# Population size estimation using the modified Horvitz-Thompson estimator with estimated sighting probability

## Date

1996

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## Abstract

Wildlife aerial population surveys usually use a two-stage sampling technique. The first stage involves dividing the whole survey area into smaller land units, which we called the primary units, and then taking a sample from those. In the second stage, an aerial survey of the selected units is made in an attempt to observe (count) every animal. Some animals, usually occurring in groups, are not observed for a variety of reasons. Estimates from these surveys are plagued with two major sources of errors, namely, errors due to sampling variation in both stages. The first error may be controlled by choosing a suitable sampling plan for the first stage. The second error is also termed "visibility bias", which acknowledges that only a portion of the groups in a sampled land unit will be enumerated. The objective of our study is to provide improved variance estimators over those provided by Steinhorst and Samuel (1989) and to evaluate performances of various corresponding interval procedures for estimating population size. For this purpose, we have found an asymptotically unbiased estimator for the approximate variance of the population size estimator when sighting probabilities of groups are unknown and fitted with a logistic model. We have broken down the approximate variance term into three components, namely, error due to sampling of primary units, error due to sighting of groups in second stage sampling and error due all three components separately in order to get a better insight to error control. Simplified versions of variance estimators are provided when all primary units are surveyed and for stratified random sampling of primary units. Third central moment of population size estimator was also obtained. Simulation studies were conducted to evaluate performances of our asymptotically unbiased variance estimators and of confidence interval procedures such as the large sample procedure, with and without transformation, for constructing 90% and 95% confidence intervals for the population size. Confidence intervals for the population size were also constructed by assuming that the distribution of log(T-T) is normally distributed, where f is the population size estimate and T is the number of animals seen in a sample obtained from a population survey. From our simulation results, we observed that the population size is estimated with negligible bias (according to Cochran's (1977) working rule) with a sample of at least 100 groups of elk obtained from a population survey when sighting probabilities are known. When sighting probabilities are unknown, one needs to conduct a sightability survey to obtain a sample, independent of the sample obtained from a population survey, for fitting a logistic model to estimate sighting probabilities of sighted groups in the sample obtained from the population survey. In this case, the population size is also estimated with negligible bias when the sample size of both samples is at least 100 groups of elk. We also observed that when sighting probabilities are known, we needed a sample of at least 348 groups of elk from a population survey to obtain reasonable coverage rates of the true population size. When sighting probabilities are unknown and estimated via logistic regression, the size of both samples is at least 428 groups of elk for obtaining reasonable coverage rates of the true population size. Among all these confidence intervals, we found that those approximate confidence intervals constructed based on the assumption that log (T-T) is normally distributed and using the delta method have better coverage rates and shorter estimated expected interval widths. Confidence intervals for the population size using bootstrapping were also evaluated. We were unable to find an existing bootstrapping procedure which could be directly applied to our problem. We have, therefore, proposed a couple of bootstrapping procedures for obtaining a sample to fit a logistic model and a couple of bootstrapping procedures for obtaining a sample to construct a population size estimate. With 1000 pairs of independent samples from a sightability survey and a population survey, each sample of size 107 groups of elk and using 500 bootstrap iterations, we obtained reasonable coverage rates of the true population size. Our other problem is model selection of a logistic model for the unknown sighting probabilities. We evaluated the performance of the population size estimator and our variance estimator when we fit a simpler model. For this purpose, we have derived theoretical expressions for the bias of the population size estimator and the mean-squared-error. We found, from our simulation results of fitting a couple of models simpler than the full model, that the population size was still well estimated for the fitted model based only on group size but was severely overestimated for the fitted model based only on percent of vegetation cover. For both fitted models, our variance estimator overestimated the observed variance of 1000 simulated population size estimates. We also found that the approximate expression of the expected value of the population size estimator we derived for a fitted model simpler than the full model has negligible bias (by Cochran's (1977) working rule) relative to the average of those 1000 simulated population size estimates. The approximate expression of the variance of the population size estimator we derived for this case somewhat underestimated the observed variance of those 1000 simulated population size estimates. Both approximate expressions apparently give us an idea of the expected size of the population size estimate and its variance when the fitted model is not the full model.

## Description

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Print version deaccessioned 2024.

Print version deaccessioned 2024.

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## Subject

Population density -- Mathematical models

Animal populations -- Mathematical models