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.     

Sample size determination in fixed‐dose combination drug studies

In this paper, the application of the intersection–union test method in fixed‐dose combination drug studies is discussed. An approximate sample size formula for the problem of testing the efficacy of a combination drug using intersection–union tests is proposed. The sample sizes obtained from the formula are found to be reasonably accurate in terms of attaining the target power 1?β for a specified β. Copyright © 2003 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.     

Power and Sample Size for Fixed-Effects Inference in Reversible Linear Mixed Models

ABSTRACT

Despite the popularity of the general linear mixed model for data analysis, power and sample size methods and software are not generally available for commonly used test statistics and reference distributions. Statisticians resort to simulations with homegrown and uncertified programs or rough approximations which are misaligned with the data analysis. For a wide range of designs with longitudinal and clustering features, we provide accurate power and sample size approximations for inference about fixed effects in the linear models we call reversible. We show that under widely applicable conditions, the general linear mixed-model Wald test has noncentral distributions equivalent to well-studied multivariate tests. In turn, exact and approximate power and sample size results for the multivariate Hotelling–Lawley test provide exact and approximate power and sample size results for the mixed-model Wald test. The calculations are easily computed with a free, open-source product that requires only a web browser to use. Commercial software can be used for a smaller range of reversible models. Simple approximations allow accounting for modest amounts of missing data. A real-world example illustrates the methods. Sample size results are presented for a multicenter study on pregnancy. The proposed study, an extension of a funded project, has clustering within clinic. Exchangeability among the participants allows averaging across them to remove the clustering structure. The resulting simplified design is a single-level longitudinal study. Multivariate methods for power provide an approximate sample size. All proofs and inputs for the example are in the supplementary materials (available online).     

Cancer immunotherapy trial design with delayed treatment effect

A challenge arising in cancer immunotherapy trial design is the presence of a delayed treatment effect wherein the proportional hazard assumption no longer holds true. As a result, a traditional survival trial design based on the standard log‐rank test, which ignores the delayed treatment effect, will lead to substantial loss of statistical power. Recently, a piecewise weighted log‐rank test is proposed to incorporate the delayed treatment effect into consideration of the trial design. However, because the sample size formula was derived under a sequence of local alternative hypotheses, it results in an underestimated sample size when the hazard ratio is relatively small for a balanced trial design and an inaccurate sample size estimation for an unbalanced design. In this article, we derived a new sample size formula under a fixed alternative hypothesis for the delayed treatment effect model. Simulation results show that the new formula provides accurate sample size estimation for both balanced and unbalanced designs.     

Approximating sample size requirements for s2 charts

Sample size determination is one of the most important considerations in the design of a control chart. The optimal sample size will provide control over both type I and type II errors. The optimal sample size for an S2 chart can be determined exactly using an iterative procedure. Duncan presented a procedure to approximate the required sample size. The accuracy of Duncan's approximation is examined and an improved approximation is proposed.     

Approximate Inference for the Factor Loading of a Simple Factor Analysis Model

We study a factor analysis model with two normally distributed observations and one factor. Two approximate conditional inference procedures for the factor loading are developed. The first proposal is a very simple procedure but it is not very accurate. The second proposal gives extremely accurate results even for very small sample size. Moreover, the calculations require only the signed log-likelihood ratio statistic and a measure of the standardized maximum likelihood departure. Simulations are used to study the accuracy of the proposed procedures.     

A note on effective sample size for constructing confidence intervals for the difference of two proportions

Proportion differences are often used to estimate and test treatment effects in clinical trials with binary outcomes. In order to adjust for other covariates or intra-subject correlation among repeated measures, logistic regression or longitudinal data analysis models such as generalized estimating equation or generalized linear mixed models may be used for the analyses. However, these analysis models are often based on the logit link which results in parameter estimates and comparisons in the log-odds ratio scale rather than in the proportion difference scale. A two-step method is proposed in the literature to approximate the calculation of confidence intervals for the proportion difference using a concept of effective sample sizes. However, the performance of this two-step method has not been investigated in their paper. On this note, we examine the properties of the two-step method and propose an adjustment to the effective sample size formula based on Bayesian information theory. Simulations are conducted to evaluate the performance and to show that the modified effective sample size improves the coverage property of the confidence intervals.     

Sample size determinations when two binomial proportions are very small

Sample size determination for testing the hypothesis of equality of proportions with a specified type I and type I1 error probabilitiesis of ten based on normal approximation to the binomial distribution. When the proportionsinvolved are very small, the exact distribution of the test statistic may not follow the assumed distribution. Consequently, the sample size determined by the test statistic may not result in the sespecifiederror probabilities. In this paper the author proposes a square root formula and compares it with several existing sample size approximation methods. It is found that with small proportion (p≦.01) the squar eroot formula provides the closest approximation to the exact sample sizes which attain a specified type I and type II error probabilities. Thes quare root formula is simple inform and has the advantage that equal differencesare equally detectable.     

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.     

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.     

Optimal sample size planning for the Wilcoxon–Mann–Whitney and van Elteren tests under cost constraints

Sampling cost is a crucial factor in sample size planning, particularly when the treatment group is more expensive than the control group. To either minimize the total cost or maximize the statistical power of the test, we used the distribution-free Wilcoxon–Mann–Whitney test for two independent samples and the van Elteren test for randomized block design, respectively. We then developed approximate sample size formulas when the distribution of data is abnormal and/or unknown. This study derived the optimal sample size allocation ratio for a given statistical power by considering the cost constraints, so that the resulting sample sizes could minimize either the total cost or the total sample size. Moreover, for a given total cost, the optimal sample size allocation is recommended to maximize the statistical power of the test. The proposed formula is not only innovative, but also quick and easy. We also applied real data from a clinical trial to illustrate how to choose the sample size for a randomized two-block design. For nonparametric methods, no existing commercial software for sample size planning has considered the cost factor, and therefore the proposed methods can provide important insights related to the impact of cost constraints.     

Homogeneity test, sample size determination and interval construction of difference of two proportions in stratified bilateral-sample designs

In stratified otolaryngologic (or ophthalmologic) studies, the misleading results may be obtained when ignoring the confounding effect and the correlation between responses from two ears. Score statistic and Wald-type statistic are presented to test equality in a stratified bilateral-sample design, and their corresponding sample size formulae are given. Score statistic for testing homogeneity of difference between two proportions and score confidence interval of the common difference of two proportions in a stratified bilateral-sample design are derived. Empirical results show that (1) score statistic and Wald-type statistic based on dependence model assumption outperform other statistics in terms of the type I error rates; (2) score confidence interval demonstrates reasonably good coverage property; (3) sample size formula via Wald-type statistic under dependence model assumption is rather accurate. A real example is used to illustrate the proposed methodologies.     

Small sample asymptotic inference for the coefficient of variation: normal and nonnormal models

In applied statistics, the coefficient of variation is widely calculated and interpreted even when the sample size of the data set is very small. However, confidence intervals for the coefficient of variation are rarely reported. One of the reasons is the exact confidence interval for the coefficient of variation, which is given in Lehmann (Testing Statistical Hypotheses, 2nd Edition, Wiley, New York, 1996), is very difficult to calculate. Various asymptotic methods have been proposed in literature. These methods, in general, require the sample size to be large. In this article, we will apply a recently developed small sample asymptotic method to obtain approximate confidence intervals for the coefficient of variation for both normal and nonnormal models. These small sample asymptotic methods are very accurate even for very small sample size. Numerical examples are given to illustrate the accuracy of the proposed method.     

多阶段抽样比率估计

目前 ,单阶段、两阶段的抽样理论较丰富 (见文献 [1])。三阶段以上的抽样理论不完善。在这篇文章中 ,我们讨论多阶段抽样比率估计 ,给出了样本估计抽样均方误差及其无偏估计的公式。首先将总体Ｍ个单位分为Ｒ个一级组 ,第ｉ1(1≤ｉ1≤Ｒ)个一级组包含Ｒｉ1个二级组 ,第ｉ1一级组中第ｉ2 (1≤ｉ2 ≤Ｒｉ1)个二级组包含Ｒｉ1,ｉ2 个三级组 ,… ,直到前ｎ - 2级组分别是第ｉ1,ｉ2 ,… ,ｉｎ -2 个的ｎ- 2级组包含Ｒｉ1,… ,ｉｎ-2 个ｎ - 1级组。第ｉｎ -1个 (1≤ｉｎ -1≤Ｒｉ1,… ,ｉｎ -2 )ｎ - 1级组包含Ｒｉ1,… ,ｉｎ -1个总体单位。则Ｍ…     

Comparison of disease prevalence in two populations under double-sampling scheme with two fallible classifiers

A disease prevalence can be estimated by classifying subjects according to whether they have the disease. When gold-standard tests are too expensive to be applied to all subjects, partially validated data can be obtained by double-sampling in which all individuals are classified by a fallible classifier, and some of individuals are validated by the gold-standard classifier. However, it could happen in practice that such infallible classifier does not available. In this article, we consider two models in which both classifiers are fallible and propose four asymptotic test procedures for comparing disease prevalence in two groups. Corresponding sample size formulae and validated ratio given the total sample sizes are also derived and evaluated. Simulation results show that (i) Score test performs well and the corresponding sample size formula is also accurate in terms of the empirical power and size in two models; (ii) the Wald test based on the variance estimator with parameters estimated under the null hypothesis outperforms the others even under small sample sizes in Model II, and the sample size estimated by this test is also accurate; (iii) the estimated validated ratios based on all tests are accurate. The malarial data are used to illustrate the proposed methodologies.     

A computationally fast alternative to cross-validation in penalized Gaussian graphical models

We study the problem of selecting a regularization parameter in penalized Gaussian graphical models. When the goal is to obtain a model with good predictive power, cross-validation is the gold standard. We present a new estimator of Kullback–Leibler loss in Gaussian Graphical models which provides a computationally fast alternative to cross-validation. The estimator is obtained by approximating leave-one-out-cross-validation. Our approach is demonstrated on simulated data sets for various types of graphs. The proposed formula exhibits superior performance, especially in the typical small sample size scenario, compared to other available alternatives to cross-validation, such as Akaike's information criterion and Generalized approximate cross-validation. We also show that the estimator can be used to improve the performance of the Bayesian information criterion when the sample size is small.     

Some results on bootstrap prediction intervals

We investigate the construction of a BC_a-type bootstrap procedure for setting approximate prediction intervals for an efficient estimator θ_m of a scalar parameter θ, based on a future sample of size m. The results are also extended to nonparametric situations, which can be used to form bootstrap prediction intervals for a large class of statistics. These intervals are transformation-respecting and range-preserving. The asymptotic performance of our procedure is assessed by allowing both the past and future sample sizes to tend to infinity. The resulting intervals are then shown to be second-order correct and second-order accurate. These second-order properties are established in terms of min(m, n), and not the past sample size n alone.     

Index method of determining the sample size

To quantify how worthy and reliable an interval estimate of a parameter, an index is defined. This index is derived for estimating the mean μ of a normal population. Using this index, a method is devised for determining the sample size formula. Advantages of this new sample size formula are pointed out.     

A note on the determination of sample sizes for hypergeometric distributions

Sample size determination for testing the hypothesis of equality of two proportions against an alternative with specified type I and type II error probabilities is considered for two finite populations. When two finite populations involved are quite different in sizes, the equal size assumption may not be appropriate. In this paper, we impose a balanced sampling condition to determine the necessary samples taken without replacement from the finite populations. It is found that our solution requires smaller samples as compared to those using binomial distributions. Furthermore, our solution is consistent with the sampling with replacement or when population size is large. Finally, three examples are given to show the application of the derived sample size formula.