# residuals.cjs: Residuals for CJS Model In mra: Mark-Recapture Analysis

## Description

Residual extraction routine for a CJS object. Returns Pearson or deviance residuals of a CJS capture-recapture model.

## Usage

 ```1 2``` ```## S3 method for class 'cjs' residuals(object, type="pearson", ...) ```

## Arguments

 `object` a CJS (Cormack-Jolly-Seber capture-recapture) object, which is usually the result of calling F.cjs.estim `type` string indicating type of residual to return. Either "pearson" for Pearson residuals (i.e., (o - e)/sqrt(e*(1-e))) or "deviance" for deviance residuals (i.e., 2*sign(o-e)*sqrt(o*log(o/e) + (1-o)*log((1-o)/(1-e))) ) `...` Additional arguments to other functions. Not used, but must be here for compatibility with the generic `residuals` function.

## Details

In almost all cases, a CJS model fitted by F.cjs.estim already has a `\$residuals` component. This routine either extracts this component, or computes residuals of the component if not found.

Observed component (o(ij)) in formulas above is the capture indicator for animal i during occasion j. If animal i was seen during occasion j, o(ij) = 1. Otherwise, o(ij) = 0.

Expected component (e(ij)) in formula above is the expected value of the capture indicator for animal i during occasion j. In other words, o(ij) is a binomial random variable with expected value e(ij). Under the assumptions of a CJS model, e(ij) is computed as phi(i(1)) * phi(i(2)) * ... * phi(i(j-1)) * p(ij), where p(ij) is the estimated capture probability of animal i during occasion j, and phi(i(1)) is estimated survival during the first interval following initial capture of the animal, phi(i(2)) is survival during the second interval after initial capture, and phi(i(j-1)) is survival during the interval just prior to occasion j.

## Value

A NAN X NS matrix of residuals, where NAN = number of animals and NS = number of capture occasions. Residuals in the non-active cells are set to NA. Non-active cells are those prior to and including the initial capture, and after the occasion on which an animal is known to have died.

If type = "pearson", the residual for active cell (i,j) is (o(ij) - e(ij)) / sqrt(e(ij) * (1 - e(ij))).

If type = "deviance", the residual for active cell (i,j) is 2 * sign(o(ij) - e(ij)) * sqrt(o(ij)*log(o(ij) / e(ij)) + (1 - o(ij)) * log((1 - o(ij)) / (1 - e(ij)))).

Observed (o(ij)) and expected (e(ij)) are defined in Details.

## Author(s)

Trent McDonald

`F.cjs.estim`, `predict.cjs`
 ```1 2 3 4 5 6``` ```# Fit CJS model to dipper data, time-varying capture and survivals. data(dipper.histories) xy <- F.cjs.covars( nrow(dipper.histories), ncol(dipper.histories) ) for(j in 1:ncol(dipper.histories)){ assign(paste("x",j,sep=""), xy\$x[,,j]) } dipper.cjs <- F.cjs.estim( ~x2+x3+x4+x5+x6, ~x1+x2+x3+x4+x5, dipper.histories ) residuals(dipper.cjs) ```