Improving the visualization, interpretation and analysis of two-sample summary data Mendelian randomization via the Radial plot and Radial regression

Abstract

Background: data furnishing a two-sample Mendelian randomization (MR) study are often visualized with the aid of a scatter plot, in which single-nucleotide polymorphism (SNP)-outcome associations are plotted against the SNP-exposure associations to provide an immediate picture of the causal-effect estimate for each individual variant. It is also convenient to overlay the standard inverse-variance weighted (IVW) estimate of causal effect as a fitted slope, to see whether an individual SNP provides evidence that supports, or conflicts with, the overall consensus. Unfortunately, the traditional scatter plot is not the most appropriate means to achieve this aim whenever SNP-outcome associations are estimated with varying degrees of precision and this is reflected in the analysis.

Methods: We propose instead to use a small modification of the scatter plot-the Galbraith Radial plot-for the presentation of data and results from an MR study, which enjoys many advantages over the original method. On a practical level, it removes the need to recode the genetic data and enables a more straightforward detection of outliers and influential data points. Its use extends beyond the purely aesthetic, however, to suggest a more general modelling framework to operate within when conducting an MR study, including a new form of MR-Egger regression.

Results: We illustrate the methods using data from a two-sample MR study to probe the causal effect of systolic blood pressure on coronary heart disease risk, allowing for the possible effects of pleiotropy. The Radial plot is shown to aid the detection of a single outlying variant that is responsible for large differences between IVW and MR-Egger regression estimates. Several additional plots are also proposed for informative data visualization.

Conclusions: The Radial plot should be considered in place of the scatter plot for visualizing, analysing and interpreting data from a two-sample summary data MR study. Software is provided to help facilitate its use.

Figure 1.

Traditional scatter plot of SNP–CHD associations ${\widehat{\Gamma }}_j$ vs SNP–SBP associations ${\widehat{\gamma }}_j$. SNP rs17249754 is shown as a square symbol.

Figure 2.

Top: Individual variant contributions to Cochran’s heterogeneity statistic. The contribution of SNP rs17249754 (labelled Q₈) is shown as a square. Bottom left: Cook’s distance for each genetic variant in the SBP–CHD data, with standard influence threshold (4/#SNPs) indicated by a dashed line. Bottom right: Studentized residuals for each variant in the SBP–CHD data with standard 5% significance thresholds (solid black lines) and Bonferroni-corrected significance thresholds (5%/#SNPs, dashed lines). SNP rs17249754 is again shown as a square.

Figure 3.

Left: Radial MR plot of the blood-pressure data. IVW and Radial MR-Egger regression slopes calculated using first-order weights are overlaid. The square-root contribution of SNP rs17249754 to Cochran’s Q statistic ($\sqrt{Q_8}$) is denoted by the vertical dashed line from the IVW slope. The square-root contribution of a separate SNP to Rücker’s $Q^{\prime }$ statistic ($\sqrt{Q_{11}}$) is denoted by the vertical dashed line from the Radial MR-Egger slope. Right: Generalized funnel plot of same data with first-order IVW and Radial MR-Egger regression slopes (and 95% confidence intervals) shown. SNP rs17249754 is shown as a square.

Figure 4.

Radial MR-Egger funnel plot. Horizontal dashed lines link the position of data in the standard funnel plot (circles) to their implied position under a Radial MR-Egger analysis (triangles).

Figure 5.

Leave-one-out sensitivity analysis of the data, showing the values of Q and $Q^{\prime }$ when each variant is left out of the analysis in turn. Points are overlaid on the Rücker decision space that governs which of four model choices should be favoured. It assumes a significance threshold of δ = 0.05 to affect the model selection.

Figure 6.

Radial plots of the blood-pressure data produced using the RadialMR package. Top: Only the IVW estimate shown, Radial lines joining each data point back to the origin. Bottom: Radial MR-Egger and IVW model fits shown.