Extending Zelterman's Approach for Robust Estimation of Population Size to Zero‐truncated Clustered Data

Abstract

Estimation of population size with missing zero-class is an important problem that is encountered in epidemiological assessment studies. Fitting a Poisson model to the observed data by the method of maximum likelihood and estimation of the population size based on this fit is an approach that has been widely used for this purpose. In practice, however, the Poisson assumption is seldom satisfied. Zelterman (1988) has proposed a robust estimator for unclustered data that works well in a wide class of distributions applicable for count data. In the work presented here, we extend this estimator to clustered data. The estimator requires fitting a zero-truncated homogeneous Poisson model by maximum likelihood and thereby using a Horvitz-Thompson estimator of population size. This was found to work well, when the data follow the hypothesized homogeneous Poisson model. However, when the true distribution deviates from the hypothesized model, the population size was found to be underestimated. In the search of a more robust estimator, we focused on three models that use all clusters with exactly one case, those clusters with exactly two cases and those with exactly three cases to estimate the probability of the zero-class and thereby use data collected on all the clusters in the Horvitz-Thompson estimator of population size. Loss in efficiency associated with gain in robustness was examined based on a simulation study. As a trade-off between gain in robustness and loss in efficiency, the model that uses data collected on clusters with at most three cases to estimate the probability of the zero-class was found to be preferred in general. In applications, we recommend obtaining estimates from all three models and making a choice considering the estimates from the three models, robustness and the loss in efficiency.

On the design of closed recapture experiments

“…There are several routes for further work. The first regards exploration of closed form expressions for additional relevant cases, including lower bound estimates (Chao, 1987;Zelterman, 1988;Navaratna et al, 2008), persistence models (Ramsey and Usner, 2003), more general behavioral models (Yang and Chao, 2005;Farcomeni, 2011Farcomeni, , 2016Alunni Fegatelli and Tardella, 2016). The second regards extension of the approach to continuous-time closed recapture experiments (Yip et al, 1996;Farcomeni and Scacciatelli, 2013).…”

On the question of proportionality of the count of observed Scrapie cases and the size of holding

“…It would seem natural to expect that the case count rose linearly if not proportional with the holding size and, had this been correct, it could be used beneficially in estimating the hidden burden of the scrapie epidemic. The offset-model (proportionality hypothesis) was considered in [ 11 ] and needs to be revised on the basis of these findings here. For example, for SND data and 2002 and 2004, the proportionality model returns estimates of the scrapie-affected population twice as large as those returned by the model with the best fit, that with the quadratic term.…”

“…Previous studies have applied capture-recapture (CRC) techniques to obtain adjusted estimates of the prevalence of scrapie-affected holdings in Great Britain [ 9 ]. Recently, two approaches [ 10 , 11 ] were pursued for the incorporation of observed holding-specific variability in their CRC models: via an extension of Zelterman's (1988) estimator either as a covariate in a logistic model [ 10 ] or as a proportional term [ 11 ]. The form of this relationship influences the estimates of the scrapie-affected holding population and requires further analyses.…”