A Nonparametric Estimator of Heterogeneity Variance with Applications to SMR- and Proportion-Data

• Published in: Biometrical journal Vol. 42; no. 3; pp. 321 - 334
• Main Authors: Böhning, Dankmar, Sarol Jr, Jesus
• Format: Journal Article
• Language: English
• Published: Berlin WILEY-VCH Verlag Berlin GmbH 01.07.2000; WILEY‐VCH Verlag Berlin GmbH; Wiley-VCH
• Subjects: Confidence interval estimation adjusted for unobserved heterogeneity; Exact sciences and technology; Mathematics; Moment estimator; Multivariate analysis; Nonparametric inference; Population heterogeneity; Probability and statistics; Random effects model; Sciences and techniques of general use; Statistics; Variance separation; Heterogeneity; Extreme value; Non parametric estimation; Binomial distribution; Iterative method; Arithmetic mean; Optimal estimation; Poisson distribution; Variance analysis; Mean estimation; Confidence interval
• ISSN: 0323-3847; 1521-4036
• DOI: 10.1002/1521-4036(200007)42:3<321::AID-BIMJ321>3.0.CO;2-Q

In this paper the situation of extra population heterogeneity is discussed from a analysis of variance point of view. We first provide a non‐iterative way of estimating the variance of the heterogeneity distribution without estimating the heterogeneity distribution itself for Poisson and binomial counts. The consequences of the presence of heterogeneity in the estimation of the mean are discussed. We show that if the homogeneity assumption holds, the pooled mean is optimal while in the presence of strong heterogeneity, the simple (arithmetic) mean is an optimal estimator of the mean SMR or mean proportion. These results lead to the problem of finding an optimal estimator for situations not represented by these two extreme cases. We propose an iterative solution to this problem. Illustrations for the application of these findings are provided with examples from various areas.