A new conditional performance score for the evaluation of adaptive group sequential designs with sample size recalculation

In standard clinical trial designs, the required sample size is fixed in the planning stage based on initial parameter assumptions. It is intuitive that the correct choice of the sample size is of major importance for an ethical justification of the trial. The required parameter assumptions should be based on previously published results from the literature. In clinical practice, however, historical data often do not exist or show highly variable results. Adaptive group sequential designs allow a sample size recalculation after a planned unblinded interim analysis in order to adjust the sample size during the ongoing trial. So far, there exist no unique standards to assess the performance of sample size recalculation rules. Single performance criteria commonly reported are given by the power and the average sample size; the variability of the recalculated sample size and the conditional power distribution are usually ignored. Therefore, the need for an adequate performance score combining these relevant performance criteria is evident. To judge the performance of an adaptive design, there exist two possible perspectives, which might also be combined: Either the global performance of the design can be addressed, which averages over all possible interim results, or the conditional performance is addressed, which focuses on the remaining performance conditional on a specific interim result. In this work, we give a compact overview of sample size recalculation rules and performance measures. Moreover, we propose a new conditional performance score and apply it to various standard recalculation rules by means of Monte-Carlo simulations.     

Sample size and power for pair-matched case-control studies

A new sample size formula is derived herein for matched-pair case-control studies. This result is based on an unconditional approach to the use of McNemar's statistic. We contrast this method for determining sample size and power with Schlesselman's conditional approach and compare the results numerically in a Monte Carlo study.     

Sample size and power determination for clustered repeated measurements

It is common in epidemiological and clinical studies that each subject has repeated measurements on a single common variable, while the subjects are also 'clustered'. To compute sample size or power of a test, we have to consider two types of correlation: correlation among repeated measurements within the same subject, and correlation among subjects in the same cluster. We develop, based on generalized estimating equations, procedures for computing sample size and power with clustered repeated measurements. Explicit formulae are derived for comparing two means, two slopes and two proportions, under several simple correlation structures.     

Sample size and power for prospective analysis of relative risk

In a placebo-controlled vaccine efficacy trial of a trial of equivalence of vaccines, one may wish to show that relative risk of disease is less than a specified value R₀, not equal to one. This paper compares three methods for estimating relative risk in the binomial setting, based on a logarithmic transformation, likelihood scores, and a Poisson approximation. Exact power and size of test are calculated by enumeration of possible binomial outcomes, and power is approximated from asymptotic formulations. Although the score method is generally preferable, for most studies of practical interest the log and score methods are comparable, and the Poisson method is also appropriate for small risks, up to about 0.05. When true and null relative risks are less than one, unequal allocation of study individuals can increase power, and the asymptotic formula for the log method may substantially understimate power; in such a study the power approximation for the score method is more reliable, even if the log method is used in analysis. Exact power calculations are helpful in planning studies. The log and Poisson methods, but not the score method, apply readily in the case of unequal follow-up.     

Sample size and power for McNemar's test with clustered data

McNemar's test is used to compare the distribution of two paired binary random variables. When the data are clustered adjustment is needed to ensure that it is still a valid test. This article presents two approximations for calculating the power and sample size for the adjusted McNemar's test for clustered data, working with a particular adjustment. A simulation study is conducted to demonstrate the accuracy of these approximations. The method is also applied to the design of a study involving positron emission tomography in detecting metastatic colorectal cancer and sensitivity of sample size computations to the design parameters are explored in this context.     

Sample size and power for case-control studies when exposures are continuous

In estimating the sample size for a case-control study, epidemiologic texts present formulae that require a binary exposure of interest. Frequently, however, important exposures are continuous and dichotomization may result in a 'not exposed' category that has little practical meaning. In addition, if risks vary monotonically with exposure, then dichotomization will obscure risk effects and require a greater number of subjects to detect differences in the exposure distributions among cases and controls. Starting from the usual score statistic to detect differences in exposure, this paper develops sample size formulae for case-control studies with arbitrary exposure distributions; this includes both continuous and dichotomous exposure measurements as special cases. The score statistic is appropriate for general differentiable models for the relative odds, and, in particular, for the two forms commonly used in prospective disease occurrence models: (1) the odds of disease increase linearly with exposure; or (2) the odds increase exponentially with exposure. Under these two models we illustrate calculation of sample sizes for a hypothetical case-control study of lung cancer among non-smokers who are exposed to radon decay products at home.     

Sample size and power calculations for correlations between bivariate longitudinal data

The analysis of a baseline predictor with a longitudinally measured outcome is well established and sample size calculations are reasonably well understood. Analysis of bivariate longitudinally measured outcomes is gaining in popularity and methods to address design issues are required. The focus in a random effects model for bivariate longitudinal outcomes is on the correlations that arise between the random effects and between the bivariate residuals. In the bivariate random effects model, we estimate the asymptotic variances of the correlations and we propose power calculations for testing and estimating the correlations. We compare asymptotic variance estimates to variance estimates obtained from simulation studies and compare our proposed power calculations for correlations on bivariate longitudinal data to power calculations for correlations on cross‐sectional data. Copyright © 2010 John Wiley & Sons, Ltd.     

Sample size and power based on the population attributable fraction.

Most methods for calculating sample size use the relative risk (RR) to indicate the strength of the association between exposure and disease. For measuring the public health importance of a possible association, the population attributable fraction (PAF)--the proportion of disease incidence in a population that is attributable to an exposure--is more appropriate. We determined sample size and power for detecting a specified PAF in both cohort and case-control studies and compared the results with those obtained using conventional estimates based on the relative risk. When an exposure is rare, a study that has little power to detect a small RR often has adequate power to detect a small PAF. On the other hand, for common exposures, even a relatively large study may have inadequate power to detect a small PAF. These comparisons emphasize the importance of selecting the most pertinent measure of association, either relative risk or population attributable fraction, when calculating power and sample size.     

An examination of methods for sample size recalculation during an experiment

In designing experiments, investigators frequently can specify an important effect that they wish to detect with high power, without the ability to provide an equally certain assessment of the variance of the response. If the experiment is designed based on a guess of the variance, an under-powered study may result. To remedy this problem, there have been several procedures proposed that obtain estimates of the variance from the data as they accrue and then recalculate the sample size accordingly. One class of procedures is fully sequential in that it assesses after each response whether the current sample size yields the desired power based on the current estimate of the variance. This approach is efficient, but it is not practical or advisable in many situations. Another class of procedures involves only two or three stages of sampling and recalculates the sample size based on the observed variance at designated times, perhaps coinciding with interim efficacy analyses. The two-stage approach can result in substantial oversampling, but it is feasible in many situations, whereas the three-stage approach corrects the problem of oversampling, but is less feasible. We propose a procedure that aims to combine the advantages of both the fully sequential and the two-stage approaches. This quasi-sequential procedure involves only two stages of sampling and it applies the stopping rule from the fully sequential procedure to data beyond the initial sample which we obtain via multiple imputation. We show through simulations that when the initial sample size is substantially less than the correct sample size, the mean squared error of the final sample size calculated from the quasi-sequential procedure can be considerably less than that from the two-stage procedure. We compare the distributions of these recalculated sample sizes and discuss our findings for alternative procedures, as well. © 1997 by John Wiley & Sons, Ltd.     

Sample size determinations using logistic regression with pilot data

Suppose the goal of a projected study is to estimate accurately the value of a ‘prediction’ proportion p that is specific to a given set of covariates. Available pilot data show that (1) the covariates are influential in determining the value of p and (2) their relationship to p can be modelled as a logistic regression. A sample size justification for the projected study can be based on the logistic model; the resulting sample sizes not only are more reasonable than the usual binomial sample size values from a scientific standpoint (since they are based on a model that is more realistic), but also give smaller prediction standard errors than the binomial approach with the same sample size. In appropriate situations, the logistic-based sample sizes could make the difference between a feasible proposal and an unfeasible, binomial-based proposal. An example using pilot study data of dental radiographs demonstrates the methods.     

Operating characteristics of sample size re-estimation with futility stopping based on conditional power

Various methods have been described for re-estimating the final sample size in a clinical trial based on an interim assessment of the treatment effect. Many re-weight the observations after re-sizing so as to control the pursuant inflation in the type I error probability alpha. Lan and Trost (Estimation of parameters and sample size re-estimation. Proceedings of the American Statistical Association Biopharmaceutical Section 1997; 48-51) proposed a simple procedure based on conditional power calculated under the current trend in the data (CPT). The study is terminated for futility if CPT < or = CL, continued unchanged if CPT > or = CU, or re-sized by a factor m to yield CPT = CU if CL < CPT < CU, where CL and CU are pre-specified probability levels. The overall level alpha can be preserved since the reduction due to stopping for futility can balance the inflation due to sample size re-estimation, thus permitting any form of final analysis with no re-weighting. Herein the statistical properties of this approach are described including an evaluation of the probabilities of stopping for futility or re-sizing, the distribution of the re-sizing factor m, and the unconditional type I and II error probabilities alpha and beta. Since futility stopping does not allow a type I error but commits a type II error, then as the probability of stopping for futility increases, alpha decreases and beta increases. An iterative procedure is described for choice of the critical test value and the futility stopping boundary so as to ensure that specified alpha and beta are obtained. However, inflation in beta is controlled by reducing the probability of futility stopping, that in turn dramatically increases the possible re-sizing factor m. The procedure is also generalized to limit the maximum sample size inflation factor, such as at m max = 4. However, doing so then allows for a non-trivial fraction of studies to be re-sized at this level that still have low conditional power. These properties also apply to other methods for sample size re-estimation with a provision for stopping for futility. Sample size re-estimation procedures should be used with caution and the impact on the overall type II error probability should be assessed.     

Sample size and power calculations for open cohort longitudinal cluster randomized trials

When calculating sample size or power for stepped wedge or other types of longitudinal cluster randomized trials, it is critical that the planned sampling structure be accurately specified. One common assumption is that participants will provide measurements in each trial period, that is, a closed cohort, and another is that each participant provides only one measurement during the course of the trial. However some studies have an “open cohort” sampling structure, where participants may provide measurements in variable numbers of periods. To date, sample size calculations for longitudinal cluster randomized trials have not accommodated open cohorts. Feldman and McKinlay (1994) provided some guidance, stating that the participant-level autocorrelation could be varied to account for the degree of overlap in different periods of the study, but did not indicate precisely how to do so. We present sample size and power formulas that allow for open cohorts and discuss the impact of the degree of “openness” on sample size and power. We consider designs where the number of participants in each cluster will be maintained throughout the trial, but individual participants may provide differing numbers of measurements. Our results are a unification of closed cohort and repeated cross-sectional sample results of Hooper et al (2016), and indicate precisely how participant autocorrelation of Feldman and McKinlay should be varied to account for an open cohort sampling structure. We discuss different types of open cohort sampling schemes and how open cohort sampling structure impacts on power in the presence of decaying within-cluster correlations and autoregressive participant-level errors.     

Sample size/power calculation for stratified case–cohort design

The case–cohort (CC) study design usually has been used for risk factor assessment in epidemiologic studies or disease prevention trials for rare diseases. The sample size/power calculation for a stratified CC (SCC) design has not been addressed before. This article derives such result based on a stratified test statistic. Simulation studies show that the proposed test for the SCC design utilizing small sub‐cohort sampling fractions is valid and efficient for situations where the disease rate is low. Furthermore, optimization of sampling in the SCC design is discussed and compared with proportional and balanced sampling techniques. An epidemiological study is provided to illustrate the sample size calculation under the SCC design. Copyright © 2014 John Wiley & Sons, Ltd.     

A two-stage sample size recalculation procedure for placebo- and active-controlled non-inferiority trials

Many non-inferiority trials of a test treatment versus an active control may also, if ethical, incorporate a placebo arm. Inclusion of a placebo arm enables a direct assessment of assay sensitivity. It also allows construction of a non-inferiority test that avoids the problematic specification of an absolute non-inferiority margin, and instead evaluates whether the test treatment preserves a pre-specified proportion of the effect of the active control over placebo. We describe a two-stage procedure for sample size recalculation in such a setting that maintains the desired power more closely than a fixed sample approach when the magnitude of the effect of the active control differs from that anticipated. We derive an allocation rule for randomization under which the procedure preserves the type I error rate, and show that this coincides with that previously presented for optimal allocation of the sample size among the three treatment arms.     

Sample size needed to detect gene-gene interactions using association designs

It is likely that many complex diseases result from interactions among several genes, as well as environmental factors. The presence of such interactions poses challenges to investigators in identifying susceptibility genes, understanding biologic pathways, and predicting and controlling disease risks. Recently, Gauderman (Am J Epidemiol 2002;155:478-84) reported results from the first systematic analysis of the statistical power needed to detect gene-gene interactions in association studies. However, Gauderman used different statistical models to model disease risks for different study designs, and he assumed a very low disease prevalence to make different models more comparable. In this article, assuming a logistic model for disease risk for different study designs, the authors investigate the power of population-based and family-based association designs to detect gene-gene interactions for common diseases. The results indicate that population-based designs are more powerful than family-based designs for detecting gene-gene interactions when disease prevalence in the study population is moderate.     

Sample size evaluation for a multiply matched case-control study using the score test from a conditional logistic (discrete Cox PH) regression model

The conditional logistic regression model (Biometrics 1982; 38:661-672) provides a convenient method for the assessment of qualitative or quantitative covariate effects on risk in a study with matched sets, each containing a possibly different number of cases and controls. The conditional logistic likelihood is identical to the stratified Cox proportional hazards model likelihood, with an adjustment for ties (J. R. Stat. Soc. B 1972; 34:187-220). This likelihood also applies to a nested case-control study with multiply matched cases and controls, selected from those at risk at selected event times. Herein the distribution of the score test for the effect of a covariate in the model is used to derive simple equations to describe the power of the test to detect a coefficient theta (log odds ratio or log hazard ratio) or the number of cases (or matched sets) and controls required to provide a desired level of power. Additional expressions are derived for a quantitative covariate as a function of the difference in the assumed mean covariate values among cases and controls and for a qualitative covariate in terms of the difference in the probabilities of exposure for cases and controls. Examples are presented for a nested case-control study and a multiply matched case-control study.     

阶梯整群随机对照试验的样本量和功效计算方法

目的 介绍阶梯整群随机对照试验（SW-CRT）的样本量和功效计算方法。 方法 参考既往的相关方法学研究,并配合实际案例,介绍SW-CRT两种情景下的样本量和功效计算方法、具体实现步骤和实现工具。 结果 利用所介绍的公式能方便计算已知每群调查样本量或已知调查总群数的两种情景下满足样本功效要求的样本量,推荐使用Stata软件提供的steppedwedge程序进行样本功效的计算。结论 本研究能够为相关研究者的研究设计和结果检验工作提供具有实际操作意义的方法选择。