Prediction of 2 × 2 tables of change from repeat cluster sampling of marginal counts

Abstract

Repeat cluster sampling of a binary (0,1) attribute at time 1 (Y1) and time 2 (Y2) in a finite population of discrete units is considered. All clusters contain m units and a cluster provides the marginal count of ones and zeroes at the two time points only. From these counts, we seek to predict a 2 × 2 table of the rates of no change (π11 = E[Y1Y2], π00 = E[(1 – Y1)(1 – Y2)]) and change (π10 = E[Y1(1 – Y2)], –01 = E[(1 – Y1)Y2]). Two predictors are proposed; one is derived from the temporal correlation of marginal counts and the second from the odds ratio of no change that maximizes a (pseudo-) likelihood of a non-central, hypergeometric distribution. The bias of the first is positive when there is a positive intracluster correlation of Y1, Y2, and Y1Y2, while the bias of the second is negative when the odds ratio of no change is >1. A proposed combined estimator worked well in three examples of change analysis with paired, classified Landsat images of forest cover type and cluster sampling with 3 × 3 arrays of 30 m × 30 m units (pixels). 2 × 2 tables obtained from marginal counts were superior, in terms of mean absolute error, to estimates based on a direct unit-by-unit count when the time 2 image had a root mean square registration error of 0.5 pixel relative to the time 1 image. The proposed method is intended for settings where a direct unit-by-unit estimation of the 2 × 2 table is either compromised or when data (by design) consist of marginal counts from a repeat cluster sampling.