Commentary: To cause or not to cause confusion vs transparency with Mendelian Randomization

Appendix 1: sample size requirements for instrumental variable analysis with genetic instruments For given Type 1 and Type 2 error probabilities and study design, the sample size required to detect an effect of size d is proportional to 1/d 2 V, where V is the Fisher information (expectation of minus the second derivative of the log-likelihood of the effect size) from a single observation. For a single observation from a logistic regression model (in which the effect is measured as the log OR), the Fisher information is f(1Àf)v, where f is the probability of being a case, and v is the variance of the predictor variable. For a cohort design testing for association of a rare disease (f close to 0) with a quantitative trait that is scaled to have variance of 1, the Fisher information is simply the total number n of cases yielded by the cohort study. For a case–control design with N cases and N controls (f ¼ 0.5), testing for an effect on disease risk of genotype (coded as 0, 1, 2) at an SNP with allele frequency p, the Fisher information is 2NÂ0:5Â0:5Â2pð1 À pÞ ¼ Np 1 À p ð Þ. For allele frequency 0.2, this evaluates to N/6.25. Thus, in this situation the number N of cases required for a case– control study to detect the effect (measured as log OR associated with one extra copy of the disease-associated allele) is 6.25 times larger than the number n of cases required for a cohort study to detect an effect of the same size (measured as log OR associated with change of 1 SD) of a continuous trait on disease risk. In reality, the size of the genotypic effect a g on the intermediate phenotype is usually modest: typically no more than 0.25 SD for each extra copy of the trait-raising allele. As sample size scales inversely with the square of the effect size, this implies that the case–control collection would have to be 100 (16 Â 6.25) times larger (in terms of number of cases) than the cohort study for the effect of genotype on disease to be detected in a conventional significance test. For a Bayesian hypothesis test, the sample size requirements for the case–control study of genotype–disease association are similar. Bayesian sample size requirements for an experiment comparing two hypotheses can be calculated by specifying the expected log-likelihood ratio (ELOD) favouring …