Power and sample size when multiple endpoints are considered

Abstract

A common approach to analysing clinical trials with multiple outcomes is to control the probability for the trial as a whole of making at least one incorrect positive finding under any configuration of true and false null hypotheses. Popular approaches are to use Bonferroni corrections or structured approaches such as, for example, closed-test procedures. As is well known, such strategies, which control the family-wise error rate, typically reduce the type I error for some or all the tests of the various null hypotheses to below the nominal level. In consequence, there is generally a loss of power for individual tests. What is less well appreciated, perhaps, is that depending on approach and circumstances, the test-wise loss of power does not necessarily lead to a family wise loss of power. In fact, it may be possible to increase the overall power of a trial by carrying out tests on multiple outcomes without increasing the probability of making at least one type I error when all null hypotheses are true. We examine two types of problems to illustrate this. Unstructured testing problems arise typically (but not exclusively) when many outcomes are being measured. We consider the case of more than two hypotheses when a Bonferroni approach is being applied while for illustration we assume compound symmetry to hold for the correlation of all variables. Using the device of a latent variable it is easy to show that power is not reduced as the number of variables tested increases, provided that the common correlation coefficient is not too high (say less than 0.75). Afterwards, we will consider structured testing problems. Here, multiplicity problems arising from the comparison of more than two treatments, as opposed to more than one measurement, are typical. We conduct a numerical study and conclude again that power is not reduced as the number of tested variables increases.

Implementing optimal allocation in clinical trials with multiple endpoints

“…When RAR procedures are used in this case, the graphical approach is potentially helpful to understand and define the objective of the problem and write out the objective function in the optimization problem. Fourth, Senn and Bretz (2007) discussed different types of powers, and innovative approaches such as using a latent variable were proposed. When RAR procedures are used in the above scenarios, we need to write out different objective or utility functions in the optimality problems.…”

“…Moreover, we can also use RAR procedures to achieve some desirable features such as assigning more patients to the better treatment even if they are not from formal optimality problems. The control of type I error rate could be addressed using the theoretical results from Hu and Zhang (2004) and the innovative methods proposed for the scenario of interest such as Bretz et al (2009) and Senn and Bretz (2007). Finally, responses are assumed to be available immediately in this paper.…”

Sample Size Considerations in Clinical Trials When Comparing Two Interventions Using Multiple Co-Primary Binary Relative Risk Contrasts

“…For this reason, co-primary endpoints are becoming a common design feature in many clinical trials. However, these co-primary endpoints are potentially correlated creating complexities in the evaluation of power and sample size in the designing of such clinical trials, specifically relating to control of the Type I or Type II error rate (Gong et al, 2000; Sen and Bretz, 2007; Hung and Wang, 2009; Dmitrienko et al, 2010). It is important to note the distinction between multiple co-primary endpoints and multiple primary endpoints to which many researchers are unaware (Offen et al (2007)).…”

“…If each endpoint is evaluated using a one-sided test at the same significance level of a , then the rejection regions of H 0 are [{ Z 1 < − z α } ∩ … ∩ { Z K < − z α }], where z α is the (1 − α)th percentile of the standard normal distribution. Therefore, for the true relative risks R k , for large samples, using straightforward algebra and substituting population parameters for estimates, provides the approximate overall power: $$1-\beta =\text{Pr}true[\underset{k=1}{\overset{K}{\cap }}false\{Z_k<-z_{\alpha }false\}true|{\mathrm{H}}_{1}true]\approx \text{Pr}true[\underset{k=1}{\overset{K}{\cap }}false\{Z_k^{*}<-c_k^{*}false\}true|{\mathrm{H}}_{1}true],$$ which is referred to as “conjunctive (or complete) power” (Senn and Bretz, 2007), where $$Z_k^{*}=true(\text{log}{\hat{R}}_k-\text{log}{R}_{k}true)/\sqrt{\frac{1}{N}(\frac{1-R_k{p}_{\mathrm{C}k}}{r{R}_k{p}_{\mathrm{C}k}}+\frac{1-p_{\mathrm{C}k}}{(1-r)p_{\mathrm{C}k}})}\text{and}{c}_k^{*}=true(z_{\alpha }\sqrt{\frac{1-{\overline{p}}_k}{N{\overline{p}}_k}(\frac{1}{r}+\frac{1}{1-r})}+\text{log}{R}_{k}true)/\sqrt{\frac{1}{N}(\frac{1-R_k{p}_{\mathrm{C}k}}{r{R}_k{p}_{\mathrm{C}k}}+\frac{1-p_{\mathrm{C}k}}{(1-r)p_{\mathrm{C}k}})}$$ with p̄ k = rp T k + (1 − r ) p C k . The overall power (2) can be evaluated by using the distribution function of the K -variate normal distribution with zero mean vector and correlation matrix …”

“…In order to provide a more practical sample size, methods for clinical trials with co-primary endpoints have been discussed for fixed sample size designs. Methodologies to address multiple co-primary continuous endpoints are well-developed (Chuang-Stein et al, 2007; Dmitrienko et al, 2010; Eaton and Muirhead, 2007; Hung and Wang, 2009; Julious and McIntyre, 2012; Kordzakhia et al, 2010; Offen et al, 2007; Senn and Bretz, 2007; Sozu et al, 2006; Sugimoto et al, 2012; Xiong et al, 2005). When evaluating relative risks with time-to-event outcomes, Hamasaki et al (2013) and Sugimoto et al (2013) have developed methods for sizing clinical trials, focusing on the hazard ratio and logrank test statistics.…”