2014, Dickhaus, Thorsten, Stange, Jens, Demirhan, Haydar

We are concerned with statistical inference for 2 x 2 x K contingency tables in the context of genetic case-control association studies. Multivariate methods based on asymptotic Gaussianity of vectors of test statistics require information about the asymptotic correlation structure among these test statistics under the global null hypothesis. We show that for a wide variety of test statistics this asymptotic correlation structure is given by the linkage disequilibrium matrix of the K loci under investigation. Three popular choices of test statistics are discussed for illustration.

2014, Gün, Onur, Yilmaz, Atilla

Stochastic encounter-mating (SEM) models describe monogamous permanent pair formation in finite zoological populations of multitype females and males. In this article we study SEM with Poisson firing times. We prove that an infinite population corresponds to a fluid limit, i.e., the stochastic dynamics converges to a deterministic system governed by coupled ODEs. Moreover, we establish a functional central limit theorem and give a diffusion approximation for the model. Next, we convert the fluid limit ODEs to the well-known Lotka-Volterra and replicator equations from population dynamics. Under the so-called fine balance condition, which characterizes panmixia for finite populations, we solve the corresponding replicator equations and give an exact expression for the fluid limit. Finally, we consider the case with two types of females and males. Without the fine balance assumption, but under some symmetry conditions, we give an explicit formula for the limiting mating pattern, and then use it to fully characterize assortative mating.

2014, Gün, Onur, Yilmaz, Atilla

We propose a new model of permanent monogamous pair formation in zoological populations comprised of kge 2 types of females and males, which unifies and generalizes the encounter-mating models of Gimelfarb (1988). In our model, animals randomly encounter members of the opposite sex at their so-called firing times to form temporary pairs which then become permanent if mating happens. Given the distributions of the firing times and the mating preferences upon encounter, which depend on the sex and the type of the animals, we analyze the contingency table Q(t) of permanent pair types at any time tge 0. First, we consider definite mating upon encounter and provide a formula for the distribution of Q(t). In particular, at the terminal time T, the so-called mating pattern Q(T) has a multiple hypergeometric distribution. This implies panmixia which means that female and male types are uncorrelated in the expected mating pattern. Next, when the firing times come from Poisson and Bernoulli point processes, we formulate conditions that characterize panmixia. Moreover, when these conditions are satisfied, the underlying parameters of the model can be changed to yield definite mating upon encounter, and our results for the latter case carry over. Finally, when k=2, we fully characterize heterogamy/panmixia/homogamy, i.e., negative/zero/positive correlation of same type females and males in the expected mating pattern. We thereby rigorously prove, strengthen and generalize Gimelfarb's results.