## Introduction

### Data

In ring-recovery studies, wild birds are ringed and released each year and records kept of reported rings from dead birds; see Balmer *et al.* (2008). Ring-recovery data can be used to obtain estimates of survival probability which is a key demographic parameter of interest for wild animal populations. Such data are especially important for estimating the survival of birds during the first year of life. We focus on the analysis of data from birds ringed as young, in the nest. However, the modelling can also be applied to data from birds marked as adults. See Greenwood (2009) for relevant historical material. An illustrative recovery data set on grey herons, *Ardea cinerea*, is provided in Supporting Information Table S1, which is a subset of data available at http://www.tibs.org, analysed by North & Morgan (1979); we analyse the full data set later in the article.

Especially when ringing is carried out by many volunteer ringers at a national level, the number of birds ringed and released in a given year, which we refer to as cohort size, may be unavailable or unreliable. For example, the British Trust for Ornithology (BTO) has recently computerised cohort size numbers for some 20 common bird species, but only since 1985. Computerising records is very time-consuming, and of the order of 15 million remain to be computerised at the BTO. These are entered on paper in ring number order, so selecting out individual species is not practicable, especially for the smaller passerine species, on which the majority of rings are placed. It will take approximately 25 years to complete the task. Most European ringing schemes will have similar data as is also true in North America and Australia.

We use the heron data set to illustrate the work of this study as cohort sizes are known in this case, which allows us to make comparisons between analyses with and without cohort numbers. We refer to data without cohort sizes as conditional data, as when constructing appropriate probabilities we condition on the fact that we only use information from individuals that have been recovered dead. Model parameters are appropriate probabilities of annual survival, probabilities of reporting dead birds and where appropriate regression coefficients, which can be used to incorporate covariate information into the model. Lebreton (2001) provides a useful overview of the use of ring-recovery models to estimate survival and includes a discussion of the conditional ring-recovery models, which are the focus of this study.

Recent years have seen a decline in the reporting probability of many species of wild birds; see Baillie & Green (1987), Robinson, Grantham & Clark (2009), Guillemain *et al.* (2011) and Mazzetta (2010), and as yet no models for conditional ring-recovery data have been proposed which can incorporate this decline in recovery probability. The motivation for this study is the need for a proper description of large bodies of conditional ring-recovery data which may currently be modelled inappropriately.

### Multinomial Models and Conditional Analysis

If we assume that birds behave independently, multinomial models can be constructed for each cohort of ringed birds, parameterised in terms of survival and reporting probabilities. Data from multiple cohorts can be simultaneously analysed by using a product-multinomial model, which dates from Seber (1970, 1971) ; for more recent work, see Freeman & Morgan (1992) and Catchpole & Morgan (1991). Chapter 16 of Williams, Nichols & Conroy (2002) provides a useful overview of modelling this type of data. For illustration we shall assume two age classes for (true) survival, with respective annual survival probabilities *φ*_{1}(*t*), for birds in the first year of life at time *t*, and *φ*_{a}(*t*), for older birds, for *t* = 1,…,*T*. In the simulation studies given later, these probabilities are taken as constant, while for the analyses of real data, each survival probability is logistically regressed on an annual weather covariate, *ω*_{t}, denoting the number of days below freezing in central England in year t (see Besbeas *et al.* 2002), to give

The form of the logistic regression ensures that the estimates of probabilities are constrained between 0 and 1. We denote the reporting probability in year *t* by *λ*_{t}. Thus, for example, for cohort *c* the multinomial cell probability *p*_{j}, which denotes the probability an individual released at time *c* is recovered dead at time *j*, is given by

for *j* = *c*,…,*T* and . Here, *p*_{T+1} denotes the probability of not being observed following marking and *S* is the probability of being recovered at some point in the study.

We assume that the different cohorts are independent, so that the likelihood is product multinomial over cohorts *c* = 1,…,*C*. When cohort sizes are not available, we obtain a conditional analysis, where, for example, for the cohort released at time *c*, the cell probabilities are given by for *j* = *c*,…,*T*. The cell probability is the probability of being recovered dead in year *j* for an individual released in cohort *c*, given that it is recovered within the study.

The widely used computer package MARK (White & Burnham 1999) only allows a constant reporting probability for the conditional analysis of ring-recovery data (model denoted *constant* in the results below). In the corresponding cell probabilities for the constant model, the recovery probability multiplies terms in both the numerator and denominator and cancels, so that the recovery probability cannot be estimated. This was first observed by Seber (1971). Robinson, Baillie & Crick (2007) state that this is the only possibility for conditional analysis; however, Cole & Morgan (2010) demonstrate that time-dependent recovery probability models for conditional data are parameter redundant, but showed that models incorporating time regressions of the reporting probability are full rank and hence may in principle be fitted in conditional analysis of ring-recovery data. We show in this study that such an approach is not straightforward and offer alternative models for conditional ring-recovery data.