Sample size determination for p-charts

Two types of decision errors can be made when using a quality control chart for non-conforming units (p-chart). A Type I error occurs when the process is not out of control but a search for an assignable cause is performed unnecessarily. A Type II error occurs when the process is out of control but a search for an assignable cause is not performed. The probability of a Type I error is under direct control of the decision-maker while the probability of a Type II error depends, in part, on the sample size. A simple sample size formula is presented for determining the required sample size for a p-chart with specified probabilities of Type I and Type II errors.     

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.     

Likelihood-based confidence limits for the common odds ratio

In this paper we derive two likelihood-based procedures for the construction of confidence limits for the common odds ratio in K 2 × 2 contingency tables. We then conduct a simulation study to compare these procedures with a recently proposed procedure by Sato (Biometrics 46 (1990) 71–79), based on the asymptotic distribution of the Mantel-Haenszel estimate of the common odds ratio. We consider the situation in which the number of strata remains fixed (finite), but the sample sizes within each stratum are large. Bartlett's score procedure based on the conditional likelihood is found to be almost identical, in terms of coverage probabilities and average coverage lengths, to the procedure recommended by Sato, although the score procedure has some edge, in some instances, in terms of average coverage lengths. So, for ‘fixed strata and large sample’ situation Bartlett's score procedure can be considered as an alternative to the procedure proposed by Sato, based on the asymptotic distribution of the Mantel-Haenszel estimator of the common odds ratio.     

Sample size determination for clustered right-censored data using copula models

Determination of an adequate sample size is critical to the design of research ventures. For clustered right-censored data, Manatunga and Chen [Sample size estimation for survival outcomes in cluster-randomized studies with small cluster sizes. Biometrics. 2000;56(2):616–621] proposed a sample size calculation based on considering the bivariate marginal distribution as Clayton copula model. In addition to the Clayton copula, other important family of copulas, such as Gumbel and Frank copulas are also well established in multivariate survival analysis. However, sample size calculation based on these assumptions has not been fully investigated yet. To broaden the scope of Manatunga and Chen [Sample size estimation for survival outcomes in cluster-randomized studies with small cluster sizes. Biometrics. 2000;56(2):616–621]'s research and achieve a more flexible sample size calculation for clustered right-censored data, we extended the work by assuming the marginal distribution as bivariate Gumbel and Frank copulas. We evaluate the performance of the proposed method and investigate the impacts of the accrual times, follow-up times and the within-clustered correlation effect of the study. The proposed method is applied to two real-world studies, and the R code is made available to users.     

Sample size determination of steel's nonparametric many-one test

We derive sample size formulas for the many-one test of Steel (1959) when the all-pairs power is preassigned. In this large sample approach we replace, similar to Noether (1987), the unknown variances and also the unknown correlation coefficients in the power expressions by their known values under the null hypotheses. We then obtain least favorable configurations for one-and two-sided comparisons. The reliability of our formulas is examined in computer simulations for different alternatives with various distributions.     

Sample size determination for 2k-r experiments with a binomial response

This paper provides closed form expressions for the sample size for two-level factorial experiments when the response is the number of defectives. The sample sizes are obtained by approximating the two-sided test for no effect through tests for the mean of a normal distribution, and borrowing the classical sample size solution for that problem. The proposals are appraised relative to the exact sample sizes computed numerically, without appealing to any approximation to the binomial distribution, and the use of the sample size tables provided is illustrated through an example.     

Sample size determination for the parallel model in a survey with sensitive questions

Recently, a new non-randomized parallel design is proposed by Tian (2013) for surveys with sensitive topics. However, the sample size formulae associated with testing hypotheses for the parallel model are not yet available. As a crucial component in surveys, the sample size formulae with the parallel design are developed in this paper by using the power analysis method for both the one- and two-sample problems. We consider both the one- and two-sample problems. The asymptotic power functions and the corresponding sample size formulae for both the one- and two-sided tests based on the large-sample normal approximation are derived. The performance is assessed through comparing the asymptotic power with the exact power and reporting the ratio of the sample sizes with the parallel model and the design of direct questioning. We numerically compare the sample sizes needed for the parallel design with those required for the crosswise and triangular models. Two theoretical justifications are also provided. An example from a survey on ‘sexual practices’ in San Francisco, Las Vegas and Portland is used to illustrate the proposed methods.     

Sample size determination for estimating multivariate process capability indices based on lower confidence limits

With the advent of modern technology, manufacturing processes have become very sophisticated; a single quality characteristic can no longer reflect a product's quality. In order to establish performance measures for evaluating the capability of a multivariate manufacturing process, several new multivariate capability (NMC) indices, such as NMC _p and NMC _pm , have been developed over the past few years. However, the sample size determination for multivariate process capability indices has not been thoroughly considered in previous studies. Generally, the larger the sample size, the more accurate an estimation will be. However, too large a sample size may result in excessive costs. Hence, the trade-off between sample size and precision in estimation is a critical issue. In this paper, the lower confidence limits of NMC _p and NMC _pm indices are used to determine the appropriate sample size. Moreover, a procedure for conducting the multivariate process capability study is provided. Finally, two numerical examples are given to demonstrate that the proper determination of sample size for multivariate process indices can achieve a good balance between sampling costs and estimation precision.     

Bivariate odds ratio and association measures

In this paper, we consider a class of bivariate distributions by forming the odds of failure of a two component system. The properties of this odds function and the association between the two variables are investigated by studying the local dependence function and the association measure defined by Clayton (Biometrika 65:141–151, 1978) and Oakes (J Am Stat Assoc 84:487–493, 1989). We also study the effect of the association parameter on the failure rate of a series system and the regression mean residual life function of a parallel system. Some stochastic comparisons with respect to the association parameter are also studied.     

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.     

Sample size determination for the confidence interval of mean comparison adjusted by multiple covariates

The current method of determining sample size for confidence intervals does not accommodate multiple covariate adjustment. Under the normality assumption, the effect of multiple covariate adjustment on the standard error of the mean comparison is related to a Hotelling T ² statistic. Sample size can be calculated to obtain a desired probability of achieving a predetermined width in the confidence interval of the mean comparison with multiple covariate adjustment, given that the confidence interval includes the population parameter.     

A pattern-mixture odds ratio model for incomplete categorical data

Most models for incomplete data are formulated within the selection model framework. Pattern-mixture models are increasingly seen as a viable alternative, both from an interpretational as well as from a computational point of view (Little 1993, Hogan and Laird 1997, Ekholm and Skinner 1998). Whereas most applications are either for continuous normally distributed data or for simplified categorical settings such as contingency tables, we show how a multivariate odds ratio model (Molenberghs and Lesaffre 1994, 1998) can be used to fit pattern-mixture models to repeated binary outcomes with continuous covariates. Apart from point estimation, useful methods for interval estimation are presented and data from a clinical study are analyzed to illustrate the methods.     

Methods improving the estimate of diagnostic odds ratio

Diagnostic odds ratio is defined as the ratio of the odds of the positivity of a diagnostic test results in the diseased population relative to that in the non-diseased population. It is a function of sensitivity and specificity, which can be seen as an indicator of the diagnostic accuracy for the evaluation of a biomarker/test. The naïve estimator of diagnostic odds ratio fails when either sensitivity or specificity is close to one, which leads the denominator of diagnostic odds ratio equal to zero. We propose several methods to adjust for such situation. Agresti and Coull’s adjustment is a common and straightforward way for extreme binomial proportions. Alternatively, estimation methods based on a more advanced sampling design can be applied, which systematically selects samples from underlying population based on judgment ranks. Under such design, the odds can be estimated by the sum of indicator functions and thus avoid the situation of dividing by zero and provide a valid estimation. The asymptotic mean and variance of the proposed estimators are derived. All methods are readily applied for the confidence interval estimation and hypothesis testing for diagnostic odds ratio. A simulation study is conducted to compare the efficiency of the proposed methods. Finally, the proposed methods are illustrated using a real dataset.     

Sample size determination for quality control process using a multiple components tolerance interval

ABSTRACT

In the past, a tolerance interval was used for the statistical quality control process on raw materials and/or the final product. In the traditional concept of the tolerance interval, the variance from the measurements is a single component. However, we can find examples about several components that could vary in their measurements, so an approximate method must be found to modify the traditional tolerance interval. Now we employ a tolerance interval considering multiple components in the variance from the measurements to deal with quality control process. In our paper, the proposed method is used to solve the sample size determination for a two-sided tolerance interval approach considering multiple components on the variance of measurements.     

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.     

Test for conditional odds ratio in matching pairs inverse sampling design

Conditional odds ratio is often used in matched pair design to assess the equivalence of two treatments for both prospective and retrospective clinical trials. In this context some test procedures are available under binomial sampling scheme. The present study provides an alternative approach to the same problem under an inverse binomial sampling scheme. Various performance characteristics of the proposed procedure are examined through exact and simulation studies.     

Sample size determination for testing equality in frequency data under an incomplete block crossover design

When there are more than two treatments under comparison, we may consider the use of the incomplete block crossover design (IBCD) to save the number of patients needed for a parallel groups design and reduce the duration of a crossover trial. We develop an asymptotic procedure for simultaneously testing equality of two treatments versus a control treatment (or placebo) in frequency data under the IBCD with two periods. We derive a sample size calculation procedure for the desired power of detecting the given treatment effects at a nominal-level and suggest a simple ad hoc adjustment procedure to improve the accuracy of the sample size determination when the resulting minimum required number of patients is not large. We employ Monte Carlo simulation to evaluate the finite-sample performance of the proposed test, the accuracy of the sample size calculation procedure, and that with the simple ad hoc adjustment suggested here. We use the data taken as a part of a crossover trial comparing the number of exacerbations between using salbutamol or salmeterol and a placebo in asthma patients to illustrate the sample size calculation procedure.     

Sample size determination in step-down and step-up multiple tests for comparing treatments with a control

We address the problem of sample size determination in multiple comparisons of k treatments with a control for step-down and step-up testing, assuming normal data and homogeneous variances. We define power as the probability of correctly rejecting all hypotheses for which the treatment vs. control difference exceeds a specified value. Our paper supplements papers by Hayter and Tamhane (J. Statist. Plann. Inference 27 (1991) 271–290) who solved the problem for one-sided comparisons using the step-down procedure and by Liu (J. Statist. Plann. Inference 62 (1997b) 255–261) who considered the two-sided case using the single-step method. We provide expressions that allow computer evaluation of the power and necessary sample sizes for one- and two-sided tests using either step-down or step-up procedures. Tables are given from which sample sizes to guarantee a specified power can be determined.