Prediction Theory for Multinomial Proportions Using Two-stage Cluster Samples

In a two-stage clusters sampling setup for categorical data, it is well known that the so-called best prediction of the category based proportions involves computing the conditional means of the non-sampled multinomial variables conditional on the sampled multinomial responses. This computation is however not easy mainly due to the complex cluster correlations among multinomial responses within a cluster. The independence assumption based approach or any linear model approach for cluster correlated data those used so far in the existing studies are not valid for the computation of such conditional means in the prediction function for multinomial data. As opposed to these ‘working’ independence or linear models based approaches, in this paper we first develop a cluster correlation structure for multinomial data and exploit this structure to compute theoretically valid formulas for the conditional means of non-sampled hypothetical responses. Next because these conditional means or equivalently the prediction function contains the regression and clustered variance/correlation parameters, we estimate these parameters using the survey sampling weights based conditional likelihood approach, whereas the existing studies mostly use the independence assumption based likelihood or moment approaches which are invalid or inadequate in a correlation setup. The proposed conditional likelihood estimators are shown to be consistent for their respective parameters leading to the consistent estimation of the prediction function for the multinomial proportions.