To see how the covariance equation translates into Hamilton’s kin-selection equation, begin with \(w\Delta g = \texttt{Cov}(w,g)\) where \(g\) is the breeding value that determines the level of altruism.

The least-squares multiple regression that predicts fitness, \(w\), can be written as \(w = \alpha + g\beta_{wg'ag'} + g'\beta_{wg''ag+\epsilon}\) where \(g'\) is the average \(g\) value of an individual’s social neighbors, \(\alpha\) is a constant, and \(\epsilon\) is the residual which is uncorrelated with \(g\) and \(g'\). The \(\beta\) are partial regression coefficients that summarize costs and benefits in the following way: \(\beta_{wg'ag'}\) is the effect of an individual’s breeding value has on its own fitness in the presence of neighbors’ \(g'\), and \(\beta_{wg''ag}\) is the effect of an individual’s breeding value on the fitness of its neighbors.

Solving for the condition under which \(w\Delta g > 0\) gives Hamilton’s inclusive fitness equation (\(rB > C\)) in the following form: \(\beta_{wg'ag'}+\beta_{g'g}\beta_{wg''ag} > 0\), where \(\beta_{g'g}\) is the regression coefficient of relatedness.