## Introduction

Detection and explanation of spatial and temporal patterns in abundance lie at the heart of ecology (Krebs 2001) as well as of its applications such as conservation biology, pest management or monitoring science (Caughley 1994; Norris 2004). Abundance *N* (also called local abundance *N*_{i} at site *i*) is *the* key state variable for describing populations, but it can hardly ever be measured without error due to imperfect detection probability *p*– in most situations, some individuals will be missed, that is, detectability *p* < 1. Hence, simple counts *C* are not equivalent to *N* but are related to abundance by the well-known relationship *E*(*C*) = *Np* (Williams, Nichols & Conroy 2002); they are only *indices* to abundance with the expected value *E* of a count being a proportion *p* of *N*. Absent double counting, simple counts almost always underestimate abundance. In addition, spatial and temporal patterns in simple counts will be due to both patterns in abundance *and* patterns in detection (Kéry 2008). Hence, when unbiased estimates of abundance are required or when population trends need to be assessed free from possible distorting patterns in detectability, abundance must be estimated separately from detectability (MacKenzie & Kendall 2002; Kéry & Schmid 2004; Kéry & Schmidt 2008).

Over the last decades, an armada of protocols and associated statistical models have been developed to ‘adjust’ simple counts by an estimate of detection probability and thus arrive at an estimate of abundance. Examples include distance sampling (Buckland *et al.* 2001) and a large array of capture–recapture protocols (Borchers, Buckland & Zucchini 2002; Williams *et al.* 2002). Distance sampling uses the distribution of detection distances to provide information about detection probability, while capture–recapture uses the pattern of detection/non-detection over replicated surveys from a period during which a population can be assumed static or closed. In both frameworks, the estimate of detection probability provides the direct link between the observed count and the estimate of population size.

Much of ecology and its applications is concerned with comparisons of abundance in space, and consequently abundance is frequently assessed at multiple sites and using similar protocols at each. This, in essence, represents a metapopulation design (Royle 2004b). Analysis of abundance and detection on a site-by-site basis would be highly inefficient or sometimes impossible due to locally small sample size or even zero counts. Instead, an integrated analysis is required for the most efficient use of available information and to directly model patterns of abundance.

Recently, hierarchical models have been developed for inference about abundance and detection that explicitly account for a metapopulation design (Royle & Dorazio 2006, 2008; also see Borchers *et al.* 2002). These models have one stochastic component to describe spatial and possibly temporal variation in abundance and another stochastic component that specifies the stochastic outcome of the observation process. The beauty of these models is that, conceptually, they provide a truly mechanistic rendering of how counts of organisms arise as a result of two linked stochastic processes, an ecological process and a dependent observation process. Different descriptions of these two processes can simply be combined in a modular fashion as needed for the particular study at hand. This means that a large variety of animal sampling protocols can all be subsumed into a generic hierarchical model by simply using different stochastic descriptions of the observation process (Royle 2004b). Examples include distance sampling (Royle, Dawson & Bates 2004), point counts (Royle 2004a), removal sampling (Dorazio, Jelks & Jordan 2005), detection–non-detection (a.k.a. ‘presence–absence’) sampling (MacKenzie *et al.* 2002; Royle & Nichols 2003; Dorazio 2007; Royle & Kéry 2007) and capture–recapture proper (Royle *et al.* 2007; Webster, Pollock & Simons 2008).

Furthermore, these hierarchical models are just variations of the Poisson generalized linear model (GLM) or generalized linear mixed model (GLMM), both of which are commonly used to model population change from count data (Link & Sauer 2002; Gregory *et al.* 2005; Link, Sauer & Niven 2006). Thus, all what is known from this important class of models (see e.g. McCulloch & Searle 2001; Lee, Nelder & Pawitan 2006) could be directly carried over to these hierarchical models. Examples include the introduction of covariate information by means of a link function (McCullagh & Nelder 1989), the introduction of random effects and correlation among parameters (Lee *et al.* 2006) or the extension to additive models with smooth terms (Wood 2006).

In this article, we compare two classes of hierarchical models that are particularly useful for inference about abundance in metapopulation designs, that is, when observations of ‘individuals’ are replicated in space *and* time; the binomial (Royle 2004a) and the multinomial mixture model (Royle *et al.* 2007). ‘Individuals’ may be any individually recognizable units, such as individual animals or plants, breeding pairs or territories, or even species (Kéry 2009). Here, we use territory-mapping data from the Swiss Breeding bird survey MHB (Schmid, Zbinden & Keller 2004) for two species; hence, individuals represent individual territories. First, in the binomial mixture model, we regard the data as independent binomial counts and inference is based on a product-binomial/Poisson hierarchical model. Secondly, we use the more complex detection–non-detection data for each territory to form encounter history frequencies for each site, and our analysis is based on a multinomial/Poisson hierarchical model. As the data for the former are just an aggregated form of the more detailed data format used in the latter, we expect very similar inferences under these two models. We hypothesize better precision for the multinomial model because it uses a more detailed (i.e. information-rich) format of the data. However, the data collection assumptions are somewhat more strict as we describe subsequently.

Importantly, in our comparison of the binomial and the multinomial mixture models, we extend both models to directly estimate population trends over multiple years (see also Royle & Dorazio 2008, pp. 4–7 and Kéry *et al.* 2009 for similar models). That is, our models enable one to estimate population trends corrected for any (parallel) time trends that might exist in detectability. We believe that the ability to directly model population dynamics (here, a simple log-linear population trend) embedded within a framework that fully accounts for the observation process (here, imperfect detection) will be of great value to monitoring and ecological studies alike.