Sample size estimation for a two-group comparison of repeated count outcomes using GEE

Randomized clinical trials with count measurements as the primary outcome are common in various medical areas such as seizure counts in epilepsy trials, or relapse counts in multiple sclerosis trials. Controlled clinical trials frequently use a conventional parallel-group design that assigns subjects randomly to one of two treatment groups and repeatedly evaluates them at baseline and intervals across a treatment period of a fixed duration. The primary interest is to compare the rates of change between treatment groups. Generalized estimating equations (GEEs) have been widely used to compare rates of change between treatment groups because of its robustness to misspecification of the true correlation structure. In this paper, we derive a sample size formula for comparing the rates of change between two groups in a repeatedly measured count outcome using GEE. The sample size formula incorporates general missing patterns such as independent missing and monotone missing, and general correlation structures such as AR(1) and compound symmetry (CS). The performance of the sample size formula is evaluated through simulation studies. Sample size estimation is illustrated by a clinical trial example from epilepsy.     

Effects of correlation and missing data on sample size estimation in longitudinal clinical trials

In longitudinal clinical trials, a common objective is to compare the rates of changes in an outcome variable between two treatment groups. Generalized estimating equation (GEE) has been widely used to examine if the rates of changes are significantly different between treatment groups due to its robustness to misspecification of the true correlation structure and randomly missing data. The sample size formula for repeated outcomes is based on the assumption of missing completely at random and a large sample approximation. A simulation study is conducted to investigate the performance of GEE sample size formula with small sample sizes, damped exponential family of correlation structure and non‐ignorable missing data. Copyright © 2008 John Wiley & Sons, Ltd.     

Sample Size Calculation and Power Analysis of Changes in Mean Response Over Time

Despite tremendous effort on different designs with cross-sectional data, little research has been conducted for sample size calculation and power analyses under repeated measures design. In addition to time-averaged difference, changes in mean response over time (CIMROT) is the primary interest in repeated measures analysis. We generalized sample size calculation and power analysis equations for CIMROT to allow unequal sample size between groups for both continuous and binary measures, through simulation, evaluated the performance of proposed methods, and compared our approach to that of a two-stage model formulization. We also created a software procedure to implement the proposed methods.     

Sample size estimation for comparing rates of change in K-group repeated count outcomes

Sample size estimation for comparing the rates of change in two-arm repeated measurements has been investigated by many investigators. In contrast, the literature has paid relatively less attention to sample size estimation for studies with multi-arm repeated measurements where the design and data analysis can be more complex than two-arm trials. For continuous outcomes, Jung and Ahn (2004 Jung, S., Ahn, C. (2004). K-sample test and sample size calculation for comparing slopes in data with repeated measurements. Biometrical J. 46(5):554–564.[Crossref], [Web of Science ®] , [Google Scholar]) and Zhang and Ahn (2013 Zhang, S., Ahn, C. (2013). Sample size calculation for comparing time-averaged responses in k-group repeated measurement studies. Comput. Stat. Data Anal. 58:283–291.[Crossref], [PubMed], [Web of Science ®] , [Google Scholar]) have presented sample size formulas to compare the rates of change and time-averaged responses in multi-arm trials, using the generalized estimating equation (GEE) approach. To our knowledge, there has been no corresponding development for multi-arm trials with count outcomes. We present a sample size formula for comparing the rates of change in multi-arm repeated count outcomes using the GEE approach that accommodates various correlation structures, missing data patterns, and unbalanced designs. We conduct simulation studies to assess the performance of the proposed sample size formula under a wide range of designing configurations. Simulation results suggest that empirical type I error and power are maintained close to their nominal levels. The proposed method is illustrated using an epileptic clinical trial example.     

A note on effective sample size for constructing confidence intervals for the difference of two proportions

Proportion differences are often used to estimate and test treatment effects in clinical trials with binary outcomes. In order to adjust for other covariates or intra-subject correlation among repeated measures, logistic regression or longitudinal data analysis models such as generalized estimating equation or generalized linear mixed models may be used for the analyses. However, these analysis models are often based on the logit link which results in parameter estimates and comparisons in the log-odds ratio scale rather than in the proportion difference scale. A two-step method is proposed in the literature to approximate the calculation of confidence intervals for the proportion difference using a concept of effective sample sizes. However, the performance of this two-step method has not been investigated in their paper. On this note, we examine the properties of the two-step method and propose an adjustment to the effective sample size formula based on Bayesian information theory. Simulations are conducted to evaluate the performance and to show that the modified effective sample size improves the coverage property of the confidence intervals.     

Sample size calculation for a proportional hazards mixture cure model with nonbinary covariates

Sample size calculation is a critical issue in clinical trials because a small sample size leads to a biased inference and a large sample size increases the cost. With the development of advanced medical technology, some patients can be cured of certain chronic diseases, and the proportional hazards mixture cure model has been developed to handle survival data with potential cure information. Given the needs of survival trials with potential cure proportions, a corresponding sample size formula based on the log-rank test statistic for binary covariates has been proposed by Wang et al. [25]. However, a sample size formula based on continuous variables has not been developed. Herein, we presented sample size and power calculations for the mixture cure model with continuous variables based on the log-rank method and further modified it by Ewell's method. The proposed approaches were evaluated using simulation studies for synthetic data from exponential and Weibull distributions. A program for calculating necessary sample size for continuous covariates in a mixture cure model was implemented in R.     

Sample size determination for cluster randomization phase II trials of dichotomous data

One of the main goals for a phase II trial is to screen and select the best treatment to proceed onto further studies in a phase III trial. Under the flexible design proposed elsewhere, we discuss for cluster randomization trials sample size calculation with a given desired probability of correct selection to choose the best treatment when one treatment is better than all the others. We develop exact procedures for calculating the minimum required number of clusters with a given cluster size (or the minimum number of patients with a given number of repeated measurements) per treatment. An approximate sample size and the evaluation of its performance for two arms are also given. To help readers employ the results presented here, tables are provided to summarize the resulting minimum required sample sizes for cluster randomization trials with two arms and three arms in a variety of situations. Finally, to illustrate the sample size calculation procedures developed here, we use the data taken from a cluster randomization trial to study the association between the dietary sodium and the blood pressure.     

Study design of single‐arm phase II immunotherapy trials with long‐term survivors and random delayed treatment effect

In the traditional study design of a single‐arm phase II cancer clinical trial, the one‐sample log‐rank test has been frequently used. A common practice in sample size calculation is to assume that the event time in the new treatment follows exponential distribution. Such a study design may not be suitable for immunotherapy cancer trials, when both long‐term survivors (or even cured patients from the disease) and delayed treatment effect are present, because exponential distribution is not appropriate to describe such data and consequently could lead to severely underpowered trial. In this research, we proposed a piecewise proportional hazards cure rate model with random delayed treatment effect to design single‐arm phase II immunotherapy cancer trials. To improve test power, we proposed a new weighted one‐sample log‐rank test and provided a sample size calculation formula for designing trials. Our simulation study showed that the proposed log‐rank test performs well and is robust of misspecified weight and the sample size calculation formula also performs well.     

Cancer immunotherapy trial design with long-term survivors

Cancer immunotherapy often reflects the improvement in both short-term risk reduction and long-term survival. In this scenario, a mixture cure model can be used for the trial design. However, the hazard functions based on the mixture cure model between two groups will ultimately crossover. Thus, the conventional assumption of proportional hazards may be violated and study design using standard log-rank test (LRT) could lose power if the main interest is to detect the improvement of long-term survival. In this paper, we propose a change sign weighted LRT for the trial design. We derived a sample size formula for the weighted LRT, which can be used for designing cancer immunotherapy trials to detect both short-term risk reduction and long-term survival. Simulation studies are conducted to compare the efficiency between the standard LRT and the change sign weighted LRT.     

Power and sample size for GEE analysis of incomplete paired outcomes in 2 × 2 crossover trials

The 2 × 2 crossover trial uses subjects as their own control to reduce the intersubject variability in the treatment comparison, and typically requires fewer subjects than a parallel design. The generalized estimating equations (GEE) methodology has been commonly used to analyze incomplete discrete outcomes from crossover trials. We propose a unified approach to the power and sample size determination for the Wald Z-test and t-test from GEE analysis of paired binary, ordinal and count outcomes in crossover trials. The proposed method allows misspecification of the variance and correlation of the outcomes, missing outcomes, and adjustment for the period effect. We demonstrate that misspecification of the working variance and correlation functions leads to no or minimal efficiency loss in GEE analysis of paired outcomes. In general, GEE requires the assumption of missing completely at random. For bivariate binary outcomes, we show by simulation that the GEE estimate is asymptotically unbiased or only minimally biased, and the proposed sample size method is suitable under missing at random (MAR) if the working correlation is correctly specified. The performance of the proposed method is illustrated with several numerical examples. Adaption of the method to other paired outcomes is discussed.     

Properties of the weighted log‐rank test in the design of confirmatory studies with delayed effects

Proportional hazards are a common assumption when designing confirmatory clinical trials in oncology. This assumption not only affects the analysis part but also the sample size calculation. The presence of delayed effects causes a change in the hazard ratio while the trial is ongoing since at the beginning we do not observe any difference between treatment arms, and after some unknown time point, the differences between treatment arms will start to appear. Hence, the proportional hazards assumption no longer holds, and both sample size calculation and analysis methods to be used should be reconsidered. The weighted log‐rank test allows a weighting for early, middle, and late differences through the Fleming and Harrington class of weights and is proven to be more efficient when the proportional hazards assumption does not hold. The Fleming and Harrington class of weights, along with the estimated delay, can be incorporated into the sample size calculation in order to maintain the desired power once the treatment arm differences start to appear. In this article, we explore the impact of delayed effects in group sequential and adaptive group sequential designs and make an empirical evaluation in terms of power and type‐I error rate of the of the weighted log‐rank test in a simulated scenario with fixed values of the Fleming and Harrington class of weights. We also give some practical recommendations regarding which methodology should be used in the presence of delayed effects depending on certain characteristics of the trial.     

Missing data in clinical trials: control‐based mean imputation and sensitivity analysis

In some randomized (drug versus placebo) clinical trials, the estimand of interest is the between‐treatment difference in population means of a clinical endpoint that is free from the confounding effects of “rescue” medication (e.g., HbA1c change from baseline at 24 weeks that would be observed without rescue medication regardless of whether or when the assigned treatment was discontinued). In such settings, a missing data problem arises if some patients prematurely discontinue from the trial or initiate rescue medication while in the trial, the latter necessitating the discarding of post‐rescue data. We caution that the commonly used mixed‐effects model repeated measures analysis with the embedded missing at random assumption can deliver an exaggerated estimate of the aforementioned estimand of interest. This happens, in part, due to implicit imputation of an overly optimistic mean for “dropouts” (i.e., patients with missing endpoint data of interest) in the drug arm. We propose an alternative approach in which the missing mean for the drug arm dropouts is explicitly replaced with either the estimated mean of the entire endpoint distribution under placebo (primary analysis) or a sequence of increasingly more conservative means within a tipping point framework (sensitivity analysis); patient‐level imputation is not required. A supplemental “dropout = failure” analysis is considered in which a common poor outcome is imputed for all dropouts followed by a between‐treatment comparison using quantile regression. All analyses address the same estimand and can adjust for baseline covariates. Three examples and simulation results are used to support our recommendations.     

Empirical evaluation of the implementation of the EMA guideline on missing data in confirmatory clinical trials: Specification of mixed models for longitudinal data in study protocols

In confirmatory clinical trials, the prespecification of the primary analysis model is a universally accepted scientific principle to allow strict control of the type I error. Consequently, both the ICH E9 guideline and the European Medicines Agency (EMA) guideline on missing data in confirmatory clinical trials require that the primary analysis model is defined unambiguously. This requirement applies to mixed models for longitudinal data handling missing data implicitly. To evaluate the compliance with the EMA guideline, we evaluated the model specifications in those clinical study protocols from development phases II and III submitted between 2015 and 2018 to the Ethics Committee at Hannover Medical School under the German Medicinal Products Act, which planned to use a mixed model for longitudinal data in the confirmatory testing strategy. Overall, 39 trials from different types of sponsors and a wide range of therapeutic areas were evaluated. While nearly all protocols specify the fixed and random effects of the analysis model (95%), only 77% give the structure of the covariance matrix used for modeling the repeated measurements. Moreover, the testing method (36%), the estimation method (28%), the computation method (3%), and the fallback strategy (18%) are given by less than half the study protocols. Subgroup analyses indicate that these findings are universal and not specific to clinical trial phases or size of company. Altogether, our results show that guideline compliance is to various degrees poor and consequently, strict type I error rate control at the intended level is not guaranteed.     

Some sampling effects of pairwise correlated observations on likelihood ratio tests for the difference between two means

We study the effects of the inclusion of pairs of correlated observations in a sample on likelihood ratio tests for the difference in two means. In particular, we assess how the inclusion of correlated data pairs (e.g., such as data inadvertently collected from sib-pairs) affects the sample size requirements necessary for the implementation of a Likelihood Ratio (LR) test for the difference between two means. Our results suggest that correlated data pairs beneficially or adversely effect sample size requirements for an LR test to a degree functionally related to the mixture parameters dictating their relative frequencies in the larger sample on which the test will be performed, the strength of the correlation between the observations, and the size of imbalances in the sample with respect to the number of observations in each group. The relevance and implications of the results for genetic and epidemiologic research are discussed.     

Sample Size Calculation and Blinded Sample Size Recalculation in Clinical Trials Where the Treatment Effect is Measured by the Relative Risk

In clinical trials with binary endpoints, the required sample size does not depend only on the specified type I error rate, the desired power and the treatment effect but also on the overall event rate which, however, is usually uncertain. The internal pilot study design has been proposed to overcome this difficulty. Here, nuisance parameters required for sample size calculation are re-estimated during the ongoing trial and the sample size is recalculated accordingly. We performed extensive simulation studies to investigate the characteristics of the internal pilot study design for two-group superiority trials where the treatment effect is captured by the relative risk. As the performance of the sample size recalculation procedure crucially depends on the accuracy of the applied sample size formula, we firstly explored the precision of three approximate sample size formulae proposed in the literature for this situation. It turned out that the unequal variance asymptotic normal formula outperforms the other two, especially in case of unbalanced sample size allocation. Using this formula for sample size recalculation in the internal pilot study design assures that the desired power is achieved even if the overall rate is mis-specified in the planning phase. The maximum inflation of the type I error rate observed for the internal pilot study design is small and lies below the maximum excess that occurred for the fixed sample size design.     

Effect of heteroscedasticity between treatment groups on mixed‐effects models for repeated measures

Mixed‐effects models for repeated measures (MMRM) analyses using the Kenward‐Roger method for adjusting standard errors and degrees of freedom in an “unstructured” (UN) covariance structure are increasingly becoming common in primary analyses for group comparisons in longitudinal clinical trials. We evaluate the performance of an MMRM‐UN analysis using the Kenward‐Roger method when the variance of outcome between treatment groups is unequal. In addition, we provide alternative approaches for valid inferences in the MMRM analysis framework. Two simulations are conducted in cases with (1) unequal variance but equal correlation between the treatment groups and (2) unequal variance and unequal correlation between the groups. Our results in the first simulation indicate that MMRM‐UN analysis using the Kenward‐Roger method based on a common covariance matrix for the groups yields notably poor coverage probability (CP) with confidence intervals for the treatment effect when both the variance and the sample size between the groups are disparate. In addition, even when the randomization ratio is 1:1, the CP will fall seriously below the nominal confidence level if a treatment group with a large dropout proportion has a larger variance. Mixed‐effects models for repeated measures analysis with the Mancl and DeRouen covariance estimator shows relatively better performance than the traditional MMRM‐UN analysis method. In the second simulation, the traditional MMRM‐UN analysis leads to bias of the treatment effect and yields notably poor CP. Mixed‐effects models for repeated measures analysis fitting separate UN covariance structures for each group provides an unbiased estimate of the treatment effect and an acceptable CP. We do not recommend MMRM‐UN analysis using the Kenward‐Roger method based on a common covariance matrix for treatment groups, although it is frequently seen in applications, when heteroscedasticity between the groups is apparent in incomplete longitudinal data.     

Determination of hazard ratio for progression‐free survival considering the tumor assessment schedule in sample size calculation

Progression‐free survival is recognized as an important endpoint in oncology clinical trials. In clinical trials aimed at new drug development, the target population often comprises patients that are refractory to standard therapy with a tumor that shows rapid progression. This situation would increase the bias of the hazard ratio calculated for progression‐free survival, resulting in decreased power for such patients. Therefore, new measures are needed to prevent decreasing the power in advance when estimating the sample size. Here, I propose a novel calculation procedure to assume the hazard ratio for progression‐free survival using the Cox proportional hazards model, which can be applied in sample size calculation. The hazard ratios derived by the proposed procedure were almost identical to those obtained by simulation. The hazard ratio calculated by the proposed procedure is applicable to sample size calculation and coincides with the nominal power. Methods that compensate for the lack of power due to biases in the hazard ratio are also discussed from a practical point of view.     

Sample size calculation based on risk ratio under multiple matching

To increase the efficiency of comparisons between treatments in clinical trials, we may consider the use of a multiple matching design, in which, for each patient receiving the experimental treatment, we match with more than one patient receiving the standard treatment. To assess the efficacy of the experimental treatment, the risk ratio (RR) of patient responses between two treatments is certainly one of the most commonly used measures. Because the probability of patient responses in clinical trial is often not small, the odds ratio (OR), of which the practical interpretation is not easily understood, cannot approximate RR well. Thus, all sample size formulae in terms of OR for case-control studies with multiple matched controls per case can be of limited use here. In this paper, we develop three sample size formulae based on RR for randomized trials with multiple matching. We propose a test statistic for testing the equality of RR under multiple matching. On the basis of Monte Carlo simulation, we evaluate the performance of the proposed test statistic with respect to Type I error. To evaluate the accuracy and usefulness of the three sample size formulae developed in this paper, we further calculate their simulated powers and compare them with those of the sample size formula ignoring matching and the sample size formula based on OR for multiple matching published elsewhere. Finally, we include an example that employs the multiple matching study design about the use of the supplemental ascorbate in the supportive treatment of terminal cancer patients to illustrate the use of these formulae.     

Comparative performance of tests of normality in detecting mixtures of parallel regression lines

This paper addresses the problem of detecting a mixture of parallel regression lines when information about group member¬ship of individual cases is not given. The problem is approached as a missing variable problem, with the missing variables being the dummy variables that code for groups. If a mixture of par¬allel regression lines with normally distributed error terms is present, a simple regression model without dummy variables will produce residuals that follow approximately a mixed normal dis¬tribution. In a simulation studyr several goodness-of-fit tests of normality were used to test the residuals obtained from mis-specified models that excluded dummy variables, Factors varied in the simulation included the number and the separation of the parallel lines and the sample size, The goodness-of-fit test based on the sample kurtosis (82) was overall most powerful in detecting mixtures of parallel regression lines, Applications are discussed.     

An extension of the placebo‐based pattern‐mixture model

Pattern‐mixture models provide a general and flexible framework for sensitivity analyses of nonignorable missing data in longitudinal studies. The placebo‐based pattern‐mixture model handles missing data in a transparent and clinically interpretable manner. We extend this model to include a sensitivity parameter that characterizes the gradual departure of the missing data mechanism from being missing at random toward being missing not at random under the standard placebo‐based pattern‐mixture model. We derive the treatment effect implied by the extended model. We propose to utilize the primary analysis based on a mixed‐effects model for repeated measures to draw inference about the treatment effect under the extended placebo‐based pattern‐mixture model. We use simulation studies to confirm the validity of the proposed method. We apply the proposed method to a clinical study of major depressive disorders. Copyright © 2013 John Wiley & Sons, Ltd.