Power calculation for the general two-sample Mendelian randomization analysis

Mendelian randomization (MR) study has become a powerful approach to assess the potential causal effect of a risk exposure on an outcome. Most current MR studies are conducted under the two-sample setting by combining summary data from two separate genome-wide association studies (GWAS), with one providing measures on associations between genetic markers and the risk exposure, and the other on associations between genetic markers and the outcome. We develop a power calculation procedure for the general two-sample MR study, allowing for the use of multiple genetic markers, and shared participants between the two GWAS. This procedure requires a few easy-to-interpret parameters and is validated through extensive simulation studies.     

Mendelian randomization with fine‐mapped genetic data: Choosing from large numbers of correlated instrumental variables

Mendelian randomization uses genetic variants to make causal inferences about the effect of a risk factor on an outcome. With fine‐mapped genetic data, there may be hundreds of genetic variants in a single gene region any of which could be used to assess this causal relationship. However, using too many genetic variants in the analysis can lead to spurious estimates and inflated Type 1 error rates. But if only a few genetic variants are used, then the majority of the data is ignored and estimates are highly sensitive to the particular choice of variants. We propose an approach based on summarized data only (genetic association and correlation estimates) that uses principal components analysis to form instruments. This approach has desirable theoretical properties: it takes the totality of data into account and does not suffer from numerical instabilities. It also has good properties in simulation studies: it is not particularly sensitive to varying the genetic variants included in the analysis or the genetic correlation matrix, and it does not have greatly inflated Type 1 error rates. Overall, the method gives estimates that are less precise than those from variable selection approaches (such as using a conditional analysis or pruning approach to select variants), but are more robust to seemingly arbitrary choices in the variable selection step. Methods are illustrated by an example using genetic associations with testosterone for 320 genetic variants to assess the effect of sex hormone related pathways on coronary artery disease risk, in which variable selection approaches give inconsistent inferences.     

弓形虫感染与精神分裂症的两样本孟德尔随机化研究

目的 采用两样本孟德尔随机化方法探究弓形虫感染和精神分裂症之间的因果关系.方法 利用汇总的大样本GWAS数据提取与弓形虫血清抗体密切相关的遗传位点作为工具变量,分别运用MR-Egger回归、加权中位数和逆方差加权法进行孟德尔随机化分析,以OR值及95％CI评价弓形虫感染与精神分裂症之间是否存在关联.采用Egger-in...     

Bias in causal estimates from Mendelian randomization studies with weak instruments

Mendelian randomization studies using genetic instrumental variables (IVs) are now being commonly used to estimate the causal association of a phenotype on an outcome. Even when the necessary underlying assumptions are valid, estimates from analyses using IVs are biased in finite samples. The source and nature of this bias appear poorly understood in the epidemiological field. We explain why the bias is in the direction of the confounded observational association, with magnitude relating to the statistical strength of association between the instrument and phenotype. We comment on the size of the bias, from simulated data, showing that when multiple instruments are used, although the variance of the IV estimator decreases, the bias increases. We discuss ways to analyse Mendelian randomization studies to alleviate the problem of weak instrument bias.     

A comparison of robust Mendelian randomization methods using summary data

The number of Mendelian randomization (MR) analyses including large numbers of genetic variants is rapidly increasing. This is due to the proliferation of genome-wide association studies, and the desire to obtain more precise estimates of causal effects. Since it is unlikely that all genetic variants will be valid instrumental variables, several robust methods have been proposed. We compare nine robust methods for MR based on summary data that can be implemented using standard statistical software. Methods were compared in three ways: by reviewing their theoretical properties, in an extensive simulation study, and in an empirical example. In the simulation study, the best method, judged by mean squared error was the contamination mixture method. This method had well-controlled Type 1 error rates with up to 50% invalid instruments across a range of scenarios. Other methods performed well according to different metrics. Outlier-robust methods had the narrowest confidence intervals in the empirical example. With isolated exceptions, all methods performed badly when over 50% of the variants were invalid instruments. Our recommendation for investigators is to perform a variety of robust methods that operate in different ways and rely on different assumptions for valid inferences to assess the reliability of MR analyses.     

东亚人群血压与心力衰竭的因果关联研究:两样本双向孟德尔随机化分析

目的 利用两样本双向孟德尔随机化（MR）研究设计,探究东亚人群中三种血压指标与心力衰竭风险的关联。方法 从发表的全基因组关联研究（GWAS）中提取汇总数据进行分析,收缩压和舒张压的遗传工具变量来自韩国基因组与流行病学研究,脉压的遗传工具变量来自日本生物银行,心力衰竭的遗传工具变量来自一项包含五个日本队列的GWAS研究。采用单变量MR、双向MR和多变量MR方法分析三种血压指标与心力衰竭风险的关联。 结果 逆方差加权法显示收缩压（每升高1 mm Hg,OR=1.52; 95% CI: 1.25~1.84）、舒张压（1.62; 1.34~1.95）和脉压（1.85; 1.27~2.69）升高均可能增加患心力衰竭的风险,而心力衰竭对三种血压指标没有潜在的因果影响（P＞0.05）。多变量分析显示脉压经调整收缩压（1.25; 0.77~2.05）或舒张压（1.46; 0.95~2.23）后与心力衰竭风险没有显著关联。结论 在东亚人群中血压对心力衰竭可能存在单向的因果关联,脉压对心力衰竭风险不存在独立于收缩压和舒张压的直接效应。     

Mendelian randomization analysis of case-control data using structural mean models

'Instrumental Variable' (IV) methods provide a basis for estimating an exposure's causal effect on the risk of disease. In Mendelian randomization studies, where genetic information plays the role of the IV, IV analyses are routinely performed on case-control data, rather than prospectively collected observational data. Although it is a well-appreciated fact that ascertainment bias may invalidate such analyses, ad hoc assumptions and approximations are made to justify their use. In this paper we attempt to explain and clarify why they may fail and show how they can be adjusted for improved performance. In particular, we propose consistent estimators of the causal relative risk and odds ratio if a priori knowledge is available regarding either the population disease prevalence or the population distribution of the IV (e.g. population allele frequencies). We further show that if no such information is available, approximate estimators can be obtained under a rare disease assumption. We illustrate this with matched case-control data from the recently completed EPIC study, from which we attempt to assess the evidence for a causal relationship between C-reactive protein levels and the risk of Coronary Artery Disease.     

Mendelian randomization incorporating uncertainty about pleiotropy

Mendelian randomization (MR) requires strong assumptions about the genetic instruments, of which the most difficult to justify relate to pleiotropy. In a two‐sample MR, different methods of analysis are available if we are able to assume, M₁: no pleiotropy (fixed effects meta‐analysis), M₂: that there may be pleiotropy but that the average pleiotropic effect is zero (random effects meta‐analysis), and M₃: that the average pleiotropic effect is nonzero (MR‐Egger). In the latter 2 cases, we also require that the size of the pleiotropy is independent of the size of the effect on the exposure. Selecting one of these models without good reason would run the risk of misrepresenting the evidence for causality. The most conservative strategy would be to use M₃ in all analyses as this makes the weakest assumptions, but such an analysis gives much less precise estimates and so should be avoided whenever stronger assumptions are credible. We consider the situation of a two‐sample design when we are unsure which of these 3 pleiotropy models is appropriate. The analysis is placed within a Bayesian framework and Bayesian model averaging is used. We demonstrate that even large samples of the scale used in genome‐wide meta‐analysis may be insufficient to distinguish the pleiotropy models based on the data alone. Our simulations show that Bayesian model averaging provides a reasonable trade‐off between bias and precision. Bayesian model averaging is recommended whenever there is uncertainty about the nature of the pleiotropy.     

Extending the MR‐Egger method for multivariable Mendelian randomization to correct for both measured and unmeasured pleiotropy

Methods have been developed for Mendelian randomization that can obtain consistent causal estimates while relaxing the instrumental variable assumptions. These include multivariable Mendelian randomization, in which a genetic variant may be associated with multiple risk factors so long as any association with the outcome is via the measured risk factors (measured pleiotropy), and the MR‐Egger (Mendelian randomization‐Egger) method, in which a genetic variant may be directly associated with the outcome not via the risk factor of interest, so long as the direct effects of the variants on the outcome are uncorrelated with their associations with the risk factor (unmeasured pleiotropy). In this paper, we extend the MR‐Egger method to a multivariable setting to correct for both measured and unmeasured pleiotropy. We show, through theoretical arguments and a simulation study, that the multivariable MR‐Egger method has advantages over its univariable counterpart in terms of plausibility of the assumption needed for consistent causal estimation and power to detect a causal effect when this assumption is satisfied. The methods are compared in an applied analysis to investigate the causal effect of high‐density lipoprotein cholesterol on coronary heart disease risk. The multivariable MR‐Egger method will be useful to analyse high‐dimensional data in situations where the risk factors are highly related and it is difficult to find genetic variants specifically associated with the risk factor of interest (multivariable by design), and as a sensitivity analysis when the genetic variants are known to have pleiotropic effects on measured risk factors.     

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. © 2015 The Authors. Statistics in Medicine published by John Wiley & Sons Ltd.     

A framework for the investigation of pleiotropy in two‐sample summary data Mendelian randomization

Mendelian randomization (MR) uses genetic data to probe questions of causality in epidemiological research, by invoking the Instrumental Variable (IV) assumptions. In recent years, it has become commonplace to attempt MR analyses by synthesising summary data estimates of genetic association gleaned from large and independent study populations. This is referred to as two‐sample summary data MR. Unfortunately, due to the sheer number of variants that can be easily included into summary data MR analyses, it is increasingly likely that some do not meet the IV assumptions due to pleiotropy. There is a pressing need to develop methods that can both detect and correct for pleiotropy, in order to preserve the validity of the MR approach in this context. In this paper, we aim to clarify how established methods of meta‐regression and random effects modelling from mainstream meta‐analysis are being adapted to perform this task. Specifically, we focus on two contrastin g approaches: the Inverse Variance Weighted (IVW) method which assumes in its simplest form that all genetic variants are valid IVs, and the method of MR‐Egger regression that allows all variants to violate the IV assumptions, albeit in a specific way. We investigate the ability of two popular random effects models to provide robustness to pleiotropy under the IVW approach, and propose statistics to quantify the relative goodness‐of‐fit of the IVW approach over MR‐Egger regression. © 2017 The Authors. Statistics in Medicine Published by JohnWiley & Sons Ltd     

Mendelian randomization analysis of a time‐varying exposure for binary disease outcomes using functional data analysis methods

A Mendelian randomization (MR) analysis is performed to analyze the causal effect of an exposure variable on a disease outcome in observational studies, by using genetic variants that affect the disease outcome only through the exposure variable. This method has recently gained popularity among epidemiologists given the success of genetic association studies. Many exposure variables of interest in epidemiological studies are time varying, for example, body mass index (BMI). Although longitudinal data have been collected in many cohort studies, current MR studies only use one measurement of a time‐varying exposure variable, which cannot adequately capture the long‐term time‐varying information. We propose using the functional principal component analysis method to recover the underlying individual trajectory of the time‐varying exposure from the sparsely and irregularly observed longitudinal data, and then conduct MR analysis using the recovered curves. We further propose two MR analysis methods. The first assumes a cumulative effect of the time‐varying exposure variable on the disease risk, while the second assumes a time‐varying genetic effect and employs functional regression models. We focus on statistical testing for a causal effect. Our simulation studies mimicking the real data show that the proposed functional data analysis based methods incorporating longitudinal data have substantial power gains compared to standard MR analysis using only one measurement. We used the Framingham Heart Study data to demonstrate the promising performance of the new methods as well as inconsistent results produced by the standard MR analysis that relies on a single measurement of the exposure at some arbitrary time point.     

Bayesian methods for meta‐analysis of causal relationships estimated using genetic instrumental variables

Genetic markers can be used as instrumental variables, in an analogous way to randomization in a clinical trial, to estimate the causal relationship between a phenotype and an outcome variable. Our purpose is to extend the existing methods for such Mendelian randomization studies to the context of multiple genetic markers measured in multiple studies, based on the analysis of individual participant data. First, for a single genetic marker in one study, we show that the usual ratio of coefficients approach can be reformulated as a regression with heterogeneous error in the explanatory variable. This can be implemented using a Bayesian approach, which is next extended to include multiple genetic markers. We then propose a hierarchical model for undertaking a meta‐analysis of multiple studies, in which it is not necessary that the same genetic markers are measured in each study. This provides an overall estimate of the causal relationship between the phenotype and the outcome, and an assessment of its heterogeneity across studies. As an example, we estimate the causal relationship of blood concentrations of C‐reactive protein on fibrinogen levels using data from 11 studies. These methods provide a flexible framework for efficient estimation of causal relationships derived from multiple studies. Issues discussed include weak instrument bias, analysis of binary outcome data such as disease risk, missing genetic data, and the use of haplotypes. Copyright © 2010 John Wiley & Sons, Ltd.     

Improving bias and coverage in instrumental variable analysis with weak instruments for continuous and binary outcomes

Causal estimates can be obtained by instrumental variable analysis using a two-stage method. However, these can be biased when the instruments are weak. We introduce a Bayesian method, which adjusts for the first-stage residuals in the second-stage regression and has much improved bias and coverage properties. In the continuous outcome case, this adjustment reduces median bias from weak instruments to close to zero. In the binary outcome case, bias from weak instruments is reduced and the estimand is changed from a marginal population-based effect to a conditional effect. The lack of distributional assumptions on the posterior distribution of the causal effect gives a better summary of uncertainty and more accurate coverage levels than methods that rely on the asymptotic distribution of the causal estimate. We discuss these properties in the context of Mendelian randomization.     

Sample size formulae for intervention studies with the cluster as unit of randomization

This paper presents sample size formulae for both continuous and dichotomous endpoints obtained from intervention studies that use the cluster as the unit of randomization. The formulae provide the required number of clusters or the required number of individuals per cluster when the other number is given. The proposed formulae derive from Student's t-test with use of cluster summary measures and a variance that consists of within and between cluster components. Power contours are provided to help in the design of intervention studies that use cluster randomization. Sample size formulae for designs with and without stratification of clusters appear separately.     

A Bayesian approach for instrumental variable analysis with censored time‐to‐event outcome

Instrumental variable (IV) analysis has been widely used in economics, epidemiology, and other fields to estimate the causal effects of covariates on outcomes, in the presence of unobserved confounders and/or measurement errors in covariates. However, IV methods for time‐to‐event outcome with censored data remain underdeveloped. This paper proposes a Bayesian approach for IV analysis with censored time‐to‐event outcome by using a two‐stage linear model. A Markov chain Monte Carlo sampling method is developed for parameter estimation for both normal and non‐normal linear models with elliptically contoured error distributions. The performance of our method is examined by simulation studies. Our method largely reduces bias and greatly improves coverage probability of the estimated causal effect, compared with the method that ignores the unobserved confounders and measurement errors. We illustrate our method on the Women's Health Initiative Observational Study and the Atherosclerosis Risk in Communities Study. Copyright © 2014 John Wiley & Sons, Ltd.     

Genetic random effects model for family data with long-term survivors: analysis of diabetic nephropathy in type 1 diabetes

A shared and additive genetic variance component-long-term survivor (LTS) model for familial aggregation studies of complex diseases with variable age-at-onset phenotype and non-susceptible subjects in the study cohort is proposed. LTS has been used from the early 1970s, especially in epidemiological studies of cancer. The LTS model utilizes information on the age at onset (survival) distribution to make inference on partially latent susceptibility. Bayesian modeling with uninformative priors is used and estimates of the posterior distribution of age at onset and susceptibility parameters of interest have been obtained using Bayesian Markov chain Monte Carlo (MCMC) methods with OpenBugs program. A simulation study confirms that we obtain posterior estimates of the model parameters on shared and genetic variance components of age at onset and susceptibility with good coverage rates. Further, we analyze familial aggregation of diabetic nephropathy (DN) in large Finnish cohort of 528 sibships with type 1 diabetes (T1D). According to the variance components estimated a substantial familial variation in the susceptibility to DN exist among families, while time to DN is less influenced by shared familial factors.