Estimates of the causal effect using all the genetic variants are calculated using individual-level data with the two-stage least squares (2SLS) method [Baum et al., 2003 ], and using summarized data with the IVW (equations 1 and 2 ) and likelihood-based (equation 3 ) methods. The first-stage model in the 2SLS was taken as additive in the variants throughout, and as such the genetic model was misspecified when there were gene–gene interactions. Summarized associations were obtained by ordinary least squares (OLS) linear regression of the risk factor and outcome on each variant in separate regression models. The likelihood-based analyses were performed in R (http://www.r-project.org ) using the optim command to directly maximize the likelihood.
An estimate of the correlation between the genetic associations with risk factor and outcome of was used based on the approximate observational correlation between the risk factor and outcome. Estimates were not especially sensitive to moderate (±0.2) changes in this correlation. (A sensitivity analysis for this parameter is shown later for an applied example.)
In each scenario, results from 10,000 simulated datasets for the comparison of the individual-level and summarized data methods are given. We present the mean and median estimates across simulations, the standard deviation (SD) of estimates, the mean standard error (SE), the coverage of the 95% confidence interval for the causal effect (the proportion of simulated datasets for which the 95% confidence interval included the true value of ), and the empirical power at a 5% significance level (the proportion of simulated datasets for which the 95% confidence interval excluded the null value of ). The Monte Carlo standard error (representing the variation in estimates due to the finite number of simulations) was approximately 0.001 for the mean estimate (0.004 for the final scenario with gene–gene interactions) and 0.2% for the coverage. In each set of simulations, the mean value of the F statistic in the regression of the risk factor on the IVs is given.
An estimate of the correlation between the genetic associations with risk factor and outcome of was used based on the approximate observational correlation between the risk factor and outcome. Estimates were not especially sensitive to moderate (±0.2) changes in this correlation. (A sensitivity analysis for this parameter is shown later for an applied example.)
In each scenario, results from 10,000 simulated datasets for the comparison of the individual-level and summarized data methods are given. We present the mean and median estimates across simulations, the standard deviation (SD) of estimates, the mean standard error (SE), the coverage of the 95% confidence interval for the causal effect (the proportion of simulated datasets for which the 95% confidence interval included the true value of ), and the empirical power at a 5% significance level (the proportion of simulated datasets for which the 95% confidence interval excluded the null value of ). The Monte Carlo standard error (representing the variation in estimates due to the finite number of simulations) was approximately 0.001 for the mean estimate (0.004 for the final scenario with gene–gene interactions) and 0.2% for the coverage. In each set of simulations, the mean value of the F statistic in the regression of the risk factor on the IVs is given.
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