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Group testing has been extensively studied as an efficient way to classify units as defective or satisfactory when the proportion (p) of defectives is small. It can also be used to estimate p, often substantially reducing the mean squared error (MSE) of p and cost per unit information. Group testing is useful for larger p in the estimation problem than in the classification problem, but for larger p more care must be taken in choosing the group size (k); k being too large not only increases MSE (p), but adversely affects the robustness of p to both errors in testing (misclassification) and errors in the assumed binomial model. Procedures that retest units from defective groups, if even feasible, are shown to reduce cost per unit information very little in the estimation problem, but can provide useful information for testing the model. Methods are given for using data from tests of unequal-sized groups to estimate p and for testing the validity of the binomial model.

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For the one-way classification random model with unbalanced data, we compare five estimators of cr.' and cf, the among-and within-treatments variance components: analysis of variance (ANOVA), maximum likelihood (ML), restricted maximum likelihood (REML), and two minimum variance quadratic unbiased (MIVQUE) estimators. MIVQUE (0) is MIVQUE with a priori values 62 = 0 and 5: = 1; MIVQUE(A) is MIVQUE with the ANOVA estimates used as a priori's, We enforce nonnegativity for all estimators, setting any negative estimate to zero in accord with usual practice. The estimators are compared through their biases and MSE's, estimated by Monte Carlo simulation. Our results indicate that the ANOVA estimators perform well, except with seriously unbalanced data when u.'/u: > 1; ML is excellent when u:/c: < 0.5, and MIVQUE(A) is adequate; further iteration to the REML estimates is unnecessary. When u:/u: 2 1, MIVQUE(0) (the default for SASS PROCEDURE VAR-COMP) is poor for estimating 0,' and very poor for u:, even for just mildly unbalanced data.

In binomial group testing, unlike one-at-a-time testing, the test unit consists of a group of individuals, and each group is declared to be defective or nondefective. A defective group is one that is presumed to include one or more defective (e.g., infected, positive) individuals and a nondefective group to contain only nondefective individuals. The usual binomial model considers the individuals being grouped as independent and identically distributed Bernoulli random variables. Under the binomial model and presuming that groups are tested and classified without error, it has been shown that, when the proportion of defective individuals is low, group testing is often preferable to individual testing for identifying infected individuals and for estimating proportions of defectives. We discuss the robustness of group testing for estimating proportions when the underlying assumptions of (i) no testing errors and (ii) independent individuals are violated. To evaluate the effect of these model violations, two dilution-effect models and a serial correlation model are considered. Group testing proved to be quite robust to serial correlation. In the presence of a dilution effect, smaller group sizes should be used, but most of the benefits of group testing can still be realized.