In all simulation scenarios, causal effect estimates were obtained using established MR methods (multiplicative random effects IVW,7 (link) multiplicative random effects MR-Egger regression7 (link) and weighted median, all implemented using inverse-variance weights calculated under NOME), as well as the simple and the weighted MBEs. Each version of the MBE was evaluated using weights calculated with and without making the NOME assumption, thus yielding four MBEs. Each of these four methods was evaluated for two values of the tuning parameter , totalling eight versions of the MBE method. Parametric bootstrap was used to estimate the standard errors of the MBE using the median absolute deviation from the median (multiplied by 1.4826 for asymptotically normal consistency) of the bootstrap distribution of causal effect estimates. These were used to derive symmetrical confidence intervals.
In each scenario, coverage, power and average causal effect estimates, standard errors, and statistics (which quantify the magnitude of violation of the NOME assumption in IVW and MR-Egger regression estimates, respectively7 (link),13 (link)) were obtained across 10 000 simulated datasets. Power was defined as the proportion of times that 95% confidence intervals excluded zero, and coverage as the proportion of times that 95% confidence intervals included the true causal effect.
MR methods were also applied to estimate the causal effect of plasma lipid fractions and urate levels on CHD risk. The magnitude of regression dilution bias in IVW and MR-Egger regression was assessed by the and statistics, respectively. Cochran’s Q test was used to test for the presence of horizontal pleiotropy (under the assumption that this is the only source of heterogeneity between s other than chance).20 (link) All simulations and analyses were performed using R 3.3.1 [www.r-project.org ]. R code for implementing the MBE is provided in Supplementary Methods (available as Supplementary data at IJE online).
In each scenario, coverage, power and average causal effect estimates, standard errors, and statistics (which quantify the magnitude of violation of the NOME assumption in IVW and MR-Egger regression estimates, respectively7 (link),13 (link)) were obtained across 10 000 simulated datasets. Power was defined as the proportion of times that 95% confidence intervals excluded zero, and coverage as the proportion of times that 95% confidence intervals included the true causal effect.
MR methods were also applied to estimate the causal effect of plasma lipid fractions and urate levels on CHD risk. The magnitude of regression dilution bias in IVW and MR-Egger regression was assessed by the and statistics, respectively. Cochran’s Q test was used to test for the presence of horizontal pleiotropy (under the assumption that this is the only source of heterogeneity between s other than chance).20 (link) All simulations and analyses were performed using R 3.3.1 [
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