Fisher's fundamental theorem and regression in causal analysis

TL;DR

The paper reframes Fisher's fundamental theorem as a special case of a general regression-change partition derived from the product rule for finite differences, showing that the total change in a mean outcome Δzbar decomposes into a predictor-change term at fixed coefficients and a context-change term where coefficients shift. Applied to fitness and gene frequencies, the first term equals the additive genetic variance in fitness, Var(g) = Cov(g, w), under the standard marginal-fitness definition w_i = p_i'/p_i. The approach unifies concepts across biology and economics by connecting Fisher's theorem to the Oaxaca-Blinder decomposition and the Price equation, highlighting that causality depends on the chosen counterfactual (what is held fixed) and that context-driven changes are captured by Δb. Overall, the work provides a concise, algebraic framework for causal analysis in regression across disciplines.

Abstract

Fisher's fundamental theorem describes the change caused by natural selection as the change in gene frequencies multiplied by the partial regression coefficients for the average effects of genes on fitness. Fisher's result has generated extensive controversy in biology. I show that the theorem is a simple example of a general partition for change in regression predictions across altered contexts. By that rule, the total change in a mean response is the sum of two terms. The first ascribes change to the difference in predictor variables, holding constant the regression coefficients. The second ascribes change to altered context, captured by shifts in the regression coefficients. This general result follows immediately from the product rule for finite differences applied to a regression equation. Economics widely applies this same partition, the Oaxaca-Blinder decomposition, as a fundamental tool that can in proper situations be used for causal analysis. Recognizing the underlying mathematical generality clarifies Fisher's theorem, provides a useful tool for causal analysis, and reveals connections across disciplines.