Measuring the Intensity of Natural Selection

Abstract

HALDANE¹ put forward a general measure of the intensity of natural selection. If W₀ is the chance of survival of an individual with the optimum phenotype and W̄ is the mean chance of survival in the population, then Haldane's measure is H = log_e W₀−log_e W̄. Van Valen² uses the measure I=(W₀−W̄)/W₀ which is simply the proportion of individuals that die because they do not have the optimum phenotype. It could well be called the “phenotypic load”, for the genetic load is measured in the same way except that the values of W are the fitnesses of the genotypes. If the exact forms of the distributions of individuals before and after selection are known, then I can be found directly. The frequencies are adjusted so that individuals with the optimum phenotype have the same frequency before and after selection. The two distributions are superimposed and the intensity of selection is the proportion of the distribution before selection that is missing from the distribution after selection. This method cannot often be used, however, because it is necessary to know the relative numbers of individuals before and after selection in order to find which phenotype has the best chance of survival. Haldane showed that H could be calculated from the change in variance after selection, provided the distributions before and after selection were both normal. But this assumption is unlikely to be strictly true. Van Valen^2,3 gave charts to calculate I from the observed changes in mean and variance. Essentially his charts show what proportion of the distribution before selection must be truncated by removal of extreme values in order to give the change in mean and variance. But natural selection does not operate by the death of all individuals with values greater or less than certain extremes. In natural selection all individuals have a chance of survival, but those nearer the optimum have a better chance than those further away. Van Valen's estimates of I are probably too small, for by cutting off the tail of a distribution a smaller proportion of the distribution can be removed to produce a given change than would be removed by natural selection. To obtain more realistic estimates of I, a general theory is required of the way a character x will change when the values of x determine the values of the fitness, W, in some general way. The simplest general relationship between x and W is a quadratic one: W decreases as the square of the deviation of x from some optimum phenotypic value θ. Thus we may have where K is a constant and W= 1 − α when x = θ. If the frequency density of x, f(x), is a normal frequency density with mean, x̄, and variance, V, then the mean and variance after selection, x̄′ and V′, can be found in terms of x̄ and V. If the frequency density is not normal, however, then x̄′ and V′ also depend on the third and fourth moments about the mean, µ₃ and µ₄. The equations for x̄′ and V′ can be derived by my method⁴. If µ₃ = 0 as for a symmetrical distribution and µ₄ = 3 V² as for a normal distribution, then these equations reduce to those already published⁴. If x̄′ − x̄ = Δx̄, V′ − V = ΔV and then we have Thus Thus there are two equations to estimate θ and φ. This is the estimation of the parabola of the relation between fitness and the character being measured. We have so that Thus It follows by definition that the intensity of selection is so that I can be calculated from these estimates of θ − x̄ and φ.