Sample size for a two-group comparison of repeated binary measurements using GEE

Controlled clinical trials often randomize subjects to two treatment groups and repeatedly evaluate them at baseline and intervals across a treatment period of fixed duration. A popular primary objective in these trials is to compare the change rates in the repeated measurements between treatment groups. Repeated measurements usually involve missing data and a serial correlation within each subject. The generalized estimating equation (GEE) method has been widely used to fit the time trend in repeated measurements because of its robustness to random missing and mispecification of the true correlation structure. In this paper, we propose a closed form sample size formula for comparing the change rates of binary repeated measurements using GEE for a two-group comparison. The sample size formula is derived incorporating missing patterns, such as independent missing and monotone missing, and correlation structures, such as AR(1) model. We also propose an algorithm to generate correlated binary data with arbitrary marginal means and a Markov dependency and use it in simulation studies.     

Sample size and power determination in joint modeling of longitudinal and survival data

Owing to the rapid development of biomarkers in clinical trials, joint modeling of longitudinal and survival data has gained its popularity in the recent years because it reduces bias and provides improvements of efficiency in the assessment of treatment effects and other prognostic factors. Although much effort has been put into inferential methods in joint modeling, such as estimation and hypothesis testing, design aspects have not been formally considered. Statistical design, such as sample size and power calculations, is a crucial first step in clinical trials. In this paper, we derive a closed-form sample size formula for estimating the effect of the longitudinal process in joint modeling, and extend Schoenfeld's sample size formula to the joint modeling setting for estimating the overall treatment effect. The sample size formula we develop is quite general, allowing for p-degree polynomial trajectories. The robustness of our model is demonstrated in simulation studies with linear and quadratic trajectories. We discuss the impact of the within-subject variability on power and data collection strategies, such as spacing and frequency of repeated measurements, in order to maximize the power. When the within-subject variability is large, different data collection strategies can influence the power of the study in a significant way. Optimal frequency of repeated measurements also depends on the nature of the trajectory with higher polynomial trajectories and larger measurement error requiring more frequent measurements.     

Repeated measures in clinical trials: analysis using mean summary statistics and its implications for design.

This paper explores the use of simple summary statistics for analysing repeated measurements in randomized clinical trials with two treatments. Quite often the data for each patient may be effectively summarized by a pre-treatment mean and a post-treatment mean. Analysis of covariance is the method of choice and its superiority over analysis of post-treatment means or analysis of mean changes is quantified, as regards both reduced variance and avoidance of bias, using a simple model for the covariance structure between time points. Quantitative consideration is also given to practical issues in the design of repeated measures studies: the merits of having more than one pre-treatment measurement are demonstrated, and methods for determining sample sizes in repeated measures designs are provided. Several examples from clinical trials are presented, and broad practical recommendations are made. The examples support the value of the compound symmetry assumption as a realistic simplification in quantitative planning of repeated measures trials. The analysis using summary statistics makes no such assumption. However, allowance in design for alternative non-equal correlation structures can and should be made when necessary.     

Optimal multistage designs for randomised clinical trials with continuous outcomes

Multistage designs allow considerable reductions in the expected sample size of a trial. When stopping for futility or efficacy is allowed at each stage, the expected sample size under different possible true treatment effects (δ) is of interest. The δ-minimax design is the one for which the maximum expected sample size is minimised amongst all designs that meet the types I and II error constraints. Previous work has compared a two-stage δ-minimax design with other optimal two-stage designs. Applying the δ-minimax design to designs with more than two stages was not previously considered because of computational issues. In this paper, we identify the δ-minimax designs with more than two stages through use of a novel application of simulated annealing. We compare them with other optimal multistage designs and the triangular design. We show that, as for two-stage designs, the δ-minimax design has good expected sample size properties across a broad range of treatment effects but generally has a higher maximum sample size. To overcome this drawback, we use the concept of admissible designs to find trials which balance the maximum expected sample size and maximum sample size. We show that such designs have good expected sample size properties and a reasonable maximum sample size and, thus, are very appealing for use in clinical trials.     

A new approach for sizing trials with composite binary endpoints using anticipated marginal values and accounting for the correlation between components

Composite binary endpoints are increasingly used as primary endpoints in clinical trials. When designing a trial, it is crucial to determine the appropriate sample size for testing the statistical differences between treatment groups for the primary endpoint. As shown in this work, when using a composite binary endpoint to size a trial, one needs to specify the event rates and the effect sizes of the composite components as well as the correlation between them. In practice, the marginal parameters of the components can be obtained from previous studies or pilot trials; however, the correlation is often not previously reported and thus usually unknown. We first show that the sample size for composite binary endpoints is strongly dependent on the correlation and, second, that slight deviations in the prior information on the marginal parameters may result in underpowered trials for achieving the study objectives at a pre-specified significance level. We propose a general strategy for calculating the required sample size when the correlation is not specified and accounting for uncertainty in the marginal parameter values. We present the web platform CompARE to characterize composite endpoints and to calculate the sample size just as we propose in this paper. We evaluate the performance of the proposal with a simulation study and illustrate it by means of a real case study using CompARE.     

Issues in the use of change scores in randomized trials

The literature on the measurement of change is often confusing and contradictory. Some authors advocate the use of change scores as the best approach to the analysis of treatment effects in clinical trials; others maintain that change scores should be avoided entirely. This paper reviews these arguments and demonstrates that contradictions arise in part from different definitions of change, and in part from some misunderstanding of the relationship between reliability and responsiveness to change. Conditions under which it is, and is not appropriate to use change scores in experimental designs are specified, and formulae for sample size calculations are introduced.     

D-optimal designs for multiarm trials with dropouts

Multiarm trials with follow-up on participants are commonly implemented to assess treatment effects on a population over the course of the studies. Dropout is an unavoidable issue especially when the duration of the multiarm study is long. Its impact is often ignored at the design stage, which may lead to less accurate statistical conclusions. We develop an optimal design framework for trials with repeated measurements, which takes potential dropouts into account, and we provide designs for linear mixed models where the presence of dropouts is noninformative and dependent on design variables. Our framework is illustrated through redesigning a clinical trial on Alzheimer's disease, whereby the benefits of our designs compared with standard designs are demonstrated through simulations.     

Flexible two-stage design with sample size reassessment for survival trials

Flexible trials with adaptive design modification at interim analyses have been recently proposed as an answer to cope with some limitations of traditional designs for phase III trials. Actually, the sample size and duration of fixed design trials strongly depend on the determination, prior to the study, of key parameters such as the expected treatment effect and the event rates. A misspecification of these parameters may result in an underpowered or overpowered trial. In the flexible framework, the remainder of a design can be modified at an interim analysis, while preserving the initially specified global error rates. In this work, we present a flexible design with sample size re-evaluation for survival trials and study its properties in practical settings. The results show that, if parameters are initially misspecified, the proposed method allows an improved power control with a reasonable increase in sample size, if any. Practical guidelines concerning the choice of the trial parameters are also given.     

Evaluating the adaptive performance of flexible sample size designs with treatment difference in an interval

In clinical trials, the study sample size is often chosen to provide specific power at a single point of a treatment difference. When this treatment difference is not close to the true one, the actual power of the trial can deviate from the specified power. To address this issue, we consider obtaining a flexible sample size design that provides sufficient power and has close to the 'ideal' sample size over possible values of the true treatment difference within an interval. A performance score is proposed to assess the overall performance of these flexible sample size designs. Its application to the determination of the best solution among considered candidate sample size designs is discussed and illustrated through computer simulations.     

AN ASYMPTOTIC TEST FOR THE EQUALITY OF COEFFICIENTS OF VARIATION FROM k POPULATIONS

The coefficient of variation is often used as a guide of the repeatability of measurements in clinical trials and other medical work. When possible, one makes repeated measurements on a set of individuals to calculate the relative variability of the test with the understanding that a reliable clinical test should give similar results when repeated on the same patient. There are times, however, when repeated measurements on the same patient are not possible. Under these circumstances, to combine results from different clinical trials or test sites, it is necessary to compare the coefficients of variation of several clinical trials. Using the work of Miller, we develop a general statistic for testing the hypothesis that the coefficients of variation are the same for k populations, with unequal sample sizes. This statistic is invariant under the choice of the order of the populations, and is asymptotically χ². We provide an example using data from Yang and HayGlass. We compare the size and the power of the test to that of Bennett, Doornbos and Dijkstra and a statistic based on Hedges and Olkin.     

Information‐based sample size re‐estimation in group sequential design for longitudinal trials

Group sequential design has become more popular in clinical trials because it allows for trials to stop early for futility or efficacy to save time and resources. However, this approach is less well‐known for longitudinal analysis. We have observed repeated cases of studies with longitudinal data where there is an interest in early stopping for a lack of treatment effect or in adapting sample size to correct for inappropriate variance assumptions. We propose an information‐based group sequential design as a method to deal with both of these issues. Updating the sample size at each interim analysis makes it possible to maintain the target power while controlling the type I error rate. We will illustrate our strategy with examples and simulations and compare the results with those obtained using fixed design and group sequential design without sample size re‐estimation. Copyright © 2014 John Wiley & Sons, Ltd.     

Sample size requirement for repeated measurements in continuous data.

In this paper we extend Bloch's discussion on the usefulness and the limitations in the application of repeated measurements per subject in study designs. We derive general sample size formulae for any finite number of comparison groups to calculate the required number of subjects with repeated measurements, that do not have to be conditionally independent. For fixed total cost, we discuss the optimal sample allocation for repeated measurements needed to maximize the power and the underestimation when using Bloch's sample size formula if in the hypothesis testing procedure the variance parameters are unknown. We have also included a quantitative investigation of the effectiveness of taking repeated measurements per subjects to reduced the required number of subjects for a given power at a given alpha-level.     

Optimizing the design of clinical trials where the outcome is a rate. Can estimating a baseline rate in a run‐in period increase efficiency?

It is well known that the statistical power of randomized controlled trials with a continuous outcome can be increased by using a pre-randomization baseline measure of the outcome variable as a covariate in the analysis. For a trial where the outcome measure is a rate, for example in a therapeutic trial in Alzheimer's disease, the relevant covariate is a pre-randomization measure of that rate. Obtaining this requires separating the total follow-up period into two periods. In the first 'run-in' period all patients would be 'off-treatment' to facilitate the calculation of baseline atrophy rates. In the second 'on-treatment' period half of the patients, selected at random, would be switched onto active treatment with the others remaining off treatment. In this paper we use linear mixed models to establish a methodological framework that is then used to assess the extent to which such designs can increase statistical power. We illustrate our methodology with two examples. The first is a design with three evenly spaced time points analysed with a standard random slopes model. The second is a model for repeated 'direct' measures of changes used for the analysis of imaging studies with visits at multiple time points. We show that run-in designs can materially reduce sample size provided that true between-subject variability in rates is large relative to measurement error.     

Longitudinal Assessment Design and Statistical Power for Detecting an Intervention Impact

In evaluating randomized control trials (RCTs), statistical power analyses are necessary to choose a sample size which strikes the balance between an insufficient and an excessive design, with the latter leading to misspent resources. With the growing popularity of using longitudinal data to evaluate RCTs, statistical power calculations have become more complex. Specifically, with repeated measures, the number and frequency of measurements per person additionally influence statistical power by determining the precision with which intra-individual change can be measured as well as the reliability with which inter-individual differences in change can be assessed. The application of growth mixture models has shown that the impact of universal interventions is often concentrated among a small group of individuals at the highest level of risk. General sample size calculations were consequently not sufficient to determine whether statistical power is adequate to detect the desired effect. Currently, little guidance exists to recommend a sufficient assessment design to evaluating intervention impact. To this end, Monte Carlo simulations are conducted to assess the statistical power and precision when manipulating study duration and assessment frequency. Estimates were extracted from a published evaluation of the proximal of the Good Behavior Game (GBG) on the developmental course of aggressive behavior. Results indicated that the number of time points and the frequency of assessments influence statistical power and precision. Recommendations for the assessment design of longitudinal studies are discussed.     

Design of HIV viral dynamics studies

HIV viral dynamics studies involve repeated measurement of viral load in HIV-infected individuals, to asses short-term rates of viral load change in response to interventions such as initiation or withdrawal of antiviral therapy. Such studies are an important source of information on HIV pathogenesis. This paper concerns some statistical issues arising in their design. Using a linear random-effects model to incorporate between-patient differences in rates of viral load change, I discuss the choice of number of individuals and frequency of observation per individual. I suggest an approach for calculating the optimal sample size and observation frequency, based on minimizing the total number of viral load measurements that one needs to undertake. The conclusion, using this approach, is that over a period of linear change in viral load, three to five measurements per individual is generally appropriate. I also examine the observation frequency when the number of available individuals is limited, in which case it is shown that one can use a higher frequency of measurement per individual to achieve adequate power or precision. Finally, I consider sources of data for prior specification of variance components, together with conservative designs that are insensitive to a lack of prior information about between-patient differences. © 1998 John Wiley & Sons, Ltd.     

A three-outcome design for randomized comparative phase II clinical trials

Randomized designs have been increasingly called for use in phase II oncology clinical trials to protect against potential patient selection bias. However, formal statistical comparison is rarely conducted due to the sample size restriction, despite its appeal. In this paper, we offer an approach to sample size reduction by extending the three-outcome design of Sargent et al. (Control Clin. Trials 2001; 22:117-125) for single-arm trials to randomized comparative trials. In addition to the usual two outcomes of a hypothesis testing (rejecting the null hypothesis or rejecting the alternative hypothesis), the three-outcome comparative design allows a third outcome of rejecting neither hypotheses when the testing result is in some 'grey area' and leaves the decision to the clinical judgment based on the overall evaluation of trial outcomes and other relevant factors. By allowing a reasonable region of uncertainty, the three-outcome design enables formal statistical comparison with considerably smaller sample size, compared to the standard two-outcome comparative design. Statistical formulation of the three-outcome comparative design is discussed for both the single-stage and two-stage trials. Sample sizes are tabulated for some common clinical scenarios.     

Release of data from an ongoing randomized clinical trial for sample size adjustment or planning

The determination of an appropriate sample size is a key issue in planning and designing randomized clinical trials. In settings with time-to-event or binary outcomes, the required sample size depends on the control-arm event (response) rate. An accurate estimate of this rate is not often available at the planning stage. Therefore, non-comparative control-arm or pooled-arm event rates from an ongoing trial are sometimes released for sample size adjustment or planning purposes. It is shown that such non-comparative data release may still contain information on the relative treatment benefit and may thus adversely affect the ongoing trial. A simple approach to minimizing the effect of the data release is suggested.     

The analysis of cross-over trials with baseline measurements

The statistical power of cross-over trials can be increased by taking 'baseline' measurements of the outcome variable at the start of each treatment period. Analysis of covariance (ANCOVA), rather than analysis of change scores, takes best advantage of this. However, ANCOVA can give biased treatment effect estimates in observational studies with true baseline imbalance. While in truth balanced, chance baseline imbalance is possible in individual randomized cross-over studies due to their typically small sample size. Although such chance imbalance does not cause biased estimation on average over repeated trials, this simulation study will aim to confirm the appropriateness of ANCOVA when faced with the analysis of data from an individual trial in which chance baseline imbalance is clearly apparent. Randomized cross-over trials were simulated, varying in sample size and the pattern and strength of correlation between repeated measures. Estimates from ANCOVA, change scores, and post-treatment difference were unbiased on average across each set of simulated data sets. ANCOVA and change scores could use baseline information to improve precision, but change scores could also reduce precision if baseline measures were uninformative. Change scores only were correlated with chance within-subject baseline imbalance. All three estimators could be correlated with chance between-subjects imbalance in the first period baseline measurements, the strongest associations being with the post-treatment difference. Consistent results were obtained from a real data example. In conclusion, ANCOVA took best advantage of baseline measures to improve precision, and avoided bias in the widest set of circumstances with chance imbalance in those baseline measures.     

Optimal treatment allocation for placebo‐treatment comparisons in trials with discrete‐time survival endpoints

In many randomized controlled trials, treatment groups are of equal size, but this is not necessarily the best choice. This paper provides a methodology to calculate optimal treatment allocations for longitudinal trials when we wish to compare multiple treatment groups with a placebo group, and the comparisons may have unequal importance. The focus is on trials with a survival endpoint measured in discrete time. We assume the underlying survival process is Weibull and show that values for the parameters in the Weibull distribution have an impact on the optimal treatment allocation scheme in an interesting way. Additionally, we incorporate different cost considerations at the subject and measurement levels and determine the optimal number of time periods. We also show that when many events occur at the beginning of the trial, fewer time periods are more efficient. As an application, we revisit a risperidone maintenance treatment trial in schizophrenia and use our proposed methodology to redesign it and compare merits of our optimal design. Copyright © 2015 John Wiley & Sons, Ltd.