The generalized Price equation: forces that change population statistics

The Price equation partitions the change in the expected value of a population measure. The first component describes the partial change caused by altered frequencies. The second component describes the partial change caused by altered measurements. In biology, frequency changes often associate with the direct effect of natural selection. Measure changes reflect processes during transmission that alter trait values. More broadly, the two components describe the direct forces that change population composition and the altered frame of reference that changes measured values. The classic Price equation is limited to population statistics that can expressed as the expected value of a measure. Many statistics cannot be expressed as expected values, such as the harmonic mean and the family of rescaled diversity measures. We generalize the Price equation to any population statistic that can be expressed as a function of frequencies and measurements. We obtain the generalized partition between the direct forces that cause frequency change and the altered frame of reference that changes measurements.