To help interpret PGIs, we lay out a theoretical framework. Denote individual i’s phenotype value by yi⋆. Denote individual i’s allele count at genetic variant j by xij⋆∈0,1,2. Without loss of generality, we use a mean-centred transformation of the phenotype and allele counts, such that yi≡yi⋆-Eyi⋆ and xij≡xij⋆-Exij⋆ for each SNP j. We denote the vector of mean-centered allele counts at J genetic variants by xi=xi1,xi2,…,xiJ'. As a benchmark, consider the standardized best linear predictor of the phenotype based on the allele counts: gi≡xi′γsd(xi′γ), where γ=arg min γ~Eyi-xi′γ~2. That is, the optimal weight vector γ is the vector of coefficients from the population regression of yi on xi. This population regression may also include control variables; we omit them here to avoid cluttering notation, but in the Supplementary Methods we extend the framework to include them and explain why they do not affect the results in this paper. In the User Guide (also in the Supplementary Methods), we explain how control variables do matter for the interpretation of a PGI.
