[Paper Review] Fisher's fundamental theorem and regression in causal analysis

Steven A. Frank|arXiv (Cornell University)|Jan 13, 2026

Game Theory and Applications0 citations

TL;DR

The paper shows Fisher’s fundamental theorem as a special case of a general regression change partition via the product rule for finite differences, linking it to the Oaxaca-Blinder decomposition used in causal analysis.

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. The same partition also arises in demography and thermodynamics. Recognizing the underlying mathematical generality clarifies Fisher's theorem, provides a useful tool for causal analysis, and reveals connections across disciplines.

Motivation & Objective

• Explain how the total change in a regression mean can be partitioned into changes from predictor differences and context changes.
• Demonstrate that Fisher’s fundamental theorem is an instance of a broader regression decomposition.
• Highlight the connection between Fisher’s theorem and the Oaxaca-Blinder decomposition used in economics for causal analysis.
• Clarify the role of context changes and regression coefficient shifts in causal interpretations.

Proposed method

• Derive the product rule for finite differences and extend it to regression equations.
• Express the change in the mean of a regression outcome as a sum of a predictor-difference term and a coefficient-change term.
• Define the regression model and mean in terms of predictor means and regression coefficients, then apply the finite-difference product rule.
• Specialize the general decomposition to fitness as the outcome and allele frequencies as predictors, recovering Fisher’s theorem.
• Relate the first term to additive genetic variance and the second term to changes in context via coefficient shifts.

Experimental results

Research questions

• RQ1 How can total changes in a regression mean be partitioned when the context changes are allowed to alter regression coefficients?
• RQ2 In what sense is Fisher’s fundamental theorem a specific case of a general regression decomposition?
• RQ3 How does the Oaxaca-Blinder decomposition relate to causal interpretation across disciplines?
• RQ4 What is the precise mathematical role of the regression-coefficient changes in causal analyses?
• RQ5 What does the Price equation contribute in the same product-rule framework?

Key findings

• The total change in a mean regression response equals the change in predictor means multiplied by fixed coefficients plus the change in coefficients evaluated in the new context.
• Fisher’s theorem corresponds to holding regression coefficients fixed while allele frequencies change, isolating the effect of natural selection.
• The first component of the decomposition equals the additive genetic variance in fitness for the given model.
• The second component captures context-related changes via shifts in the regression coefficients.
• The Price equation is presented as another product-rule expression for finite differences with a covariance form.
• The Oaxaca-Blinder decomposition is shown to be mathematically parallel to the Fisher-style partition and useful for causal analysis across disciplines.