Exact tests in binary data under an incomplete block crossover design.

To improve the power of a parallel groups design and reduce the time length of a crossover trial, we may consider an incomplete block crossover design. Under a distribution-free random effects logistic regression model, we derive an exact test and a Mantel-Haenszel Type of summary test procedure for testing non-equality in binary data when comparing three treatments. We employ Monte Carlo simulation to evaluate the performance of these test procedures. We find that both test procedures developed here can perform well in a variety of situations. We use the data taken as a part of the crossover trial comparing the low and high doses of an analgesic with a placebo for the relief of pain in primary dysmenorrhea to illustrate the use of the proposed test procedures.

Testing equality and interval estimation of the generalized odds ratio in ordinal data under a three-period crossover design.

The crossover design can be of use to save the number of patients or improve power of a parallel groups design in studying treatments to noncurable chronic diseases. We propose using the generalized odds ratio for paired sample data to measure the relative effects in ordinal data between treatments and between periods. We show that one can apply the commonly used asymptotic and exact test procedures for stratified analysis in epidemiology to test non-equality of treatments in ordinal data, as well as obtain asymptotic and exact interval estimators for the generalized odds ratio under a three-period crossover design. We further show that one can apply procedures for testing the homogeneity of the odds ratio under stratified sampling to examine whether there are treatment-by-period interactions. We use the data taken from a three-period crossover trial studying the effects of low and high doses of an analgesic versus a placebo for the relief of pain in primary dysmenorrhea to illustrate the use of these test procedures and estimators proposed here.

Test equality in binary data for a 4 × 4 crossover trial under a Latin-square design.

When there are four or more treatments under comparison, the use of a crossover design with a complete set of treatment-receipt sequences in binary data is of limited use because of too many treatment-receipt sequences. Thus, we may consider use of a 4 × 4 Latin square to reduce the number of treatment-receipt sequences when comparing three experimental treatments with a control treatment. Under a distribution-free random effects logistic regression model, we develop simple procedures for testing non-equality between any of the three experimental treatments and the control treatment in a crossover trial with dichotomous responses. We further derive interval estimators in closed forms for the relative effect between treatments. To evaluate the performance of these test procedures and interval estimators, we employ Monte Carlo simulation. We use the data taken from a crossover trial using a 4 × 4 Latin-square design for studying four-treatments to illustrate the use of test procedures and interval estimators developed here. Copyright © 2016 John Wiley & Sons, Ltd.

Notes on testing equality and interval estimation in Poisson frequency data under a three-treatment three-period crossover trial.

When the frequency of event occurrences follows a Poisson distribution, we develop procedures for testing equality of treatments and interval estimators for the ratio of mean frequencies between treatments under a three-treatment three-period crossover design. Using Monte Carlo simulations, we evaluate the performance of these test procedures and interval estimators in various situations. We note that all test procedures developed here can perform well with respect to Type I error even when the number of patients per group is moderate. We further note that the two weighted-least-squares (WLS) test procedures derived here are generally preferable to the other two commonly used test procedures in the contingency table analysis. We also demonstrate that both interval estimators based on the WLS method and interval estimators based on Mantel-Haenszel (MH) approach can perform well, and are essentially of equal precision with respect to the average length. We use a double-blind randomized three-treatment three-period crossover trial comparing salbutamol and salmeterol with a placebo with respect to the number of exacerbations of asthma to illustrate the use of these test procedures and estimators.

A semi-parametric approach to the frequency of occurrence under a simple crossover trial.

To analyze the frequency of occurrence for an event of interest in a crossover design, we propose a semi-parametric approach. We develop two point estimators and four interval estimators in closed forms for the treatment effect under a random effects multiplicative risk model. Using Monte Carlo simulations, we evaluate these estimators and compare the four interval estimators with the classical interval estimator suggested elsewhere in a variety of situations. We note that the point estimator using the ratio of two arithmetic averages of mean frequencies under a multiplicative risk model can be comparable to the point estimator using the ratio of two geometric averages of mean frequencies. We note that as long as the number of patients per group is large, all the four interval estimators developed here can perform well. We also note that the classical interval estimator derived under the commonly assumed Poisson distribution for the frequency data can be conservative and lose precision if the Poisson distribution assumption is violated. We use a double-blind randomized crossover trial comparing salmeterol with a placebo in exacerbations of asthma to illustrate the practical use of these estimators.

Analysis of Poisson frequency data under a simple crossover trial.

When the frequency of occurrence for an event of interest follows a Poisson distribution, we develop asymptotic and exact procedures for testing non-equality, non-inferiority and equivalence, as well as asymptotic and exact interval estimators for the ratio of mean frequencies between two treatments under a simple crossover design. Using Monte Carlo simulations, we evaluate the performance of these test procedures and interval estimators in a variety of situations. We note that all asymptotic test procedures developed here can generally perform well with respect to Type I error and can be preferable to the exact test procedure with respect to power if the number of patients per group is moderate or large. We further find that in these cases the asymptotic interval estimator with the logarithmic transformation can be more precise than the exact interval estimator without sacrificing the accuracy with respect to the coverage probability. However, the exact test procedure and exact interval estimator can be of use when the number of patients per group is small. We use a double-blind randomized crossover trial comparing salmeterol with a placebo in exacerbations of asthma to illustrate the practical use of these estimators.

Notes on testing noninferiority in multivariate binary data under the matched-pair design.

Since therapeutic efficacy is often measured by multiple endpoints, it will be of use if one can incorporate the information on various variables of response into procedures for testing noninferiority to improve power of a univariate test procedure for each individual variable. On the basis of the proposed mixed effects logistic regression model for multivariate binary data under the matched-pairs design, we develop procedures for testing noninferiority with respect to the odds ratio in multivariate binary data under the matched-pair design. We discuss use of Bonferroni's and Scheffe's methods to control the inflation in Type I error due to multiple tests. We further employ Monte Carlo simulation to evaluate and compare the performance of these test procedures. Finally, we use the data taken from a crossover clinical trial that monitored several adverse events of an antidepressive drug to illustrate the use of test procedures derived here.

Sample size determination for testing nonequality under a three-treatment two-period incomplete block crossover trial.

To reduce the lengthy duration of a crossover trial for comparing three treatments, the incomplete block design has been often considered. A sample size calculation procedure for testing nonequality between either of the two experimental treatments and a placebo under such a design is developed. To evaluate the performance of the proposed sample size calculation procedure, Monte Carlo simulation is employed. The accuracy of the sample size calculation procedure developed here is demonstrated in a variety of situations. As compared with the parallel groups design, a substantial proportional reduction in the total minimum required sample size in use of the incomplete block crossover design is found. A crossover trial comparing two different doses of formoterol with a placebo on the forced expiratory volume is applied to illustrate the use of the sample size calculation procedure.

Test equality between two binary screening tests with a confirmatory procedure restricted on screen positives.

In studies of screening accuracy, we may commonly encounter the data in which a confirmatory procedure is administered to only those subjects with screen positives for ethical concerns. We focus our discussion on simultaneously testing equality of sensitivity and specificity between two binary screening tests when only subjects with screen positives receive the confirmatory procedure. We develop four asymptotic test procedures and one exact test procedure. We derive sample size calculation formula for a desired power of detecting a difference at a given nominal [Formula: see text]-level. We employ Monte Carlo simulation to evaluate the performance of these test procedures and the accuracy of the sample size calculation formula developed here in a variety of situations. Finally, we use the data obtained from a study of the prostate-specific-antigen test and digital rectal examination test on 949 Black men to illustrate the practical use of these test procedures and the sample size calculation formula.

Testing equality and interval estimation in binary responses when high dose cannot be used first under a three-period crossover design.

When comparing two doses of a new drug with a placebo, we may consider using a crossover design subject to the condition that the high dose cannot be administered before the low dose. Under a random-effects logistic regression model, we focus our attention on dichotomous responses when the high dose cannot be used first under a three-period crossover trial. We derive asymptotic test procedures for testing equality between treatments. We further derive interval estimators to assess the magnitude of the relative treatment effects. We employ Monte Carlo simulation to evaluate the performance of these test procedures and interval estimators in a variety of situations. We use the data taken as a part of trial comparing two different doses of an analgesic with a placebo for the relief of primary dysmenorrhea to illustrate the use of the proposed test procedures and estimators.

Notes on testing noninferiority in ordinal data under the parallel groups design.

When testing the noninferiority of an experimental treatment to a standard (or control) treatment in a randomized clinical trial (RCT), we may come across the outcomes of patient response on an ordinal scale. We focus our discussion on testing noninferiority in ordinal data for an RCT under the parallel groups design. We develop simple test procedures based on the generalized odds ratio (GOR). We note that these test procedures not only can account for the information on the order of ordinal responses without assuming any specific parametric structural model, but also can be independent of any arbitrarily subjective scoring system. We further develop sample size determination based on the test procedure using the GOR. We apply Monte Carlo simulation to evaluate the performance of these test procedures and the accuracy of sample size calculation formula proposed here in a variety of situations. Finally, we employ the data taken from a trial comparing once-daily gatifloxican with three-times-daily co-amoxiclav in the treatment of community-acquired pneumonia to illustrate the use of these test procedures and sample size calculation formula.

Optimal allocation of sample sizes to multicenter clinical trials.

In this article, we discuss an approach for optimal sample size allocation in designing multicenter clinical trials. The method we studied was adapted from a stratified sampling survey design. The sample size allocated to centers is a function of the center's treatment cost, the standard deviation of the endpoint, and the availability of patients. We illustrate our approach using two hypothetical scenarios derived from our experiences in designing and conducting multicenter clinical trials. Simulation results are also presented.

Five interval estimators of the risk difference under stratified randomized clinical trials with noncompliance and repeated measurements.

We often employ stratified analysis to control the confounding effect due to centers in a multicenter trial or the confounding effect due to trials in a meta-analysis. On the basis of a general risk additive model, we focus discussion on interval estimation of the risk difference (RD) in repeated binary measurements under a stratified randomized clinical trial (RCT) in the presence of noncompliance. We develop five asymptotic interval estimators for the RD in closed form. These include the interval estimator using the weighted least-squares (WLS) estimator, the WLS interval estimator with tanh (-1)(x) transformation, the Mantel-Haenszel (MH) type interval estimator, the MH interval estimator with tanh (-1)(x) transformation, and the interval estimator using the idea of Fieller's theorem and a randomization-based variance. We employ Monte Carlo simulation to study and compare the finite-sample performance of these interval estimators in a variety of situations. We include an example studying the use of macrophage colony-stimulating factor to reduce the risk of febrile neutropenia events in acute myeloid leukaemia patients published elsewhere to illustrate the use of these estimators.

Testing and estimation of proportion (or risk) ratio under the matched-pair design with multiple binary endpoints.

The proportion ratio (PR) of responses between an experimental treatment and a control treatment is one of the most commonly used indices to measure the relative treatment effect in a randomized clinical trial. We develop asymptotic and permutation-based procedures for testing equality of treatment effects as well as derive confidence intervals of PRs for multivariate binary matched-pair data under a mixed-effects exponential risk model. To evaluate and compare the performance of these test procedures and interval estimators, we employ Monte Carlo simulation. When the number of matched pairs is large, we find that all test procedures presented here can perform well with respect to Type I error. When the number of matched pairs is small, the permutation-based test procedures developed in this paper is of use. Furthermore, using test procedures (or interval estimators) based on a weighted linear average estimator of treatment effects can improve power (or gain precision) when the treatment effects on all response variables of interest are known to fall in the same direction. Finally, we apply the data taken from a crossover clinical trial that monitored several adverse events of an antidepressive drug to illustrate the practical use of test procedures and interval estimators considered here.

Notes on interval estimation of the generalized odds ratio under stratified random sampling.

It is not rare to encounter the patient response on the ordinal scale in a randomized clinical trial (RCT). Under the assumption that the generalized odds ratio (GOR) is homogeneous across strata, we consider four asymptotic interval estimators for the GOR under stratified random sampling. These include the interval estimator using the weighted-least-squares (WLS) approach with the logarithmic transformation (WLSL), the interval estimator using the Mantel-Haenszel (MH) type of estimator with the logarithmic transformation (MHL), the interval estimator using Fieller's theorem with the MH weights (FTMH) and the interval estimator using Fieller's theorem with the WLS weights (FTWLS). We employ Monte Carlo simulation to evaluate the performance of these interval estimators by calculating the coverage probability and the average length. To study the bias of these interval estimators, we also calculate and compare the noncoverage probabilities in the two tails of the resulting confidence intervals. We find that WLSL and MHL can generally perform well, while FTMH and FTWLS can lose either precision or accuracy. We further find that MHL is likely the least biased. Finally, we use the data taken from a study of smoking status and breathing test among workers in certain industrial plants in Houston, Texas, during 1974 to 1975 to illustrate the use of these interval estimators.

Hypothesis testing and estimation in ordinal data under a simple crossover design.

Since each patient serves as his/her own control, the crossover design can be of use to improve power as compared with the parallel-groups design in studying noncurative treatments to certain chronic diseases. Although the research studies on the crossover design have been quite intensive, the discussions on analyzing ordinal data under such a design are truly limited. We propose using the generalized odds ratio (GOR) for paired sample data to measure the relative effect on patient responses for both treatment and period in ordinal data under a simple crossover trial. Assuming the treatment and period effects are multiplicative, we note that one can easily derive the maximum likelihood estimator (LE) in closed forms for the GOR of treatment and period effects. We develop asymptotic and exact procedures for testing treatment and period effects. We further derive asymptotic and exact interval estimators for the GOR of treatment and period effects. We use the data taken from a crossover trial to assess the clarity of leaflet instructions between two devices among asthma patients to illustrate the use of these test procedures and estimators developed here.

Notes on interval estimation of the gamma correlation under stratified random sampling.

We have developed four asymptotic interval estimators in closed forms for the gamma correlation under stratified random sampling, including the confidence interval based on the most commonly used weighted-least-squares (WLS) approach (CIWLS), the confidence interval calculated from the Mantel-Haenszel (MH) type estimator with the Fisher-type transformation (CIMHT), the confidence interval using the fundamental idea of Fieller's Theorem (CIFT) and the confidence interval derived from a monotonic function of the WLS estimator of Agresti's α with the logarithmic transformation (MWLSLR). To evaluate the finite-sample performance of these four interval estimators and note the possible loss of accuracy in application of both Wald's confidence interval and MWLSLR using pooled data without accounting for stratification, we employ Monte Carlo simulation. We use the data taken from a general social survey studying the association between the income level and job satisfaction with strata formed by genders in black Americans published elsewhere to illustrate the practical use of these interval estimators.

Exact sample-size determination in testing non-inferiority under a simple crossover trial.

For testing the non-inferiority (or equivalence) of an experimental treatment to a standard treatment, the odds ratio (OR) of patient response rates has been recommended to measure the relative treatment efficacy. On the basis of an exact test procedure proposed elsewhere for a simple crossover design, we develop an exact sample-size calculation procedure with respect to the OR of patient response rates for a desired power of detecting non-inferiority at a given nominal type I error. We note that the sample size calculated for a desired power based on an asymptotic test procedure can be much smaller than that based on the exact test procedure under a given situation. We further discuss the advantage and disadvantage of sample-size calculation using the exact test and the asymptotic test procedures. We employ an example by studying two inhalation devices for asthmatics to illustrate the use of sample-size calculation procedure developed here.

Interval estimation of the proportion ratio in repeated binary measurements under a stratified randomized clinical trial with noncompliance.

The proportion ratio (PR) of a positive response between an experimental treatment and a standard treatment (or placebo) is often used to measure the relative treatment efficacy in a randomized clinical trial (RCT). For ethical reasons, it is almost inevitable to encounter some patients not complying with their assigned treatment. Furthermore, when there are confounders in a RCT or meta-analysis, we commonly employ stratified analysis to control the confounding effects on interval estimation of the PR. On the basis of a general risk multiplicative model, we focus our discussion on interval estimation of the PR in repeated binary data under a stratified RCT with noncompliance. We develop seven asymptotic closed-form interval estimators for the PR. We apply Monte Carlo simulation to study the finite-sample performance of these interval estimators in a variety of situations. We note that the two interval estimators with the logarithmic transformation based on the commonly used weighted least squares (WLS) approach can be liberal, while the three interval estimators with the Mantel-Haenszel (MH) weight derived from various methods can consistently perform well. We also note that the two estimators with the estimated optimal weight defined in the context using Fieller's Theorem and a randomization-based approach may not necessarily produce a confidence interval preferable to the MH-type interval estimators for the PR with respect to accuracy and precision.

Test non-inferiority (and equivalence) based on the odds ratio under a simple crossover trial.

For testing the non-inferiority (or equivalence) of a generic drug to a standard drug, the odds ratio (OR) of patient response rates has been recommended to measure the relative treatment efficacy. On the basis of a random effects logistic regression model, we develop asymptotic test procedures for testing non-inferiority and equivalence with respect to the OR of patient response rates under a simple crossover design. We further derive exact test procedures, which are especially useful for the situations in which the number of patients in a crossover trial is small. We address sample size calculation for testing non-inferiority and equivalence based on the asymptotic test procedures proposed here. We also discuss estimation of the OR of patient response rates for both the treatment and period effects. Finally, we include two examples, one comparing two solution aerosols in treating asthma, and the other one studying two inhalation devices for asthmatics, to illustrate the use of the proposed test procedures and estimators.