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The results of quantal dose-response experiments are often summarized by an estimate of the "median lethal dose," denoted LD50, and many sequential designs have been proposed for efficient estimation of LD50. These designs strive to produce a sequence of trials at dose levels that get closer and closer to LD50. Consequently, they may not provide very good estimates of the overall shape of the dose-response curve. In this paper we propose guidelines for the design of experiments that estimate LD50 fairly efficiently and that also allow for efficient global estimation of the curve.

The standard estimator of the common odds ratio for pair-matched case-control studies, the stratified estimate, is consistent but it ignores all information from the concordant pairs. At the other extreme, the pooled estimator is more efficient as it uses all the data, but is not consistent. In order to trade between bias and precision, Liang and Zeger (1988, Biometrics 44, 1145-1156) proposed an estimator that is a compromise between the stratified and pooled estimates. In the current paper, the possibility of optimizing the trade-off is explored. Specifically, the family of weighted averages of the stratified and pooled estimates is considered, and the weight that minimizes an asymptotic approximation of mean squared error is derived. In practice, the optimal weight must be estimated from the data so that the estimator is only approximately optimal. Small-sample properties are evaluated via simulations.

Many clinical trials have a binary outcome variable. If covariate adjustment is necessary in the analysis, the logistic-regression model is frequently used. Optimal designs for allocating treatments for this model, or for any nonlinear or heteroscedastic model, are generally unbalanced with regard to overall treatment totals and totals within strata. However, all treatment-allocation methods that have been recommended for clinical trials in the literature are designed to balance treatments within strata, either directly or asymptotically. In this paper, the efficiencies of balanced sequential allocation schemes are measured relative to sequential Ds-optimal designs for the logistic model, using as examples completed trials conducted by the Eastern Cooperative Oncology Group and systematic simulations. The results demonstrate that stratified, balanced designs are quite efficient, in general. However, complete randomization is frequently inefficient, and will occasionally result in a trial that is very inefficient.

We assess the efficiency of balanced treatment allocation methods in clinical trials for comparing treatments with respect to survival. We compare optimal designs for each of three standard survival analysis techniques (maximum partial likelihood estimation, log-rank test, exponential regression) with balanced designs, over a range of hypothetical trials. Although balanced designs are not optimal, we find them to be very efficient. In view of the high efficiency demonstrated in this and in a previous paper (Begg and Kalish, 1984, Biometrics 40, 409-420), and practical difficulties in implementing an optimal design, we recommend the use of balanced allocation methods in practice.

Recent research in the design of observational epidemiologic studies has been focused on determining which design is most efficient for controlling a potential confounding factor in the analysis of the disease-exposure relationship. Typically, only two candidate designs have been considered, the matched design and the random sample design. These are merely two of an infinite variety of potential designs in which the distributions of the confounder in the comparison groups and the ratio of group sample sizes are arbitrarily chosen. Only in special cases will either of the standard designs be the most efficient or "optimal" design. We construct optimal designs by minimizing the variance of the desired estimate of effect with respect to the controllable design parameters. Construction of an optimal design depends on unknown parameters, so that in practice only an approximately optimal design, perhaps constructed sequentially, is possible. We evaluate the potential usefulness of optimal designs by identifying circumstances in which an optimal design results in large efficiency gains relative to both the matched and random sample designs. We find that there can be substantial efficiency gains in follow-up studies when both the exposure and confounder are strong risk factors. Practical issues in the implementation of these designs are discussed.