Conditional inference in cis-Mendelian randomization using weak genetic factors.

Mendelian randomization (MR) is a widely used method to estimate the causal effect of an exposure on an outcome by using genetic variants as instrumental variables. MR analyses that use variants from only a single genetic region (cis-MR) encoding the protein target of a drug are able to provide supporting evidence for drug target validation. This paper proposes methods for cis-MR inference that use many correlated variants to make robust inferences even in situations, where those variants have only weak effects on the exposure. In particular, we exploit the highly structured nature of genetic correlations in single gene regions to reduce the dimension of genetic variants using factor analysis. These genetic factors are then used as instrumental variables to construct tests for the causal effect of interest. Since these factors may often be weakly associated with the exposure, size distortions of standard t-tests can be severe. Therefore, we consider two approaches based on conditional testing. First, we extend results of commonly-used identification-robust tests for the setting where estimated factors are used as instruments. Second, we propose a test which appropriately adjusts for first-stage screening of genetic factors based on their relevance. Our empirical results provide genetic evidence to validate cholesterol-lowering drug targets aimed at preventing coronary heart disease.

A novel penalized inverse-variance weighted estimator for Mendelian randomization with applications to COVID-19 outcomes.

Mendelian randomization utilizes genetic variants as instrumental variables (IVs) to estimate the causal effect of an exposure variable on an outcome of interest even in the presence of unmeasured confounders. However, the popular inverse-variance weighted (IVW) estimator could be biased in the presence of weak IVs, a common challenge in MR studies. In this article, we develop a novel penalized inverse-variance weighted (pIVW) estimator, which adjusts the original IVW estimator to account for the weak IV issue by using a penalization approach to prevent the denominator of the pIVW estimator from being close to zero. Moreover, we adjust the variance estimation of the pIVW estimator to account for the presence of balanced horizontal pleiotropy. We show that the recently proposed debiased IVW (dIVW) estimator is a special case of our proposed pIVW estimator. We further prove that the pIVW estimator has smaller bias and variance than the dIVW estimator under some regularity conditions. We also conduct extensive simulation studies to demonstrate the performance of the proposed pIVW estimator. Furthermore, we apply the pIVW estimator to estimate the causal effects of five obesity-related exposures on three coronavirus disease 2019 (COVID-19) outcomes. Notably, we find that hypertensive disease is associated with an increased risk of hospitalized COVID-19; and peripheral vascular disease and higher body mass index are associated with increased risks of COVID-19 infection, hospitalized COVID-19, and critically ill COVID-19.

Profile-likelihood Bayesian model averaging for two-sample summary data Mendelian randomization in the presence of horizontal pleiotropy.

Two-sample summary data Mendelian randomization is a popular method for assessing causality in epidemiology, by using genetic variants as instrumental variables. If genes exert pleiotropic effects on the outcome not entirely through the exposure of interest, this can lead to heterogeneous and (potentially) biased estimates of causal effect. We investigate the use of Bayesian model averaging to preferentially search the space of models with the highest posterior likelihood. We develop a Metropolis-Hasting algorithm to perform the search using the recently developed MR-RAPS as the basis for defining a posterior distribution that efficiently accounts for pleiotropic and weak instrument bias. We demonstrate how our general modeling approach can be extended from a standard one-component causal model to a two-component model, which allows a large proportion of SNPs to violate the InSIDE assumption. We use Monte Carlo simulations to illustrate our methods and compare it to several related approaches. We finish by applying our approach to investigate the causal role of cholesterol on the development age-related macular degeneration.

A Bayesian approach to Mendelian randomisation with dependent instruments.

Mendelian randomisation (MR) is a method for establishing causality between a risk factor and an outcome by using genetic variants as instrumental variables. In practice, the association between individual genetic variants and the risk factor is often weak, which may lead to a lack of precision in the MR and even biased MR estimates. Usually, the most significant variant within a genetic region is selected to represent the association with the risk factor, but there is no guarantee that this variant will be causal or that it will capture all of the genetic association within the region. It may be advantageous to use extra variants selected from the same region in the MR. The problem is to decide which variants to select. Rather than selecting a specific set of variants, we investigate the use of Bayesian model averaging (BMA) to average the MR over all possible combinations of genetic variants. Our simulations demonstrate that the BMA version of MR outperforms classical estimation with many dependent variants and performs much better than an MR based on variants selected by penalised regression. In further simulations, we investigate robustness to violations in the model assumptions and demonstrate sensitivity to the inclusion of invalid instruments. The method is illustrated by applying it to an MR of the effect of body mass index on blood pressure using SNPs in the FTO gene.

Combining information on multiple instrumental variables in Mendelian randomization: comparison of allele score and summarized data methods.

Mendelian randomization is the use of genetic instrumental variables to obtain causal inferences from observational data. Two recent developments for combining information on multiple uncorrelated instrumental variables (IVs) into a single causal estimate are as follows: (i) allele scores, in which individual-level data on the IVs are aggregated into a univariate score, which is used as a single IV, and (ii) a summary statistic method, in which causal estimates calculated from each IV using summarized data are combined in an inverse-variance weighted meta-analysis. To avoid bias from weak instruments, unweighted and externally weighted allele scores have been recommended. Here, we propose equivalent approaches using summarized data and also provide extensions of the methods for use with correlated IVs. We investigate the impact of different choices of weights on the bias and precision of estimates in simulation studies. We show that allele score estimates can be reproduced using summarized data on genetic associations with the risk factor and the outcome. Estimates from the summary statistic method using external weights are biased towards the null when the weights are imprecisely estimated; in contrast, allele score estimates are unbiased. With equal or external weights, both methods provide appropriate tests of the null hypothesis of no causal effect even with large numbers of potentially weak instruments. We illustrate these methods using summarized data on the causal effect of low-density lipoprotein cholesterol on coronary heart disease risk. It is shown that a more precise causal estimate can be obtained using multiple genetic variants from a single gene region, even if the variants are correlated.

A weak instrument [Formula: see text]-test in linear IV models with multiple endogenous variables.

We consider testing for weak instruments in a model with multiple endogenous variables. Unlike Stock and Yogo (2005), who considered a weak instruments problem where the rank of the matrix of reduced form parameters is near zero, here we consider a weak instruments problem of a near rank reduction of one in the matrix of reduced form parameters. For example, in a two-variable model, we consider weak instrument asymptotics of the form [Formula: see text] where [Formula: see text] and [Formula: see text] are the parameters in the two reduced-form equations, [Formula: see text] is a vector of constants and [Formula: see text] is the sample size. We investigate the use of a conditional first-stage [Formula: see text]-statistic along the lines of the proposal by Angrist and Pischke (2009) and show that, unless [Formula: see text], the variance in the denominator of their [Formula: see text]-statistic needs to be adjusted in order to get a correct asymptotic distribution when testing the hypothesis [Formula: see text]. We show that a corrected conditional [Formula: see text]-statistic is equivalent to the Cragg and Donald (1993) minimum eigenvalue rank test statistic, and is informative about the maximum total relative bias of the 2SLS estimator and the Wald tests size distortions. When [Formula: see text] in the two-variable model, or when there are more than two endogenous variables, further information over and above the Cragg-Donald statistic can be obtained about the nature of the weak instrument problem by computing the conditional first-stage [Formula: see text]-statistics.

The many weak instruments problem and Mendelian randomization.

Instrumental variable estimates of causal effects can be biased when using many instruments that are only weakly associated with the exposure. We describe several techniques to reduce this bias and estimate corrected standard errors. We present our findings using a simulation study and an empirical application. For the latter, we estimate the effect of height on lung function, using genetic variants as instruments for height. Our simulation study demonstrates that, using many weak individual variants, two-stage least squares (2SLS) is biased, whereas the limited information maximum likelihood (LIML) and the continuously updating estimator (CUE) are unbiased and have accurate rejection frequencies when standard errors are corrected for the presence of many weak instruments. Our illustrative empirical example uses data on 3631 children from England. We used 180 genetic variants as instruments and compared conventional ordinary least squares estimates with results for the 2SLS, LIML, and CUE instrumental variable estimators using the individual height variants. We further compare these with instrumental variable estimates using an unweighted or weighted allele score as single instruments. In conclusion, the allele scores and CUE gave consistent estimates of the causal effect. In our empirical example, estimates using the allele score were more efficient. CUE with corrected standard errors, however, provides a useful additional statistical tool in applications with many weak instruments. The CUE may be preferred over an allele score if the population weights for the allele score are unknown or when the causal effects of multiple risk factors are estimated jointly.

Mendelian randomization analysis with multiple genetic variants using summarized data.

Genome-wide association studies, which typically report regression coefficients summarizing the associations of many genetic variants with various traits, are potentially a powerful source of data for Mendelian randomization investigations. We demonstrate how such coefficients from multiple variants can be combined in a Mendelian randomization analysis to estimate the causal effect of a risk factor on an outcome. The bias and efficiency of estimates based on summarized data are compared to those based on individual-level data in simulation studies. We investigate the impact of gene-gene interactions, linkage disequilibrium, and 'weak instruments' on these estimates. Both an inverse-variance weighted average of variant-specific associations and a likelihood-based approach for summarized data give similar estimates and precision to the two-stage least squares method for individual-level data, even when there are gene-gene interactions. However, these summarized data methods overstate precision when variants are in linkage disequilibrium. If the P-value in a linear regression of the risk factor for each variant is less than 1×10⁻5, then weak instrument bias will be small. We use these methods to estimate the causal association of low-density lipoprotein cholesterol (LDL-C) on coronary artery disease using published data on five genetic variants. A 30% reduction in LDL-C is estimated to reduce coronary artery disease risk by 67% (95% CI: 54% to 76%). We conclude that Mendelian randomization investigations using summarized data from uncorrelated variants are similarly efficient to those using individual-level data, although the necessary assumptions cannot be so fully assessed.

Lack of identification in semiparametric instrumental variable models with binary outcomes.

A parameter in a statistical model is identified if its value can be uniquely determined from the distribution of the observable data. We consider the context of an instrumental variable analysis with a binary outcome for estimating a causal risk ratio. The semiparametric generalized method of moments and structural mean model frameworks use estimating equations for parameter estimation. In this paper, we demonstrate that lack of identification can occur in either of these frameworks, especially if the instrument is weak. In particular, the estimating equations may have no solution or multiple solutions. We investigate the relationship between the strength of the instrument and the proportion of simulated data sets for which there is a unique solution of the estimating equations. We see that this proportion does not appear to depend greatly on the sample size, particularly for weak instruments (ρ(2) ≤ 0.01). Poor identification was observed in a considerable proportion of simulated data sets for instruments explaining up to 10% of the variance in the exposure with sample sizes up to 1 million. In an applied example considering the causal effect of body mass index (weight (kg)/height (m)(2)) on the probability of early menarche, estimates and standard errors from an automated optimization routine were misleading.

Narrative Restrictions and Proxies.

We compare two approaches to using information about the signs of structural shocks at specific dates within a structural vector autoregression (SVAR): imposing "narrative restrictions" (NR) on the shock signs in an otherwise set-identified SVAR; and casting the information about the shock signs as a discrete-valued "narrative proxy" (NP) to point-identify the impulse responses. The NP is likely to be "weak" given that the sign of the shock is typically known in a small number of periods, in which case the weak-proxy robust confidence intervals in Montiel Olea, Stock, and Watson are the natural approach to conducting inference. However, we show both theoretically and via Monte Carlo simulations that these confidence intervals have distorted coverage-which may be higher or lower than the nominal level-unless the sign of the shock is known in a large number of periods. Regarding the NR approach, we show that the prior-robust Bayesian credible intervals from Giacomini, Kitagawa, and Read deliver coverage exceeding the nominal level, but which converges toward the nominal level as the number of NR increases.

Strengthening Instrumental Variables Through Weighting.

Instrumental variable (IV) methods are widely used to deal with the issue of unmeasured confounding and are becoming popular in health and medical research. IV models are able to obtain consistent estimates in the presence of unmeasured confounding, but rely on assumptions that are hard to verify and often criticized. An instrument is a variable that influences or encourages individuals toward a particular treatment without directly affecting the outcome. Estimates obtained using instruments with a weak influence over the treatment are known to have larger small-sample bias and to be less robust to the critical IV assumption that the instrument is randomly assigned. In this work, we propose a weighting procedure for strengthening the instrument while matching. Through simulations, weighting is shown to strengthen the instrument and improve robustness of resulting estimates. Unlike existing methods, weighting is shown to increase instrument strength without compromising match quality. We illustrate the method in a study comparing mortality between kidney dialysis patients receiving hemodialysis or peritoneal dialysis as treatment for end-stage renal disease.

Re-evaluating associations between the Supplemental Nutrition Assistance Program participation and body mass index in the context of unmeasured confounders.

OBJECTIVE: To evaluate the association between participation in the Supplemental Nutrition Assistance Program (SNAP) and body mass index (BMI) in the presence of unmeasured confounding. METHODS: We applied new matching methods to determine whether previous reports of associations between SNAP participation and BMI were robust to unmeasured confounders. We applied near-far matching, which strengthens standard matching by combining it with instrumental variables analysis, to the nationally-representative National Household Food Acquisition and Purchasing Survey (FoodAPS, N = 10,360, years 2012-13). RESULTS: In ordinary least squares regressions controlling for individual demographic and socioeconomic characteristics, SNAP was associated with increased BMI (+1.23 kg/m2, 95% CI: 0.84, 1.63). While propensity-score-based analysis replicated this finding, using instrumental variables analysis and particularly near-far matching to strengthen the instruments' discriminatory power revealed the association between SNAP and BMI was likely confounded by unmeasured covariates (+0.21 kg/m2, 95% CI: -3.88, 4.29). CONCLUSIONS: Previous reports of an association between SNAP and obesity should be viewed with caution, and use of near-far matching may assist similar assessments of health effects of social programs.

A review of instrumental variable estimators for Mendelian randomization.

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.

Use of allele scores as instrumental variables for Mendelian randomization.

BACKGROUND: An allele score is a single variable summarizing multiple genetic variants associated with a risk factor. It is calculated as the total number of risk factor-increasing alleles for an individual (unweighted score), or the sum of weights for each allele corresponding to estimated genetic effect sizes (weighted score). An allele score can be used in a Mendelian randomization analysis to estimate the causal effect of the risk factor on an outcome. METHODS: Data were simulated to investigate the use of allele scores in Mendelian randomization where conventional instrumental variable techniques using multiple genetic variants demonstrate 'weak instrument' bias. The robustness of estimates using the allele score to misspecification (for example non-linearity, effect modification) and to violations of the instrumental variable assumptions was assessed. RESULTS: Causal estimates using a correctly specified allele score were unbiased with appropriate coverage levels. The estimates were generally robust to misspecification of the allele score, but not to instrumental variable violations, even if the majority of variants in the allele score were valid instruments. Using a weighted rather than an unweighted allele score increased power, but the increase was small when genetic variants had similar effect sizes. Naive use of the data under analysis to choose which variants to include in an allele score, or for deriving weights, resulted in substantial biases. CONCLUSIONS: Allele scores enable valid causal estimates with large numbers of genetic variants. The stringency of criteria for genetic variants in Mendelian randomization should be maintained for all variants in an allele score.