A note on point estimation and interval estimation of the relative treatment effect under a simple crossover design.

To increase power or reduce the number of patients needed for a parallel groups design, the crossover design has been often used to study treatments for noncurable chronic diseases. However, in the presence of carry-over effect caused by treatments, the commonly-used estimator which ignores the carry-over effect leads to a biased estimator for estimating the treatment effect difference. A two-stage test approach aimed to address carry-over effect proposed was found to be potentially misleading. In this paper, we propose a weighted average of the commonly-used estimator and an unbiased estimator that uses only the first period of the data. We derive an optimal weight that minimizes the mean squared error (MSE) and its modified estimator. We apply Monte Carlo simulation to evaluate the performance of the proposed estimators in a variety of situations. In the simulations, we examine the estimated MSE (EMSE), percentile interval length, and coverage probability calculated from the percentile intervals among considered estimators. Simulation results show that our proposed weighted average estimator and its modified estimator lead to smaller EMSEs on average comparing to the two commonly used estimators. The coverage probabilities using our proposed estimators are reasonably close to the nominal confidence level and the interval lengths are shorter comparing to the use of the unbiased estimator that uses only the first period of the data. We apply an example that was to evaluate the efficacy of two type of bronchodilators for asthma treatment to demonstrate the use of the proposed estimators.

Exact interval estimators for some commonly used measures of binary agreement.

We develop exact interval estimators for some commonly used classical measures of agreement in binary responses. We apply Monte Carlo simulation to evaluate the performance of these estimators. When the measure of agreement is homogeneous, we note that extending the results presented here to accommodate stratified analysis is straightforward. We use the data taken from a survey studying the agreement of religious identifications and the data taken from a study assessing the diagnostic performance of Whooley questions for major depression disorder to illustrate the use of these interval estimators.

Model-free tests of equality in binary data under an incomplete block design.

Using Prescott's model-free approach, we develop an asymptotic procedure and an exact procedure for testing equality between treatments with binary responses under an incomplete block crossover design. We employ Monte Carlo simulation and note that these test procedures can not only perform well in small-sample cases but also outperform the corresponding test procedures accounting for only patients with discordant responses published elsewhere. We use the data taken as a part of the crossover trial comparing two different doses of an analgesic with placebo for the relief of primary dysmenorrhea to illustrate the use of test procedures discussed here.

Exact and asymptotic tests under the sequential parallel comparison design.

When one studies treatments for psychological or mental diseases in a double-blind placebo-controlled trial with a high placebo response rate, the sequential parallel comparison design (SPCD) has been proposed elsewhere to improve power. All procedures for testing equality of treatments under the SPCD have been so far derived from large sample theory. If the trial size is small, asymptotic test procedures can be theoretically invalid. Thus, the development of an exact test procedure assuring type I error rate to be less than or equal to the nominal α-level is of use and interest. Using the conditional arguments to remove nuisance parameters, we derive two exact and one asymptotic procedures for testing equality of treatments for the SPCD. On the basis of Monte Carlo simulation, we find that all three test procedures can control type I error rate well in a variety of situations. We use the data taken from a double-blind placebo-controlled SPCD trial to assess the efficacy of a low dose (2 mg/day) of aripiprazole adjunctive to antidepressant therapy in the treatment of patients with major depressive disorder with a history of inadequate response to prior antidepressant therapy to illustrate the use of these test procedures.

Prescott tests of equality in binary data under a three-treatment three-period crossover design.

Three test procedures accounting for patients with tied responses based on Prescott's ideas are developed for comparing three treatments under a three-period crossover trial in binary data. Monte Carlo simulation is employed to evaluate the performance of these test procedures in a variety of situations. The test procedures proposed here are noted to have power larger than those procedures, which utilize only those patients with un-tied responses. The data taken from a three-period crossover trial comparing two different doses of an analgesic with placebo for the relief of primary dysmenorrhea are used to illustrate the use of the test procedures developed 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.

Test Equality between Three Treatments under an Incomplete Block Crossover Design.

Under a random effects linear additive risk model, we compare two experimental treatments with a placebo in continuous data under an incomplete block crossover trial. We develop three test procedures for simultaneously testing equality between two experimental treatments and a placebo, as well as interval estimators for the mean difference between treatments. We apply Monte Carlo simulations to evaluate the performance of these test procedures and interval estimators in a variety of situations. We note that the bivariate test procedure accounting for the dependence structure based on the F-test is preferable to the other two procedures when there is only one of the two experimental treatments has a non-zero effect vs. the placebo. We note further that when the effects of the two experimental treatments vs. a placebo are in the same relative directions and are approximately of equal magnitude, the summary test procedure based on a simple average of two weighted-least-squares (WLS) estimators can outperform the other two procedures with respect to power. When one of the two experimental treatments has a relatively large effect vs. the placebo, the univariate test procedure with using Bonferroni's equality can be still of use. Finally, we use the data about the forced expiratory volume in 1 s (FEV1) readings taken from a double-blind crossover trial comparing two different doses of formoterol with a placebo to illustrate the use of test procedures and interval estimators proposed here.

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.

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.

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.

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.

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.

Sample size determination for testing equality in Poisson frequency data under an AB/BA crossover trial.

Assuming that the frequency of occurrence follows the Poisson distribution, we develop sample size calculation procedures for testing equality based on an exact test procedure and an asymptotic test procedure under an AB/BA crossover design. We employ Monte Carlo simulation to demonstrate the use of these sample size formulae and evaluate the accuracy of sample size calculation formula derived from the asymptotic test procedure with respect to power in a variety of situations. We note that when both the relative treatment effect of interest and the underlying intraclass correlation between frequencies within patients are large, the sample size calculation based on the asymptotic test procedure can lose accuracy. In this case, the sample size calculation procedure based on the exact test is recommended. On the other hand, if the relative treatment effect of interest is small, the minimum required number of patients per group will be large, and the asymptotic test procedure will be valid for use. In this case, we may consider use of the sample size calculation formula derived from the asymptotic test procedure to reduce the number of patients needed for the exact test procedure. We include an example regarding a double-blind randomized crossover trial comparing salmeterol with a placebo in exacerbations of asthma to illustrate the practical use of these sample size formulae.

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.

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.

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.

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.

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.