## Introduction

Nest success – the probability that a nest will produce at least one individual – is a key vital rate affecting the evolution, ecology, and management of avian populations. Nest success has long been a focus of study for avian ecologists, and associated statistical methods have been in development for more than 50 years. Mayfield's (1961, 1975) work was a major breakthrough, developed to address the problem that apparent nest success (the proportion of sampled nests that are successful) will be a biased measure of true nest success for most sampling scenarios. Nests that are lost early in incubation tend to be underrepresented in samples, as nests are less likely to be detected after they fail. To address this, the Mayfield method instead considers daily nest survival (*S*):

where *y* is the number of nest failures observed, and *D* is exposure days – the sum across nests of days in which each nest is monitored, from initial detection to termination. However, Mayfield's method requires that the date of nest failure be known, which will not be achieved when the interval between nest checks is >1 day. Alternatively, Mayfield assumed that unsuccessful nests failed halfway through the terminal observation interval, thus allocating to exposure days half the number of days in the interval. Johnson (1979), Hensler and Nichols (1981), and Bart and Robson (1982) developed likelihood functions for daily nest survival [reviewed by Williams et al. (2002)] that addressed the problem of uncertain failure date. Subsequently, Dinsmore et al. (2002), Stephens (2003), and Shaffer (2004) developed generalized linear models allowing for flexible modeling of variation in daily nest survival [reviewed by Rotella et al. (2004)]. Bayesian developments have also been presented (Royle and Dorazio 2008; Schmidt et al. 2010). In the above cases, the mortality hazard rate is assumed to be constant over the life of the nest (i.e., age-constant survival), or if age-dependent variation in survival is of interest, the assumption is that nests can be aged without error at first detection (e.g., Dinsmore et al. 2002). The case of age- or stage-dependent survival with unknown nest age has also been considered (Heisey and Nordheim 1995; He et al. 2001; Pollock and Cornelius 2001; He 2003; Stanley 2004; Cao et al. 2008).

We focus here on the case of age-constant nest survival and extend nest survival models to handle incompletely observed temporally varying covariates. A challenge with covariates of this type arises frequently in survival analysis under mark–recapture designs, when temporally varying covariates associated with an individual (e.g., body mass, reproductive condition) cannot be observed when the individual is not captured. Modeling the impact of such covariates on survival, then, has long been a technical challenge (Pollock 2002), and a handful of solutions have emerged (Nichols et al. 1992; Bonner and Schwarz 2006; Catchpole et al. 2008; Langrock and King 2013). We developed an extension of nest survival models to handle incompletely observed temporally varying covariates. Our extension was motivated by a case study involving nest survival in whooping cranes (*Grus americana*).

The reintroduction of the Eastern Migratory Population of whooping cranes to central Wisconsin (largely on Necedah National Wildlife Refuge; NNWR) is a cornerstone effort in whooping crane conservation. Adding an additional 1 or 2 whooping crane populations is a goal of the Whooping Crane International Recovery Plan (Canadian Wildlife Service & U.S. Fish and Wildlife Service 2005). Many indicators of success for this reintroduced population, established with releases of captive-reared birds beginning in 2001, are good (Urbanek et al. 2009; Converse et al. 2012). However, reproductive success has been poor (only 20 of 109 nests, through 2012, produced a hatchling), largely due to a high rate of nest abandonment.

In 2008, RP Urbanek posed the hypothesis that nest abandonment is caused by harassment of cranes by blood-feeding black flies of the genus *Simulium* (Urbanek et al. 2010). Since that time, regular collection of insect index data has been conducted on NNWR using carbon dioxide traps. However, insect sampling is logistically challenging (e.g., transporting dry ice to remote trap locations) and time-consuming (e.g., sample processing). Therefore, sampling is conducted less than once per day. While the intensive sampling of this small, reintroduced, whooping crane population largely obviates the original motivation for development of the Mayfield method – essentially all nests are located within 1–2 days of initiation – the focus of estimation is on daily nest survival because the temporal pattern of nest failure may be key to understanding the cause of nest failure. To carry out the analysis for the biting-insect hypothesis, we developed a novel daily nest survival model to account for missing insect population indices from carbon dioxide traps. The goal of this article is to describe and demonstrate the model with a subset of the whooping crane nest survival data (2009–2010). We also demonstrate the use of Bayesian model selection, which allows us to distinguish among a relatively large set of potentially predictive insect population metrics. The method described herein will be implemented for the full nest survival data set for this population upon the completion of ongoing monitoring and experimentation.