The -2 log likelihood of the saturated model is needed to compute the deviance of a model. The deviance is then used to estimate the amount of over-dispersion in the data via c-hat.
The saturated model is loosely defined as the model where the number of parameters equals the number of data points. In the context of MARK, a data point is the observed outcome of a binomial variable, e.g., the number of animals that survive (y) given that n animals were available. However, this simple approach does not work for most of the models in MARK, so that other methods are used. In the following, the method used to compute the saturated model likelihood is described for each type of data. This is the method used when no individual covariates are included in the analysis. Individual covariates cause a different method to be used for any data type.
Live Encounters Model. For the live encounters model (Cormack-Jolly-Seber model), the encounter histories within each attribute group are treated as a multinomial. Given n animals are released on occasion i, then the number observed for encounter history j [n(j)] divided by n is the parameter estimate for the history. The -2 log likelihood for the saturated model is computed as the sum of all groups and encounter histories. For each encounter history, the quantity n(j)*log[n(j)/n] is computed, and then these values are summed across all encounter histories and groups.
Dead Encounters Model. The method used is identical to the live encounters model. For this type of data, the saturated model can be calculated by specifying a different value in every PIM entry. The resulting -2 log likelihood value for this model should be identical to the saturated model value.
Joint Live and Dead Encounters Model. The method used is identical to the live encounters model.
Known Fate Model. The known fate data type uses the group*time model as the saturated model. For each occasion and each group, the number of animals alive at the end of the interval divided by the number of animals alive at the start of the interval is the parameter estimate. The -2 log likelihood value for the saturated model is the same as the -2 log likelihood value computed for the group*time model.
Closed Captures Model. The saturated model for this type of data includes an additional term over the live encounters model, which is the term for the binomial coefficient portion of the likelihood for N-hat. For the saturated model, N-hat is the number of animals known to be alive [M(t+1)], so the log of N-hat factorial is added to the likelihood for each group.
Robust Design Model. The saturated model for this data type is the same as the closed captures model, but each closed-captures trapping session contributes to the log likelihood.
Multi-strata Model. The saturated model for this data type is the same as for the live encounters model.
Dead Encounters Model with S and f coding. The saturated model for this data type is the same as the usual dead encounters model.
BTO Ring Recovery Model. The saturated model for this data type is the same as for the live encounters data.
Joint Live and Dead Encounters, Barker's Model. The method used is identical to the live encounters model.
Pradel and Link-Barker Models. These models assume that an animal can enter the study on any occasion, so the saturated model is computed with the parameter estimate as the number of animals with the encounter history divided by the total number of animals encountered. The same procedure is used for the Burnham Jolly-Seber model and the POPAN model, but because these data types include population size, complications result.
All Data Types with Individual Covariates. For any of the models with individual covariates, the sample size for each encounter history is 1. The saturated model then has a -2 log likelihood value of zero. The deviance for any model with individual covariates is then just its -2 log likelihood value.