An appraisal of methods for the analysis of longitudinal categorical data with MAR drop-outs.

A number of methods for analysing longitudinal ordinal categorical data with missing-at-random drop-outs are considered. Two are maximum-likelihood methods (MAXLIK) which employ marginal global odds ratios to model associations. The remainder use weighted or unweighted generalized estimating equations (GEE). Two of the GEE use Cholesky-decomposed standardized residuals to model the association structure, while another three extend methods developed for longitudinal binary data in which the association structures are modelled using either Gaussian estimation, multivariate normal estimating equations or conditional residuals. Simulated data sets were used to discover differences among the methods in terms of biases, variances and convergence rates when the association structure is misspecified. The methods were also applied to a real medical data set. Two of the GEE methods, referred to as Cond and ML-norm in this paper and by their originators, were found to have relatively good convergence rates and mean squared errors for all sample sizes (80, 120, 300) considered, and one more, referred to as MGEE in this paper and by its originators, worked fairly well for all but the smallest sample size, 80.

The evolution of strategy variation: will an ESS evolve?

Evolutionarily stable strategy (ESS) models are widely viewed as predicting the strategy of an individual that when monomorphic or nearly so prevents a mutant with any other strategy from entering the population. In fact, the prediction of some of these models is ambiguous when the predicted strategy is "mixed", as in the case of a sex ratio, which may be regarded as a mixture of the subtraits "produce a daughter" and "produce a son." Some models predict only that such a mixture be manifested by the population as a whole, that is, as an "evolutionarily stable state"; consequently, strategy monomorphism or polymorphism is consistent with the prediction. The hawk-dove game and the sex-ratio game in a panmictic population are models that make such a "degenerate" prediction. We show here that the incorporation of population finiteness into degenerate models has effects for and against the evolution of a monomorphism (an ESS) that are of equal order in the population size, so that no one effect can be said to predominate. Therefore, we used Monte Carlo simulations to determine the probability that a finite population evolves to an ESS as opposed to a polymorphism. We show that the probability that an ESS will evolve is generally much less than has been reported and that this probability depends on the population size, the type of competition among individuals, and the number of and distribution of strategies in the initial population. We also demonstrate how the strength of natural selection on strategies can increase as population size decreases. This inverse dependency underscores the incorrectness of Fisher's and Wright's assumption that there is just one qualitative relationship between population size and the intensity of natural selection.

A new statistical test of fitness set data from reciprocal transplant experiments involving intermediate phenotypes.

Experimental biologists use reciprocal transplant experiments (RTEs) involving divergent forms to test hypotheses about fitness trade-offs across, and local adaptation to, native environments. Additional evolutionary hypotheses about diversifying selection, the evolution of specialization, and the coexistence of specialists and generalists are only testable when the RTE also includes intermediate (or alternatively generalist) forms. Environmental variation makes such RTEs challenging, and so strategies that increase their effectiveness are useful. Here, we focus on improvements to the efficiency of RTEs involving intermediate forms with respect to the experimental design and the analysis of the resulting data. We provide a likelihood ratio-based test that offers increased statistical power and robustness relative to another test involving nonlinear regression, when used both for simulated data sets and for data from a study of two divergent fish species and their hybrids transplanted between two lake habitats. The test can be used with unequal numbers of observations, unequal variances, and binomial-type survival data and other nonnormal data. Simulations suggest that having equal numbers of experimental units in each phenotype-environment combination is reasonable. The intentional pairing of observations between environmental conditions (by using clones, full sibs, or half-sibs) is beneficial when paired observations have fitnesses that are negatively related between conditions but is detrimental with positive relatedness. Our methods can be extended to study more than two divergent forms.