Mendelian randomization with invalid instruments: effect estimation and bias detection through Egger regression

Abstract

Background: The number of Mendelian randomization 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. However, some genetic variants may not be valid instrumental variables, in particular due to them having more than one proximal phenotypic correlate (pleiotropy).

Methods: We view Mendelian randomization with multiple instruments as a meta-analysis, and show that bias caused by pleiotropy can be regarded as analogous to small study bias. Causal estimates using each instrument can be displayed visually by a funnel plot to assess potential asymmetry. Egger regression, a tool to detect small study bias in meta-analysis, can be adapted to test for bias from pleiotropy, and the slope coefficient from Egger regression provides an estimate of the causal effect. Under the assumption that the association of each genetic variant with the exposure is independent of the pleiotropic effect of the variant (not via the exposure), Egger's test gives a valid test of the null causal hypothesis and a consistent causal effect estimate even when all the genetic variants are invalid instrumental variables.

Results: We illustrate the use of this approach by re-analysing two published Mendelian randomization studies of the causal effect of height on lung function, and the causal effect of blood pressure on coronary artery disease risk. The conservative nature of this approach is illustrated with these examples.

Conclusions: An adaption of Egger regression (which we call MR-Egger) can detect some violations of the standard instrumental variable assumptions, and provide an effect estimate which is not subject to these violations. The approach provides a sensitivity analysis for the robustness of the findings from a Mendelian randomization investigation.

Keywords: MR-Egger test; Mendelian randomization; invalid instruments; meta-analysis; pleiotropy; small study bias.

Figure 1.

Illustrative diagram showing the standard instrumental variable assumptions for genetic variant $G_j$ (solid lines) with potential violations of the assumptions shown by dotted lines (which are marked with a ‘cross’). The genetic effect on the exposure $X$ is ${\gamma }_j$, the direct genetic effect on the outcome $Y$ is ${\alpha }_j$ and the causal effect of the exposure $X$ on the outcome $Y$ is $\beta$.

Figure 2.

Plot of the gene–outcome ($\widehat{\Gamma }$) vs gene–exposure ($\widehat{\gamma }$) regression coefficients for a fictional Mendelian randomization analysis with 15 genetic variants. The true slope is shown by a dotted line, the inverse-variance weighted (IVW) estimate by a red line, and the MR-Egger regression estimate by a blue line. Refer to text for explanation of points (i) and (ii).

Figure 3.

Genetic associations with height and lung function from 180 variants measured in the ALSPAC dataset. Left: scatter plot of genetic associations with forced vital capacity (${\widehat{\Gamma }}_j$) against associations with height (${\widehat{\gamma }}_j$), with causal estimate of height on lung function estimated by inverse-variance weighted method. Right: funnel plot of minor allele frequency corrected genetic associations with height (${\widehat{\gamma }}_j^C$) against causal estimates based on each genetic variant individually (${\widehat{\beta }}_j$).

Figure 4.

Genetic associations with blood pressure and coronary artery disease risk from 29 variants—funnel plots of minor allele frequency corrected genetic associations with blood pressure (${\widehat{\gamma }}_j^C$) against causal estimates of blood pressure on CAD based on each genetic variant individually (${\widehat{\beta }}_j$). Left: funnel plot for systolic blood pressure. Right: funnel plot for diastolic blood pressure. The inverse-variance weighted (IVW) and MR-Egger causal effect estimates are also shown.

Figure 5.

Funnel plots of minor allele frequency corrected genetic associations with exposure (${\widehat{\gamma }}_j^C$) against causal estimates based on each genetic variant individually (${\widehat{\beta }}_j$) for 50 IV estimates in four scenarios: (a) no pleiotropy; (b) balanced pleiotropy; (c) directional pleiotropy, InSIDE assumption satisfied; and (d) directional pleiotropy, InSIDE assumption not satisfied. The inverse-variance weighted (IVW, red) and MR-Egger (blue) causal effect estimates are also shown.