Design and Estimation in Two-Way Stratification

Abstract
One may encounter populations for which there are two effective stratifying criteria, both of which are desirable in a sample design. However, the number of permitted observations may be less than the number of strata formed by the usual double stratification technique. A method which will permit estimation in these cases is presented. Both biased and unbiased estimators are considered. It is shown that if the stratification effects are additive in the analysis-of-variance sense the method is particularly effective. Also, if the substrata sizes (formed from the two-way classification of the population according to the stratifying criteria) are proportional to the product of the corresponding one-way strata sizes, the possible loss in efficiency compared to single stratification is trivial. Even in populations in which substrata disproportion is great one can still use the method effectively by employing a method of allocating, with certainty, some of the sample observations to the substrata. Variances of both biased and unbiased estimators are given, along with a method for obtaining essentially unbiased estimates of the variances.