Analysis of Multinomial Models With Unknown Index Using Data Augmentation

Abstract

Multinomial models with unknown index ("sample size") arise in many practical settings. In practice, Bayesian analysis of such models has proved difficult because the dimension of the parameter space is not fixed, being in some cases a function of the unknown index. We describe a data augmentation approach to the analysis of this class of models that provides for a generic and efficient Bayesian implementation. Under this approach, the data are augmented with all-zero detection histories. The resulting augmented dataset is modeled as a zero-inflated version of the complete-data model where an estimable zero-inflation parameter takes the place of the unknown multinomial index. Interestingly, data augmentation can be justified as being equivalent to imposing a discrete uniform prior on the multinomial index. We provide three examples involving estimating the size of an animal population, estimating the number of diabetes cases in a population using the Rasch model, and the motivating example of estimating the number of species in an animal community with latent probabilities of species occurrence and detection.

Inference About Species Richness and Community Structure Using Species-Specific Occupancy Models in the National Swiss Breeding Bird Survey MHB

“…They used these to develop predictions of missing z(i, j) variables, and functions of those variables such as community similarity and richness. Here we provide a fully Bayesian analysis of the model applied to the MHB data, using the data augmentation parameterization described by Royle et al (2007a). This formulation was also used in the analysis presented in Dorazio et al (2006).…”

“…For instance, in all three previous applications (Dorazio and Royle 2005;Dorazio et al 2006;Royle et al 2007a), it was assumed that quadrat effects α j and β j are constant and hence, logit(ψ(i)) = μ i + α and logit( p(i)) = ν i + β. Since μ i and ν i were assumed to be draws from zero-mean normal distributions, α and β were the mean logit-scale parameters of occupancy and detection probability.…”

“…This computational burden, and the search for a convenient Bayesian treatment of the problem, motivated the data augmentation approach described by Royle et al (2007a), which we adopt for our analysis of the MHB data. With data augmentation, the observed data comprised of encounter histories on n individuals is physically augmented with a large number of "all zero" encounter histories, say M -n, yielding a fixed total number of available "species" of M. This comprises n observed species and M -n pseudo-species represented by all zeros.…”

“…The interpretation of M is that of a super-population of species from which the actual sampled community (the species exposed to sampling) was drawn by some hypothetical random sampling mechanism; see Royle et al (2007a) for additional details and discussion of the data augmentation idea. The main consideration in setting a fixed M is that it must be large enough so as to avoid truncation of the posterior distribution.…”

“…Here, we present a statistical model for the estimation of species richness and related community measures that is applicable in the frequent case when replicate sites (=communities) representing a metacommunity are sampled repeatedly over a short time (Dorazio and Royle 2005;Royle et al 2007a). Our model consists of component models for site-occupancy of individual species tied together by random effects assumptions about some of the species-specific parameters and is a multispecies generalisation of single-species site-occupancy models (MacKenzie et al 2002(MacKenzie et al , 2006Royle and Kéry 2007).…”