An Analysis of Variance Framework for Matrix Sampling

Abstract

This paper, a discussion of the methodology of matrix sampling, and the empirical and theoretical research on matrix sampling, attempts to demonstrate the f llowing points: 1. Matrix sampling can be viewed as a simple two factor, random model analysis of variance design, the matrix sampling formulas for estimating the mean and variance being simply the point estimate formulas for estimating components of the underlying linear model. 2. These formulas can be based on the weakest possible set of assumptions, viz., random and independent sampling of examinees and items. No assumptions about the statistical nature of the data need be made. 3. The literature is unclear about what effect the above sampling assumptions have upon matrix sampling in the estimation of the mean and, especially, the variance. 4. Of the three alternative procedures suggested for dealing with negative variance estimates in multiple matrix sampling--equating the negative estimate to zero, Winsorizing the distribution of estimates, or treating all estimates alike regardless of sign--the third procedure appears to be most promising. A simulation study is necessary to determine the shape of the distribution of variance component estimates for matrix sampling as well as the relative efficiency of the three methods for handling negative estimates.