-4 Live and dead +4 Alive and dead
4.1 Introduction
-In this fourth chapter, you will learn about the Cormack-Jolly-Seber model that allows estimating survival based on capture-recapture data. You will also see how to deal with covariates to try and explain temporal and/or individual variation in survival. This chapter will also be the opportunity to illustrate how to incorporate prior information to improve model inference.
+In this fourth chapter, you will learn about the Cormack-Jolly-Seber model that allows estimating survival based on capture-recapture data. You will also see how to deal with covariates to try and explain temporal and/or individual variation in survival. This chapter will also be the opportunity to introduce tools to compare models and assess their quality of fit to data.
4.2 The Cormack-Jolly-Seber (CJS) model
-In chapter 3, we introduced a capture-recapture model with constant survival and detection probabilities which we formulated as a HMM and fitted to data in NIMBLE. Historically, however, it was a slightly more complicated model that was first proposed – the so-called Cormack-Jolly-Seber (CJS) model – in which survival and recapture probabilities are time-varying. This feature of the CJS model is useful to account for variation due to environmental conditions in survival or to sampling effort in detection. Schematically the CJS model can be represented this way:
+In Chapter 3, we introduced a capture-recapture model with constant survival and detection probabilities which we formulated as a HMM and fitted to data in NIMBLE. Historically, however, it was a slightly more complicated model that was first proposed – the so-called Cormack-Jolly-Seber (CJS) model – in which survival and recapture probabilities are time-varying. This feature of the CJS model is useful to account for variation due to environmental conditions in survival or to sampling effort in detection. Schematically the CJS model can be represented this way:
Note that the states (in gray) and the observations (in white) do not change. We still have \(z = 1\) for alive, \(z = 2\) for dead, \(y = 1\) for non-detected, and \(y = 2\) for detected.
Parameters are now indexed by time. The survival probability is defined as the probability of staying alive (or “ah, ha, ha, ha, stayin’ alive” like the Bee Gees would say) over the interval between \(t\) and \(t+1\), that is \(\phi_t = \Pr(z_{t+1} = 1 | z_t = 1)\). The detection probability is defined as the probability of being observed at \(t\) given you’re alive at \(t\), that is \(p_t = \Pr(y_{t} = 1 | z_t = 1)\). It is important to bear in mind that survival operates over an interval while detection occurs at a specific time (see Section 4.8).
-The CJS model is named after three statisticians who each published independently a paper introducing more or less the same approach, a year apart ! In fact, Richard Cormack and George Jolly were working in the same corridor in Scotland back in the 1960’s. They would meet every day at coffee and would play some game together, but never mention work and were not aware of each other’s work.
+The CJS model is named after the three statisticians – Richard Cormack, George Jolly and George Seber – who each published independently a paper introducing more or less the same approach, a year apart ! In fact, Richard Cormack and George Jolly were working in the same corridor in Scotland back in the 1960’s. They would meet every day at coffee and would play some game together, but never mention work and were not aware of each other’s work.