On the Use of the Upper Confidence Limit for the Variance from a Pilot Sample for Sample Size Determination

In a recent paper, Browne (1995) investigated the use of a pilot sample for sample size calculation. Monte Carlo simulations indicated that using a 100 · (1 — γ) per cent upper one-sided confidence limit on the population varíance σ² leads to a sample size that guarantees the planned power with a probability of at least 1 ? γ. The purpose of this paper is to get further insight into the results of Browne by analytical considerations. Furthermore, the expected power is investigated when applying the strategy and recommendations for the choice of the pilot sample size are given.     

Sample Size Calculation for Clinical Studies on the Efficacy of a New Contraceptive Method

The current guideline of the European Agency for the Evaluation of Medicinal Products (EMEA) on the clinical investigation of steroid contraceptives in women, which was prepared by the Committee for Proprietary Medicinal Products (CPMP), calls for the calculation of a confidence interval for the Pearl Index, a widely used measure to describe the reliability of a contraceptive method. The key studies should be large enough to give a Pearl Index with a 95% confidence interval such that the difference between the upper limit of the confidence interval and the corresponding point estimate does not exceed a given margin. As a consequence of this guidance, the success probability of a Pearl Index study is given by its capability, i.e. probability to fulfil this criterion. The resulting power function based on the necessary exposure time does not increase strictly. The minimum exposure time T_min should be calculated such that this function exceeds a given probability for all T ≥ T_min. In this paper, the underlying model is discussed, and definitions and formulae are given for the assumption of a Poisson model. The necessary total exposure time is calculated as a function of a given true pregnancy rate. In addition, some simulations were conducted for the model where a drop‐out mechanism is incorporated into the process. Assuming an exponential distribution for the time to drop‐out the phenomenon that the power does not increase strictly with exposure time is less pronounced. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Sample Size Estimation in Cluster Randomized Studies with Varying Cluster Size

Cluster randomized studies are common in community trials. The standard method for estimating sample size for cluster randomized studies assumes a common cluster size. However often in cluster randomized studies, size of the clusters vary. In this paper, we derive sample size estimation for continuous outcomes for cluster randomized studies while accounting for the variability due to cluster size. It is shown that the proposed formula for estimating total cluster size can be obtained by adding a correction term to the traditional formula which uses the average cluster size. Application of these results to the design of a health promotion educational intervention study is discussed.     

Evaluation of Noether's Method of Sample Size Determination for the Wilcoxon-Mann-Whitney Test

NOETHER (1987) proposed a method of sample size determination for the Wilcoxon-Mann-Whitney test. To obtain a sample size formula, he restricted himself to alternatives that differ only slightly from the null hypothesis, so that the unknown variance o² of the Mann-Whitney statistic can be approximated by the known variance under the null hypothesis which depends only on n. This fact is frequently forgotten in statistical practice. In this paper, we compare Noether's large sample solution against an alternative approach based on upper bounds of σ² which is valid for any alternatives. This comparison shows that Noether's approximation is sufficiently reliable with small and large deviations from the null hypothesis.     

Sample Size Calculation for Intraclass Correlation in the Presence of Random Loss of Sampled Patients

When a trial involves an invasive laboratory test procedure or requires patients to make a commitment to follow a restrictive test schedule, we can often lose a great proportion of our sampled patients due to refusal of participation into our study. Therefore, incorporating the possible loss of patients into sample size calculation is certainly important in the planning stage of a study. In this paper, we have generalized the sample size calculation procedure for intraclass correlation by accounting for the random loss of patients in the beginning of a trial. We have demonstrated that the simple ad hoc procedure, that raises the estimated sample size in the absence of loss of patients by the factor 1/p_o, where p_o is the retention probability for a randomly selected patient, is adequate when p_o is large (=0.80). When p_o is small (i.e., a high refusal rate), however, use of this simple ad hoc procedure tends to underestimate the required sample size. Furthermore, we have found that if the individual retention probability varied substantially among patients, then the magnitude of the above underestimation could even be critical and therefore, the application of the simple direct adjustment procedure in this situation should be avoided.     

Sample Size Requirements in Studies of the Etiologic Fraction

This paper outlines methods of determining sample size for epidemiologic research in studies of the etiologic fraction. The basic model with a dichotomous disease and a single dichotomous exposure factor is considered. To determine sample size, the researcher must specify: the magnitude of the etiologic fraction ε to be detected as statistically significant, the level of significance α, the power 1 - β of the test, p the proportion of the population exposed to the risk factor and R the proportion of the population with the disease. Sample size formulas and tables are presented for the case-control, cohort and cross-sectional designs. Optimal allocation considerations are examined to minimize cost for a specified power. Extensive use is made of Walter's results concerning the asymptotic variance of the maximum likelihood estimator of the etiologic fraction for the three epidemiologic study designs.     

Estimating Population Size for Continuous-Time Capture-Recapture Models Via Sample Coverage

An estimation procedure using the idea of sample coverage is proposed to estimate population size for capture-recapture experiments in continuous time. The capture rates (intensity) are allowed to vary by time and individuals (heterogeneity). Only capture frequency history are sufficient for estimating population size while capture times and sequential orders of animals caught are irrelevant for the analysis. An example is given for illustration. The performance of the proposed estimation procedure is also investigated by simulation.     

Sample size re‐estimation in clinical trials

Adaptive clinical trials are becoming very popular because of their flexibility in allowing mid‐stream changes of sample size, endpoints, populations, etc. At the same time, they have been regarded with mistrust because they can produce bizarre results in very extreme settings. Understanding the advantages and disadvantages of these rapidly developing methods is a must. This paper reviews flexible methods for sample size re‐estimation when the outcome is continuous.     

Sample Size Reassessment in Adaptive Clinical Trials Using a Bias Corrected Estimate

Point estimation in group sequential and adaptive trials is an important issue in analysing a clinical trial. Most literature in this area is only concerned with estimation after completion of a trial. Since adaptive designs allow reassessment of sample size during the trial, reliable point estimation of the true effect when continuing the trial is additionally needed. We present a bias adjusted estimator which allows a more exact sample size determination based on the conditional power principle than the naive sample mean does.     

Minimum Sample Size Ensuring Validity of Classical Confidence Intervals for Means of Skewed and Platykurtic Distributions

COCHRAN (1953) and BARTCH (1957) gave formulae for the magnitude of the sample size (n) ensuring the validity of the limiting normal distribution of the sample mean x(ⁿ) obtained from a non-normal distribution with marked asymmetry and kurtosis. These formulae have been checked empirically in this paper using (a) simulated data with given asymmetry and kurtosis and (b) real data gathered from a coronary heart disease study. We find that our results are in general agreement with Bartch's formula. However, in a number of cases, the asymptotic normal distribution is attained for smaller sample size than that required by Bartch's formula.     

Efficient Bayesian Sample Size Calculation for Designing a Clinical Trial with Multi‐Cluster Outcome Data

Health care utilization and outcome studies call for hierarchical approaches. The objectives were to predict major complications following percutaneous coronary interventions by health providers, and to compare Bayesian and non‐Bayesian sample size calculation methods. The hierarchical data structure consisted of: (1) Strata: PGY4, PGY7, and physician assistant as providers with varied experiences; (2) Clusters: k_s providers per stratum; (3) Individuals: n_s patients reviewed by each provider. The main outcome event illustrated was mortality modeled by a Bayesian beta‐binomial model. Pilot information and assumptions were utilized to elicit beta prior distributions. Sample size calculations were based on the approximated average length, fixed at 1%, of 95% posterior intervals of the mean event rate parameter. Necessary sample sizes by both non‐Bayesian and Bayesian methods were compared. We demonstrated that the developed Bayesian methods can be efficient and may require fewer subjects to satisfy the same length criterion.     

Sample Size Determination for Grouped Exponential Observations: A Cost Function Approach

Calculating the required sample size for a desired power at a given type I error level, we often assume that we know the exact time of all subject responses whenever they occur during our study period. It is very common, however, in practice that we only monitor subjects periodically and, therefore, we know only whether responses occur or not during an interval. This paper includes a quantitative discussion of the effect resulting from data grouping or interval censoring on the required sample size when we have two treatment groups. Furthermore, with the goal of exploring the optimum in the number of subjects, the number of examinations per subject for test responses, and the total length of a study time period, this paper also provides a general guideline about how to determine these to minimize the total cost of a study for a desired power at a given α-level. A specified linear cost function that incorporates the costs of obtaining subjects, periodic examinations for test responses of subjects, and the total length of a study period, is assumed, primarily for illustrative purpose.     

Sample Size Determination for Designing a Strata-Matched Case-Control Study to Detect Multiple Risk Factors

Investigations of sample size for planning case-control studies have usually been limited to detecting a single factor. In this paper, we investigate sample size for multiple risk factors in strata-matched case-control studies. We construct an omnibus statistic for testing M different risk factors based on the jointly sufficient statistics of parameters associated with the risk factors. The statistic is non-iterative, and it reduces to the Cochran statistic when M = 1. The asymptotic power function of the test is a non-central chi-square with M degrees of freedom and the sample size required for a specific power can be obtained by the inverse relationship. We find that the equal sample allocation is optimum. A Monte Carlo experiment demonstrates that an approximate formula for calculating sample size is satisfactory in typical epidemiologic studies. An approximate sample size obtained using Bonferroni's method for multiple comparisons is much larger than that obtained using the omnibus test. Approximate sample size formulas investigated in this paper using the omnibus test, as well as the individual tests, can be useful in designing case-control studies for detecting multiple risk factors.     

Sample Size Considerations for GEE Analyses of Three‐Level Cluster Randomized Trials

Summary Cluster randomized trials in health care may involve three instead of two levels, for instance, in trials where different interventions to improve quality of care are compared. In such trials, the intervention is implemented in health care units (“clusters”) and aims at changing the behavior of health care professionals working in this unit (“subjects”), while the effects are measured at the patient level (“evaluations”). Within the generalized estimating equations approach, we derive a sample size formula that accounts for two levels of clustering: that of subjects within clusters and that of evaluations within subjects. The formula reveals that sample size is inflated, relative to a design with completely independent evaluations, by a multiplicative term that can be expressed as a product of two variance inflation factors, one that quantifies the impact of within‐subject correlation of evaluations on the variance of subject‐level means and the other that quantifies the impact of the correlation between subject‐level means on the variance of the cluster means. Power levels as predicted by the sample size formula agreed well with the simulated power for more than 10 clusters in total, when data were analyzed using bias‐corrected estimating equations for the correlation parameters in combination with the model‐based covariance estimator or the sandwich estimator with a finite sample correction.     

Testing the Intraclass Version of Kappa Coeffcient of Agreement with Binary Scale and Sample Size Determination

The intraclass version of kappa coefficient has been commonly applied as a measure of agreement for two ratings per subject with binary outcome in reliability studies. We present an efficient statistic for testing the strength of kappa agreement using likelihood scores, and derive asymptotic power and sample size formula. Exact evaluation shows that the score test is generally conservative and more powerful than a method based on a chi‐square goodness‐of‐fit statistic (Donner and Eliasziw , 1992, Statistics in Medicine 11 , 1511–1519). In particular, when the research question is one directional, the one‐sided score test is substantially more powerful and the reduction in sample size is appreciable.     

On Sample Size of the Kruskal–Wallis Test with Application to a Mouse Peritoneal Cavity Study

Summary As the nonparametric generalization of the one‐way analysis of variance model, the Kruskal–Wallis test applies when the goal is to test the difference between multiple samples and the underlying population distributions are nonnormal or unknown. Although the Kruskal–Wallis test has been widely used for data analysis, power and sample size methods for this test have been investigated to a much lesser extent. This article proposes new power and sample size calculation methods for the Kruskal–Wallis test based on the pilot study in either a completely nonparametric model or a semiparametric location model. No assumption is made on the shape of the underlying population distributions. Simulation results show that, in terms of sample size calculation for the Kruskal–Wallis test, the proposed methods are more reliable and preferable to some more traditional methods. A mouse peritoneal cavity study is used to demonstrate the application of the methods.     

Asymmetric regression models with limited responses with an application to antibody response to vaccine

We develop regression models for limited and censored data based on the mixture between the log‐power‐normal and Bernoulli‐type distributions. A likelihood‐based approach is implemented for parameter estimation and a small‐scale simulation study is conducted to evaluate parameter recovery, with emphasis on bias estimation. The main conclusion is that the approach is very much satisfactory for moderate and large sample sizes. A real data example, the safety and immunogenecity study of measles vaccine in Haiti, is presented to illustrate how different models can be used to fit this type of data. As shown, the asymmetric models considered seem to present the best fit for the data set under study, revealing significance of the explanatory variable sex, which is not found significant with the log‐normal model.     

Statistical Inference for a Two‐Stage Outcome‐Dependent Sampling Design with a Continuous Outcome

Summary The two‐stage case–control design has been widely used in epidemiology studies for its cost‐effectiveness and improvement of the study efficiency ( White, 1982 , American Journal of Epidemiology 115, 119–128; Breslow and Cain, 1988 , Biometrika 75, 11–20). The evolution of modern biomedical studies has called for cost‐effective designs with a continuous outcome and exposure variables. In this article, we propose a new two‐stage outcome‐dependent sampling (ODS) scheme with a continuous outcome variable, where both the first‐stage data and the second‐stage data are from ODS schemes. We develop a semiparametric empirical likelihood estimation for inference about the regression parameters in the proposed design. Simulation studies were conducted to investigate the small‐sample behavior of the proposed estimator. We demonstrate that, for a given statistical power, the proposed design will require a substantially smaller sample size than the alternative designs. The proposed method is illustrated with an environmental health study conducted at National Institutes of Health.