Power evaluation of small drug and vaccine experiments with binary outcomes

Approximations of the power functions for Wald, likelihood ratio, and score tests and their applications to linear and logistic regressions

Traditionally, asymptotic tests are studied and applied under local alternative (Aivazian, et al., 1985). There exists a widespread opinion that the Wald, likelihood ratio, and score tests are asymptotically equivalent. We dispel this myth by showing that These tests have different statistical power in the presence of nuisance parameters. The local properties of the tests are described in terms of the first and second derivative evaluated at the null hypothesis. The comparison of the tests are illustrated with two popular regression models: linear regression with random predictor and logistic regression with binary covariate. We study the aberrant behavior of the tests when the distance between the null and alternative does not vanish with the sample size. We demonstrate that these tests have different asymptotic power. In particular, the score test is generally asymptotically biased but slightly superior for linear regression in a close neighborhood of the null. The power approximations are confirmed through simulations.

Optimal designs for two-arm, phase II clinical trial design with multiple constraints

Multi-stage Phase II trials are often employed in practice but may not be the best approach when the endpoint of interest is not obtained soon after enrollment and/or when a control arm is desired. We present a new design in which sample size determination includes a control arm and allows for the estimation of response for each treatment as well as estimation of the difference in the response rates. We evaluate this design under varying allocation schemes to treatment arms and response rates for each treatment.

Sample size determination for logistic regression revisited

There is no consensus on the approach to compute the power and sample size with logistic regression. Some authors use the likelihood ratio test; some use the test on proportions; some suggest various approximations to handle the multivariate case. We advocate the use of the Wald test since the Z-score is routinely used for statistical significance testing of regression coefficients. The null-variance formula became popular from early studies, which contradicts modern software, which utilizes the method of maximum likelihood estimation (MLE), when the variance of the MLE is estimated at the MLE, not at the null. We derive general Wald-based power and sample size formulas for logistic regression and then apply them to binary exposure and confounder to obtain a closed-form expression. These formulas are applied to minimize the total sample size in a case-control study to achieve a given power by optimizing the ratio of controls to cases. Approximately, the optimal number of controls to cases is equal to the square root of the alternative odds ratio. Our sample size and power calculations can be carried out online at www.dartmouth.edu/ approximately eugened.

Subject allocation and study curtailment for fixed event comparative poisson trials

Comparative Poisson trials of prophylactic interventions, such as vaccines, can be lengthy and costly. We evaluate two easily implemented approaches to reduce numbers of disease cases and person years of follow up (N(u+t)) for comparative Poisson trials with fixed numbers of cases (T); (i) altering k the portion of N(u+t) allocated to treatment and (ii) curtailed stopping before T cases if numbers of cases in the treatment or control group indicate that H(0) has already been rejected or will not be rejected at T cases. Normal and arcsine approximations as well as discrete exact tests are evaluated. For studies not stopped early, allocating about 1/(1+\sqrt r_a) of person years to treatment roughly minimizes T needed for given size and power (where r(a) is the alternative hypothesized relative disease incidence in treated subjects used to power the study). This reduces T moderately vs equal allocation (k=0.5) by 2-3 cases in our examples with exact tests. However, the common practice of allocating k=0.5 of the person years to treatment may be the overall best strategy to minimize N(u+t) for studies that are not stopped early. For studies analysed by exact test and planned with a one-sided alpha ranging from 0.005 to 0.025, beta from 0.1 to 0.2 and r(a) from 0.2 to 0.5, curtailed stopping reduces both the number of disease cases and N(u+t) by 6-40 per cent depending on true treatment benefit. With curtailed stopping, allocating k=0.5 person years to treatment approximately minimizes the numbers of cases and person years under most conditions, although k as large as 0.6 often performs comparably well. If a specific localized k is selected to minimize disease cases for a study analysed by exact test, the study may be underpowered should the final allocation deviate even slightly from that k when it is conducted.