Combining an Internal Pilot with an Interim Analysis for Single Degree of Freedom Tests

An internal pilot with interim analysis (IPIA) design combines interim power analysis (an internal pilot) with interim data analysis (two stage group sequential). We provide IPIA methods for single df hypotheses within the Gaussian general linear model, including one and two group t tests. The design allows early stopping for efficacy and futility while also re-estimating sample size based on an interim variance estimate. Study planning in small samples requires the exact and computable forms reported here. The formulation gives fast and accurate calculations of power, type I error rate, and expected sample size.     

Combining an Internal Pilot with an Interim Analysis for Single Degree of Freedom Tests

An internal pilot with interim analysis (IPIA) design combines interim power analysis (an internal pilot) with interim data analysis (two-stage group sequential). We provide IPIA methods for single df hypotheses within the Gaussian general linear model, including one and two group t tests. The design allows early stopping for efficacy and futility while also re-estimating sample size based on an interim variance estimate. Study planning in small samples requires the exact and computable forms reported here. The formulation gives fast and accurate calculations of power, Type I error rate, and expected sample size.     

Blinded sample size re‐estimation in superiority and noninferiority trials: bias versus variance in variance estimation

The internal pilot study design allows for modifying the sample size during an ongoing study based on a blinded estimate of the variance thus maintaining the trial integrity. Various blinded sample size re‐estimation procedures have been proposed in the literature. We compare the blinded sample size re‐estimation procedures based on the one‐sample variance of the pooled data with a blinded procedure using the randomization block information with respect to bias and variance of the variance estimators, and the distribution of the resulting sample sizes, power, and actual type I error rate. For reference, sample size re‐estimation based on the unblinded variance is also included in the comparison. It is shown that using an unbiased variance estimator (such as the one using the randomization block information) for sample size re‐estimation does not guarantee that the desired power is achieved. Moreover, in situations that are common in clinical trials, the variance estimator that employs the randomization block length shows a higher variability than the simple one‐sample estimator and in turn the sample size resulting from the related re‐estimation procedure. This higher variability can lead to a lower power as was demonstrated in the setting of noninferiority trials. In summary, the one‐sample estimator obtained from the pooled data is extremely simple to apply, shows good performance, and is therefore recommended for application. Copyright © 2013 John Wiley & Sons, Ltd.     

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.     

Bootstrap power of the generalized correlation coefficient

We present a bootstrap Monte Carlo algorithm for computing the power function of the generalized correlation coefficient. The proposed method makes no assumptions about the form of the underlying probability distribution and may be used with observed data to approximate the power function and pilot data for sample size determination. In particular, the bootstrap power functions of the Pearson product moment correlation and the Spearman rank correlation are examined. Monte Carlo experiments indicate that the proposed algorithm is reliable and compares well with the asymptotic values. An example which demonstrates how this method can be used for sample size determination and power calculations is provided.     

Seme distributions and their implications for an internal pilot study with a univariate linear model

In planning a study, the choice of sample size may depend on a variance value based on speculation or obtained from an earlier study. Scientists may wish to use an internal pilot design to protect themselves against an incorrect choice of variance. Such a design involves collecting a portion of the originally planned sample and using it to produce a new variance estimate. This leads to a new power analysis and increasing or decreasing sample size. For any general linear univariate model, with fixed predictors and Gaussian errors, we prove that the uncorrected fixed sample F-statistic is the likelihood ratio test statistic. However, the statistic does not follow an F distribution. Ignoring the discrepancy may inflate test size. We derive and evaluate properties of the components of the likelihood ratio test statistic in order to characterize and quantify the bias. Most notably, the fixed sample size variance estimate becomes biased downward. The bias may inflate test size for any hypothesis test, even if the parameter being tested was not involved in the sample size re-estimation. Furthermore, using fixed sample size methods may create biased confidence intervals for secondary parameters and the variance estimate.     

Distribution of the two‐sample t‐test statistic following blinded sample size re‐estimation

We consider the blinded sample size re‐estimation based on the simple one‐sample variance estimator at an interim analysis. We characterize the exact distribution of the standard two‐sample t‐test statistic at the final analysis. We describe a simulation algorithm for the evaluation of the probability of rejecting the null hypothesis at given treatment effect. We compare the blinded sample size re‐estimation method with two unblinded methods with respect to the empirical type I error, the empirical power, and the empirical distribution of the standard deviation estimator and final sample size. We characterize the type I error inflation across the range of standardized non‐inferiority margin for non‐inferiority trials, and derive the adjusted significance level to ensure type I error control for given sample size of the internal pilot study. We show that the adjusted significance level increases as the sample size of the internal pilot study increases. Copyright © 2016 John Wiley & Sons, Ltd.     

Sample size re‐estimation incorporating prior information on a nuisance parameter

Prior information is often incorporated informally when planning a clinical trial. Here, we present an approach on how to incorporate prior information, such as data from historical clinical trials, into the nuisance parameter–based sample size re‐estimation in a design with an internal pilot study. We focus on trials with continuous endpoints in which the outcome variance is the nuisance parameter. For planning and analyzing the trial, frequentist methods are considered. Moreover, the external information on the variance is summarized by the Bayesian meta‐analytic‐predictive approach. To incorporate external information into the sample size re‐estimation, we propose to update the meta‐analytic‐predictive prior based on the results of the internal pilot study and to re‐estimate the sample size using an estimator from the posterior. By means of a simulation study, we compare the operating characteristics such as power and sample size distribution of the proposed procedure with the traditional sample size re‐estimation approach that uses the pooled variance estimator. The simulation study shows that, if no prior‐data conflict is present, incorporating external information into the sample size re‐estimation improves the operating characteristics compared to the traditional approach. In the case of a prior‐data conflict, that is, when the variance of the ongoing clinical trial is unequal to the prior location, the performance of the traditional sample size re‐estimation procedure is in general superior, even when the prior information is robustified. When considering to include prior information in sample size re‐estimation, the potential gains should be balanced against the risks.     

Sequential design approaches for bioequivalence studies with crossover designs

The planning of bioequivalence (BE) studies, as for any clinical trial, requires a priori specification of an effect size for the determination of power and an assumption about the variance. The specified effect size may be overly optimistic, leading to an underpowered study. The assumed variance can be either too small or too large, leading, respectively, to studies that are underpowered or overly large. There has been much work in the clinical trials field on various types of sequential designs that include sample size reestimation after the trial is started, but these have seen only little use in BE studies. The purpose of this work was to validate at least one such method for crossover design BE studies. Specifically, we considered sample size reestimation for a two-stage trial based on the variance estimated from the first stage. We identified two methods based on Pocock's method for group sequential trials that met our requirement for at most negligible increase in type I error rate.     

Sample size of 12 per group rule of thumb for a pilot study

When designing a clinical trial an appropriate justification for the sample size should be provided in the protocol. However, there are a number of settings when undertaking a pilot trial when there is no prior information to base a sample size on. For such pilot studies the recommendation is a sample size of 12 per group. The justifications for this sample size are based on rationale about feasibility; precision about the mean and variance; and regulatory considerations. The context of the justifications are that future studies will use the information from the pilot in their design. Copyright © 2005 John Wiley & Sons, Ltd.     

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.     

Some factors affecting statistical power of approximate tests in the linear mixed model for longitudinal data

Different longitudinal study designs require different statistical analysis methods and different methods of sample size determination. Statistical power analysis is a flexible approach to sample size determination for longitudinal studies. However, different power analyses are required for different statistical tests which arises from the difference between different statistical methods. In this paper, the simulation-based power calculations of F-tests with Containment, Kenward-Roger or Satterthwaite approximation of degrees of freedom are examined for sample size determination in the context of a special case of linear mixed models (LMMs), which is frequently used in the analysis of longitudinal data. Essentially, the roles of some factors, such as variance–covariance structure of random effects [unstructured UN or factor analytic FA0], autocorrelation structure among errors over time [independent IND, first-order autoregressive AR1 or first-order moving average MA1], parameter estimation methods [maximum likelihood ML and restricted maximum likelihood REML] and iterative algorithms [ridge-stabilized Newton-Raphson and Quasi-Newton] on statistical power of approximate F-tests in the LMM are examined together, which has not been considered previously. The greatest factor affecting statistical power is found to be the variance–covariance structure of random effects in the LMM. It appears that the simulation-based analysis in this study gives an interesting insight into statistical power of approximate F-tests for fixed effects in LMMs for longitudinal data.     

Sample size recalculation for binary data in internal pilot study designs

For binary endpoints, the required sample size depends not only on the known values of significance level, power and clinically relevant difference but also on the overall event rate. However, the overall event rate may vary considerably between studies and, as a consequence, the assumptions made in the planning phase on this nuisance parameter are to a great extent uncertain. The internal pilot study design is an appealing strategy to deal with this problem. Here, the overall event probability is estimated during the ongoing trial based on the pooled data of both treatment groups and, if necessary, the sample size is adjusted accordingly. From a regulatory viewpoint, besides preserving blindness it is required that eventual consequences for the Type I error rate should be explained. We present analytical computations of the actual Type I error rate for the internal pilot study design with binary endpoints and compare them with the actual level of the chi‐square test for the fixed sample size design. A method is given that permits control of the specified significance level for the chi‐square test under blinded sample size recalculation. Furthermore, the properties of the procedure with respect to power and expected sample size are assessed. Throughout the paper, both the situation of equal sample size per group and unequal allocation ratio are considered. The method is illustrated with application to a clinical trial in depression. Copyright © 2004 John Wiley & Sons Ltd.     

Blinded continuous monitoring in clinical trials with recurrent event endpoints

In studies with recurrent event endpoints, misspecified assumptions of event rates or dispersion can lead to underpowered trials or overexposure of patients. Specification of overdispersion is often a particular problem as it is usually not reported in clinical trial publications. Changing event rates over the years have been described for some diseases, adding to the uncertainty in planning. To mitigate the risks of inadequate sample sizes, internal pilot study designs have been proposed with a preference for blinded sample size reestimation procedures, as they generally do not affect the type I error rate and maintain trial integrity. Blinded sample size reestimation procedures are available for trials with recurrent events as endpoints. However, the variance in the reestimated sample size can be considerable in particular with early sample size reviews. Motivated by a randomized controlled trial in paediatric multiple sclerosis, a rare neurological condition in children, we apply the concept of blinded continuous monitoring of information, which is known to reduce the variance in the resulting sample size. Assuming negative binomial distributions for the counts of recurrent relapses, we derive information criteria and propose blinded continuous monitoring procedures. The operating characteristics of these are assessed in Monte Carlo trial simulations demonstrating favourable properties with regard to type I error rate, power, and stopping time, ie, sample size.     

On the distribution of summary statistics for missing data

Under an assumption that missing values occur randomly in a matrix, formulae are developed for the expected value and variance of six statistics that summarize the number and location of the missing values. For a seventh statistic, a regression model based on simulated data yields an estimate of the expected value. The results can be used in the development of methods to control the Type I error and approximate power and sample size for multilevel and longitudinal studies with missing data.     

Using pilot study information to increase efficiency in clinical trials

It is often necessary to conduct a pilot study to determine the sample size required for a clinical trial. Due to differences in sampling environments, the pilot data are usually discarded after sample size calculation. This paper tries to use the pilot information to modify the subsequent testing procedure when a two-sided t$t$-test or a regression model is used to compare two treatments. The new test maintains the required significance level regardless of the dissimilarity between the pilot and the target populations, but increases the power when the two are similar. The test is constructed based on the posterior distribution of the parameters given the pilot study information, but its properties are investigated from a frequentist's viewpoint. Due to the small likelihood of an irrelevant pilot population, the new approach is a viable alternative to the current practice.     

Efficient sample size allocation with cost constraints for heterogeneous-variance group comparison

When conducting research with controlled experiments, sample size planning is one of the important decisions that researchers have to make. However, current methods do not adequately address this issue with regard to variance heterogeneity with some cost constraints for comparing several treatment means. This paper proposes a sample size allocation ratio in the fixed-effect heterogeneous analysis of variance when group variances are unequal and in cases where the sampling and/or variable cost has some constraints. The efficient sample size allocation is determined for the purpose of minimizing total cost with a designated power or maximizing the power with a given total cost. Finally, the proposed method is verified by using the index of relative efficiency and the corresponding total cost and the total sample size needed. We also apply our method in a pain management trial to decide an efficient sample size. Simulation studies also show that the proposed sample size formulas are efficient in terms of statistical power. SAS and R codes are provided in the appendix for easy application.     

Designing clinical trials with uncertain estimates of variability

An estimated sample size is a function of three components: the required power, the predetermined Type I error rate, and the specified effect size. For Normal data the standardized effect size is taken as the difference between two means divided by an estimate of the population standard deviation. However, in early phase trials one may not have a good estimate of the population variance as it is often based on the results of a few relatively small trials. The imprecision of this estimate should be taken into account in sample size calculations. When estimating a trial sample size this paper recommends that one should investigate the sensitivity of the trial to the assumptions made about the variance and consider being adaptive in one's trial design. Copyright © 2004 John Wiley & Sons Ltd.     

Practical Methods for Bounding Type I Error Rate with an Internal Pilot Design

New analytic forms for distributions at the heart of internal pilot theory solve many problems inherent to current techniques for linear models with Gaussian errors. Internal pilot designs use a fraction of the data to re-estimate the error variance and modify the final sample size. Too small or too large a sample size caused by an incorrect planning variance can be avoided. However, the usual hypothesis test may need adjustment to control the Type I error rate. A bounding test achieves control of Type I error rate while providing most of the advantages of the unadjusted test. Unfortunately, the presence of both a doubly truncated and an untruncated chi-square random variable complicates the theory and computations. An expression for the density of the sum of the two chi-squares gives a simple form for the test statistic density. Examples illustrate that the new results make the bounding test practical by providing very stable, convergent, and much more accurate computations. Furthermore, the new computational methods are effectively never slower and usually much faster. All results apply to any univariate linear model with fixed predictors and Gaussian errors, with the t-test a special case.     

Two separate effects of variance heterogeneity on the validity and power of significance tests of location

Heterogeneity of variances of treatment groups influences the validity and power of significance tests of location in two distinct ways. First, if sample sizes are unequal, the Type I error rate and power are depressed if a larger variance is associated with a larger sample size, and elevated if a larger variance is associated with a smaller sample size. This well-established effect, which occurs in t and F tests, and to a lesser degree in nonparametric rank tests, results from unequal contributions of pooled estimates of error variance in the computation of test statistics. It is observed in samples from normal distributions, as well as non-normal distributions of various shapes. Second, transformation of scores from skewed distributions with unequal variances to ranks produces differences in the means of the ranks assigned to the respective groups, even if the means of the initial groups are equal, and a subsequent inflation of Type I error rates and power. This effect occurs for all sample sizes, equal and unequal. For the t test, the discrepancy diminishes, and for the Wilcoxon–Mann–Whitney test, it becomes larger, as sample size increases. The Welch separate-variance t test overcomes the first effect but not the second. Because of interaction of these separate effects, the validity and power of both parametric and nonparametric tests performed on samples of any size from unknown distributions with possibly unequal variances can be distorted in unpredictable ways.