  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.
