Segregation of noisy Mendelian traits and the effect of age-dependence: a prolegomenon

A discussion of the primordial confusion between the multiplicative Galtonian ("lognormal") trait and the imperfectly segregating Mendelian trait is laid out from a probabilistic standpoint. Several criteria used in comparing them (bimodality; bitangentiality; goodness of fit to the multinomialized form of the distribution; cumulants of the distributions) are reviewed and the inadequacy of their probabilistic properties discussed in some detail. The logical asymmetry of the normalized score ("Roberts" correction") and hence its invalidity as a criterion for distinguishing between the models is pointed out. The form of a mixture of two Gaussian distributions with fixed and equal variances but with differing age-dependent means ("the Platt model") is explored. The epistemological implications are exhibited. As a first step to restoring symmetry, a general model is proposed of which these and other models in wide use emerge as special cases. No attempt is made to deal here with the statistical aspects of the problem.