Notes on testing equality in binary data under a three period crossover design

https://doi.org/10.1016/j.csda.2014.06.015Get rights and content

Abstract

Under a random effects logistic regression model, asymptotic and exact test procedures in closed form for testing equality of binary responses are developed for comparing three treatments in a three-period crossover trial. Monte Carlo simulation is employed to evaluate the performance of these test procedures in a variety of situations. Interval estimators for the relative treatment effects are provided. The commonly-used procedures for testing the homogeneity of odds ratio (OR) is shown to be applicable for testing whether there is an interaction between treatments and periods. Finally, 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 the proposed test procedures and estimators.

Introduction

When studying treatments for non-curable chronic diseases, such as angina pectoris, epilepsy, hypertension or asthma, we may often consider use of the crossover design to improve power without recruiting additional patients into a trial (Smith et al., 1985, Rhind et al., 1985). This is because each patient serves as his/her own control and we can increase the efficiency of test procedures through reducing the variation of patient responses between treatments (Fleiss, 1986, Hills and Armitage, 1979, Senn, 2002). However, if the treatment being evaluated has a long-lasting effect, the crossover design will not be appropriate, because the effect of a treatment administered at the latter period can be confounded with the residual effects of the treatment administered at the earlier period. To alleviate this concern of carry-over effect, one commonly applies an adequate washout period between administering treatments to assure that patients are weaned off the effects of earlier treatments. Research on crossover trial has been actually extensive. Senn, 2002, Senn, 2006 provided an excellent review of the literature on crossover designs. Based on a fixed effects logistic regression model (Cox and Snell, 1989), Gart (1969) discussed testing equality of two treatment effects and developed an exact test procedure. Using the linear additive risk model (LARM) proposed by Grizzle (1965) and Zimmermann and Rahlfs (1978) discussed testing equality in patient response rates under a two-period crossover design. For the simple crossover design, Ezzet and Whitehead (1992) proposed a random effects logistic regression model with assuming random effects to follow a normal distribution and discussed estimation of treatment and period effects. Schouten and Kester (2010) also used the LARM to develop tests and sample size calculation for dichotomous data in a simple crossover design. Recently, Lui and Chang (2011) have discussed testing both non-inferiority and equivalence with respect to the odds ratio (OR) under a simple crossover trial. All of these papers concentrated attention on comparison of two treatments under a two-period crossover trial. As noted by Jones and Kenward (1987), it is not uncommon to encounter trials with more than two treatments and more than two periods. For example, consider the three-period crossover trial comparing two different doses of an analgesic with a placebo for the relief of primary dysmenorrhea in dichotomous responses (Jones and Kenward, 1987). We summarize in Table 1 the data published elsewhere (Jones and Kenward, 1987). Note that the responses taken from the same patient over three periods are likely correlated and there can be period effects on the patient response. Thus, it would not be appropriate if we did not account for the intraclass correlation between responses within subjects and the period effect on patient responses in data analysis. Except for assuming a specific structural dependence between responses at the three periods (Jones and Kenward, 1987), the research on comparing three treatments over a three-period crossover trial in binary responses is limited.

In this paper, we propose a random effects logistic regression model for testing equality between treatments in dichotomous responses under a three-period crossover trial. We derive both the asymptotic and exact test procedures in closed forms for testing equality between treatments. We employ Monte Carlo simulation to evaluate the performance of these test procedures. We provide interval estimators for the relative treatment effects. We further show that test procedures for testing the homogeneity of OR can also be used to examine whether there is an interaction between treatments and periods. Finally, we use the data (Table 1) taken from a trial comparing two different doses of an analgesic with a placebo for the relief of primary dysmenorrhea (Jones and Kenward, 1987) to illustrate the use the proposed test procedures and interval estimators.

Section snippets

Notation, model assumptions and methods

Consider comparing two experimental treatments A and B with a placebo (P) in a three-period crossover design. For clarity, we use the treatment-receipt sequence XYZ to denote that a patient receives treatments X,Y and Z at periods 1, 2 and 3, respectively. Suppose that we randomly assign ng patients to group g=1 with P–A–B treatment-receipt sequence; = 2 with P–B–A treatment-receipt sequence; = 3 with A–P–B treatment receipt sequence; = 4 with A–B–P treatment-receipt sequence; = 5 with B–P–A

Monte Carlo simulation

To evaluate the performance of asymptotic test procedures TAP(WLS), TBP(WLS), TBA(WLS), TAP(MH), TBP(MH), TBA(MH) and the exact test procedure defined by Eqs. (14), (15), we employ Monte Carlo simulation. We consider the situations in which the period effects γ1 and γ2 are arbitrarily set equal to 0.10 and 0.15; the random effects μi(g) are assumed to follow the normal distribution with mean 0 and standard deviation σ=0.1, 1.0, 3.0; the effects for treatments A and B versus placebo, η1=0.0,

Results

We summarize in Table 2  the estimated Type I error (in boldface) and power for test procedures TAP(WLS),TBP(WLS),TBA(WLS), TAP(MH),TBP(MH),TBA(MH) and the exact test procedure defined by Eqs. (14), (15) at the 0.05-level. We can see that the estimated Type I error of these test procedures are all less than or approximately equal to the nominal 0.05-level (Table 2). We note that the estimated Type I error for the MH procedure is generally closer to the nominal 0.05-level than the WLS procedure.

An example

Consider the data in Table 1 taken from a three-period crossover trial comparing the low dose (treatment A) and high dose (treatment B) of analgesic with placebo (P) for the relief (yes  =  1, no  =  0) of primary dysmenorrhea. A wash-out period of one month was applied to separate each treatment period (Jones and Kenward, 1987). There were 86 patients randomly divided into six different groups, corresponding to the six possible ways of ordering the three treatments over the three periods. Each

Discussion

The test procedures and interval estimators proposed here do not need to assume any parametric distribution for the random effects in model (1) and hence are, as noted before, semi-parametric. On the other hand, if we are willing to assume that the random effects independently follow a normal distribution, we may use proc GLIMMIX in SAS (2009) to obtain PMLEs based on the pseudo-likelihood. If the normality distribution for the random effects can be satisfied by data, the test procedure using

Acknowledgments

The authors wish to thank the two referees for many valuable and useful comments to improve the contents and clarity of this paper.

References (28)

  • J.L. Fleiss

    A critique of recent research on the two-treatment crossover design

    Controlled Clin Trials

    (1989)
  • A. Agresti

    Categorical Data Analysis

    (1990)
  • N.E. Breslow et al.

    Statistical Methods in Cancer Research, Volume 1. The Analysis of Case-Control Studies

    (1980)
  • D.R. Cox et al.

    Analysis of Binary Data

    (1989)
  • A. Ejigou et al.

    Testing the homogeneity of the relative risk under multiple matching

    Biometrika

    (1984)
  • F. Ezzet et al.

    A random effects model for binary data from crossover clinical trials

    Appl. Stat.

    (1992)
  • J.L. Fleiss

    Statistical Methods for Rates and Proportion

    (1981)
  • J.L. Fleiss

    The Design and Analysis of Clinical Experiments

    (1986)
  • J.J. Gart

    An exact test for comparing matched proportions in crossover designs

    Biometrika

    (1969)
  • J.J. Gart

    Point and interval estimation of the common odds ratio in the combination of 2×2 tables with fixed margins

    Biometrika

    (1970)
  • J.E. Grizzle

    The two-period change-over design and its use in clinical trials

    Biometrics

    (1965)
  • M. Hills et al.

    The two-period cross-over clinical trial

    Br. J. Clin. Pharmacol.

    (1979)
  • B. Jones et al.

    Modelling binary data from a three-period cross-over trial

    Stat. Med.

    (1987)
  • B. Jones et al.

    Design and Analysis of Cross-Over Trials

    (1989)
  • Cited by (0)

    View full text