Practical guide to sample size calculations: superiority trials

A sample size justification is a vital part of any investigation. However, estimating the number of participants required to give meaningful results is not always straightforward. A number of components are required to facilitate a suitable sample size calculation. In this paper, the steps for conducting sample size calculations for superiority trials are summarised. Practical advice and examples are provided illustrating how to carry out the calculations by hand and using the app SampSize. Copyright © 2015 John Wiley & Sons, Ltd.     

Practical guide to sample size calculations: an introduction

A sample size justification is a vital step when designing any trial. However, estimating the number of participants required to give a meaningful result is not always straightforward. A number of components are required to facilitate a suitable sample size calculation. In this paper, the general steps are summarised for conducting sample size calculations with practical advice and guidance on how to utilise the app SampSize. Copyright © 2015 John Wiley & Sons, Ltd.     

Sample Size

Conventionally, sample size calculations are viewed as calculations determining the right number of subjects needed for a study. Such calculations follow the classical paradigm: “for a difference X, I need sample size Y.” We argue that the paradigm “for a sample size Y, I get information Z” is more appropriate for many studies and reflects the information needed by scientists when planning a study. This approach applies to both physiological studies and Phase I and II interventional studies. We provide actual examples from our own consulting work to demonstrate this. We conclude that sample size should be viewed not as a unique right number, but rather as a factor needed to assess the utility of a study.     

Sample Size and Power Calculations for Left-Truncated Normal Distribution

Abstract

Sample size calculation is an important component in designing an experiment or a survey. In a wide variety of fields—including management science, insurance, and biological and medical science—truncated normal distributions are encountered in many applications. However, the sample size required for the left-truncated normal distribution has not been investigated, because the distribution of the sample mean from the left-truncated normal distribution is complex and difficult to obtain. This paper compares an ad hoc approach to two newly proposed methods based on the Central Limit Theorem and on a high degree saddlepoint approximation for calculating the required sample size with the prespecified power. As shown by use of simulations and an example of health insurance cost in China, the ad hoc approach underestimates the sample size required to achieve prespecified power. The method based on the high degree saddlepoint approximation provides valid sample size and power calculations, and it performs better than the Central Limit Theorem. When the sample size is not too small, the Central Limit Theorem also provides a valid, but relatively simple tool to approximate that sample size.     

Approximations for a fixed sample size selection procedure for negative binomial populations

A large sample approximation of the least favorable configuration for a fixed sample size selection procedure for negative binomial populations is proposed. A normal approximation of the selection procedure is also presented. Optimal sample sizes required to be drawn from each population and the bounds for the sample sizes are tabulated. Sample sizes obtained using the approximate least favorable configuration are compared with those obtained using the exact least favorable configuration. Alternate form of the normal approximation to the probability of correct selection is also presented. The relation between the required sample size and the number of populations involved is studied.     

Optimal One-Way Random Effects Designs for the Intraclass Correlation Based on Confidence Intervals

ABSTRACT

Confidence intervals for the intraclass correlation coefficient (ρ) are used to determine the optimal allocation of experimental material in one-way random effects models. Designs that produce narrow intervals are preferred since they provide greater precision to estimate ρ. Assuming the total cost and the relative cost of the two stages of sampling are fixed, the authors investigate the number of classes and the number of individuals per class required to minimize the expected length of confidence intervals. We obtain results using asymptotic theory and compare these results to those obtained using exact calculations. The best design depends on the unknown value of ρ. Minimizing the maximum expected length of confidence intervals guards against worst-case scenarios. A good overall recommendation based on asymptotic results is to choose a design having classes of size 2 + √4 + 3r, where r is the relative cost of sampling at the class-level compared to the individual-level. If r = 0, then the overall cost is the sample size and the recommendation reduces to a design having classes of size 4.     

Sample size requirements for rare or abundant attribute sampling

Two approximation procedures to determine required sample size for a Fixed width binomial confidence interval are given and compared to exact calculations as well as the normal and Poisson approximations. The approximation procedures are found to be quite simple but very accurate for estimating sample sizes for either rare or abundant attributes.     

Calculating power for the comparison of dependent κ-coefficients

Summary. In the psychosocial and medical sciences, some studies are designed to assess the agreement between different raters and/or different instruments. Often the same sample will be used to compare the agreement between two or more assessment methods for simplicity and to take advantage of the positive correlation of the ratings. Although sample size calculations have become an important element in the design of research projects, such methods for agreement studies are scarce. We adapt the generalized estimating equations approach for modelling dependent κ -statistics to estimate the sample size that is required for dependent agreement studies. We calculate the power based on a Wald test for the equality of two dependent κ -statistics. The Wald test statistic has a non-central χ ²-distribution with non-centrality parameter that can be estimated with minimal assumptions. The method proposed is useful for agreement studies with two raters and two instruments, and is easily extendable to multiple raters and multiple instruments. Furthermore, the method proposed allows for rater bias. Power calculations for binary ratings under various scenarios are presented. Analyses of two biomedical studies are used for illustration.     

A convenient formula for sample size calculations in clinical trials with multiple co-primary continuous endpoints

The clinical efficacy of a new treatment may often be better evaluated by two or more co-primary endpoints. Recently, in pharmaceutical drug development, there has been increasing discussion regarding establishing statistically significant favorable results on more than one endpoint in comparisons between treatments, which is referred to as a problem of multiple co-primary endpoints. Several methods have been proposed for calculating the sample size required to design a trial with multiple co-primary correlated endpoints. However, because these methods require users to have considerable mathematical sophistication and knowledge of programming techniques, their application and spread may be restricted in practice. To improve the convenience of these methods, in this paper, we provide a useful formula with accompanying numerical tables for sample size calculations to design clinical trials with two treatments, where the efficacy of a new treatment is demonstrated on continuous co-primary endpoints. In addition, we provide some examples to illustrate the sample size calculations made using the formula. Using the formula and the tables, which can be read according to the patterns of correlations and effect size ratios expected in multiple co-primary endpoints, makes it convenient to evaluate the required sample size promptly.     

A Two-stage minimax procedure with screening for selecting the largest normal mean

The problem of selecting the normal population with the largest population mean when the populations have a common known variance is considered. A two-stage procedure is proposed which guarantees the same probability requirement using the indifference-zone approach as does the single-stage procedure of Bechhofer (1954). The two-stage procedure has the highly desirable property that the expected total number of observations required by the procedure is always less than the total number of observations required by the corresponding single-stage procedure, regardless of the configuration of the population means. The saving in expected total number of observations can be substantial, particularly when the configuration of the population means is favorable to the experimenter. The saving is accomplished by screening out “non-contending” populations in the first stage, and concentrating sampling only on “contending” populations in the second stage.

The two-stage procedure can be regarded as a composite one which uses a screening subset-type approach (Gupta (1956), (1965)) in the first stage, and an indifference-zone approach (Bechhofer (1954)) applied to all populations retained in the selected sub-set in the second stage. Constants to implement the procedure for various k and P? are provided, as are calculations giving the saving in expected total sample size if the two-stage procedure is used in place of the corresponding single-stage procedure.     

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.     

Multiple-arm superiority and non-inferiority designs with various endpoints

Multiple-arm dose-response superiority trials are widely studied for continuous and binary endpoints, while non-inferiority designs have been studied recently in two-arm trials. In this paper, a unified asymptotic formulation of a sample size calculation for k-arm (k>0) trials with different endpoints (continuous, binary and survival endpoints) is derived for both superiority and non-inferiority designs. The proposed method covers the sample size calculation for single-arm and k-arm (k> or =2) designs with survival endpoints, which has not been covered in the statistic literature. A simple, closed form for power and sample size calculations is derived from a contrast test. Application examples are provided. The effect of the contrasts on the power is discussed, and a SAS program for sample size calculation is provided and ready to use.     

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.     

Sample size planning for testing significance of curves

Smoothing methods for curve estimation have received considerable attention in statistics with a wide range of applications. However, to our knowledge, sample size planning for testing significance of curves has not been discussed in the literature. This paper focuses on sample size calculations for nonparametric regression and partially linear models based on local linear estimators. We describe explicit procedures for sample size calculations based on non- and semi-parametric F-tests. Data examples are provided to demonstrate the use of the procedures.     

Graphical displays for clarifying how allocation ratio affects total sample size for the two sample logrank test

For time‐to‐event data, the power of the two sample logrank test for the comparison of two treatment groups can be greatly influenced by the ratio of the number of patients in each of the treatment groups. Despite the possible loss of power, unequal allocations may be of interest due to a need to collect more data on one of the groups or to considerations related to the acceptability of the treatments to patients. Investigators pursuing such designs may be interested in the cost of the unbalanced design relative to a balanced design with respect to the total number of patients required for the study. We present graphical displays to illustrate the sample size adjustment factor, or ratio of the sample size required by an unequal allocation compared to the sample size required by a balanced allocation, for various survival rates, treatment hazards ratios, and sample size allocation ratios. These graphical displays conveniently summarize information in the literature and provide a useful tool for planning sample sizes for the two sample logrank test. Copyright © 2010 John Wiley & Sons, Ltd.     

A two- sample partially sequential probability ratio test

A two-sample partially sequential probability ratio test (PSPRT) is considered for the two-sample location problem with one sample fixed and the other sequential. Observations are assumed to come from two normal poptilatlons with equal and known variances. Asymptotically in the fixed-sample size the PSPRT is a truncated Wald one sample sequential probability test. Brownian motion approximations for boundary-crossing probabilities and expected sequential sample size are obtained. These calculations are compared to values obtained by Monte Carlo simulation.     

Blinded sample size recalculation for clinical trials with normal data and baseline adjusted analysis

Baseline adjusted analyses are commonly encountered in practice, and regulatory guidelines endorse this practice. Sample size calculations for this kind of analyses require knowledge of the magnitude of nuisance parameters that are usually not given when the results of clinical trials are reported in the literature. It is therefore quite natural to start with a preliminary calculated sample size based on the sparse information available in the planning phase and to re‐estimate the value of the nuisance parameters (and with it the sample size) when a portion of the planned number of patients have completed the study. We investigate the characteristics of this internal pilot study design when an analysis of covariance with normally distributed outcome and one random covariate is applied. For this purpose we first assess the accuracy of four approximate sample size formulae within the fixed sample size design. Then the performance of the recalculation procedure with respect to its actual Type I error rate and power characteristics is examined. The results of simulation studies show that this approach has favorable properties with respect to the Type I error rate and power. Together with its simplicity, these features should make it attractive for practical application. Copyright © 2009 John Wiley & Sons, Ltd.     

Testing bioequivalence for multiple formulations with power and sample size calculations

Bioequivalence (BE) trials play an important role in drug development for demonstrating the BE between test and reference formulations. The key statistical analysis for BE trials is the use of two one‐sided tests (TOST), which is equivalent to showing that the 90% confidence interval of the relative bioavailability is within a given range. Power and sample size calculations for the comparison between one test formulation and the reference formulation has been intensively investigated, and tables and software are available for practical use. From a statistical and logistical perspective, it might be more efficient to test more than one formulation in a single trial. However, approaches for controlling the overall type I error may be required. We propose a method called multiplicity‐adjusted TOST (MATOST) combining multiple comparison adjustment approaches, such as Hochberg's or Dunnett's method, with TOST. Because power and sample size calculations become more complex and are difficult to solve analytically, efficient simulation‐based procedures for this purpose have been developed and implemented in an R package. Some numerical results for a range of scenarios are presented in the paper. We show that given the same overall type I error and power, a BE crossover trial designed to test multiple formulations simultaneously only requires a small increase in the total sample size compared with a simple 2 × 2 crossover design evaluating only one test formulation. Hence, we conclude that testing multiple formulations in a single study is generally an efficient approach. The R package MATOST is available at https://sites.google.com/site/matostbe/ . Copyright © 2012 John Wiley & Sons, Ltd.     

A modified least-failures sampling procedure for bernoulli subset selection

We restrict attention to a class of Bernoulli subset selection procedures which take observations one-at-a-time and can be compared directly to the Gupta-Sobel single-stage procedure. For the criterion of minimizing the expected total number of observations required to terminate experimentation, we show that optimal sampling rules within this class are not of practical interest. We thus turn to procedures which, although not optimal, exhibit desirable behavior with regard to this criterion. A procedure which employs a modification of the so-called least-failures sampling rule is proposed, and is shown to possess many desirable properties among a restricted class of Bernoulli subset selection procedures. Within this class, it is optimal for minimizing the number of observations taken from populations excluded from consideration following a subset selection experiment, and asymptotically optimal for minimizing the expected total number of observations required. In addition, it can result in substantial savings in the expected total num¬ber of observations required as compared to a single-stage procedure, thus it may be de¬sirable to a practitioner if sampling is costly or the sample size is limited.     

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.