Density Estimation
Density Estimation
A density estimator has been incorporated into MARK that uses radio-tracking data to correct the population estimate from mark-recapture sampling. The density estimator is derived from the Huggins estimator of a closed population.
Derivation of the Estimator. The basic Huggins estimator of population size can be written as Nhat = M(t+1)/p*, where M(t+1) is the number of marked animals in the study (i.e., the total number of individual animals captured), and p* is the probability that an animal is captured one or more times during the study. The Huggins estimator can be re-written to incorporate individual characteristics as Nhat = sum over i =1 to M(t+1) of 1/p*(i). To estimate density, 1 in the number can be converted to the proportion of its time an animal spends on the trapping grid. So defining ptilde as the proportion of time on the grid, the density estimator is Dhat = sum over i =1 to M(t+1) of ptilde(i)/p*(i) divided by the area of the grid. The value of ptilde(i) is estimated as the proportion of radio-tracking locations for animal i that fall on the grid, with the radios attached to animals during the trapping session, and the radio-tracking taking place after the traps and bait are removed so that the grid is no longer an attractant.
Instead of using the observed proportion of an animal’s locations that fall on the trapping grid, a logistic regression model can be used to estimate this proportion based on the animal’s trapping locations on the grid. Create the covariate distance to edge (DTE) that is the mean distance of the animal’s trapping locations to the nearest edge of the trapping grid, and include this covariate in a logistic regression model: logit[ptilde(i)] = B0 + B1(DTE). By using the logistic regression model to estimate ptilde, not all animals have to have radios attached.
Encounter Histories Format. Because additional data are required for this estimator besides the typical encounter history, the format of the encounter histories file is different. The number of locations on the grid and the total number of radio-tracking locations must be included. These values are inserted between the encounter history and the frequency or count of animals:
encounter_history num_locs_on_grid num_locs_taken frequency individual_covariate(s);
An example for 5 trapping occasions would be:
01101 6 10 1 30.5;
where the single animal was captured on occasions 2, 3, and 5, 6 of the 10 radio locations were on the trapping grid, and the mean distance to the edge of the grid (an individual covariate) was 30.5. For an animal that did not receive a radio, the entry in the encounter history file would be:
01101 . . 1 30.5;
where dots replace the number of radio locations on the grid and the total number of radio locations taken.
Huggins Estimators. When the density estimator is selected as the data type for analysis, a pop-up window asks the user to specify which of the 4 Huggins estimators is to be the default data type for estimating p*. If either of the Pledger estimators is selected, the number of mixtures must also be specified.
Grid Area. The area of the trapping grid (for each attribute group) is entered when the MARK initial values dialog is filled in. An additional button appears on the screen that allows the user to enter the area for each group. However, this button can only be clicked once the number of groups and the group labels have been entered.
Observed versus Estimated ptilde. The estimate of density is produced as a derived parameter in MARK. Real parameters are the parameters of the Huggins estimator plus the ptilde estimates. However, the option is available to not use the estimated ptilde value to estimate density, but rather use the observed values of ptilde for each animal. A check box, Use observed ptilde, appears in the right-hand column of the Run Window that the user can check to use the observed ptilde values instead of the estimates predicted by the logistic model. However, coefficients of the logistic model are still estimated, and appear as beta parameters, and ptilde still appears in the real parameters. Thus, the AIC and deviance of the model is the same regardless of whether the Use observed ptilde box is checked or not. Only the derived parameter estimates change. Typically, the estimates of density change little when every animal is given a radio regardless of whether the estimated or observed values are used to estimate density.