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.     

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.     

Sample sizes required for binomial trials when the true response rates are estimated

Binomial trial sample size specification depends upon the values of the unknown response rate parameters, as well as upon the size and power of the resulting test. In practice, the values assumed for these parameters are based upon the results of previous or pilot trials, or upon the investigator's prior knowledge or belief. In either case, there is some uncertainty associated with these values that should be taken into account if the sample sizes are to be specified realistically. This paper describes a procedure for incorporating this uncertainty explicitly into the sample size determination on the basis of joint confidence distributions obtained from the pilot or prior information.     

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.     

Some aspects of the design and analysis of cluster randomization trials

Trials which randomize intact social groups, or clusters, to different interventions are becoming increasingly widespread. Although statistically less efficient than trials which randomize individuals, such designs are often preferred from a practical or ethical point of view, particularly in the evaluation of health care or educational strategies. We discuss selected issues that arise in the conduct of such trials, including the choice of design, ethical implications, sample size estimation and approaches to the analysis. The discussion is closely tied to methodological issues that have arisen in a recent evaluation trial of a new antenatal care programme, as sponsored by the Special Programme of Research, Development and Research Training in Human Reproduction of the World Health Organization.     

Sample size re-estimation without unblinding for normally distributed outcomes with unknown variance

Monitoring clinical trials in nonfatal diseases where ethical considerations do not dictate early termination upon demonstration of efficacy often requires examining the interim findings to assure that the protocol-specified sample size will provide sufficient power against the null hypothesis when the alternative hypothesis is true. The sample size may be increased, if necessary to assure adequate power. This paper presents a new method for carrying out such interim power evaluations for observations from normal distributions without unblinding the treatment assignments or discernably affecting the Type 1 error rate. Simulation studies confirm the expected performance of the method.     

Sample size calculation for testing non-inferiority and equivalence under Poisson distribution

When counting the number of chemical parts in air pollution studies or when comparing the occurrence of congenital malformations between a uranium mining town and a control population, we often assume Poisson distribution for the number of these rare events. Some discussions on sample size calculation under Poisson model appear elsewhere, but all these focus on the case of testing equality rather than testing equivalence. We discuss sample size and power calculation on the basis of exact distribution under Poisson models for testing non-inferiority and equivalence with respect to the mean incidence rate ratio. On the basis of large sample theory, we further develop an approximate sample size calculation formula using the normal approximation of a proposed test statistic for testing non-inferiority and an approximate power calculation formula for testing equivalence. We find that using these approximation formulae tends to produce an underestimate of the minimum required sample size calculated from using the exact test procedure. On the other hand, we find that the power corresponding to the approximate sample sizes can be actually accurate (with respect to Type I error and power) when we apply the asymptotic test procedure based on the normal distribution. We tabulate in a variety of situations the minimum mean incidence needed in the standard (or the control) population, that can easily be employed to calculate the minimum required sample size from each comparison group for testing non-inferiority and equivalence between two Poisson populations.     

Approaches to sample size calculation for clinical trials in rare diseases

We discuss 3 alternative approaches to sample size calculation: traditional sample size calculation based on power to show a statistically significant effect, sample size calculation based on assurance, and sample size based on a decision‐theoretic approach. These approaches are compared head‐to‐head for clinical trial situations in rare diseases. Specifically, we consider 3 case studies of rare diseases (Lyell disease, adult‐onset Still disease, and cystic fibrosis) with the aim to plan the sample size for an upcoming clinical trial. We outline in detail the reasonable choice of parameters for these approaches for each of the 3 case studies and calculate sample sizes. We stress that the influence of the input parameters needs to be investigated in all approaches and recommend investigating different sample size approaches before deciding finally on the trial size. Highly influencing for the sample size are choice of treatment effect parameter in all approaches and the parameter for the additional cost of the new treatment in the decision‐theoretic approach. These should therefore be discussed extensively.     

Sample size calculations for time-averaged difference of longitudinal binary outcomes

In clinical trials with repeated measurements, the responses from each subject are measured multiple times during the study period. Two approaches have been widely used to assess the treatment effect, one that compares the rate of change between two groups and the other that tests the time-averaged difference (TAD). While sample size calculations based on comparing the rate of change between two groups have been reported by many investigators, the literature has paid relatively little attention to the sample size estimation for time-averaged difference (TAD) in the presence of heterogeneous correlation structure and missing data in repeated measurement studies. In this study, we investigate sample size calculation for the comparison of time-averaged responses between treatment groups in clinical trials with longitudinally observed binary outcomes. The generalized estimating equation (GEE) approach is used to derive a closed-form sample size formula, which is flexible enough to account for arbitrary missing patterns and correlation structures. In particular, we demonstrate that the proposed sample size can accommodate a mixture of missing patterns, which is frequently encountered by practitioners in clinical trials. To our knowledge, this is the first study that considers the mixture of missing patterns in sample size calculation. Our simulation shows that the nominal power and type I error are well preserved over a wide range of design parameters. Sample size calculation is illustrated through an example.     

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 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.     

Sample sizes for trials involving multiple correlated must-win comparisons

In Clinical trials involving multiple comparisons of interest, the importance of controlling the trial Type I error is well-understood and well-documented. Moreover, when these comparisons are themselves correlated, methodologies exist for accounting for the correlation in the trial design, when calculating the trial significance levels. However, less well-documented is the fact that there are some circumstances where multiple comparisons affect the Type II error rather than the Type I error, and failure to account for this, can result in a reduction in the overall trial power. In this paper, we describe sample size calculations for clinical trials involving multiple correlated comparisons, where all the comparisons must be statistically significant for the trial to provide evidence of effect, and show how such calculations have to account for multiplicity in the Type II error. For the situation of two comparisons, we provide a result which assumes a bivariate Normal distribution. For the general case of two or more comparisons we provide a solution using inflation factors to increase the sample size relative to the case of a single outcome. We begin with a simple case of two comparisons assuming a bivariate Normal distribution, show how to factor in correlation between comparisons and then generalise our findings to situations with two or more comparisons. These methods are easy to apply, and we demonstrate how accounting for the multiplicity in the Type II error leads, at most, to modest increases in the sample size.     

Sample size allocation in multiregional equivalence studies

With the increasing globalization of drug development, the multiregional clinical trial (MRCT) has gained extensive use. The data from MRCTs could be accepted by regulatory authorities across regions and countries as the primary sources of evidence to support global marketing drug approval simultaneously. The MRCT can speed up patient enrollment and drug approval, and it makes the effective therapies available to patients all over the world simultaneously. However, there are many challenges both operationally and scientifically in conducting a drug development globally. One of many important questions to answer for the design of a multiregional study is how to partition sample size into each individual region. In this paper, two systematic approaches are proposed for the sample size allocation in a multiregional equivalence trial. A numerical evaluation and a biosimilar trial are used to illustrate the characteristics of the proposed approaches.     

Sample size calculation for an agreement study

It is often necessary to compare two measurement methods in medicine and other experimental sciences. This problem covers a broad range of data. Many authors have explored ways of assessing the agreement of two sets of measurements. However, there has been relatively little attention to the problem of determining sample size for designing an agreement study. In this paper, a method using the interval approach for concordance is proposed to calculate sample size in conducting an agreement study. The philosophy behind this is that the concordance is satisfied when no more than the pre‐specified k discordances are found for a reasonable large sample size n since it is much easier to define a discordance pair. The goal here is to find such a reasonable large sample size n. The sample size calculation is based on two rates: the discordance rate and tolerance probability, which in turn can be used to quantify an agreement study. The proposed approach is demonstrated through a real data set. Copyright © 2009 John Wiley & Sons, Ltd.     

Power of the adjusted Q statistic to evaluate heterogeneity in meta-analyses of cluster randomized trials

Because of its simplicity, the Q statistic is frequently used to test the heterogeneity of the estimated intervention effect in meta-analyses of individually randomized trials. However, it is inappropriate to apply it directly to the meta-analyses of cluster randomized trials without taking clustering effects into account. We consider the properties of the adjusted Q statistic for testing heterogeneity in the meta-analyses of cluster randomized trials with binary outcomes. We also derive an analytic expression for the power of this statistic to detect heterogeneity in meta-analyses, which can be useful when planning a meta-analysis. A simulation study is used to assess the performance of the adjusted Q statistic, in terms of its Type I error rate and power. The simulation results are compared to that obtained from the proposed formula. It is found that the adjusted Q statistic has a Type I error rate close to the nominal level of 5%, as compared to the unadjusted Q statistic commonly used to test for heterogeneity in the meta-analyses of individually randomized trials with an inflated Type I error rate. Data from a meta-analysis of four cluster randomized trials are used to illustrate the procedures.     

Subject‐level matching for imbalance in cluster randomized trials with a small number of clusters

In a cluster randomized controlled trial (RCT), the number of randomized units is typically considerably smaller than in trials where the unit of randomization is the patient. If the number of randomized clusters is small, there is a reasonable chance of baseline imbalance between the experimental and control groups. This imbalance threatens the validity of inferences regarding post‐treatment intervention effects unless an appropriate statistical adjustment is used. Here, we consider application of the propensity score adjustment for cluster RCTs. For the purpose of illustration, we apply the propensity adjustment to a cluster RCT that evaluated an intervention to reduce suicidal ideation and depression. This approach to adjusting imbalance had considerable bearing on the interpretation of results. A simulation study demonstrates that the propensity adjustment reduced well over 90% of the bias seen in unadjusted models for the specifications examined. Copyright © 2013 John Wiley & Sons, Ltd.     

Design and analysis considerations for comparing dynamic treatment regimens with binary outcomes from sequential multiple assignment randomized trials

In behavioral, educational and medical practice, interventions are often personalized over time using strategies that are based on individual behaviors and characteristics and changes in symptoms, severity, or adherence that are a result of one's treatment. Such strategies that more closely mimic real practice, are known as dynamic treatment regimens (DTRs). A sequential multiple assignment randomized trial (SMART) is a multi-stage trial design that can be used to construct effective DTRs. This article reviews a simple to use ‘weighted and replicated’ estimation technique for comparing DTRs embedded in a SMART design using logistic regression for a binary, end-of-study outcome variable. Based on a Wald test that compares two embedded DTRs of interest from the ‘weighted and replicated’ regression model, a sample size calculation is presented with a corresponding user-friendly applet to aid in the process of designing a SMART. The analytic models and sample size calculations are presented for three of the more commonly used two-stage SMART designs. Simulations for the sample size calculation show the empirical power reaches expected levels. A data analysis example with corresponding code is presented in the appendix using data from a SMART developing an effective DTR in autism.     

Sample size estimation for non-inferiority trials of time-to-event data

We consider the problem of sample size calculation for non-inferiority based on the hazard ratio in time-to-event trials where overall study duration is fixed and subject enrollment is staggered with variable follow-up. An adaptation of previously developed formulae for the superiority framework is presented that specifically allows for effect reversal under the non-inferiority setting, and its consequent effect on variance. Empirical performance is assessed through a small simulation study, and an example based on an ongoing trial is presented. The formulae are straightforward to program and may prove a useful tool in planning trials of this type.     

Sample size re-estimation for survival data in clinical trials with an adaptive design

In clinical trials with survival data, investigators may wish to re-estimate the sample size based on the observed effect size while the trial is ongoing. Besides the inflation of the type-I error rate due to sample size re-estimation, the method for calculating the sample size in an interim analysis should be carefully considered because the data in each stage are mutually dependent in trials with survival data. Although the interim hazard estimate is commonly used to re-estimate the sample size, the estimate can sometimes be considerably higher or lower than the hypothesized hazard by chance. We propose an interim hazard ratio estimate that can be used to re-estimate the sample size under those circumstances. The proposed method was demonstrated through a simulation study and an actual clinical trial as an example. The effect of the shape parameter for the Weibull survival distribution on the sample size re-estimation is presented.     

Sample size determination for the parallel model in a survey with sensitive questions

Recently, a new non-randomized parallel design is proposed by Tian (2013) for surveys with sensitive topics. However, the sample size formulae associated with testing hypotheses for the parallel model are not yet available. As a crucial component in surveys, the sample size formulae with the parallel design are developed in this paper by using the power analysis method for both the one- and two-sample problems. We consider both the one- and two-sample problems. The asymptotic power functions and the corresponding sample size formulae for both the one- and two-sided tests based on the large-sample normal approximation are derived. The performance is assessed through comparing the asymptotic power with the exact power and reporting the ratio of the sample sizes with the parallel model and the design of direct questioning. We numerically compare the sample sizes needed for the parallel design with those required for the crosswise and triangular models. Two theoretical justifications are also provided. An example from a survey on ‘sexual practices’ in San Francisco, Las Vegas and Portland is used to illustrate the proposed methods.