In multi-stage sampling, there are two or more stages of sampling and the simplest version, which the chapter emphasizes is called two-stage sampling. In two-stage sampling, an initial first-stage sample of n primary units (or clusters) is selected. Then, at the second stage of sampling, m
i subunits are selected from the M
i subunits in the selected primary units. First- and second-stage units may be selected with equal or unequal probabilities and a wide variety of estimators may be used to estimate totals within selected primary units and to estimate the total of the target variable in the finite population. Illustrative sample spaces are provided for equal sized two-stage cluster sampling with SRS selection at both stages, and for two-stage unequal size cluster sampling, with clusters selected by PPSWOR and units within clusters selected by SRS. Sampling variance is shown to originate from two sources: variation between primary unit totals or means (first-stage variance), and errors of estimation of primary units totals (second-stage variance). Topics of optimal allocation and net relative efficiency are addressed in the two-stage context with equal and unequal size clusters. General expressions for sampling variance are presented for three or more stages of sampling. The multi-stage framework can take powerful advantage of all of the concepts and sampling designs considered in previous chapters and the ecologist or natural resource scientist can apply everything he/she knows about an ecological or natural resource setting to guide development of an intelligent multi-stage sampling strategy.
STUDY OF GENERAL FAMILIES OF ESTIMATORS IN TWO-STAGE SAMPLING WITH UNEQUAL SIZE CLUSTERS UNDER NON-RESPONSE