The method

We know from Lande and Arnold's results that the change in multivariate phenotype from one generation to the next, $\Delta\bar{\bf z}$, can be written as

$$\Delta\bar{\bf z} = {\bf G\beta} \quad ,$$

where $\bf G$ is the genotypic variance-covariance matrix, ${\bf\beta} = {\bf P}^{-1}{\bf s}$ is the set of partial regression coefficients describing the direct effect of each character on relative fitness.⁵ If we are willing to assume that G remains constant, then the total change in a character subject to selection for $n$ generations is

$$\sum_{k=1}^n \Delta\bar{\bf z} = {\bf G}\sum_{k=1}^n\beta \quad .$$

Thus, $\sum_{k=1}^n\beta$ can be regarded as the cumulative selection differential associated with a particular observed change, and it can be estimated as

$$\sum_{k=1}^n\beta = {\bf G}^{-1}\sum_{k=1}^n \Delta\bar{\bf z}\quad .$$