson SG. Bias in causal estimates from Mendelian randomization studies with weak instruments. Stat Med 2011; 30: 1312–1323.

Crossref

PubMed

ISI

Google Scholar

20. Sussman JB, Hayward RA. An IV for the RCT: using instrumental variables to adjust for treatment contamination in randomised controlled trials. Br Med J 2010; 340: c2073.

Crossref

PubMed

Google Scholar

21. Frangakis CE, Rubin DB. Principal stratification in causal inference. Biometrics 2002; 58: 21–29.

Crossref

PubMed

ISI

Google Scholar

22. Sheehan NA, Didelez V, Burton PR, et al. Mendelian randomisation and causal inference in observational epidemiology. PLoS Med 2008; 5: e177.

Crossref

PubMed

ISI

Google Scholar

23. Kamstrup PR, Tybjaerg-Hansen A, Steffensen R, et al. Genetically elevated lipoprotein(a) and increased risk of myocardial infarction. J Am Med Assoc 2009; 301: 2331–2339.

Crossref

PubMed

ISI

Google Scholar

24. Balke A, Pearl J. Bounds on treatment effects from studies with imperfect compliance. J Am Stat Assoc 1997; 92: 1171–1176.

Crossref

ISI

Google Scholar

25. Chesher A. Nonparametric identification under discrete variation. Econometrica 2005; 73: 1525–1550.

Crossref

ISI

Google Scholar

26. Bound J, Jaeger D and Baker R. The cure can be worse than the disease: a cautionary tale regarding instrumental variables. NBER Working Paper No. 137. Cambridge, MA: NBER, 1993.

Google Scholar

27. Imbens GW, Angrist JD. Identification and estimation of local average treatment effects. Econometrica 1994; 62: 467–475.

Crossref

ISI

Google Scholar

28. Yau LH, Little RJ. Inference for the complier-average causal effect from longitudinal data subject to noncompliance and missing data, with application to a job training assessment for the unemployed. J Am Stat Assoc 2001; 96: 1232–1244.

Crossref

ISI

Google Scholar

29. Swanson S, Hernán M. Commentary: how to report instrumental variable analyses (suggestions welcome). Epidemiology 2013; 24: 370–374.

Crossref

PubMed

ISI

Google Scholar

30. Robins JM The analysis of randomized and nonrandomized AIDS treatment trials using a new approach to causal inference in longitudinal studies. In: Sechrest L, Freeman HE, Mulley AG (eds). Health service research methodology: a focus on AIDS, Washington, DC: National Center for Health Services Research, 1989, pp. 113–159.

Google Scholar

31. Pang M, Kaufman JS, Platt RW. Studying noncollapsibility of the odds ratio with marginal structural and logistic regression models. Stat Meth Med Res. 2016; 25: 1925–1937.

ISI

Google Scholar

32. VanderWeele T, Tchetgen Tchetgen E, Cornelis M, et al. Methodological challenges in Mendelian randomization. Epidemiology 2014; 25: 427–435.

Crossref

PubMed

ISI

Google Scholar

33. Wald A. The fitting of straight lines if both variables are subject to error. Ann Math Stat 1940; 11: 284–300.

Crossref

Google Scholar

34. Burgess S, Davies N, Thompson S, et al. Instrumental variable analysis with a nonlinear exposure–outcome relationship. Epidemiology 2014; 25: 877–885.

Crossref

PubMed

ISI

Google Scholar

35. Greenland S, Thomas DC. On the need for the rare disease assumption in case-control studies. Am J Epidemiol 1982; 116: 547–553.

Crossref

PubMed

ISI

Google Scholar

36. Collett D. Modelling binary data, Boca Raton, FL: Chapman and Hall/CRC Press, 2003.

Google Scholar

37. Minelli C, Thompson J, Tobin M, et al. An integrated approach to the meta-analysis of genetic association studies using Mendelian randomization. Am J Epidemiol 2004; 160: 445–452.

Crossref

PubMed

ISI

Google Scholar

38. Bowden J, Vansteelandt S. Mendelian randomisation analysis of case-control data using structural mean models. Stat Med 2011; 30: 678–694.

Crossref

PubMed

ISI

Google Scholar

39. Thomas DC, Lawlor DA, Thompson JR. Re: Estimation of bias in nongenetic observational studies using “Mendelian triangulation” by Bautista et al. Ann Epidemiol 2007; 17: 511–513.

Crossref

PubMed

ISI

Google Scholar

40. Fieller E. Some problems in interval estimation. J R Stat Soc Ser B 1954; 16: 175–185.

Google Scholar

41. Buonaccorsi J Fieller’s theorem. In: Armitage P, Colton T (eds). Encyclopedia of biostatistics, London, UK: Wiley, 2005, pp. 1951–1952.

Crossref

Google Scholar

42. Burgess S, Thompson SG. Improvement of bias and coverage in instrumental variable analysis with weak instruments for continuous and binary outcomes. Stat Med 2012; 31: 1582–1600.

Crossref

PubMed

ISI

Google Scholar

43. Efron B, Tibshirani R. An introduction to the bootstrap, Boca Raton, FL: Chapman & Hall/CRC Press, 1993.

Crossref

Google Scholar

44. Imbens G, Rosenbaum P. Robust, accurate confidence intervals with a weak instrument: quarter of birth and education. J R Stat Soc Ser A 2005; 168: 109–126.

Crossref

ISI

Google Scholar

45. Moreira M, Porter J, Suarez G. Bootstrap validity for the score test when instruments may be weak. J Econometr 2009; 149: 52–64.

Crossref

ISI

Google Scholar

46. Anderson TW, Rubin H. Estimators of the parameters of a single equation in a complete set of stochastic equations. Ann Math Stat 1949; 21: 570–582.

Crossref

Google Scholar

47. Moreira M. A conditional likelihood ratio test for structural models. Econometrica 2003; 71: 1027–1048.

Crossref

ISI

Google Scholar

48. Newey WK, Windmeijer F. Generalized method of moments with many weak moment conditions. Econometrica 2009; 77: 687–719.

Crossref

ISI

Google Scholar

49. Mikusheva A. Robust confidence sets in the presence of weak instruments. J Econometr 2010; 157: 236–247.

Crossref

ISI

Google Scholar

50. Davidson R, MacKinnon J. Confidence sets based on inverting Anderson–Rubin tests. Econometr J 2014; 17: S39–S58.

Crossref

ISI

Google Scholar

51. Mikusheva A, Poi BP. Tests and confidence sets with correct size when instruments are potentially weak. Stata J 2006; 6: 335–347.

Crossref

ISI

Google Scholar

52. Small D. ivpack: instrumental variable estimation. R package version 1.1, http://cran.r-project.org/package=ivpack (2014, accessed 24 July 2015).

Google Scholar

53. Kinal TW. The existence of moments of k-class estimators. Econometrica 1980; 48: 241–249.

Crossref

ISI

Google Scholar

54. Hahn J, Hausman JA, Kuersteiner GM. Estimation with weak instruments: accuracy of higher-order bias and MSE approximations. Econometr J 2004; 7: 272–306.

Crossref

Google Scholar

55. Angrist JD, Graddy K, Imbens GW. The interpretation of instrumental variables estimators in simultaneous equations models with an application to the demand for fish. Rev Econ Stud 2000; 67: 499–527.

Crossref

ISI

Google Scholar

56. Angrist JD, Pischke JS. Chapter 4: Instrumental variables in action: sometimes you get what you need. Mostly harmless econometrics: an empiricist’s companion, Princeton, NJ: Princeton University Press, 2009, pp. 113–220.

Crossref

Google Scholar

57. Wooldridge JM. Chapter 9: Simultaneous equations models. In: Econometric analysis of cross section and panel data, Cambridge, MA: MIT Press, 2002, pp. 239–280.

Google Scholar

58. Frost C, Thompson SG. Correcting for regression dilution bias: comparison of methods for a single predictor variable. J R Stat Soc Ser A 2000; 163: 173–189.

Crossref

ISI

Google Scholar

59. Pierce B, VanderWeele T. The effect of non-differential measurement error on bias, precision and power in Mendelian randomization studies. Int J Epidemiol 2012; 41: 1383–1393.

Crossref

PubMed

ISI

Google Scholar

60. Hardin J, Schmiediche H, Carroll R. Instrumental variables, bootstrapping, and generalized linear models. Stata J 2003; 3: 351–360.

Crossref

Google Scholar

61. Foster E. Instrumental variables for logistic regression: an illustration. Soc Sci Res 1997; 26: 487–504.

Crossref

ISI

Google Scholar

62. Greenland S, Robins J, Pearl J. Confounding and collapsibility in causal inference. Stat Sci 1999; 14: 29–46.

Crossref

ISI

Google Scholar

63. Ducharme GR, LePage Y. Testing collapsibility in contingency tables. J R Stat Soc Ser B 1986; 48: 197–205.

Google Scholar

64. Burgess S, CCGC (CRP CHD Genetics Collaboration). Identifying the odds ratio estimated by a two-stage instrumental variable analysis with a logistic regression model. Stat Med 2013; 32: 4726–4747.

Crossref

PubMed

ISI

Google Scholar

65. Nagelkerke N, Fidler V, Bernsen R, et al. Estimating treatment effects in randomized clinical trials in the presence of non-compliance. Stat Med 2000; 19: 1849–1864.

Crossref

PubMed

ISI

Google Scholar

66. Terza JV, Basu A, Rathouz PJ. Two-stage residual inclusion estimation: addressing endogeneity in health econometric modeling. J Health Econ 2008; 27: 531–543.

Crossref

PubMed

ISI

Google Scholar

67. Palmer T, Thompson J, Tobin M, et al. Adjusting for bias and unmeasured confounding in Mendelian randomization studies with binary responses. Int J Epidemiol 2008; 37: 1161–1168.

Crossref

PubMed

ISI

Google Scholar

68. Cai B, Small DS, Ten Have TR. Two-stage instrumental variable methods for estimating the causal odds ratio: analysis of bias. Stat Med 2011; 30: 1809–1824.

Crossref

PubMed

ISI

Google Scholar

69. Davidson R, MacKinnon J. Chapter 18: Simultaneous equation models. In: Estimation and inference in econometrics, Oxford University Press, 1993.

Google Scholar

70. Zellner A, Theil H. Three-stage least squares: simultaneous estimation of simultaneous equations. Econometrica 1962; 30: 54–78.

Crossref

ISI

Google Scholar

71. Hayashi F. Econometrics, Princeton, NJ: Princeton University Press, 2000.

Google Scholar

72. Kleibergen F, Zivot E. Bayesian and classical approaches to instrumental variable regression. J Econometr 2003; 114: 29–72.

Crossref

ISI

Google Scholar

73. Jones E, Thompson J, Didelez V, et al. On the choice of parameterisation and priors for the Bayesian analyses of Mendelian randomisation studies. Stat Med 2012; 31: 1483–1501.

Crossref

PubMed

ISI

Google Scholar

74. Spiegelhalter D, Thomas A, Best N, et al. WinBUGS version 1.4 user manual. Cambridge, UK: MRC Biostatistics Unit, 2003.

Google Scholar

75. Nagar A. The bias and moment matrix of the general k-class estimators of the parameters in simultaneous equations. Econometrica 1959; 27: 575–595.

Crossref

ISI

Google Scholar

76. Stock JH, Yogo M Asymptotic distributions of instrumental variables statistics with many instruments. In: Andrews DW, Stock JH (eds). Identification and inference for econometric models: essays in honor of Thomas Rothenberg, New York: Cambridge University Press, 2005, pp. 109–119.

Crossref

Google Scholar

77. Fuller WA. Some properties of a modification of the limited information estimator. Econometrica 1977; 45: 939–953.

Crossref

ISI

Google Scholar

78. Clarke P and Windmeijer F. Instrumental variable estimators for binary outcomes. Working Paper No. 10/239. University of Bristol, UK: Centre for Market and Public Organisation, 2010.

Google Scholar

79. Johnston K, Gustafson P, Levy A, et al. Use of instrumental variables in the analysis of generalized linear models in the presence of unmeasured confounding with applications to epidemiological research. Stat Med 2008; 27: 1539–1556.

Crossref

PubMed

ISI

Google Scholar

80. Pearl J. Causality: models, reasoning, and inference, New York: Cambridge University Press, 2000.

Google Scholar

81. Hansen LP. Large sample properties of generalized method of moments estimators. Econometrica 1982; 50: 1029–1054.

Crossref

ISI

Google Scholar

82. Hansen LP, Heaton J, Yaron A. Finite-sample properties of some alternative GMM estimators. J Business Econ Stat 1996; 14: 262–280.

ISI

Google Scholar

83. Davies N, von Hinke Kessler Scholder S, Farbmacher H, et al. The many weak instrument problem and Mendelian randomization. Stat Med 2015; 34: 454–468.

Crossref

PubMed

ISI

Google Scholar

84. Robins J. A new approach to causal inference in mortality studies with a sustained exposure period–application to control of the healthy worker survivor effect. Math Model 1986; 7: 1393–1512.

Crossref

ISI

Google Scholar

85. Greenland S, Lanes S, Jara M. Estimating effects from randomized trials with discontinuations: the need for intent-to-treat design and G-estimation. Clin Trial 2008; 5: 5–13.

Crossref

PubMed

ISI

Google Scholar

86. Robins JM. Correcting for non-compliance in randomized trials using structural nested mean models. Commun Stat – Theor Meth 1994; 23: 2379–2412.

Crossref

ISI

Google Scholar

87. Fischer-Lapp K, Goetghebeur E. Practical properties of some structural mean analyses of the effect of compliance in randomized trials. Control Clin Trial 1999; 20: 531–546.

Crossref

PubMed

Google Scholar

88. Dunn G, Maracy M, Tomenson B. Estimating treatment effects from randomized clinical trials with noncompliance and loss to follow-up: the role of instrumental variable methods. Stat Meth Med Res 2005; 14: 369–395.

Crossref

PubMed

ISI

Google Scholar

89. Clarke P, Palmer T and Windmeijer F. Estimating structural mean models with multiple instrumental variables using the generalised method of moments. Working Paper No. 11/266. University of Bristol, UK: Centre for Market and Public Organisation, 2011.

Google Scholar

90. Vansteelandt S, Goetghebeur E. Causal inference with generalized structural mean models. J R Stat Soc Ser B 2003; 65: 817–835.

Crossref

Google Scholar

91. Robins J Marginal structural models versus structural nested models as tools for causal inference. In: Halloran ME, Berry D (eds). Statistical models in epidemiology: the environment and clinical trials, New York, NY: Springer, 1999, pp. 95–134.

Google Scholar

92. Pearl J. Causality: models, reasoning, and inference, 2nd ed. New York: Cambridge University Press, 2000.

Google Scholar

93. Burgess S, Granell R, Palmer TM, et al. Lack of identification in semi-parametric instrumental variable models with binary outcomes. Am J Epidemiol 2014; 180: 111–119.

Crossref

PubMed

ISI

Google Scholar

94. Baum CF, Schaffer ME, Stillman S. Enhanced routines for instrumental variables/generalized method of moments estimation and testing. Stata J 2007; 7: 465–506.

Crossref

ISI

Google Scholar

95. Stock J, Wright J, Yogo M. A survey of weak instruments and weak identification in generalized method of moments. J Business Econ Stat 2002; 20: 518–529.

Crossref

ISI

Google Scholar

96. Bound J, Jaeger D, Baker R. Problems with instrumental variables estimation when the correlation between the instruments and the endogenous explanatory variable is weak. J Am Stat Assoc 1995; 90: 443–450.

Crossref

ISI

Google Scholar

97. Staiger D, Stock J. Instrumental variables regression with weak instruments. Econometrica 1997; 65: 557–586.

Crossref

ISI

Google Scholar

98. Bekker P. Alternative approximations to the distributions of instrumental variable estimators. Econometrica 1994; 62: 657–681.

Crossref

ISI

Google Scholar

99. Stock J, Yogo M Testing for weak instruments in linear IV regression. In: Andrews DW, Stock JH (eds). Identification and inference for econometric models: essays in honor of Thomas Rothenberg, New York: Cambridge University Press, 2005, pp. 80–108.

Crossref

Google Scholar

100. Burgess S, Thompson SG, CCGC (CRP CHD Genetics Collaboration). Avoiding bias from weak instruments in Mendelian randomization studies. Int J Epidemiol 2011; 40: 755–764.

Crossref

PubMed

ISI

Google Scholar

101. Small DS, Rosenbaum PR. War and wages: the strength of instrumental variables and their sensitivity to unobserved biases. J Am Stat Assoc 2008; 103: 924–933.

Crossref

ISI

Google Scholar

102. Burgess S, Thompson S. Use of allele scores as instrumental variables for Mendelian randomization. Int J Epidemiol 2013; 42: 1134–1144.

Crossref

PubMed

ISI

Google Scholar

103. Angrist JD, Imbens G, Krueger AB. Jackknife instrumental variables estimation. J Appl Econometr 1999; 14: 57–67.

Crossref

ISI

Google Scholar

104. Burgess S, Thompson S. Multivariable Mendelian randomization: the use of pleiotropic genetic variants to estimate causal effects. Am J Epidemiol 2015; 181: 251–260.

Crossref

PubMed

ISI

Google Scholar

105. Mogstad M and Wiswall M. Linearity in instrumental variables estimation: problems and solutions. Bonn, Germany: Forschungsinstitut zur Zukunft der Arbeit, 2010.

Google Scholar

106. Horowitz JL. Applied nonparametric instrumental variables estimation. Econometrica 2011; 79: 347–394.

Crossref

ISI

Google Scholar

107. Newey WK, Powell JL. Instrumental variable estimation of nonparametric models. Econometrica 2003; 71: 1565–1578.

Crossref

ISI

Google Scholar

108. Silverwood RJ, Holmes MV, Dale CE, et al. Testing for non-linear causal effects using a binary genotype in a Mendelian randomization study: application to alcohol and cardiovascular traits. Int J Epidemiol 2014; 43: 1781–1790.

Crossref

PubMed

ISI

Google Scholar

109. Blundell R, Powell J. Endogeneity in nonparametric and semiparametric regression models. Advances in economics and econometrics: theory and applications, Cambridge, UK: Cambridge University Press, 2005, pp. 312–357. 8th World Congress of the Econometric Society.

Google Scholar

110. Small DS. Commentary: Interpretation and sensitivity analysis for the localized average causal effect curve. Epidemiology 2014; 25: 886–888.

Crossref

PubMed

ISI

Google Scholar

111. Cole SR, Platt RW, Schisterman EF, et al. Illustrating bias due to conditioning on a collider. Int J Epidemiol 2010; 39: 417–420.

Crossref

PubMed

ISI

Google Scholar

112. Wooldridge JM. Chapter 15: Instrumental variables estimation and two stage least squares. In: Introductory econometrics: a modern approach, Nashville, TN: South-Western, 2009, pp. 506–545.

Google Scholar

113. Price AL, Patterson NJ, Plenge RM, et al. Principal components analysis corrects for stratification in genome-wide association studies. Nature Genet 2006; 38: 904–909.

Crossref

PubMed

ISI

Google Scholar

114. Sargan J. The estimation of economic relationships using instrumental variables. Econometrica 1958; 26: 393–415.

Crossref

ISI

Google Scholar

115. Angrist JD. Instrumental variables methods in experimental criminological research: what, why and how. J Exp Criminol 2006; 2: 23–44.

Crossref

Google Scholar

116. Burgess S, Freitag D, Khan H, et al. Using multivariable Mendelian randomization to disentangle the causal effects of lipid fractions. PLoS One 2014; 9: e108891.

Crossref

PubMed

ISI

Google Scholar

117. Wehby G, Ohsfeldt R, Murray J. “Mendelian randomization” equals instrumental variable analysis with genetic instruments. Stat Med 2008; 27: 2745–2749.

Crossref

PubMed

ISI

Google Scholar

118. Baum C, Schaffer M, Stillman S. Instrumental variables and GMM: estimation and testing. Stata J 2003; 3: 1–31.

Crossref

Google Scholar

119. Small DS. Sensitivity analysis for instrumental variables regression with overidentifying restrictions. J Am Stat Assoc 2007; 102: 1049–1058.

Crossref

ISI

Google Scholar

120. Bautista LE, Smeeth L, Hingorani AD, et al. Estimation of bias in nongenetic observational studies using “Mendelian triangulation”. Ann Epidemiol 2006; 16: 675–680.

Crossref

PubMed

ISI

Google Scholar

121. Ebrahim S, Davey Smith G. Mendelian randomization: can genetic epidemiology help redress the failures of observational epidemiology? Human Genet 2008; 123: 15–33.

Crossref

PubMed

ISI

Google Scholar

122. Wooldridge JM. Chapter 5: Multiple regression analysis – OLS asymptotics. In: Introductory econometrics: a modern approach, Nashville, TN: South-Western, 2009, pp. 167–183.

Google Scholar

123. Freeman G, Cowling B, Schooling M. Power and sample size calculations for Mendelian randomization studies. Int J Epidemiol 2013; 42: 1157–1163.

Crossref

PubMed

ISI

Google Scholar

124. Brion MJ, Shakhbazov K, Visscher P. Calculating statistical power in Mendelian randomization studies. Int J Epidemiol 2013; 42: 1497–1501.

Crossref

PubMed

ISI

Google Scholar

125. Burgess S. Sample size and power calculations in Mendelian randomization with a single instrumental variable and a binary outcome. Int J Epidemiol 2014; 43: 922–929.

Crossref

PubMed

ISI

Google Scholar

126. Burgess S, Butterworth A, Thompson SG. Mendelian randomization analysis with multiple genetic variants using summarized data. Genet Epidemiol 2013; 37: 658–665.

Crossref

PubMed

ISI

Google Scholar

127. Johnson T. Conditional and joint multiple-SNP analysis of GWAS summary statistics identifies additional variants influencing complex traits. Queen Mary University of London, http://webspace.qmul.ac.uk/tjohnson/gtx/outline2.pdf (2011, accessed 24 July 2015).

Google Scholar

128. Pierce B, Burgess S. Efficient design for Mendelian randomization studies: subsample and two-sample instrumental variable estimators. Am J Epidemiol 2013; 178: 1177–1184.

Crossref

PubMed

ISI

Google Scholar

129. Inoue A, Solon G. Two-sample instrumental variables estimators. Rev Econ Stat 2010; 92: 557–561.

Crossref

ISI

Google Scholar

130. Angrist JD, Krueger AB. The effect of age at school entry on educational attainment: an application of instrumental variables with moments from two samples. J Am Stat Assoc 1992; 87: 328–336.

Crossref

ISI

Google Scholar

131. Schunkert H, König I, Kathiresan S, et al. Large-scale association analysis identifies 13 new susceptibility loci for coronary artery disease. Nature Genet 2011; 43: 333–338.

Crossref

PubMed

ISI

Google Scholar

132. Morris A, Voight B, Teslovich T, et al. Large-scale association analysis provides insights into the genetic architecture and pathophysiology of type 2 diabetes. Nature Genet 2012; 44: 981–990.

Crossref

PubMed

ISI

Google Scholar

133. Burgess S, Scott R, Timpson N, et al. Using published data in Mendelian randomization: a blueprint for efficient identification of causal risk factors. Eur J Epidemiol. 2015; 30: 543–552.

ISI

Google Scholar

134. Borenstein M, Hedges LV, Higgins JPT, et al. Chapter 34: Generality of the basic inverse-variance method. Introduction to meta-analysis, Wiley, 2009.

Crossref

Google Scholar

135. Burgess S, Thompson SG, CCGC (CRP CHD Genetics Collaboration). Methods for meta-analysis of individual participant data from Mendelian randomization studies with binary outcomes. Stat Meth Med Res.. 2016; 25: 272–293.

Google Scholar

136. Han C. Detecting invalid instruments using L1-GMM. Econ Lett 2008; 101: 285–287.

Crossref

ISI

Google Scholar

137. Bowden J. Median based estimation for Mendelian randomization with multiple instruments. MRC Biostatistics Unit, http://www.mrc-bsu.cam.ac.uk/wp-content/uploads/SimpleWeightedMedian.pdf (2015, accessed 24 July 2015).

Google Scholar

138. Kang H, Zhang A, Cai T, et al. Instrumental variables estimation with some invalid instruments, and its application to Mendelian randomisation. J Am Stat Assoc 2015. available online before print.

ISI

Google Scholar

139. Kang H, Cai T and Small D. Robust confidence intervals for causal effects with possibly invalid instruments. arXiv 2015: 1504.03718.

Google Scholar

140. Kang H. sisVIVE: Some invalid some valid instrumental variables estimator; R package version 1.1, http://cran.r-project.org/package=sisVIVE (2015, accessed 24 July 2015).

Google Scholar

141. Bowden J, Davey Smith G, Burgess S. Mendelian randomization with invalid instruments: effect estimation and bias detection through Egger regression. Int J Epidemiol 2015; 44: 512–525.

Crossref

PubMed

ISI

Google Scholar

142. Egger M, Smith GD, Schneider M, et al. Bias in meta-analysis detected by a simple, graphical test. Br Med J 1997; 315: 629–634.

Crossref

PubMed

Google Scholar

143. Rücker G, Schwarzer G, Carpenter JR, et al. Treatment-effect estimates adjusted for small-study effects via a limit meta-analysis. Biostatistics 2011; 12: 122–142.

Crossref

PubMed

ISI

Google Scholar

144. Kolesár M, Chetty R, Friedman J, et al. Identification and inference with many invalid instruments. J Business Econ Stat. 2015; 33: 474–484.

ISI

Google Scholar

Cite article

If you have citation software installed, you can download article citation data to the citation manager of your choice

Information, rights and permissions

Information

Published In

Statistical Methods in Medical Research

Volume 26, Issue 5

Pages: 2333 - 2355

Article first published online: August 17, 2015

Issue published: October 2017

Keywords

Rights and permissions

This article is distributed under the terms of the Creative Commons Attribution 3.0 License (http://www.creativecommons.org/licenses/by/3.0/) which permits any use, reproduction and distribution of the work without further permission provided the original work is attributed as specified on the SAGE and Open Access pages (https://us.sagepub.com/en-us/nam/open-access-at-sage).

PubMed: 26282889

Authors

Affiliations

Stephen Burgess

Department of Public Health and Primary Care, University of Cambridge, Cambridge, UK

[email protected]

View all articles by this author

Dylan S Small

Department of Statistics, The Wharton School, University of Pennsylvania, PA, USA

View all articles by this author

Simon G Thompson

Department of Public Health and Primary Care, University of Cambridge, Cambridge, UK

View all articles by this author

Notes

Stephen Burgess, Strangeways Research Laboratory, 2 Worts Causeway, Cambridge, CB1 8RN, UK. Email: [email protected].

Metrics and citations

Metrics

This article was published in Statistical Methods in Medical Research.

VIEW ALL JOURNAL METRICS

Total views and downloads: 29627

^*Article usage tracking started in December 2016

Receive email alerts when this article is cited

Web of Science: 658 view articles Opens in new tab

Crossref: 709

Causal effects of gut microbiota on sepsis and sepsis-related death: i...

Go to citation Crossref Google Scholar
Gut microbiota, circulating cytokines and dementia: a Mendelian random...

Go to citation Crossref Google Scholar
Integrating genetic regulation and single-cell expression with GWAS pr...

Go to citation Crossref Google Scholar
Genetically determined circulating micronutrients and the risk of nona...

Go to citation Crossref Google Scholar
The causal relationship between immune cells and ankylosing spondyliti...

Go to citation Crossref Google Scholar
Exploring potential causal associations between autoimmune diseases an...

Go to citation Crossref Google Scholar
The genetic etiology of body fluids on chronic obstructive airways dis...

Go to citation Crossref Google Scholar
A systematic two-sample and bidirectional MR process highlights a unid...

Go to citation Crossref Google Scholar
Association between psychiatric disorders and glioma risk: evidence fr...

Go to citation Crossref Google Scholar
The association between adipokines and pulmonary diseases: a mendelian...

Go to citation Crossref Google Scholar
CRP, IL-1α, IL-1β, and IL-6 levels and the risk of breast cancer: a tw...

Go to citation Crossref Google Scholar
Bidirectional two-sample Mendelian randomization analysis identifies c...

Go to citation Crossref Google Scholar
Effects of nonalcoholic fatty liver disease on sarcopenia: evidence fr...

Go to citation Crossref Google Scholar
Association between inflammatory bowel disease and frailty: a two-samp...

Go to citation Crossref Google Scholar
The causal effects of genetically predicted alcohol consumption on end...

Go to citation Crossref Google Scholar
Proteome-wide mendelian randomization investigates potential associati...

Go to citation Crossref Google Scholar
Determinants of household food expenditure in Tanzania: implications o...

Go to citation Crossref Google Scholar
Exploring the causal relationship between immune cells and idiopathic ...

Go to citation Crossref Google Scholar
Statins as a risk factor for diabetic retinopathy: a Mendelian randomi...

Go to citation Crossref Google Scholar
Phenome-wide Mendelian randomisation analysis of 378,142 cases reveals...

Go to citation Crossref Google Scholar
Investigating the causal role of the gut microbiota in esophageal canc...

Go to citation Crossref Google Scholar
The causal relationship between thoracic aortic aneurysm and immune ce...

Go to citation Crossref Google Scholar
Iron status and sarcopenia-related traits: a bi-directional Mendelian ...

Go to citation Crossref Google Scholar
Shared genetic factors and causal association between chronic hepatiti...

Go to citation Crossref Google Scholar
Identification of plasma protein markers of allergic disease risk: a m...

Go to citation Crossref Google Scholar
Metabolome-wide Mendelian randomization for age at menarche and age at...

Go to citation Crossref Google Scholar
Bidirectional two-sample Mendelian randomization analyses support caus...

Go to citation Crossref Google Scholar
Causal relationship between chronic obstructive pulmonary disease and ...

Go to citation Crossref Google Scholar
Associations between genetically predicted concentrations of circulati...

Go to citation Crossref Google Scholar
Hypothyroidism's effect on stroke limited to specific subtypes: A Mend...

Go to citation Crossref Google Scholar
Causal relationships between gut microbiota and dementia: A two-sample...

Go to citation Crossref Google Scholar
Association between tea consumption and risk of kidney stones: results...

Go to citation Crossref Google Scholar
Unveiling challenges in Mendelian randomization for gene–environment i...

Go to citation Crossref Google Scholar
GWAS advancements to investigate disease associations and biological m...

Go to citation Crossref Google Scholar
Valid instrumental variable selection method using negative control ou...

Go to citation Crossref Google Scholar
The immune cells have complex causal regulation effects on cancers

Go to citation Crossref Google Scholar
Effects of immune cells on ischemic stroke and the mediating roles of ...

Go to citation Crossref Google Scholar
Causal role of immune cells in hypertension: a bidirectional Mendelian...

Go to citation Crossref Google Scholar
Frailty and risk of systemic atherosclerosis: A bidirectional Mendelia...

Go to citation Crossref Google Scholar
Osteoporosis and coronary heart disease: a bi-directional Mendelian ra...

Go to citation Crossref Google Scholar
A Mendelian randomization study on the causal effects of cigarette smo...

Go to citation Crossref Google Scholar
Assessment of the relationship between gut microbiota and bone mineral...

Go to citation Crossref Google Scholar
Genetic associations in ankylosing spondylitis: circulating proteins a...

Go to citation Crossref Google Scholar
Causal association between type 2 diabetes mellitus and acute suppurat...

Go to citation Crossref Google Scholar
Genetic associations between Rapid Eye Movement (REM) sleep behavior d...

Go to citation Crossref Google Scholar
Assessing the causal role of immune traits in rheumatoid arthritis by ...

Go to citation Crossref Google Scholar
The gut microbiota and post-traumatic major depression disorder: insig...

Go to citation Crossref Google Scholar
Investigating the Bidirectional Association of Rheumatoid Arthritis an...

Go to citation Crossref Google Scholar
Dissecting causal links between gut microbiota, inflammatory cytokines...

Go to citation Crossref Google Scholar
Causal role of immune cells in Hashimoto’s thyroiditis: Mendelian rand...

Go to citation Crossref Google Scholar
Obesity and Smoking are causal factors for meniscal injury: A mendelia...

Go to citation Crossref Google Scholar
Causality of Helicobacter pylori infection on eosinophilic esophagitis...

Go to citation Crossref Google Scholar
A Causal Relationship Between Type 1 Diabetes and Risk of Osteoporosis...

Go to citation Crossref Google Scholar
The causal relationship between gut microbiota and lymphoma: a two-sam...

Go to citation Crossref Google Scholar
Dietary preference and susceptibility to type 2 diabetes mellitus: a w...

Go to citation Crossref Google Scholar
Critical COVID-19, Victivallaceae abundance, and celiac disease: A med...

Go to citation Crossref Google Scholar
Association of immune cells and the risk of esophageal cancer: A Mende...

Go to citation Crossref Google Scholar
A two-stage Bridge estimator for regression models with endogeneity ba...

Go to citation Crossref Google Scholar
The association between the basal metabolic rate and cardiovascular di...

Go to citation Crossref Google Scholar
Causal role of myeloid cells in Parkinson’s disease: Mendelian randomi...

Go to citation Crossref Google Scholar
Acute pancreatitis and metabolic syndrome: genetic correlations and ca...

Go to citation Crossref Google Scholar
The causal role of circulating immunity‐inflammation in preeclampsia: ...

Go to citation Crossref Google Scholar
Genetic association between gut microbiota and the risk of Guillain-Ba...

Go to citation Crossref Google Scholar
Systematic characterization of multi-omics landscape between gut micro...

Go to citation Crossref Google Scholar
Impact of ambient air pollution on colorectal cancer risk and survival...

Go to citation Crossref Google Scholar
Genetically predicted hypotaurine levels mediate the relationship betw...

Go to citation Crossref Google Scholar
Genetically predicted blood metabolites mediate the association betwee...

Go to citation Crossref Google Scholar
Causal effects of hypertensive disorders of pregnancy on future gyneco...

Go to citation Crossref Google Scholar
Causal associations between gut microbiota and synovitis–tenosynovitis...

Go to citation Crossref Google Scholar
The causal effects of genetically determined immune cells on gynecolog...

Go to citation Crossref Google Scholar
The role of dietary preferences in osteoarthritis: a Mendelian randomi...

Go to citation Crossref Google Scholar
Causal effects of immune cells in glioblastoma: a Bayesian Mendelian R...

Go to citation Crossref Google Scholar
Interactions between circulating inflammatory factors and autism spect...

Go to citation Crossref Google Scholar
New target-HMGCR inhibitors for the treatment of primary sclerosing ch...

Go to citation Crossref Google Scholar
Disentangling the relationships of body mass index and circulating sex...

Go to citation Crossref Google Scholar
An exploration of causal relationships between nine neurological disea...

Go to citation Crossref Google Scholar
How do stochastic processes and genetic threshold effects explain inco...

Go to citation Crossref Google Scholar
Inflammatory cytokines may mediate the causal relationship between gut...

Go to citation Crossref Google Scholar
Elevated Blood Pressure: A Genetically Determined Risk Factor for Cere...

Go to citation Crossref Google Scholar
Combination of multidisciplinary approaches reveals potential causal a...

Go to citation Crossref Google Scholar
Assessing the causal associations of different types of statins use an...

Go to citation Crossref Google Scholar
The causal relationship between obesity, obstructive sleep apnea and a...

Go to citation Crossref Google Scholar
Mendelian randomization: causal inference leveraging genetic data

Go to citation Crossref Google Scholar
Exploring inflammatory bowel disease therapy targets through druggabil...

Go to citation Crossref Google Scholar
Mendelian Randomization Analysis of the Causal Effect of Cigarette Smo...

Go to citation Crossref Google Scholar
Plasma proteomics of diabetic kidney disease among Asians with younger...

Go to citation Crossref Google Scholar
Association between body fat distribution and age at menarche: a two s...

Go to citation Crossref Google Scholar
Causal role of immune cells in inflammatory bowel disease: A Mendelian...

Go to citation Crossref Google Scholar
Sjögren’s syndrome and Parkinson’s Disease: A bidirectional two-sample...

Go to citation Crossref Google Scholar
Association between gut microbiota and gastric cancers: a two-sample M...

Go to citation Crossref Google Scholar
Exploring the role of gut microbiota in migraine risk: a two-sample Me...

Go to citation Crossref Google Scholar
Genetic evidence suggests a genetic association between major depressi...

Go to citation Crossref Google Scholar
Assessing causality between chronic obstructive pulmonary disease with...

Go to citation Crossref Google Scholar
Effects of childhood obesity on heart failure and its associated risk ...

Go to citation Crossref Google Scholar
Lipoprotein lipids and apolipoproteins in primary immune thrombocytope...

Go to citation Crossref Google Scholar
The enigmatic interplay of immune cells and abnormal spermatozoa throu...

Go to citation Crossref Google Scholar
A bidirectional Mendelian randomization study of gut microbiota and ce...

Go to citation Crossref Google Scholar
Causal Estimation of Long-term Intervention Cost-effectiveness Using G...

Go to citation Crossref Google Scholar Pub Med
Causal association between asthma and periodontitis: A t...

Go to citation Crossref Google Scholar
LDHB contributes to the regulation of lactate levels and basal insulin...

Go to citation Crossref Google Scholar
Evidence for causal effects of neuropsychiatric conditions on risk for...

Go to citation Crossref Google Scholar
The causal impact of complement C3d receptor 2 on head and neck cancer...

Go to citation Crossref Google Scholar
Mendelian randomization analysis using GWAS ...

Go to citation Crossref Google Scholar
The causal effect of iron status on risk of anxiety disorders: A two-s...

Go to citation Crossref Google Scholar
Causal role of immune cells in digestive system cancers: A Mendelian r...

Go to citation Crossref Google Scholar
Causal role of immune cells on risk of Parkinson’s disease: a Mendelia...

Go to citation Crossref Google Scholar
Genetic correlation between circulating metabolites and chalazion: a t...

Go to citation Crossref Google Scholar
Association between multiple sclerosis and cancer risk: A two-sample M...

Go to citation Crossref Google Scholar
The causal relationship between gut microbiota and biliary tract cance...

Go to citation Crossref Google Scholar
Evaluating the impact of childhood BMI on the risk of coronavirus dise...

Go to citation Crossref Google Scholar
GENIUS-MAWII: for robust Mendelian randomization with many weak invali...

Go to citation Crossref Google Scholar
Causal relationship between genetically predicted type 2 diabetes mell...

Go to citation Crossref Google Scholar
Treg cells as a protective factor for Hashimoto`s thyroiditis: a Mende...

Go to citation Crossref Google Scholar
Approaches to estimate bidirectional causal effects using Mendelian ra...

Go to citation Crossref Google Scholar
Causal relationship between PCSK9 inhibitor and primary glomerular dis...

Go to citation Crossref Google Scholar
Peripheral immune cell traits and Parkinson’s disease: A Mendelian ran...

Go to citation Crossref Google Scholar
Genetic evidence strengthens the bidirectional connection between gut ...

Go to citation Crossref Google Scholar
Causal effects of diabetic retinopathy on depression, anxiety and bipo...

Go to citation Crossref Google Scholar
Glycemic traits and colorectal cancer survival in a cohort of South Ko...

Go to citation Crossref Google Scholar
Major depressive disorder and the risk of irritable bowel syndrome: A ...

Go to citation Crossref Google Scholar
Investigating the causal relationship between human blood/urine metabo...

Go to citation Crossref Google Scholar
Association between autoimmunity-related disorders and prostate cancer...

Go to citation Crossref Google Scholar
Causal role of immune cells in ovarian dysfunction :a mendelian random...

Go to citation Crossref Google Scholar
Association of lipid-lowering drugs with osteoarthritis outcomes from ...

Go to citation Crossref Google Scholar
Brain imaging derived phenotypes: a biomarker for the onset of inflamm...

Go to citation Crossref Google Scholar
Association of immune cell traits with Parkinson’s disease: a Mendelia...

Go to citation Crossref Google Scholar
Investigating the potential causal association between consumption of ...

Go to citation Crossref Google Scholar
Investigating the shared genetic architecture between frailty and inso...

Go to citation Crossref Google Scholar
Associations between gut microbiota and adverse neurodevelopmental out...

Go to citation Crossref Google Scholar
Bacteroidaceae, Bacteroides, and Veillonella: emerging protectors agai...

Go to citation Crossref Google Scholar
A data-centric perspective on exposomics data analysis

Go to citation Crossref Google Scholar
Causal Effects of Gut Microbiota on Oral Cavity Cancer— A Two-Sample M...

Go to citation Crossref Google Scholar
Inflammatory cytokines and oral lichen planus: a Mendelian randomizati...

Go to citation Crossref Google Scholar
Unraveling the mystery: a Mendelian randomized exploration of gut micr...

Go to citation Crossref Google Scholar
Plasma Proteome-Wide Mendelian Randomization Analysis Reveals Biomarke...

Go to citation Crossref Google Scholar
BMI and Cardiometabolic Traits in Japanese: A Mendelian Randomization ...

Go to citation Crossref Google Scholar
Causal Relationship Between Immune Cells and Endometrial Cancer: A Two...

Go to citation Crossref Google Scholar
Bias and mean squared error in Mendelian randomization with invalid in...

Go to citation Crossref Google Scholar
Analysis of the correlation between immune cell characteristics and in...

Go to citation Crossref Google Scholar
Coffee and the risk of osteoarthritis: a two-sample, two-step multivar...

Go to citation Crossref Google Scholar
The causality between systemic inflammatory regulators and chronic res...

Go to citation Crossref Google Scholar
Impact of weight loss on cancer-related proteins in serum: results fro...

Go to citation Crossref Google Scholar
Genetic relationship between ageing and coronary heart disease: a Mend...

Go to citation Crossref Google Scholar
Causal effects of gut microbiota in the development of lung cancer and...

Go to citation Crossref Google Scholar
Systems genetics analysis reveals the common genetic basis for pain se...

Go to citation Crossref Google Scholar
Causal inference between serum bilirubin levels and juvenile idiopathi...

Go to citation Crossref Google Scholar
Proteome-wide Mendelian randomization identifies causal plasma protein...

Go to citation Crossref Google Scholar
Investigation of causal relationships between cortical structure and o...

Go to citation Crossref Google Scholar
Graves’ disease and systemic lupus erythematosus: a Mendelian randomiz...

Go to citation Crossref Google Scholar
Navigating Growth: A Formalized Approach to Small Firm Revenue and Pro...

Go to citation Crossref Google Scholar
Causal relationship between gut microbiota and myasthenia gravis: a tw...

Go to citation Crossref Google Scholar
Exploring causality between bone mineral density and frailty: A bidire...

Go to citation Crossref Google Scholar
Association Between Neurodegenerative Diseases and an Increased Risk o...

Go to citation Crossref Google Scholar
Association between inflammatory bowel disease and all-cause dementia:...

Go to citation Crossref Google Scholar
The impact of age-specific childhood body-mass index on adult cardiome...

Go to citation Crossref Google Scholar
Causal relationship between gut microflora and dementia: a Mendelian r...

Go to citation Crossref Google Scholar
The association between celiac disease and digestive system cancers: A...

Go to citation Crossref Google Scholar
Causal relationship between PCOS and related sex hormones with oral in...

Go to citation Crossref Google Scholar
Investigating Causal Effects of Hematologic Traits on Lung Cancer: A M...

Go to citation Crossref Google Scholar
Exploring the intestinal ecosystem: from gut microbiota to association...

Go to citation Crossref Google Scholar
Absence of causal relationship between Parkinson’s disease and subsequ...

Go to citation Crossref Google Scholar
Effects of Participating in Religious Groups on Mental Health Issues: ...

Go to citation Crossref Google Scholar
Causal relationship between asthma and inflammatory bowel disease : A ...

Go to citation Crossref Google Scholar
The bidirectional causal association between psoriasis and psychologic...

Go to citation Crossref Google Scholar
Mendelian Randomization as a Tool for Cardiovascular Research

Go to citation Crossref Google Scholar
Associations between Serum Mineral Nutrients, Gut Microbiota, and Risk...

Go to citation Crossref Google Scholar
The Impact of Different Intensities of Physical Activity on Serum Urat...

Go to citation Crossref Google Scholar
Causal Association between Tea Intake and Acute Cerebrovascular Events...

Go to citation Crossref Google Scholar
Wilcoxon-Mann-Whitney statistics in randomized trials with non-complia...

Go to citation Crossref Google Scholar
The Effect of the COVID-19 Pandemic on Non–COVID-19 Deaths: Population...

Go to citation Crossref Google Scholar
Addressing unmeasured confounders in cohort studies: Instrumental vari...

Go to citation Crossref Google Scholar
IFN-γ, SCF, MIP1b and IL-16 Were Associated with Risk of Diabetic Neph...

Go to citation Crossref Google Scholar
Exploring the Diagnostic Value of PANoptosis-Related Genes in Kidney R...

Go to citation Crossref Google Scholar
Comparison of instrumental variable methods with continuous exposure a...

Go to citation Crossref Google Scholar
The causal relationship between osteoarthritis and bladder cancer: A M...

Go to citation Crossref Google Scholar
Mendelian Randomization and Transcriptomic Analysis Reveal the Protect...

Go to citation Crossref Google Scholar
Psoriasis and gut microbiota: A Mendelian randomization study

Go to citation Crossref Google Scholar
Exploring the impact of computer game playing on cognitive function, A...

Go to citation Crossref Google Scholar
The causal relationship between immune cells and different kidney dise...

Go to citation Crossref Google Scholar
Association of modifiable risk factors with obstructive sleep apnea: a...

Go to citation Crossref Google Scholar
Causal relationship between asthma and ulcerative colitis and the medi...

Go to citation Crossref Google Scholar
Genetic analysis of the causal relationship between gut microbiota and...

Go to citation Crossref Google Scholar
Causal relationship between Helicobacter pylori antibodies and gastroe...

Go to citation Crossref Google Scholar
Large-scale genome-wide association study to identify causal relations...

Go to citation Crossref Google Scholar
Association of COVID-19 with Risk and Progression of Alzheimer’s Disea...

Go to citation Crossref Google Scholar
Causal relationships between serum matrix metalloproteinases and estro...

Go to citation Crossref Google Scholar
Genetic association of telomere length, obesity and tobacoo smoking wi...

Go to citation Crossref Google Scholar
Mendelian randomisation study of body composition and depression in pe...

Go to citation Crossref Google Scholar
Causal relationship between gut microbiota and cancers: a two-sample M...

Go to citation Crossref Google Scholar
Impact of the gut microbiota and associated metabolites on cardiometab...

Go to citation Crossref Google Scholar
Causality of genetically determined metabolites and metabolic pathways...

Go to citation Crossref Google Scholar
Mendelian randomization indicates that atopic dermatitis contributes t...

Go to citation Crossref Google Scholar
Resting heart rate and antisocial behaviour: a Mendelian randomisation...

Go to citation Crossref Google Scholar
BMI and well-being in people of East Asian and European ancestry: a Me...

Go to citation Crossref Google Scholar
Causal association between body mass index and temporomandibular disor...

Go to citation Crossref Google Scholar
Causal effect of gallstone disease on the risk of coronary heart disea...

Go to citation Crossref Google Scholar
Relationship of Klotho with cognition and dementia: Results from the N...

Go to citation Crossref Google Scholar
Causal associations between female reproductive behaviors and psychiat...

Go to citation Crossref Google Scholar
Causal association between body mass index and autoimmune thyroiditis:...

Go to citation Crossref Google Scholar
Investigating the causal interplay between sleep traits and risk of ac...

Go to citation Crossref Google Scholar
Causal associations of 25-hydroxyvitamin D with functional gastrointes...

Go to citation Crossref Google Scholar
A Mendelian randomization-based approach to explore the relationship b...

Go to citation Crossref Google Scholar
The serum lipid profiles in immune thrombocytopenia: Mendelian randomi...

Go to citation Crossref Google Scholar
Causal relationship between PCSK9 inhibitor and autoimmune diseases: a...

Go to citation Crossref Google Scholar
Causal role of immune cells in schizophrenia: Mendelian randomization ...

Go to citation Crossref Google Scholar
The association between walking pace and hand grip strength with the r...

Go to citation Crossref Google Scholar
Genetic variants for smoking behaviour and risk of skin cancer

Go to citation Crossref Google Scholar
The causal relationship between thyroid function, autoimune thyroid dy...

Go to citation Crossref Google Scholar
Body mass index and inflammation in depression and treatment-resistant...

Go to citation Crossref Google Scholar
TIE1 and TEK signalling, intraocular pressure, and primary open-angle ...

Go to citation Crossref Google Scholar
Mendelian randomization suggests a causal relationship between gut dys...

Go to citation Crossref Google Scholar
Causal association of gut microbiota and esophageal cancer: a Mendelia...

Go to citation Crossref Google Scholar
Gamma-glutamyl transferase and calculus of kidney incidence: a Mendeli...

Go to citation Crossref Google Scholar
Cannabis use and atherosclerotic cardiovascular disease: a Mendelian r...

Go to citation Crossref Google Scholar
Causal relationships between circulating inflammatory cytokines and di...

Go to citation Crossref Google Scholar
Evaluating the effect of childhood sunburn on the risk of cutaneous me...

Go to citation Crossref Google Scholar
Mendelian randomization study reveals the relationship between dietary...

Go to citation Crossref Google Scholar
Systemic lupus erythematosus Association between Osteomyelitis: A two-...

Go to citation Crossref Google Scholar
The relationship between gut microbiota and insomnia: a bi-directional...

Go to citation Crossref Google Scholar
Mendelian randomization reveals no correlations between herpesvirus in...

Go to citation Crossref Google Scholar
Gut microbiota and risk of lower respiratory tract infections: a bidir...

Go to citation Crossref Google Scholar
Causal link between gut microbiota and four types of pancreatitis: a g...

Go to citation Crossref Google Scholar
The causal relationship between gut microbiota and leukemia: a two-sam...

Go to citation Crossref Google Scholar
New insight into the causal relationship between Graves’ disease liabi...

Go to citation Crossref Google Scholar
Depression and sarcopenia-related traits: A Mendelian randomization st...

Go to citation Crossref Google Scholar
Direction of dependence analysis for pre-post assessments using non-Ga...

Go to citation Crossref Google Scholar
Systems genetics approaches for understanding complex traits with rele...

Go to citation Crossref Google Scholar
Association of coffee and caffeine consumption with risk and prognosis...

Go to citation Crossref Google Scholar
Instrumental Variable Analysis of Racial Discrimination and Blood Pres...

Go to citation Crossref Google Scholar
Serum uric acid levels and risk of cardiovascular disease in type 2 di...

Go to citation Crossref Google Scholar
Convergent application of traditional Chinese medicine and gut microbi...

Go to citation Crossref Google Scholar
Identification of blood protein biomarkers associated with prostate ca...

Go to citation Crossref Google Scholar
The relationship between Alzheimer disease and thyroiditis: A two-samp...

Go to citation Crossref Google Scholar
The causal association between thyroid disease and gout: A Mendelian r...

Go to citation Crossref Google Scholar
Associations between gut microbiota and Parkinson disease: A bidirecti...

Go to citation Crossref Google Scholar
The Role of Polyunsaturated Fatty Acids in Osteoarthritis: Insights fr...

Go to citation Crossref Google Scholar
Assessing the causal relationship between immune traits and systemic l...

Go to citation Crossref Google Scholar
Association of BMI and waist circumference with diabetic microvascular...

Go to citation Crossref Google Scholar
Association between the gut microbiota and nonalcoholic fatty liver di...

Go to citation Crossref Google Scholar
Bidirectional relationship between 56 peripheral inflammatory regulato...

Go to citation Crossref Google Scholar
Osteoarthritis and risk of type 2 diabetes: A two‐sample...

Go to citation Crossref Google Scholar
Impact of blood pressure and antihypertensive drug classes on intracra...

Go to citation Crossref Google Scholar
Socioeconomic status and severe mental disorders: a bidirectional mult...

Go to citation Crossref Google Scholar
Genetically predicted causal associations between periodontitis and ps...

Go to citation Crossref Google Scholar
Association between genetically predicted rheumatoid arthritis and alo...

Go to citation Crossref Google Scholar
Unraveling the role of VLDL in the relationship between type 2 diabete...

Go to citation Crossref Google Scholar
Genetically predicted levels of circulating cytokines and the risk of ...

Go to citation Crossref Google Scholar
Causal effects of gut microbiota on erectile dysfunction: a two-sample...

Go to citation Crossref Google Scholar
Causal associations between blood lipids and brain structures: a Mende...

Go to citation Crossref Google Scholar
Causal effects of walking pace on osteoarthritis: a two-sample mendeli...

Go to citation Crossref Google Scholar
Maternal and fetal origins of offspring blood pressure: statistical an...

Go to citation Crossref Google Scholar
Design and quality control of large-scale two-sample Mendelian randomi...

Go to citation Crossref Google Scholar
Borderline personality disorder and thyroid diseases: a Mendelian rand...

Go to citation Crossref Google Scholar
Heritable Traits and Lung Cancer Risk: A Two-Sample Mendelian Randomiz...

Go to citation Crossref Google Scholar
Causal relationship between body mass index, type 2 diabetes and bone ...

Go to citation Crossref Google Scholar
Evidence for a causal effect of major depressive disorder, anxiety on ...

Go to citation Crossref Google Scholar
Increased transferrin protects from thrombosis in Chuvash erythrocytos...

Go to citation Crossref Google Scholar
Association between Gastric Cancer and 12 Autoimmune Diseases: A Mende...

Go to citation Crossref Google Scholar
Causal Relationship between Gut Microbiota and Gout: A Two-Sample Mend...

Go to citation Crossref Google Scholar
Mendelian randomization study of diabetes and dementia in the Million ...

Go to citation Crossref Google Scholar
Causal inference on microbiome-metabolome relations in observational h...

Go to citation Crossref Google Scholar
Off-farm income promotes energy transition in the Pan-Third Pole cross...

Go to citation Crossref Google Scholar
Metabolomic Investigation of Major Depressive Disorder Identifies a Po...

Go to citation Crossref Google Scholar
Dissecting Causal Relationships Between Gut Microbiota, Blood Metaboli...

Go to citation Crossref Google Scholar
Causal effects of gut microbiota on the risk of bipolar disorder: a Me...

Go to citation Crossref Google Scholar
The causal association between polycystic ovary syndrome and susceptib...

Go to citation Crossref Google Scholar
Causal relationship between cannabis use and cancer: a genetically inf...

Go to citation Crossref Google Scholar
A Large Genetic Causal Analysis of the Gut Microbiota and Urological C...

Go to citation Crossref Google Scholar
Genome-wide Association Studies of Retinal Vessel Tortuosity Identify ...

Go to citation Crossref Google Scholar
An epidemiological introduction to human metabolomic investigations

Go to citation Crossref Google Scholar
Effect of obesity, lipids and adipokines on allergic rhinitis risk: a ...

Go to citation Crossref Google Scholar
The association of COVID-19 with diffuse large B-cell lymphoma: a Mend...

Go to citation Crossref Google Scholar
COVID−19 hospitalization increases the risk of developing glioblastoma...

Go to citation Crossref Google Scholar
Causal association between serum 25-Hydroxyvitamin D levels and cutane...

Go to citation Crossref Google Scholar
Mendelian randomization to evaluate the causal relationship between li...

Go to citation Crossref Google Scholar
Proteome‐Wide Mendelian Randomization Analysis Identified Potential Dr...

Go to citation Crossref Google Scholar
Iron status and sarcopenia-related traits: A bi-directional Mendelian ...

Go to citation Crossref Google Scholar
Statistical Learning Methods for Neuroimaging Data Analysis with Appli...

Go to citation Crossref Google Scholar
Genetic influences of the effect of circulating inflammatory cytokines...

Go to citation Crossref Google Scholar
Sex-specific Mendelian randomisation to assess the causality of sex di...

Go to citation Crossref Google Scholar
Accounting for antihypertensive medication in Mendelian randomization ...

Go to citation Crossref Google Scholar
Two-stage multivariate Mendelian randomization on multiple outcomes wi...

Go to citation Crossref Google Scholar Pub Med
Why caution should be applied when interpreting and promoting findings...

Go to citation Crossref Google Scholar
Causal relationship between osteoarthritis with atrial fibrillation an...

Go to citation Crossref Google Scholar
COVID-19 and the risk of neuromyelitis optica spectrum disorder: a Men...

Go to citation Crossref Google Scholar
The association between gut microbiome and PCOS: evidence from meta-an...

Go to citation Crossref Google Scholar
Assessing causal relationships between gut microbiota and asthma: evid...

Go to citation Crossref Google Scholar
Genetic association of hypertension and several other metabolic disord...

Go to citation Crossref Google Scholar
Causal inference with invalid instruments: post-selection problems and...

Go to citation Crossref Google Scholar
Phenotypic and Genetic Characteristics of Retinal Vascular Parameters ...

Go to citation Crossref Google Scholar
Age at Menarche, age at Natural Menopause, and Risk of Lung and Colore...

Go to citation Crossref Google Scholar
Potential protection of computer gaming against mental health issues: ...

Go to citation Crossref Google Scholar
Association between genetically determined telomere length and health‐...

Go to citation Crossref Google Scholar
Bidirectional Causality Between Immunoglobulin G N-Glycosylation and M...

Go to citation Crossref Google Scholar
Genetic variants in HFE are associated with non-alcoholic fatty liver ...

Go to citation Crossref Google Scholar
Prioritization of nasal polyp-associated genes by integrating GWAS and...

Go to citation Crossref Google Scholar
A Bayesian approach for two‐stage multivariate Mendelian randomization...

Go to citation Crossref Google Scholar
Downregulation of interleukin 6 signaling might reduce the risk of per...

Go to citation Crossref Google Scholar
Relationship between genetically proxied vitamin D and psoriasis risk:...

Go to citation Crossref Google Scholar
Impact of altering referral threshold from out-of-hours primary care t...

Go to citation Crossref Google Scholar
How Does Enterprise Digital Transformation Affect Total Factor Product...

Go to citation Crossref Google Scholar
Genome-wide association meta-analysis of spontaneous coronary artery d...

Go to citation Crossref Google Scholar
Socioeconomic status and asthma: A bidirectional Mendelian randomizati...

Go to citation Crossref Google Scholar
MR‐BOIL: Causal inference in one‐sample Mendelian randomization for bi...

Go to citation Crossref Google Scholar
Causal associations between insulin-like growth factor 1 and vitamin D...

Go to citation Crossref Google Scholar
Causal effects of gut microbiota on sepsis: a two-sample Mendelian ran...

Go to citation Crossref Google Scholar
Multiple sclerosis and breast cancer risk: a meta-analysis of observat...

Go to citation Crossref Google Scholar
GALC variants affect galactosylceramidase enzymatic acti...

Go to citation Crossref Google Scholar
Circulating proteins and peripheral artery disease risk: observational...

Go to citation Crossref Google Scholar
Association of Mitochondrial DNA Copy Number With Brain MRI Markers an...

Go to citation Crossref Google Scholar
Partner choice, confounding and trait convergence all contribute to ph...

Go to citation Crossref Google Scholar
Cardiovascular Safety in Type 2 Diabetes With Sulfonylureas as Second-...

Go to citation Crossref Google Scholar
Assessing the Causal Association between Biological Aging Biomarkers a...

Go to citation Crossref Google Scholar
A Positive Causal Relationship between Noodle Intake and Metabolic Syn...

Go to citation Crossref Google Scholar
Causality assessment of circulating Vitamin D level on venous thromboe...

Go to citation Crossref Google Scholar
Plasma protein and venous thromboembolism: prospective cohort and mend...

Go to citation Crossref Google Scholar
Population‐based discovery and Mendelian randomization analysis identi...

Go to citation Crossref Google Scholar
Genetically predicted alterations in thyroid function are associated w...

Go to citation Crossref Google Scholar
Instrumental variable regression via kernel maximum moment loss

Go to citation Crossref Google Scholar
Multi-Threshold Structural Equation Model

Go to citation Crossref Google Scholar
Genetic correlation and gene-based pleiotropy analysis for four major ...

Go to citation Crossref Google Scholar
Celiac Disease Is a Risk Factor for Mature T and NK Cell Lymphoma: A M...

Go to citation Crossref Google Scholar
Metformin Treatment Reduces the Incidence of Rheumatoid Arthritis: A T...

Go to citation Crossref Google Scholar
The Impact of the Digital Economy on High-Quality Development of Agric...

Go to citation Crossref Google Scholar
New proposal to address mediation analysis interrogations by using gen...

Go to citation Crossref Google Scholar
Exploring the relationship between socioeconomic deprivation index and...

Go to citation Crossref Google Scholar
Dissecting the causal effect between gut microbiota, DHA, and urate me...

Go to citation Crossref Google Scholar
Ankylosing spondylitis and glaucoma in European population: A Mendelia...

Go to citation Crossref Google Scholar
Cortisol and periodontitis: Prospective observational and Mendelian ra...

Go to citation Crossref Google Scholar
The role of plasma cortisol in dementia, epilepsy, and multiple sclero...

Go to citation Crossref Google Scholar
A bi-directional Mendelian randomization study of sarcopenia-related t...

Go to citation Crossref Google Scholar
Association between brain structures and migraine: A bidirectional Men...

Go to citation Crossref Google Scholar
Among-Hospital Variation in Intensive Care Unit Admission Practices an...

Go to citation Crossref Google Scholar
Sleep behaviors and Parkinson's disease: A bidirectional Mendelian ran...

Go to citation Crossref Google Scholar
Roles of gut microbiome in epilepsy risk: A Mendelian randomization st...

Go to citation Crossref Google Scholar
Impact of genetically predicted characterization of mitochondrial DNA ...

Go to citation Crossref Google Scholar
The causal role of gastroesophageal reflux disease in anxiety disorder...

Go to citation Crossref Google Scholar
Circulatory proteins relate cardiovascular disease to cognitive perfor...

Go to citation Crossref Google Scholar
Association of Body Mass Index and the Risk of Gastro-Esophageal Cance...

Go to citation Crossref Google Scholar
Causal associations between dried fruit intake and cardiovascular dise...

Go to citation Crossref Google Scholar
Causal effects of maternal circulating amino acids on offspring birthw...

Go to citation Crossref Google Scholar
A Causal and Inverse Relationship between Plant-Based Diet Intake and ...

Go to citation Crossref Google Scholar
Can Intelligence Affect Alcohol-, Smoking-, and Physical Activity-Rela...

Go to citation Crossref Google Scholar
Statistical methods for cis ‐Mendelian ran...

Go to citation Crossref Google Scholar
Investigating the causal relationships between excess adiposity and ca...

Go to citation Crossref Google Scholar
Examining the Lancet Commission risk factors for dementia using Mendel...

Go to citation Crossref Google Scholar
Assessment of bidirectional relationships between circulating cytokine...

Go to citation Crossref Google Scholar
Multi-omic association study identifies DNA methylation-mediated genot...

Go to citation Crossref Google Scholar
Human blood metabolites and lacunar stroke: A Mendelian randomization ...

Go to citation Crossref Google Scholar Pub Med
Genetically predicted tobacco consumption and risk of intracranial ane...

Go to citation Crossref Google Scholar
Instrumental variable estimation of causal effects with applying some ...

Go to citation Crossref Google Scholar
Mendelian randomization study on the causal effects of systemic lupus ...

Go to citation Crossref Google Scholar
Psychosocial Disorders and Gastroesophageal Reflux Disease: A Bidirect...

Go to citation Crossref Google Scholar
Causal Relationships of General and Abdominal Adiposity on Osteoarthri...

Go to citation Crossref Google Scholar
The Association between Blood Lipids and Systemic Lupus Erythematosus:...

Go to citation Crossref Google Scholar
Associations of Lipids and Lipid-Lowering Drugs with Risk of Vascular ...

Go to citation Crossref Google Scholar
The effect of systemic antihypertensive drugs on the risk of primary o...

Go to citation Crossref Google Scholar
Vitamin D and suicidality: a Chinese early adolescent cohort and Mende...

Go to citation Crossref Google Scholar
Mendelian randomization and the association of fibroblast growth facto...

Go to citation Crossref Google Scholar
Overall Obesity Not Abdominal Obesity Has a Causal Relationship with O...

Go to citation Crossref Google Scholar
An Explanation of Path Analysis and Recommendations for Best Practice

Go to citation Crossref Google Scholar
Losing our way? A critical examination of path analysis in accounting ...

Go to citation Crossref Google Scholar
Association Between Endometritis and Endometrial Polyp: A Mendelian Ra...

Go to citation Crossref Google Scholar
Interpretation of Mendelian randomization using a single measure of an...

Go to citation Crossref Google Scholar
Contributions of obesity to kidney health and disease: insights from M...

Go to citation Crossref Google Scholar
Estimating causal effects of genetically predicted type 2 diabetes on ...

Go to citation Crossref Google Scholar
Genetically predicted iron status was associated with the risk of pros...

Go to citation Crossref Google Scholar
Circulating lipoprotein(a) levels and health outcomes: Phenome-wide Me...

Go to citation Crossref Google Scholar
Dissecting the association between psychiatric disorders and neurologi...

Go to citation Crossref Google Scholar
Robust Mendelian randomization in the presence of residual population ...

Go to citation Crossref Google Scholar
The genetic architecture of pneumonia susceptibility implicates mucin ...

Go to citation Crossref Google Scholar
Genome-wide associations of aortic distensibility suggest causality fo...

Go to citation Crossref Google Scholar
Quantifying the role of transcript levels in mediating DNA methylation...

Go to citation Crossref Google Scholar
Mendelian randomisation analyses of UK Biobank and published data sugg...

Go to citation Crossref Google Scholar
GIVE statistic for goodness of fit in instrumental variables models wi...

Go to citation Crossref Google Scholar
Absence of causal association between Vitamin D and bone mineral densi...

Go to citation Crossref Google Scholar
Mendelian randomization study reveals a causal relationship between ad...

Go to citation Crossref Google Scholar
Reporting methodological issues of the mendelian randomization studies...

Go to citation Crossref Google Scholar
Exploring the causal effect of maternal pregnancy adiposity on offspri...

Go to citation Crossref Google Scholar
Associations of genetically predicted IL-6 signaling with cardiovascul...

Go to citation Crossref Google Scholar
Investigating causal relations between sleep duration and risks of adv...

Go to citation Crossref Google Scholar
Mendelian randomization analysis of factors related to ovulation and r...

Go to citation Crossref Google Scholar
Health effects of milk consumption: phenome-wide Mendelian randomizati...

Go to citation Crossref Google Scholar
Exploring genetic association of insomnia with allergic disease and as...

Go to citation Crossref Google Scholar
Exploring the association between birthweight and breast cancer using ...

Go to citation Crossref Google Scholar
CSF sTREM2 in neurological diseases: a two-sample Mendelian randomizat...

Go to citation Crossref Google Scholar
Genetic insights into the risk of snoring on stroke and ischemic strok...

Go to citation Crossref Google Scholar
Causal relationship between type 1 diabetes and hypothyroidism: A Mend...

Go to citation Crossref Google Scholar
Statistical power of transcriptome‐wide association studies

Go to citation Crossref Google Scholar
Determining the Relationship Between Blood Pressure, Kidney Function, ...

Go to citation Crossref Google Scholar
Educational attainment and endometrial cancer: A Mendelian randomizati...

Go to citation Crossref Google Scholar
The impact of fatty acids biosynthesis on the risk of cardiovascular d...

Go to citation Crossref Google Scholar
Causal association between inflammatory bowel disease and IgA nephropa...

Go to citation Crossref Google Scholar
Milk intake and incident stroke and CHD in populations of European des...

Go to citation Crossref Google Scholar
Intelligence, education level, and risk of Parkinson’s disease in Euro...

Go to citation Crossref Google Scholar
Causal associations between CD40/CD40L and aortic diseases: A mendelia...

Go to citation Crossref Google Scholar
Assessing the causal role of hypertension on left atrial and left vent...

Go to citation Crossref Google Scholar
Type 2 diabetes mellitus and the risk of abnormal spermatozoa: A Mende...

Go to citation Crossref Google Scholar
Causal association between heart failure and bone mineral density: Ins...

Go to citation Crossref Google Scholar
Alcohol consumption and smoking in relation to psoriasis: a Mendelian ...

Go to citation Crossref Google Scholar
Bidirectional Association between Major Depressive Disorder and Gastro...

Go to citation Crossref Google Scholar
Investigating Causal Associations of Circulating Micronutrients Concen...

Go to citation Crossref Google Scholar
The impacts of non-farming income on rural household energy choices: E...

Go to citation Crossref Google Scholar
Deep mendelian randomization: Investigating the causal knowledge of ge...

Go to citation Crossref Google Scholar
The immune factors have complex causal regulation effects on bone mine...

Go to citation Crossref Google Scholar
Constructing an atlas of associations between polygenic scores from ac...

Go to citation Crossref Google Scholar
Physical activity, sedentary time and breast cancer risk: a Mendelian ...

Go to citation Crossref Google Scholar
A Guide for Understanding and Designing Mendelian Randomization Studie...

Go to citation Crossref Google Scholar
Osteoarthritis & stroke: a bidirectional mendelian randomization study

Go to citation Crossref Google Scholar
Alcohol consumption and colorectal cancer risk: A mendelian randomizat...

Go to citation Crossref Google Scholar
The causal impact of childhood obesity on bone mineral density and fra...

Go to citation Crossref Google Scholar
Simultaneous Maximum Likelihood Estimation for Piecewise Linear Instru...

Go to citation Crossref Google Scholar
Causal associations of alcohol consumption with cardiovascular disease...

Go to citation Crossref Google Scholar
Identification and single-base gene-editing functional validation of a...

Go to citation Crossref Google Scholar
Causal inference on neuroimaging data with Mendelian randomisation

Go to citation Crossref Google Scholar
Causal relationships between rheumatism and dyslipidemia: A two-sample...

Go to citation Crossref Google Scholar
Investigating Causal Relations between Genetic-Related Intermediate En...

Go to citation Crossref Google Scholar
Systematic evaluation for the causal effects of blood metabolites on o...

Go to citation Crossref Google Scholar
A mendelian randomization study with populations of European ancestry ...

Go to citation Crossref Google Scholar
Estimating the causal effect of frailty index on vestibular disorders:...

Go to citation Crossref Google Scholar
Assessment of the Causal Effect of IgG N-Glycosylation Level on Risk o...

Go to citation Crossref Google Scholar
Using multivariable Mendelian randomization to estimate the causal eff...

Go to citation Crossref Google Scholar
Identifying factors contributing to increased susceptibility to COVID-...

Go to citation Crossref Google Scholar
Genetic Evidence for a Causal Role of Serum Phosphate in Coronary Arte...

Go to citation Crossref Google Scholar
An Overview of Methods and Exemplars of the Use of Mendelian Randomisa...

Go to citation Crossref Google Scholar
Causal Effect of Age at Menarche on the Risk for Depression: Results F...

Go to citation Crossref Google Scholar
Polyunsaturated Fatty Acids Level and Bone Mineral Density: A Two-Samp...

Go to citation Crossref Google Scholar
Estimating Malaria Vaccine Efficacy in the Absence of a Gold Standard ...

Go to citation Crossref Google Scholar
Causal mediation analysis with multiple causally non-ordered and order...

Go to citation Crossref Google Scholar Pub Med
Genetically predicted physical activity is associated with lower serum...

Go to citation Crossref Google Scholar
High-density lipoprotein, low-density lipoprotein and triglyceride lev...

Go to citation Crossref Google Scholar
Cannabis use and the risk of periodontitis: A two‐sample Mendelian ran...

Go to citation Crossref Google Scholar
Nonparametric bounds in two‐sample summary‐data Mendelian randomizatio...

Go to citation Crossref Google Scholar
Low‐Density Lipoprotein Cholesterol Attributable Cardiovascular Diseas...

Go to citation Crossref Google Scholar
Causality of anthropometric markers associated with polycystic ovarian...

Go to citation Crossref Google Scholar
A multi-population phenome-wide association study of genetically-predi...

Go to citation Crossref Google Scholar
Large-scale Integrated Analysis of Genetics and Metabolomic Data Revea...

Go to citation Crossref Google Scholar
Doubly debiased lasso: High-dimensional inference under hidden confoun...

Go to citation Crossref Google Scholar
Review of Mendelian Randomization Studies on Endometrial Cancer

Go to citation Crossref Google Scholar
Survival Outcomes Associated With Cytoreductive Nephrectomy in Patient...

Go to citation Crossref Google Scholar
Powerful and robust inference of complex phenotypes' causal genes with...

Go to citation Crossref Google Scholar
Is Hearing Impairment Causally Associated With Falls? Evidence From a ...

Go to citation Crossref Google Scholar
Genetic associations of adult height with risk of cardioembolic and ot...

Go to citation Crossref Google Scholar
The African Female Breast Cancer Epidemiology Study Protocol

Go to citation Crossref Google Scholar
Association between plasma proteome and childhood neurodevelopmental d...

Go to citation Crossref Google Scholar
Assessing causality between osteoarthritis with urate levels and gout:...

Go to citation Crossref Google Scholar
Total Brain Volumetric Measures and Schizophrenia Risk: A Two-Sample M...

Go to citation Crossref Google Scholar
Exploration of errors in variance caused by using the first-order appr...

Go to citation Crossref Google Scholar
A Mendelian Randomization Study of the Effect of Tea Intake on Type 2 ...

Go to citation Crossref Google Scholar
Invited Commentary: Estimation and Bounds Under Data Fusion

Go to citation Crossref Google Scholar
Impact of nonrandom selection mechanisms on the causal effect estimati...

Go to citation Crossref Google Scholar
Likelihood-based Mendelian randomization analysis with automated instr...

Go to citation Crossref Google Scholar
Multiple genetic variations of chronic rhinosinusitis with nasal polyp...

Go to citation Crossref Google Scholar
Understanding the consequences of educational inequalities on periodon...

Go to citation Crossref Google Scholar
Evidence for causal effects of sleep disturbances on risk for osteoart...

Go to citation Crossref Google Scholar
Identification of causal metabolites related to multiple autoimmune di...

Go to citation Crossref Google Scholar
Total Cholesterol and APOE-Related Risk for Alzheimer’s Disease in the...

Go to citation Crossref Google Scholar
Association Between Neuroticism and Risk of Lung Cancer: Results From ...

Go to citation Crossref Google Scholar
Gut Microbiota and Psychiatric Disorders: A Two-Sample Mendelian Rando...

Go to citation Crossref Google Scholar
Elevated blood pressure, antihypertensive medications and bone health ...

Go to citation Crossref Google Scholar
Total body bone mineral density and severe COVID-19: A Mendelian rando...

Go to citation Crossref Google Scholar
Impact of transitioning from long-term to intermittent opioid therapy ...

Go to citation Crossref Google Scholar
Germline genetic variability in pancreatic cancer risk and prognosis

Go to citation Crossref Google Scholar
Mendelian randomization in the multivariate general linear model frame...

Go to citation Crossref Google Scholar
Causal Association Between Heart Failure and Alzheimer’s Disease: A Tw...

Go to citation Crossref Google Scholar
Causal Inference: Efficacy and Mechanism Evaluation

Go to citation Crossref Google Scholar
Use of twin studies to make inference about causation for measured exp...

Go to citation Crossref Google Scholar
Coffee and Caffeine Consumption and Risk of Kidney Stones: A Mendelian...

Go to citation Crossref Google Scholar
Chronic Kidney Disease and Risk of Renal Cell Carcinoma: A Two-Sample ...

Go to citation Crossref Google Scholar
The Crucial Role of Immune Factors in Susceptibility to SARS-CoV-2 Inf...

Go to citation Crossref Google Scholar
Chief Editor Internatioal Journal of Cardiologyassessing Causality bet...

Go to citation Crossref Google Scholar
Morning Cortisol and Circulating Inflammatory Cytokine Levels: A Mende...

Go to citation Crossref Google Scholar
Serum Phosphate, BMI, and Body Composition of Middle-Aged and Older Ad...

Go to citation Crossref Google Scholar
Global corporate social responsibility reporting regulation

Go to citation Crossref Google Scholar
Mendelian Randomization With Repeated Measures of a Time-varying Expos...

Go to citation Crossref Google Scholar
Statistical methods for Mendelian randomization in genome-wide associa...

Go to citation Crossref Google Scholar
Causal effect of atrial fibrillation/flutter on chronic kidney disease...

Go to citation Crossref Google Scholar
Assessing Causal Relationship Between Human Blood Metabolites and Five...

Go to citation Crossref Google Scholar
Higher maternal adiposity reduces offspring birthweight if associated ...

Go to citation Crossref Google Scholar
The causal role of gut microbiota in development of osteoarthritis

Go to citation Crossref Google Scholar
Shortened leukocyte telomere length in young adults who use methamphet...

Go to citation Crossref Google Scholar
Graphical analysis for phenome-wide causal discovery in genotyped popu...

Go to citation Crossref Google Scholar
Macular thickness varies with age-related macular degeneration genetic...

Go to citation Crossref Google Scholar
Genetically determined hypoalbuminemia as a risk factor for hypertensi...

Go to citation Crossref Google Scholar
Understanding the effect of smoking and drinking behavior on Parkinson...

Go to citation Crossref Google Scholar
Modelling hospital outcome: problems with endogeneity

Go to citation Crossref Google Scholar
Instrumental variable estimation for a time-varying treatment and a ti...

Go to citation Crossref Google Scholar
A comprehensive gene-centric pleiotropic association analysis for 14 p...

Go to citation Crossref Google Scholar
Estimating causal effects of atherogenic lipid-related traits on COVID...

Go to citation Crossref Google Scholar
Using genetic variants to evaluate the causal effect of serum vitamin ...

Go to citation Crossref Google Scholar
Beyond association: successes and challenges in linking non-coding gen...

Go to citation Crossref Google Scholar
Causal relationships between genetically determined metabolites and hu...

Go to citation Crossref Google Scholar
Two‐Sample Multivariable Mendelian Randomization Analysis Using R

Go to citation Crossref Google Scholar
The use of two-sample methods for Mendelian randomization analyses on ...

Go to citation Crossref Google Scholar
Relationship Between Serum Amino Acid Levels and Bone Mineral Density:...

Go to citation Crossref Google Scholar
Untargeted Metabolome- and Transcriptome-Wide Association Study Sugges...

Go to citation Crossref Google Scholar
Mendelian randomization under the omnigenic architecture

Go to citation Crossref Google Scholar
Mendelian randomisation highlights hypothyroidism as a causal determin...

Go to citation Crossref Google Scholar
Genetically Predicted Glucose-Dependent Insulinotropic Polypeptide (GI...

Go to citation Crossref Google Scholar
Research on the Spatio-Temporal Impacts of Environmental Factors on th...

Go to citation Crossref Google Scholar
Testing the association between tobacco smoking, alcohol consumption, ...

Go to citation Crossref Google Scholar
Genetically determined selenium concentrations and risk for autoimmune...

Go to citation Crossref Google Scholar
Type 2 Diabetes Mellitus and Amyotrophic Lateral Sclerosis: Genetic Ov...

Go to citation Crossref Google Scholar
Evaluating Causal Relationship Between Metabolites and Six Cardiovascu...

Go to citation Crossref Google Scholar
Bayesian variable selection with a pleiotropic loss function in Mendel...

Go to citation Crossref Google Scholar
Genetically Predicted Longer Telomere Length May Reduce Risk of Hip Os...

Go to citation Crossref Google Scholar
Posttraumatic Stress Disorder and Cardiovascular Disease

Go to citation Crossref Google Scholar
Instrumental variable estimation of early treatment effect in randomiz...

Go to citation Crossref Google Scholar
A Chinese host genetic study discovered IFNs and causality of laborato...

Go to citation Crossref Google Scholar
From GWAS to Gene: Transcriptome-Wide Association Studies and Other Me...

Go to citation Crossref Google Scholar
Association of Methylenetetrahydrofolate Reductase C677T Gene Polymorp...

Go to citation Crossref Google Scholar
Suggestive Evidence for Causal Effect of Leptin Levels on Risk for Ano...

Go to citation Crossref Google Scholar
Food groups associated with immune-mediated inflammatory diseases: a M...

Go to citation Crossref Google Scholar
Lung function and cardiovascular disease: a two-sample Mendelian rando...

Go to citation Crossref Google Scholar
The Confidence Interval Method for Selecting Valid Instrumental Variab...

Go to citation Crossref Google Scholar
Review of Mendelian Randomization Studies on Ovarian Cancer

Go to citation Crossref Google Scholar
Cannabis Use and the Risk of Cardiovascular Diseases: A Mendelian Rand...

Go to citation Crossref Google Scholar
A mendelian randomization study on causal effects of 25(OH)vitamin D l...

Go to citation Crossref Google Scholar
Beyond factor H: The impact of genetic-risk variants for age-related m...

Go to citation Crossref Google Scholar
Assessing the Relationship Between Serum Urate and Urolithiasis Using ...

Go to citation Crossref Google Scholar
Debiased inverse-variance weighted estimator in two-sample summary-dat...

Go to citation Crossref Google Scholar
The relationship between body mass index and income: Using genetic var...

Go to citation Crossref Google Scholar
Does agricultural trade liberalization increase obesity in developing ...

Go to citation Crossref Google Scholar
Association of Genetic Variants for Plasma LRG1 With Rapid Decline in ...

Go to citation Crossref Google Scholar
Causal Effect of Adiposity Measures on Blood Pressure Traits in 2 Urba...

Go to citation Crossref Google Scholar
Depression and interleukin-6 signaling: A Mendelian Randomization stud...

Go to citation Crossref Google Scholar
Using Mendelian randomization analysis to better understand the relati...

Go to citation Crossref Google Scholar
A Causal Web between Chronotype and Metabolic Health Traits

Go to citation Crossref Google Scholar
A modifiable risk factors atlas of lung cancer: A Mendelian randomizat...

Go to citation Crossref Google Scholar
A novel Mendelian randomization method with binary risk factor and out...

Go to citation Crossref Google Scholar
Instrument Residual Estimator for Any Response Variable with Endogenou...

Go to citation Crossref Google Scholar
Examining the possible causal relationship between lung function, COPD...

Go to citation Crossref Google Scholar
Causal association of adipokines with osteoarthritis: a Mendelian rand...

Go to citation Crossref Google Scholar
Instrumental Heterogeneity in Sex-Specific Two-Sample Mendelian Random...

Go to citation Crossref Google Scholar
Educational Attainment Decreases the Risk of COVID-19 Severity in the ...

Go to citation Crossref Google Scholar
Genetically predicted serum uric acid levels and the risk of coronary ...

Go to citation Crossref Google Scholar
Methods to Address Self-Selection and Reverse Causation in Studies of ...

Go to citation Crossref Google Scholar
Genetically Predicted Serum Iron Status Is Associated with Altered Ris...

Go to citation Crossref Google Scholar
Relationship Between Blood Pressure and Incident Cardiovascular Diseas...

Go to citation Crossref Google Scholar
Associations between lifestyle and amyotrophic lateral sclerosis strat...

Go to citation Crossref Google Scholar
Estimating the causal effect of BMI on mortality risk in people with h...

Go to citation Crossref Google Scholar
Evaluation of the causal effects of blood lipid levels on gout with su...

Go to citation Crossref Google Scholar
Alcohol Consumption Is Associated with Poor Prognosis in Obese Patient...

Go to citation Crossref Google Scholar
Life course body mass index through childhood and young adulthood and ...

Go to citation Crossref Google Scholar
No Evidence for a Causal Relationship Between Cancers and Parkinson’s ...

Go to citation Crossref Google Scholar
Mendelian randomization provides evidence for a causal effect of highe...

Go to citation Crossref Google Scholar
Shared Genetic Background Between Cerebrospinal Fluid Biomarkers and R...

Go to citation Crossref Google Scholar
Tuberculosis Exposure With Risk of Behçet Disease Among Patients With ...

Go to citation Crossref Google Scholar
The Role of Conspiracy Theories in the Spread of COVID-19 across the U...

Go to citation Crossref Google Scholar
Evidence for shared genetics between physical activity, sedentary beha...

Go to citation Crossref Google Scholar
Polygenic scores for dyslipidemia: the emerging genomic model of plasm...

Go to citation Crossref Google Scholar
Tumor Necrosis Factor Inhibition and Parkinson Disease

Go to citation Crossref Google Scholar
Mendelian Randomization With Refined Instrumental Variables From Genet...

Go to citation Crossref Google Scholar
Evaluating the role of alcohol consumption in breast and ovarian cance...

Go to citation Crossref Google Scholar
The impact of homocysteine on the risk of coronary artery diseases in ...

Go to citation Crossref Google Scholar
Exploring the causal pathway from bilirubin to CVD and diabetes in the...

Go to citation Crossref Google Scholar
Higher tea consumption is associated with decreased risk of small vess...

Go to citation Crossref Google Scholar
Causal relationship between the timing of menarche and young adult bod...

Go to citation Crossref Google Scholar
Using Mendelian randomization to evaluate the effects of alcohol consu...

Go to citation Crossref Google Scholar
Dissecting the Association Between Inflammation, Metabolic Dysregulati...

Go to citation Crossref Google Scholar
Multi-trait transcriptome-wide association studies with probabilistic ...

Go to citation Crossref Google Scholar
Assessing causal relationship from gut microbiota to heel bone mineral...

Go to citation Crossref Google Scholar
Genome-wide association study of non-tuberculous mycobacterial pulmona...

Go to citation Crossref Google Scholar
The Role of Gallstones in Gallbladder Cancer in India: A Mendelian Ran...

Go to citation Crossref Google Scholar
Genetically Predicted Blood Pressure and Risk of Atrial Fibrillation

Go to citation Crossref Google Scholar
Type 2 Diabetes and Glycemic Traits Are Not Causal Factors of Osteoart...

Go to citation Crossref Google Scholar
Association Between Circulating Linoleic Acid and Risk of Ischemic Str...

Go to citation Crossref Google Scholar
Approach for Genetic Studies

Go to citation Crossref Google Scholar
Statistical approaches to rare disease analyses

Go to citation Crossref Google Scholar
Mendelian randomization for studying the effects of perturbing drug ta...

Go to citation Crossref Google Scholar
Mendelian randomization for studying the effects of perturbing drug ta...

Go to citation Crossref Google Scholar
Revisiting the Relationship between Birthweight and Breast Cancer from...

Go to citation Crossref Google Scholar
Association Between Plasma Proteome and Childhood Neurodevelopmental D...

Go to citation Crossref Google Scholar
Assessing the Associations of Growth Differentiation Factor 15 with Rh...

Go to citation Crossref Google Scholar
Causal Relationship between Adiponectin and Diabetic Retinopathy: A Me...

Go to citation Crossref Google Scholar
Lipids, Anthropometric Measures, Smoking and Physical Activity Mediate...

Go to citation Crossref Google Scholar
Conventional and Genetic Evidence on the Association between Adiposity...

Go to citation Crossref Google Scholar
Estimating the Effect of Reduced Attendance at Emergency Departments f...

Go to citation Crossref Google Scholar
Formulating causal questions and principled statistical answers

Go to citation Crossref Google Scholar
Education, biological ageing, all-cause and cause-specific mortality a...

Go to citation Crossref Google Scholar
Testing and controlling for horizontal pleiotropy with probabilistic M...

Go to citation Crossref Google Scholar
Mendelian randomization implies no direct causal association between l...

Go to citation Crossref Google Scholar
Can increasing years of schooling reduce type 2 diabetes (T2D)?: Evide...

Go to citation Crossref Google Scholar
Sparse reduced-rank regression for integrating omics data

Go to citation Crossref Google Scholar
A novel method for controlling unobserved confounding using double con...

Go to citation Crossref Google Scholar
Clinical and genetic relationships between the QTc interval and risk o...

Go to citation Crossref Google Scholar
Instrumental Variables Regression with Measurement Errors and Multicol...

Go to citation Crossref Google Scholar
Birth weight is positively associated with adult osteoporosis risk: ob...

Go to citation Crossref Google Scholar
Rare and Common Variants in GALNT3 May Affect Bone Mass Independently ...

Go to citation Crossref Google Scholar
Bidirectional Causal Associations Between Inflammatory Bowel Disease a...

Go to citation Crossref Google Scholar
Nonalcoholic Fatty Liver Disease and Estimated Insulin Resistance in O...

Go to citation Crossref Google Scholar
Association of a Novel Index of Hospital Capacity Strain with Admissio...

Go to citation Crossref Google Scholar
Assessing the Relationship Between Leukocyte Telomere Length and Cance...

Go to citation Crossref Google Scholar
Alcohol Use and Depression: A Mendelian Randomization Study From China

Go to citation Crossref Google Scholar
Integrative Genomic Analysis Reveals Four Protein Biomarkers for Plate...

Go to citation Crossref Google Scholar
Genetics of height and risk of atrial fibrillation: A Mendelian random...

Go to citation Crossref Google Scholar
Sleep, major depressive disorder, and Alzheimer disease

Go to citation Crossref Google Scholar
Estimation of high-dimensional directed acyclic graphs with surrogate ...

Go to citation Crossref Google Scholar
Assessment of causality of natriuretic peptides and atrial fibrillatio...

Go to citation Crossref Google Scholar
Interleukin‐1 receptor antagonist, interleukin‐2 receptor alpha subuni...

Go to citation Crossref Google Scholar
Milk Consumption for the Prevention of Fragility Fractures

Go to citation Crossref Google Scholar
Protein QTL analysis of IGF-I and its binding proteins provides insigh...

Go to citation Crossref Google Scholar
Relationship between birth weight and chronic kidney disease: evidence...

Go to citation Crossref Google Scholar
Using instrumental variables to estimate the attributable fraction

Go to citation Crossref Google Scholar Pub Med
Circulating interleukins in relation to coronary artery disease, atria...

Go to citation Crossref Google Scholar
Commentary: Mendelian randomization and education–Challenges remain

Go to citation Crossref Google Scholar
Impact of tacrolimus versus cyclosporin A on renal function during the...

Go to citation Crossref Google Scholar
A genetic model of ivabradine recapitulates results from randomized cl...

Go to citation Crossref Google Scholar
Integrating untargeted metabolomics, genetically informed causal infer...

Go to citation Crossref Google Scholar
Causal Association of Haptoglobin With Obesity in Mexican Children: A ...

Go to citation Crossref Google Scholar
Alcohol Drinking and Amyotrophic Lateral Sclerosis: An Instrumental Va...

Go to citation Crossref Google Scholar
Transcriptome-wide association studies: a view from Mendelian randomiz...

Go to citation Crossref Google Scholar
Birth Weight and Stroke in Adult Life: Genetic Correlation and Causal ...

Go to citation Crossref Google Scholar
Cancer Progress and Priorities: Childhood Cancer

Go to citation Crossref Google Scholar
Impact of opioid dose escalation on the development of substance use d...

Go to citation Crossref Google Scholar
Investigating the Causal Effect of Brain Expression of CCL2, NFKB1, MA...

Go to citation Crossref Google Scholar
The Role of Genetic Variation of BMI, Body Composition, and Fat Distri...

Go to citation Crossref Google Scholar
Evaluation of associations between genetically predicted circulating p...

Go to citation Crossref Google Scholar
Stochastic imputation for integrated transcriptome association analysi...

Go to citation Crossref Google Scholar Pub Med
The use of Mendelian randomisation to identify causal cancer risk fact...

Go to citation Crossref Google Scholar
Bayesian network analysis incorporating genetic anchors complements co...

Go to citation Crossref Google Scholar
Response to: ‘Role of linoleic acid in autoimmune disorders: aMendelia...

Go to citation Crossref Google Scholar
Are blood lipids risk factors for fracture? Integrative evidence from ...

Go to citation Crossref Google Scholar
Causal Inference: Efficacy and Mechanism Evaluation

Go to citation Crossref Google Scholar
Epidemiology, genetic epidemiology and Mendelian randomisation: more n...

Go to citation Crossref Google Scholar
Basic Concepts of a Mendelian Randomization Approach

Go to citation Crossref Google Scholar
Precision Medicine and Cardiovascular Health: Insights from Mendelian ...

Go to citation Crossref Google Scholar
Turning genome-wide association study findings into opportunities for ...

Go to citation Crossref Google Scholar
Causal Inference for Genetically Determined Levels of High-Density Lip...

Go to citation Crossref Google Scholar
Mendelian randomization integrating GWAS and eQTL data reveals genetic...

Go to citation Crossref Google Scholar
Birth weight is not causally associated with adult asthma: results fro...

Go to citation Crossref Google Scholar
Mendelian randomisation analyses find pulmonary factors mediate the ef...

Go to citation Crossref Google Scholar
A Phenome-Wide Mendelian Randomization Study of Pancreatic Cancer Usin...

Go to citation Crossref Google Scholar
Causal association of type 2 diabetes with amyotrophic lateral scleros...

Go to citation Crossref Google Scholar
Prioritizing putative influential genes in cardiovascular disease susc...

Go to citation Crossref Google Scholar
The Parkinson's Disease Mendelian Randomization Research Portal

Go to citation Crossref Google Scholar
Meta‐analysis and Mendelian randomization:...

Go to citation Crossref Google Scholar
Corporate board structure and foreign equity investments in weak insti...

Go to citation Crossref Google Scholar
Local average treatment effects estimation via substantive model compa...

Go to citation Crossref Google Scholar
No Association Between Vitamin D Status and Risk of Barrett's Esophagu...

Go to citation Crossref Google Scholar
Mendelian randomization in the bone field

Go to citation Crossref Google Scholar
Association of Cortisol Levels With Neuropsychiatric Functions: A Mend...

Go to citation Crossref Google Scholar
Genetic predisposition to increased serum calcium, bone mineral densit...

Go to citation Crossref Google Scholar
Causal Association Between Birth Weight and Adult Diseases: Evidence F...

Go to citation Crossref Google Scholar
On the Use of the Lasso for Instrumental Variables Estimation with Som...

Go to citation Crossref Google Scholar
Patient-reported outcomes and health-related quality of life for cetux...

Go to citation Crossref Google Scholar
Research Techniques Made Simple: Using Genetic Variants for Randomizat...

Go to citation Crossref Google Scholar
Software application profile: mrrobust—a tool for performing two-sampl...

Go to citation Crossref Google Scholar
Associations of obesity and circulating insulin and glucose with breas...

Go to citation Crossref Google Scholar
Mendelian randomization: the challenge of unobserved environmental con...

Go to citation Crossref Google Scholar
Association Between Premorbid Body Mass Index and Amyotrophic Lateral ...

Go to citation Crossref Google Scholar
Using Mendelian randomisation to assess causality in observational stu...

Go to citation Crossref Google Scholar
Elevated Platelet Count Appears to Be Causally Associated with Increas...

Go to citation Crossref Google Scholar
Two-Sample Instrumental Variable Analyses Using Heterogeneous Samples

Go to citation Crossref Google Scholar
Assessing the robustness of sisVIVE in a Mendelian randomization study...

Go to citation Crossref Google Scholar
Impact of off-farm income on household energy expenditures in China: I...

Go to citation Crossref Google Scholar
Cholesteryl Ester Transfer Protein Influences High-Density Lipoprotein...

Go to citation Crossref Google Scholar
Conducting a Reproducible Mendelian Randomization Analysis Using the R...

Go to citation Crossref Google Scholar
A Bayesian approach to Mendelian randomisation with dependent instrume...

Go to citation Crossref Google Scholar
Mendelian Randomization and the Environmental Epigenetics of Health: a...

Go to citation Crossref Google Scholar
Effect of increased body mass index on risk of diagnosis or death from...

Go to citation Crossref Google Scholar
Causal effects of blood lipids on amyotrophic lateral sclerosis: a Men...

Go to citation Crossref Google Scholar
What is the effect of alcohol consumption on the risk of chronic wides...

Go to citation Crossref Google Scholar
Evidence of a causal relationship between body mass index and psoriasi...

Go to citation Crossref Google Scholar
Mendelian Randomization Analysis in Observational Epidemiology

Go to citation Crossref Google Scholar
Polymorphisms in Manganese Transporters SLC30A10 and SLC39A8 Are Assoc...

Go to citation Crossref Google Scholar
Integrative analysis of omics summary data reveals putative mechanisms...

Go to citation Crossref Google Scholar
Genetic markers for urine haptoglobin is associated with decline in re...

Go to citation Crossref Google Scholar
Habitual coffee consumption and cognitive function: a Mendelian random...

Go to citation Crossref Google Scholar
The relationship between ferritin and urate levels and risk of gout

Go to citation Crossref Google Scholar
Detecting and correcting for bias in Mendelian randomization analyses ...

Go to citation Crossref Google Scholar
An approach to estimate bidirectional mediation effects with applicati...

Go to citation Crossref Google Scholar
Associations of Circulating Protein Levels With Lipid Fractions in the...

Go to citation Crossref Google Scholar
Genetically Determined Serum Uric Acid and Alzheimer’s Disease Risk

Go to citation Crossref Google Scholar
Understanding the Assumptions Underlying Instrumental Variable Analyse...

Go to citation Crossref Google Scholar
Disentangling associations between DNA methylation and blood lipids: a...

Go to citation Crossref Google Scholar
Evaluating the potential role of pleiotropy in Mendelian randomization...

Go to citation Crossref Google Scholar
Blood CSF1 and CXCL12 as Causal Mediators of Coronary Artery Disease

Go to citation Crossref Google Scholar
Genetic instrumental variable regression: Explaining socioeconomic and...

Go to citation Crossref Google Scholar
Common Methods for Performing Mendelian Randomization

Go to citation Crossref Google Scholar
Will farmland transfer reduce grain acreage? Evidence from Gansu provi...

Go to citation Crossref Google Scholar
Blood Eosinophil Count and Metabolic, Cardiac and Pulmonary Outcomes: ...

Go to citation Crossref Google Scholar
Circulating insulin-like growth factors and Alzheimer disease

Go to citation Crossref Google Scholar
A Primer in Mendelian Randomization Methodology with a Focus on Utiliz...

Go to citation Crossref Google Scholar
Recent Developments in Mendelian Randomization Studies

Go to citation Crossref Google Scholar
Complement C3 Associates With Incidence of Diabetes, but No Evidence o...

Go to citation Crossref Google Scholar
Type 2 diabetes, glucose, insulin, BMI, and ischemic stroke subtypes

Go to citation Crossref Google Scholar
Adiposity amplifies the genetic risk of fatty liver disease conferred ...

Go to citation Crossref Google Scholar
Genetic variation in the ADIPOQ gene, adiponectin concentrations and r...

Go to citation Crossref Google Scholar
‘Mendelian randomization’: an approach for exploring causal relations ...

Go to citation Crossref Google Scholar
Birth weight and risk of ischemic heart disease: A Mendelian randomiza...

Go to citation Crossref Google Scholar
Using Mendelian Randomization Studies to Assess Causality and Identify...

Go to citation Crossref Google Scholar
Mendelian randomization analysis of a time‐varying exposure for binary...

Go to citation Crossref Google Scholar
Estimating Marginal Healthcare Costs Using Genetic Variants as Instrum...

Go to citation Crossref Google Scholar
Bias due to participant overlap in two‐sample Mendelian randomization

Go to citation Crossref Google Scholar
Association of vitamin D levels and risk of ovarian cancer: a Mendelia...

Go to citation Crossref Google Scholar

Figures and tables

Figures & Media

Tables

View Options

View options

PDF/ePub

View PDF/ePub

Get access

If you have access to journal content via a personal subscription, university, library, employer or society, select from the options below:

Sage Journals profile

Sign in

Access personal subscriptions, purchases, paired institutional or society access and free tools such as email alerts and saved searches.

Required fields

Email:

Password:

Remember me

Forgotten your password?

Create profile

Institution

Society

Alternatively, view purchase options below:

Purchase access

Purchase 24 hour online access to view and download content.

Issue - $543.66

Article - $41.50

Subscribe to this journal

Read with DeepDyve

Need help?

Abstract

1 Introduction

2 Context of the problem

2.1 Genetic variants used in Mendelian randomization

2.2 Causal estimand

3 Ratio of coefficients method

3.1 Continuous outcome

3.2 Binary outcome

3.3 Case–control data

3.4 Confidence intervals

3.4.1 Normal approximation

3.4.2 Fieller’s theorem

3.4.3 Bootstrapping

3.4.4 Other approaches

3.5 Absence of finite moments

4 Two-stage methods

4.1 Continuous outcome – Two-stage least squares

4.2 Binary outcome

4.3 Adjusted two-stage method

5 Likelihood-based methods

5.1 Limited information maximum likelihood

5.2 Bayesian methods

5.3 Likelihood-based methods with binary outcomes

5.4 Comparison of two-stage and likelihood-based methods

5.5 k-class estimators

6 Semi-parametric methods

6.1 Generalized method of moments

6.2 Continuous updating estimator

6.3 G-estimation of structural mean models

6.4 Potential lack of unique estimate with binary outcomes

7 Weak instruments

7.1 Problems of estimates using weak instruments

7.2 Comparison of methods with single instrumental variable

7.3 Multiple instrument variables and score-based approach

8 Discussion

8.1 Non-linear relationships

8.2 Use of measured covariates

8.3 Overidentification tests

8.4 Endogeneity tests

8.5 Power of IV analyses

8.6 Summarized data

8.7 Subsample and two-sample Mendelian randomization

8.8 Meta-analysis

8.9 Robust methods

8.10 Summary

Declaration of conflicting interests

Funding

References

Cite article

Cite article

Download to reference manager

Share

Share this article

Share with email

Share on social media

Share access to this article

Information

Published In

Keywords

Rights and permissions

Authors

Affiliations

Notes

Metrics

Journals metrics

Article usage*

Articles citing this one

Figures & Media

Tables

View options

PDF/ePub

Get access

Access options

Sign in

Also from Sage

Article usage^*