PROBING ENCOUNTERED DATA, META ANALYSIS AND WEIGHTED DISTRIBUTION METHODS

Fisher (1934) recognized that the method of ascertainment can influence the form for the distribution of recorded observations, a concept that was later formalized by Rao (1965) into what is termed “weighted distribution.” This paper has two parts. The first reviews various structural properties of weighted distributions, both univariate and bivariate. The effect of weighted observations on Bayesian inference is also considered. The second part deals with three applications of weighted distributions: (1) in stochastic population dynamics, the exploited population size distribution is a weighted form of the natural (unexploited) size distribution; (2) Diaconis and Efron have introduced the double exponential family (DEF) for addressing overdispersion in data. The DEF is shown to be a weighted distribution with the unusual feature that the weight function involves the parameters (specifically, the mean) of the original distribution; (3) weight functions have been employed by Iyengar and Greenhouse (1988) to model the effects of publication bias in a meta-analysis study. We extend their model so as to account for heterogeneity as well as publication bias. For the particular data examined by Iyengar and Greenhouse, likelihood ratio tests indicate that heterogeneity, and not publication bias, is responsible for the observed pattern of effect sizes.

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