Inverse-Variance Weighted Combination of Ratio Estimates

The ratio estimate of the causal effect of X on Y using genetic variant k is . The standard error of the ratio estimate can be approximated using the delta method; the leading term is [Thomas et al., 2007]. By using this expression for the standard error, an inverse-variance weighted (IVW) estimate of the causal effect combines the ratio estimates using each variant in a fixed-effect meta-analysis model:

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The approximate standard error of the estimate is:

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Further terms from the delta method could be used to improve the estimate of the standard error of the ratio estimates. However, if the estimate of the genetic association with the risk factor is considerably more precise than the estimate of the association with the outcome, as is often the case in practice, then the leading term will dominate.

Likelihood-Based Method

Alternatively, a model can be constructed by assuming a linear relationship between the risk factor and outcome and a bivariate normal distribution for the genetic association estimates:

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The causal effect of X on Y, β, which is assumed to be the same for all genetic variants k, can be estimated by direct maximization of the likelihood or by Bayesian methods [Thompson et al., 2005]. The correlation parameter ρ, representing the correlation between genetic association estimates and obtained from a single source can be specified as the observational correlation between the risk factor and the outcome. If the estimates and are derived from independent sources, this correlation will be zero. If is the per allele change in the log-odds or the log-probability of an outcome, then β represents a log odds ratio or a log relative risk parameter, respectively.

Independence of Information on Causal Effect From Multiple Variants

By combining the estimates of association from multiple variants into a single estimate of the causal effect, an assumption is made that the variants provide independent information. There are several reasons why this may not be the case. First, the causal estimates are derived from the same data, and so will not be entirely independent. However, correlation between the estimates should be low unless the sample size is particularly small. Secondly, there may be statistical interactions between variants in their associations with the risk factor (gene–gene interactions). Thirdly, the distribution of genetic variants may be correlated (linkage disequilibrium).

We perform simulation studies to assess the impact of the assumption that multiple genetic variants provide independent information on the causal estimate, in particular in the presence of gene–gene interactions and linkage disequilibrium. We compare the causal effect estimate (and its precision) obtained from summarized data to that obtained if individual-level data on the variants, risk factor and outcome were available on the whole study population.

Simulation Study With Independently Distributed Variants

We initially assume that the genetic variants used as IVs are independently distributed. Individual data were generated for 5,000 participants, indexed by i, according to the following model:

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where the α_.. parameters represent gene–gene interactions. The causal effect of X on Y is , and U represents negative (unmeasured) confounding between X and Y. The IVs take values 0, 1, 2 representing the number of minor alleles in three independently distributed SNPs, each with a minor allele frequency of . Nine sets of values were taken for the gene–gene interactions: (no gene–gene interactions), and (gene–gene interactions present).

Simulation Study With Correlated Variants

We repeated the simulation in the absence of gene–gene interactions, but using correlated genetic variants. In order to simulate data on the variants, two random draws and were made from a zero-mean K-dimensional multivariate normal distribution for each individual i. For each draw, if the kth component was positive, a variant allele was recorded for the kth variant. The draws represent the two haplotypes for each individual [Lunn et al., 2006]. In this way, the variance-covariance matrix in the multivariate normal distribution determines the correlation between variants , which take values 0, 1, 2. Data for three genetic variants were simulated using the following model:

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where 1_. is an indicator function. We took five values of the correlation parameter corresponding to a mean squared correlation between variants (r²) of 0 (no linkage disequilibrium), 0.06, 0.13, 0.26 and 0.41.

Weak Instruments

A weak instrument is a variable that satisfies the IV assumptions, but does not explain a large proportion of variation in the risk factor, so that the statistical association between the risk factor and the IV in the dataset is ‘weak’ [Burgess et al., 2011b]. IV estimates using weak instruments are biased in the direction of the observational estimate, and the distribution of the IV estimate is poorly approximated by a normal distribution [Burgess and Thompson, 2011]. The magnitude of bias depends on the expected value of the F statistic in the regression of the risk factor on the IVs, with lower F statistics corresponding to greater bias. Bias with a single IV in moderately large datasets is typically negligible, but bias may be considerable when there are multiple IVs [Angrist and Pischke, 2009]. For this reason, we are especially interested in how the summarized data methods perform with weak instruments.

Correlated Variants

Results from the scenarios with variants in linkage disequilibrium are given in Table​Table2.2. Estimates from the individual-level and summarized data methods were close to unbiased. The coverage of estimates from the 2SLS method was close to the nominal 95% level; however, the standard errors from the summarized data methods were too small and coverage was well below 95% when variants were correlated, even when the correlation was not large. Power was not reported in this case as it is misleading when the coverage is not close to the nominal levels. This shows that variants used in a summarized analysis must be uncorrelated in order to obtain valid statistical inferences.

Comparison of Summarized Data Methods

The IVW and likelihood-based methods make different assumptions about the distribution of variables. The IVW method uses an asymptotic estimate of the standard error of the causal (ratio) estimate from each variant; this is known to underestimate the true variation in the estimate, especially when the IV is weak [Burgess and Thompson, 2012]. No allowance is made in the method for uncertainty in the genetic association with the risk factor, although this could be incorporated by including additional terms from the delta method. The likelihood-based method assumes a bivariate normal distribution for the genetic associations with the risk factor and outcome. In both summarized data methods, the variances of the association estimates are assumed to be known; this may be why coverage is consistently slightly underestimated. The likelihood-based method is more computationally complex, but allows for correlation between the genetic association estimates with the risk factor and outcome, which is ignored in the IVW method. The likelihood-based method also has a natural extension to a meta-analysis framework using a hierarchical model (Section 5.6), which may better account for heterogeneity between studies than using meta-analysed genetic association estimates directly.

Weak Instruments

With weak instruments, estimates using both of the summarized data methods demonstrated bias similar to that of the 2SLS method. If the distributions of genetic variants are uncorrelated, then each variant explains independent variation in the risk factor. For variants of equal strength, the expected F statistic in the univariate linear regression of the risk factor on one of the variants is approximately the same as the expected F statistic in the multiple regression on all of the variants. An F statistic of 10 corresponds to a P-value of around 0.001; an F statistic of 20 to a P-value of around ; and an F statistic of 30 to a P-value of around , a threshold often used for assessing GWA significance. In the example presented, as the variants were independently distributed and each had a P-value of below , the F statistic for all of the variants is at least 20; in fact, as some variants had P-values well below , the F statistic would be greater. With a sample size of 10,000, an F statistic of 10 corresponds to a coefficient of determination (R²) for each variant of around 0.1%. When sample sizes and the strength of genetic variants are limited, it may be necessary to restrict the number of variants used in a Mendelian randomization analysis in order to mitigate against bias from weak instruments.

Close to unbiased estimates with weak instruments can be obtained from individual-level data using the limited information maximum likelihood (LIML) method [Angrist and Pischke, 2009] (see Supporting Information).

Assessment of the IV Assumptions

A limitation of the use of Mendelian randomization is that the instrumental variable assumptions cannot be assessed without supplementary data. Although the IV assumptions can never be fully tested empirically, they can be assessed to some extent, for example by testing the association of the genetic variants with measured covariates to assess potential pleiotropy. This can be undertaken using individual-level data. It is also possible with summarized data, as with HDL-C and TG in the example, although genetic associations with a full range of covariates would not necessarily be measured or routinely reported by a GWA study. These associations could be checked in the literature; however, the assumption is necessary that the literature-based estimates are valid for the population under investigation. Other assessments, such as addressing population stratification by the evaluation of genetic principal components, or testing for the attenuation of genetic associations with the outcome on adjustment for the risk factor [Glymour et al., 2012], require individual-level data.

In addition, many of the parametric assumptions required by IV methods for effect estimation, such as linearity of genetic associations or of the risk factor–outcome association [Didelez and Sheehan, 2007], cannot be assessed in summarized data.

Risk Factor and Outcome Associations in Separate Samples

Not all GWA studies measure data on a large number of phenotypic variables, and so genetic associations with the risk factor and the outcome may not be available in the same sample. Estimates of the association of the genetic variants with the risk factor and the outcome may be obtained from independent sources. This is known as a two-sample IV analysis [Pierce and Burgess, 2013]. A key assumption in this case is that the genetic associations with the risk factor and outcome are of the same magnitude in both sources.

Correlation between genetic associations with the risk factor and outcome arising from estimation of the coefficients in the same source is ignored by the inverse-variance weighted method, but can be acknowledged in the likelihood-based method. If the sources for the estimates are neither identical nor disjoint, but instead overlap, then a meta-analysis approach would be recommended to correctly account for the structure in the data. In the absence of study-specific estimates, we advocate the likelihood-based method with a sensitivity analysis for the correlation parameter.

Linkage Disequilibrium

If two genetic variants are in complete linkage disequilibrium, then inclusion of both variants in an individual-level analysis model would not lead to additional information. However, if variants are in partial linkage equilibrium, although the information provided by each variant is not independent, each variant does provide additional information on the causal effect. Summarized data methods using variants in linkage disequilibrium overstate precision.

Typically, GWA studies report the association of a single lead variant in a genetic region. If this variant is not the causal variant, or if there are multiple causal variants in the region, then additional information may be obtained by considering multiple variants per region. Information on such variants could be included correctly in a summarized analysis by considering conditional effects of variants on the risk factor and outcome by adjustment for the lead variant in a regression model [Yang et al., 2012]. We have not developed these methods as suitable data are not routinely reported in applied analyses.

Extension to Multiple Studies

The likelihood-based model introduced in this paper can be naturally extended into a hierarchical (multilevel) model in order to combine summarized associations from multiple studies. We assume that genetic variant k in study is associated with an observed mean change in the risk factor with standard error , with and similarly defined for the outcome. Although the presentation of GWA results from multiple studies is common in published research, summarized genetic associations that have been meta-analysed across studies are often reported rather than study-specific associations. We present this model for investigators with access to study-level summarized estimates in the hope that study-specific results will be routinely reported in the future.

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Random-effects distributions are assumed for the genetic association parameters and the causal effect parameters ; fixed-effect models () could also be assumed. The overall causal effect of X on Y is . Such models can be estimated in a Bayesian framework with weakly informative priors on the heterogeneity parameters τ². If a particular study does not provide an estimate , the distribution of the parameter can be estimated using the relevant random-effects distribution as an implicit prior.

Winner's Curse

The winner's curse is the phenomenon that the association estimate of the variant with the strongest association from a GWA study tends to be overestimated [Göring et al., 2001]. Typically, GWA studies report the single variant from each gene region showing the strongest association with the trait of interest. If several variants have similar strength, the variant with the strongest observed association will not always truly have the strongest association with the risk factor. Data-driven selection of the ‘lead’ variant results in bias in the causal estimate, as the association of the lead variant is typically over-estimated [Burgess et al., 2011b]. This is an example of selection bias. This bias can be eliminated by choosing the lead variant for each genetic region from an independent data source.