In Mendelian randomization (MR) studies, where genetic variants are used as proxy measures for an exposure trait of interest, obtaining adequate statistical power is frequently a concern due to the small amount of variation in a phenotypic trait that is typically explained by genetic variants. A range of power estimates based on simulations and specific parameters for two-stage least squares (2SLS) MR analyses based on continuous variables has previously been published. However there are presently no specific equations or software tools one can implement for calculating power of a given MR study. Using asymptotic theory, we show that in the case of continuous variables and a single instrument, for example a single-nucleotide polymorphism (SNP) or multiple SNP predictor, statistical power for a fixed sample size is a function of two parameters: the proportion of variation in the exposure variable explained by the genetic predictor and the true causal association between the exposure and outcome variable. We demonstrate that power for 2SLS MR can be derived using the non-centrality parameter (NCP) of the statistical test that is employed to test whether the 2SLS regression coefficient is zero. We show that the previously published power estimates from simulations can be represented theoretically using this NCP-based approach, with similar estimates observed when the simulation-based estimates are compared with our NCP-based approach. General equations for calculating statistical power for 2SLS MR using the NCP are provided in this note, and we implement the calculations in a web-based application.

A genome-wide association study of educational attainment was conducted in a discovery sample of 101,069 individuals and a replication sample of 25,490. Three independent SNPs are genome-wide significant (rs9320913, rs11584700, rs4851266), and all three replicate. Estimated effects sizes are small (R2 ≈ 0.02%), approximately 1 month of schooling per allele. A linear polygenic score from all measured SNPs accounts for ≈ 2% of the variance in both educational attainment and cognitive function. Genes in the region of the loci have previously been associated with health, cognitive, and central nervous system phenotypes, and bioinformatics analyses suggest the involvement of the anterior caudate nucleus. These findings provide promising candidate SNPs for follow-up work, and our effect size estimates can anchor power analyses in social-science genetics.

Across-nation differences in the mean of complex traits such as obesity and stature are common1–8, but the reasons for these differences are not known. Here, we find evidence that many independent loci of small effect combine to create population genetic differences in height and body mass index (BMI) in a sample of 9,416 individuals across 14 European countries. Using discovery data on over 250,000 individuals and unbiased estimates of effect sizes from 17,500 sib pairs, we estimate that 24% (95% CI: 9%, 41%) and 8% (95% CI: 4%, 16%) of the captured additive genetic variance for height and BMI across Europe are attributed to among-population genetic differences. Population genetic divergence differed significantly from that expected under a null model (P <3.94e−08 for height and P<5.95e−04 for BMI), and we find an among-population genetic correlation for tall and slender nations (r = −0.80 (95% CI: −0.95, −0.60), contrasting no genetic correlation between height and BMI within populations (r = −0.016, 95% CI: −0.041, 0.001), consistent with selection on height genes that also act to reduce BMI. Observations of mean height across nations correlated with the predicted genetic means for height (r = 0.51, P<0.001), so that a proportion of observed differences in height within Europe reflect genetic factors. In contrast, observed mean BMI did not correlate with the genetic estimates (P<0.58), implying that genetic differentiation in BMI is masked by environmental differences across Europe.