Previous equations for estimating power using the non-centrality parameter in the case of continuous outcomes(^{[1]}) were adapted for binary outcomes using an approximate linear model on the observed binary (0-1) scale. The calculations below are approximations and in the absence of X-Y confounding.
Input parameters for power calculations:
(K) = proportion of cases in the (intended) study
(N) = total sample size
(OR) = True odds ratio of the outcome variable per standard deviation of the exposure variable
(R^2_{xz}) = proportion of variance in exposure variable explained by SNPs
A linear model on the 0-1 scale in the population:
(y_{01} = K + b_{01}x + e)
The probabilities of the binary outcomes (y = 0 or y =1) for x = 0 and x = 1 standard deviation above the mean are:
(Prob(disease | x = 0) = K)
(Prob(control | x = 0) = 1 - K)
(Prob(disease | x = 1) = K + b_{01})
(Prob(control | x = 1) = 1 - K - b_{01})
The odds ratio (OR = \frac{\frac{K + b_{01}}{1 - K - b_{01}}}{\frac{K}{1 - K}})
With input variables OR and K, the regression coefficient is derived on the observed scale:
(b_{01} = K(\frac{OR}{1 + K(OR - 1)} – 1))
The sampling variance of the estimate of (b_{01}) is, approximately,
(var(\hat b_{01}) = var(e) = var(y_{01}) – b^2_{01}var(x) = K(1 - K) – b^2_{01})
So the mean and sampling variance of the MR estimator on the linear scale are:
(b_{MR} = K(\frac{OR}{1 + K(OR-1)} – 1))
(var(b_{MR}) = \frac{var(e)}{N R^2_{xz}} = \frac{K(1-K) – b^2_{01}}{N R^2_{xz}})
and
(NCP = \frac{b^2_{MR}}{var(b_{MR})} = N R^2_{xz} \frac{(K(\frac{OR}{1 + K(OR-1)} – 1))^2}{K(1-K) – b^2_{01}})
References:
1 Brion M.J., Shakhbazov K. and Visscher P.M. 2013. Calculating statistical power in Mendelian randomization studies. Int J Epidemiol 42(5) 1497-1501.