Estimation of a rare sensitive attribute in probability proportional to size measures using Poisson distribution

We propose new variants of Land et al.’s [Estimation of a rare sensitive attribute using Poisson distribution. Statistics. 2011. DOI: 10.1080/02331888.2010.524300] randomized response model when a population consists of some clusters and the population is stratified with some clusters in each stratum. The estimator for the mean number of persons who possess a rare sensitive attribute, its variance, and the variance estimator are suggested when the parameter of a rare unrelated attribute is assumed to be known and unknown. The clusters are selected with and without replacement. When they are selected with replacement, the selecting probabilities for each cluster are defined depending on the cluster sizes and with equal probability. In addition, the variance comparison between a probability proportional to size (PPS) and PPS for stratification are performed. When the parameters vary in clusters, the stratified PPS has better efficiency than the PPS.     

Additive Risk Model Using Piecewise Constant Hazard Function

We study a weighted least squares estimator for Aalen's additive risk model with right-censored survival data which allows for a very flexible handling of covariates. We divide the follow-up period into intervals and assume a constant hazard rate in each interval. The model is motivated as a piecewise approximation of a hazard function composed of three parts: arbitrary nonparametric functions for some covariate effects, smoothly varying functions for others, and known (or constant) functions for yet others. The proposed estimator is an extension of the grouped data version of the Huffer and McKeague (1991 Huffer , F. W. , McKeague , I. W. ( 1991 ). Weighted least squares estimation for Aalen's additive risk model . Journal of the American Statistical Association 86 : 114 – 129 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]) estimator. For our model, since the number of parameters is finite (although large), conventional approaches (such as maximum likelihood) are easy to formulate and implement. The approach is illustrated by simulations, and is compared to the previous studies. The method is also applied to the Framingham study data.