Design and monitoring of multi-arm multi-stage clinical trials

Two-arm group sequential designs have been widely used for over 40 years, especially for studies with mortality endpoints. The natural generalization of such designs to trials with multiple treatment arms and a common control (MAMS designs) has, however, been implemented rarely. While the statistical methodology for this extension is clear, the main limitation has been an efficient way to perform the computations. Past efforts were hampered by algorithms that were computationally explosive. With the increasing interest in adaptive designs, platform designs, and other innovative designs that involve multiple comparisons over multiple stages, the importance of MAMS designs is growing rapidly. This article provides break-through algorithms that can compute MAMS boundaries rapidly thereby making such designs practical. For designs with efficacy-only boundaries the computational effort increases linearly with number of arms and number of stages. For designs with both efficacy and futility boundaries the computational effort doubles with successive increases in number of stages.

Choice of test for association in small sample unordered r x c tables

Pearson's chi-squared, the likelihood-ratio, and Fisher-Freeman-Halton's test statistics are often used to test the association of unordered r x c tables. Asymptotical, exact conditional, or exact conditional with mid-p adjustment methods are commonly used to compute the p-value. We have compared test power and significance level for these test statistics and p-value calculations in small sample r x c tables, mostly 3 x 2 and some with both r and c are greater than 2. After extensive simulations, in general we recommend using an exact conditional mid-p test with Pearson's chi-squared or Fisher-Freeman-Halton's statistic, which usually is the most powerful test yet preserve the approximate significance level. Moreover, we recommend that the asymptotic Pearson's chi-squared or other asymptotic tests not be used for small sample r x c tables.

Power and sample size for ordered categorical data

We propose a new method for computing power and sample size for linear rank tests of differences between two ordered multinomial populations. The method is flexible in that it is applicable to any general alternative hypothesis and for any choice of rank scores. We show that the method, though asymptotic, closely approximates existing exact methods. At the same time it overcomes the computational limitations of the exact methods. This advantage makes our asymptotic approach more practical for sample size computations at the planning stages of a large study. We illustrate the method with data arising from both proportional and non-proportional odds models in the two ordered multinomial setting.

Computational tools for exact conditional logistic regression

Logistic regression analyses are often challenged by the inability of unconditional likelihood-based approximations to yield consistent, valid estimates and p-values for model parameters. This can be due to sparseness or separability in the data. Conditional logistic regression, though useful in such situations, can also be computationally unfeasible when the sample size or number of explanatory covariates is large. We review recent developments that allow efficient approximate conditional inference, including Monte Carlo sampling and saddlepoint approximations. We demonstrate through real examples that these methods enable the analysis of significantly larger and more complex data sets. We find in this investigation that for these moderately large data sets Monte Carlo seems a better alternative, as it provides unbiased estimates of the exact results and can be executed in less CPU time than can the single saddlepoint approximation. Moreover, the double saddlepoint approximation, while computationally the easiest to obtain, offers little practical advantage. It produces unreliable results and cannot be computed when a maximum likelihood solution does not exist.

An exact trend test for correlated binary data

The problem of testing a dose-response relationship in the presence of exchangeably correlated binary data has been addressed using a variety of models. Most commonly used approaches are derived from likelihood or generalized estimating equations and rely on large-sample theory to justify their inferences. However, while earlier work has determined that these methods may perform poorly for small or sparse samples, there are few alternatives available to those faced with such data. We propose an exact trend test for exchangeably correlated binary data when groups of correlated observations are ordered. This exact approach is based on an exponential model derived by Molenberghs and Ryan (1999) and Ryan and Molenberghs (1999) and provides natural analogues to Fisher's exact test and the binomial trend test when the data are correlated. We use a graphical method with which one can efficiently compute the exact tail distribution and apply the test to two examples.

Power comparisons for tests of trend in dose-response studies

The Cochran-Armitage test for trend is a popular statistical procedure for detecting increasing or decreasing probabilities of response when a categorical exposure is ordered. Such associations may arise in a variety of biomedical research settings, particularly in dose-response designs such as carcinogenicity experiments. Previously, computing limitations mandated the use of the asymptotic trend test, but with the availability of new algorithms, increased computing power, and appropriate software the exact trend test is now a practical option. Nevertheless, the exact test is sometimes criticized on the grounds that it is conservative. In this paper we investigate the implications of this conservatism by comparing the true type I error and power of three alternative tests of trend - the asymptotic test, the exact test and an admissible exact test proposed by Cohen and Sackrowitz. The computations are performed by an extension to the network algorithm of Mehta et al. This allows us to make precise power comparisons between the tests under any given design without resorting to simulation. We show how this tool can guide investigators in choosing the most appropriate test by considering the design of two-year carcinogenicity studies carried out by the National Toxicology Program. We additionally compare the tests for various other combinations of sample sizes and number of groups or levels of exposure. We conclude that the asymptotic test, while more powerful where it is valid, generally does not preserve the type I error. This violation of the a priori testing level can be greatly affected by imbalance in the data or unequal spacing of dose levels.

Exact permutational tests for group sequential clinical trials

An efficient numerical algorithm is developed for computing stopping boundaries for group sequential clinical trials. Patients arrive in sequence, and are randomized to one of two treatments. The data are monitored at interim time points, with a fresh block of patients entering the study from one monitoring point to the next. The stopping boundaries are derived from the exact joint permutational distribution of the linear rank statistics observed across all the monitoring times. Specifically, the algorithm yields the exact boundary generating function, Pr(W1 < b1, W2 < b2, ..., Wi-1 < bi-1, Wi = wi), where Wj is the linear rank statistic at the jth interim time point. The distribution theory is based on assigning ranks after pooling all the patients who have entered the study, and then permuting the patients to the two treatments independently within each block of newly arrived patients. The methods are applicable for an arbitrary number of monitoring times, which need not be specified at the start of the study. The data may be continuous or categorical, and censored or uncensored. The randomization rule for treatment allocation can be adaptive. The algorithm is especially useful during the early stages of a clinical trial, when very little data have been gathered, and stopping boundaries are based on the extreme tails of the relevant boundary generating function. In that case the corresponding large-sample theory is not very reliable. To illustrate the techniques we present a group sequential analysis of a recently completed study by the Eastern Cooperative Oncology Group.