Remarks on a Question by H. Wolda

In a very interesting paper Wolda (1969) describes a remarkable cline in gene frequency in the snail Cepaea nemoralis (L.), which has remained fairly constant over a period of 12 years. He asks the question, if the environment is stationary, how can one distinguish between those cases in which the gene frequency stays constant from a regulating form of selection, and those cases in which it stays constant because of a balance resulting from changes in the direction of selection from time to time. On p. 631 he raises various difficulties, and he says 'I hope some statistician will accept the challenge'. This problem has a fairly simple solution in the terms stated. If the selective forces are not tending to produce an equilibrium, either by heterozygote advantage or by any of the other methods known (Williamson 1958) then there will be some selective forces whose action will tend to move the gene frequency continuously in one direction, which we may call 'up' and others which will tend to move it in what we may call 'down'. In the stationary environment postulated by Wolda, the net expected change of gene frequency will be the resultant of these two, and will, in all real cases, be either up or down. That is, despite fluctuations that will take place from time to time, there will be a net movement of the gene frequency in one way or the other. This change will not usually be linear, and in any case the change at any one time will be Markovian, depending on a previous point reached, so it is not valid to test this by a regression of gene frequency against time. However, there is a simple way to test the hypothesis. Let the number of observations be k + 1, taken at time to to tk, and the frequency observed by qo to qk. Calculate successive, nested, variances vi based on the values from qo to qi inclusive, with i from 1 to k. Then, by the Central Limit theorem, if the polymorphism is a balanced one the vi will show no tendency to increase and will approach a fixed value asymptotically. On the other hand, if the polymorphism is not governed, then the series vi will tend to increase. The vi are, of course, not independent, but a persistent increase in their values is nevertheless valid evidence against the hypothesis that one is observing a balanced polymorphism in a stationary environment. This method will not work unless the environment is stationary. When it is not, the problems are much more acute, and can be solved only with a knowledge either of the nature of the selective forces, or of the values of the relevant environmental variables, or both. These considerations can be ignored in examining Wolda's data, because here the variance is remarkably stationary, and we can say there is every indication that he is working with a balanced polymorphism. Without knowing how this system is brought about though, we cannot start to estimate the size of the selective forces from the observed