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.

Urinary metabolites of furan in waterpipe tobacco smokers compared to non-smokers in home settings in the US.

OBJECTIVES: Determine uptake of furan, a potential human carcinogen, in waterpipe tobacco (WPT) smokers in home settings. METHODS: We analysed data from a US convenience sample of 50 exclusive WPT smokers, mean age 25.3 years, and 25 non-smokers, mean age 25.5 years. For WPT smokers, data were collected at a home visit by research assistants during which participants smoked one WPT head of one brand for a mean of 33.1 min in their homes. Research assistants provided and prepared a WP for participants by weighing and loading 10 g of WPT in the WP head. At the completion of the smoking session, research assistants measured the remaining WPT. Cotinine and six furan metabolites were quantified in first morning urine samples provided on 2 consecutive days for non-smokers, and on the morning of a WPT smoking session and on the following morning for smokers. RESULTS: WPT smokers consumed a mean of 2.99 g WPT. In WPT smokers, urinary cotinine levels increased significantly 26.1 times the following morning; however, urinary metabolites of furan did not increase significantly. Compared to non-smokers, 2 furan metabolites, N-acetyl-S-[1-(5-acetylamino-5-carboxylpentyl)-1H-pyrrol-3-yl]-L-cysteine and N-acetyl-S-[1-(5-amino-5-carboxypentyl)-1H-pyrrol-3-yl]-L-cysteine sulfoxide, were significantly higher in WPT smokers in pre and in post WPT smoking levels. CONCLUSIONS: To enable a more rigorous assessment of furan exposure from WPT smoking, future research should determine furan concentrations in WPT smoke, quantify furan metabolites from users of various WPT brands; and extend the investigation to social settings where WPT smoking is habitually practiced.

Exact Confidence Limits on Some New Measures of Concordance and Discordance in Binary Outcomes.

Using a measure of agreement that does not distinguish the "positive" outcome from the "negative" outcome can be sometimes misleading in assessing resemblance. To alleviate this concern, some new indices, including the "positive" and "negative" conditional synchrony measures (CSM) (or the conditional discordant measures [CDM]), as well as their related measures, have been recently proposed elsewhere. We show that one can easily derive exact confidence limits for these new indices. Using Monte Carlo simulation, we find that the asymptotic interval estimator derived from the score test and these exact interval estimators can all perform well in a variety of situations, while the asymptotic interval estimator based on Wald's statistic can lose accuracy. We use the data taken from a cross-sectional validation study assessing the diagnostic performance of the Whooley questions for major depression disorder (MDD) among older adults to illustrate the use of these interval estimators developed here.

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.

Estimation of the relative difference (or relative risk reduction) under the sequential parallel comparison design.

To increase power or reduce the number of patients needed in trials studying treatments for psychiatric or mental disorders with a high placebo response rate, we may consider use of the sequential parallel comparison design proposed elsewhere. Because statistical significance does not necessarily imply that the difference between treatment and placebo is of clinical importance, it is always of importance to quantify the treatment effect in clinical trials. When the patient responses are dichotomous, the treatment and other covariates effects are not likely additive. Thus, using a weighted average of the risk differences over two phases may not be a meaningful summary index to measure the treatment effect. To alleviate this concern, we consider use of the relative difference or relative risk reduction to measure the treatment effect. We derive both point and interval estimators for the relative difference by use of the weighted-least-squares estimator and Mantel-Haenszel type estimator. We employ Monte Carlo simulation to evaluate the finite-sample performance of these estimators in a variety of situations. We also include a procedure for testing the homogeneity of the relative difference between phases under the sequential parallel comparison design. We use the placebo-controlled study to assess the efficacy of a low dose of aripiprazole adjunctive to antidepressant therapy in the treatment of patients with major depressive disorder to illustrate the use of estimators developed here.

Asymptotic and exact interval estimators of the common odds ratio under the sequential parallel comparison design.

When studying treatments for psychiatric or mental diseases in a placebo-controlled trial, we may consider use of the sequential parallel comparison design to reduce the number of patients needed through the reduction of the high placebo response rate. Under the assumption that the odds ratio of responses is constant between phases in the sequential parallel comparison design, we derive the conditional maximum likelihood estimator for the odds ratio. On the basis of the conditional likelihood, we further derive three asymptotic interval and an exact interval estimators for the odds ratio of responses. We employ Monte Carlo simulation to evaluate the performance of these interval estimators in a variety of situations. We find that the asymptotic interval and exact interval estimators developed here can all perform well. We use the double-blind, placebo-controlled study assessing the efficacy of a low dose of aripiprazole adjunctive to antidepressant therapy for treating patients with major depressive disorder to illustrate the use of these estimators.

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.

Sample Size Determination for a 3-Treatment 3-Period Crossover Trial in Frequency Data.

BACKGROUND: To reduce the number of patients needed or increase the power of hypothesis testing for the parallel groups design, the crossover design has been often employed when one is studying noncurable chronic diseases. This article focuses attention on sample size calculation for testing non-inferiority and equality in frequency data under a 3-treatment 3-period crossover trial. METHOD: Under a multiplicative mixed effects model, this article provides asymptotic sample size calculation procedures for testing non-inferiority of an experimental treatment to a control treatment, as well as for simultaneously testing either of 2 treatments versus a placebo. To improve the performance of these asymptotic procedures in small-sample cases, this article further suggests a simple ad hoc adjustment. RESULTS: On the basis of Monte Carlo simulation, we demonstrate that the asymptotic test procedures proposed here can perform well with respect to Type I error. We find that the asymptotic sample size calculation procedures can generally perform well with respect to power when the resulting sample size is moderate or large. We further find that using the simple ad hoc adjustment can improve the performance of the proposed sample size calculation procedures, which are derived from large-sample theory, in small-sample cases.

Notes on Test and Estimation in Comparison of Three Treatments under A Simple Carry-Over Three-Period Model.

Under the three-treatment three-period crossover design with simple carry-over effects, we derive the least-squares estimators for period effects, treatment effects and carry-over effects, as well as their covariance matrix in closed and explicit expressions. Using Monte Carlo simulation, we compare the test procedure adjusting carry-over with that ignoring carry-over with respect to Type I error and power. We further compare interval estimators adjusting carry-over with those ignoring carry-over with respect to the coverage probability and the average length. When the variation of responses within patients is small, the test procedure and interval estimators ignoring carry-over can lose accuracy in the presence of carry-over effects. When the variation of responses within patients is large, this loss of accuracy may become small or even minimal. We note that the loss of efficiency due to the adjustment of carry-over under the simple carry-over three-period crossover design is moderate, and is much less than that found for a two-period crossover design. We use the double-blind three-period crossover trial comparing formoterol solution aerosol and salbutamol suspension aerosol with a placebo for patients suffering from exercise-induced asthma on the forced expiratory volume in one second (FEV1) to illustrate the use of test procedures and interval estimators discussed here.

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 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 of Treatments under an Incomplete Block Crossover Design with Ordinal Responses.

The generalized odds ratio (GOR) for paired sample is considered to measure the relative treatment effect on patient responses in ordinal data. Under a three-treatment two-period incomplete block crossover design, both asymptotic and exact procedures are developed for testing equality between treatments with ordinal responses. Monte Carlo simulation is employed to evaluate and compare the finite-sample performance of these test procedures. A discussion on advantages and disadvantages of the proposed test procedures based on the GOR versus those based on Wald's tests under the normal random effects proportional odds model is provided. The data taken as a part of a crossover trial studying the effects of low and high doses of an analgesic versus a placebo for the relief of pain in primary dysmenorrhea over the first two periods are applied to illustrate the use of these test procedures.

Estimation of the treatment effect under an incomplete block crossover design in binary data - A conditional likelihood approach.

A random effects logistic regression model is proposed for an incomplete block crossover trial comparing three treatments when the underlying patient response is dichotomous. On the basis of the conditional distributions, the conditional maximum likelihood estimator for the relative effect between treatments and its estimated asymptotic standard error are derived. Asymptotic interval estimator and exact interval estimator are also developed. Monte Carlo simulation is used to evaluate the performance of these estimators. Both asymptotic and exact interval estimators are found to perform well in a variety of situations. When the number of patients is small, the exact interval estimator with assuring the coverage probability larger than or equal to the desired confidence level can be especially of use. The data taken from a crossover trial comparing the low and high doses of an analgesic with a placebo for the relief of pain in primary dysmenorrhea are used to illustrate the use of estimators and the potential usefulness of the incomplete block crossover design.

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.

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.

Testing Equality in Ordinal Data with Repeated Measurements: A Model-Free Approach.

In randomized clinical trials, we often encounter ordinal categorical responses with repeated measurements. We propose a model-free approach with using the generalized odds ratio (GOR) to measure the relative treatment effect. We develop procedures for testing equality of treatment effects and derive interval estimators for the GOR. We further develop a simple procedure for testing the treatment-by-period interaction. To illustrate the use of test procedures and interval estimators developed here, we consider two real-life data sets, one studying the gender effect on pain scores on an ordinal scale after hip joint resurfacing surgeries, and the other investigating the effect of an active hypnotic drug in insomnia patients on ordinal categories of time to falling asleep.

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.