A Bayesian analysis of the two-period crossover design for clinical trials

Statisticians have been critical of the use of the two-period crossover designs for clinical trials because the estimate of the treatment difference is biased when the carryover effects of the two treatments are not equal. In the standard approach, if the null hypothesis of equal carryover effects is not rejected, data from both periods are used to estimate and test for treatment differences; if the null hypothesis is rejected, data from the first period alone are used. A Bayesian analysis based on the Bayes factor against unequal carryover effects is given. Although this Bayesian approach avoids the "all-or-nothing" decision inherent in the standard approach, it recognizes that with small trials it is difficult to provide unequivocal evidence that the carryover effects of the two treatments are equal, and thus that the interpretation of the difference between treatment effects is highly dependent on a subjective assessment of the reality or not of equal carryover effects.     

Carryover and the two-period crossover clinical trial

The use of the two-period crossover trial for comparing two noncurative treatments in patients with a chronic disease is appealing since it eliminates the between-subject variability. However, the possibility of the existence of carryover effect leads many authors to advise that a parallel design be used whenever carryover is suspected. We examine this advice and quantify the degree of carryover required to make the parallel design preferable in terms of the power of the test of treatment effect and precision of the estimate of treatment difference. We conclude that in many situations this amount of carryover is substantial and unlikely to exist.     

Crossover Designs with Random Periods

Crossover experiments usually are modelled with fixed treatment, carryover, and period effects while the effects of subjects are assumed to be random. In actual realisations of crossover experiments however periods are quite arbitrary intervals of time, depending on administrative affairs for instance and possibly they are even varying from subject to subject. Modelling these arbitrary periods as random effects either globally for all subjects or individually for each one seems to be more adequate than the assumption of fixed period effects. This paper is concerned with the two treatments, two periods crossover design. It is described that—in spite of the somewhat involved covariance structure—the estimators and tests developed for models with fixed period effects remain valid for models with random effects for periods provided that no treatment X period interaction exists.     

On the Role of Baseline Measurements for Crossover Designs under the Self and Mixed Carryover Effects Model

Summary .  It is well known that optimal designs are strongly model dependent. In this article, we apply the Lagrange multiplier approach to the optimal design problem, using a recently proposed model for carryover effects. Generally, crossover designs are not recommended when carryover effects are present and when the primary goal is to obtain an unbiased estimate of the treatment effect. In some cases, baseline measurements are believed to improve design efficiency. This article examines the impact of baselines on optimal designs using two different assumptions about carryover effects during baseline periods and employing a nontraditional crossover design model. As anticipated, baseline observations improve design efficiency considerably for two-period designs, which use the data in the first period only to obtain unbiased estimates of treatment effects, while the improvement is rather modest for three- or four-period designs. Further, we find little additional benefits for measuring baselines at each treatment period as compared to measuring baselines only in the first period. Although our study of baselines did not change the results on optimal designs that are reported in the literature, the problem of strong model dependency problem is generally recognized. The advantage of using multiperiod designs is rather evident, as we found that extending two-period designs to three- or four-period designs significantly reduced variability in estimating the direct treatment effect contrast.     

Using the maximum test statistic in the two-period crossover clinical trial

In a two-period crossover trial where residual carryover is suspected, it is often advised that first-period data only be used in an analysis appropriate for a parallel design. However, it has been shown (Willan and Pater, 1986, Biometrics 42, 593-599) that the crossover analysis is more powerful than the parallel analysis if the residual carryover, expressed as a proportion of treatment effect, is less than 2- square root of 2(1 - rho), where rho is the intrasubject correlation coefficient. Choosing between the analyses based on the empirical evaluation of this condition is equivalent to choosing the analysis with the larger corresponding test statistic. Approximate nominal significance levels are presented that maintain the desired level when basing the analysis on the maximum test statistic. Furthermore, the power and precision of the analysis based on the maximum test statistic are compared to the crossover and parallel analyses.     

A Note on the Analysis of the Two-Period Crossover Design When the Period-Treatment Interaction is Significant

In the analysis of a two-period crossover study Grizzle (1965) suggests that, if a preliminary test for a period by treatment interaction (residual effect, carryover effect) is significant at the 10 % level, the direct effects of the treatments should be compared by performing a t-test on the data from the first period only. In this note it is shown that under Grizzle's model the comparison of the direct treatment effects is equivalent to a Behrens-Fisher problem. Depending on one's viewpoint–Bayesian, fiducial, or sampling theory–different solutions are possible. The solutions are illustrated using three well-known data sets.     

The performance of a two‐stage analysis of ABAB/BABA crossover trials

Freeman has considered the following two‐stage procedure for finding a confidence interval for the treatment difference theta, using data from an AB/BA crossover trial. In the first stage, a preliminary test of the null hypothesis that the differential carryover is zero is carried out. If this hypothesis is accepted then the confidence interval for theta is constructed assuming that the differential carryover is zero. If, on the other hand, this hypothesis is rejected then this confidence interval is constructed using only data from the first period. Freeman has shown that this confidence interval has minimum coverage probability far below nominal. He therefore concludes that this confidence interval should not be used. In the present paper, we analyze the performance of a similar two‐stage procedure for an ABAB/BABA crossover trial. This trial differs in very significant ways from an AB/BA crossover trial, including the fact that for an ABAB/BABA crossover trial there is an unbiased estimator of the differential carryover that is unaffected by between‐subject variation. Despite these great differences, we arrive at the same conclusion as Freeman. Namely, that the confidence interval resulting from the two‐stage procedure should not be used.     

Multiple Hypotheses in the Analysis of a Crossover Trial

Crossover trials are used in a variety of fields, such as medicine, biology, psychology, and some commercial goods investigations. The aim of this paper is to extend a methodology for multiple comparisons to the problem of testing in crossover trials with two treatments. These two treatments are given in two orderings, treatment A first or treatment B first. We perform inference on the effect of one treatment relative to the effect of the other, without assuming that these effects are independent of treatment ordering, using techniques from order-restricted inference and multiple comparisons, and compare to some existing multiple comparison tests.     

Optimal crossover designs in the presence of carryover effects

Under either the random patient-effect model with sequence effects or the fixed patient-effect model, the usual two-period, two-treatment crossover design, AB,BA, cannot be used to estimate the contrast between direct treatment effects when unequal carryover effects are present. If baseline observations are available, the design AB,BA can validly be used to estimate a treatment contrast. However, the design AB,BA,AA,BB with baseline observations is more efficient. In fact, we show that this design is optimal whether or not baseline observations are available. For experiments with more than two periods, universally optimal designs are found for both models, with and without carryover effects. It is shown that uncertainty about the presence of carryover effects is of little or no consequence, and the addition of baseline observations is of little or no added value for designs with three or more periods; however, if the experiment is limited to only two periods the investigator pays a heavy penalty.     

Design and Analysis of Crossover Trials for Absorbing Binary Endpoints

Summary The crossover is a popular and efficient trial design used in the context of patient heterogeneity to assess the effect of treatments that act relatively quickly and whose benefit disappears with discontinuation. Each patient can serve as her own control as within‐individual treatment and placebo responses are compared. Conventional wisdom is that these designs are not appropriate for absorbing binary endpoints, such as death or HIV infection. We explore the use of crossover designs in the context of these absorbing binary endpoints and show that they can be more efficient than the standard parallel group design when there is heterogeneity in individuals' risks. We also introduce a new two‐period design where first period “survivors” are rerandomized for the second period. This design combines the crossover design with the parallel design and achieves some of the efficiency advantages of the crossover design while ensuring that the second period groups are comparable by randomization. We discuss the validity of the new designs and evaluate both a mixture model and a modified Mantel–Haenszel test for inference. The mixture model assumes no carryover or period effects while the Mantel–Haenszel approach conditions out period effects. Simulations are used to compare the different designs and an example is provided to explore practical issues in implementation.     

Estimating individualized treatment regimes from crossover designs

The field of precision medicine aims to tailor treatment based on patient-specific factors in a reproducible way. To this end, estimating an optimal individualized treatment regime (ITR) that recommends treatment decisions based on patient characteristics to maximize the mean of a prespecified outcome is of particular interest. Several methods have been proposed for estimating an optimal ITR from clinical trial data in the parallel group setting where each subject is randomized to a single intervention. However, little work has been done in the area of estimating the optimal ITR from crossover study designs. Such designs naturally lend themselves to precision medicine since they allow for observing the response to multiple treatments for each patient. In this paper, we introduce a method for estimating the optimal ITR using data from a 2 × 2 crossover study with or without carryover effects. The proposed method is similar to policy search methods such as outcome weighted learning; however, we take advantage of the crossover design by using the difference in responses under each treatment as the observed reward. We establish Fisher and global consistency, present numerical experiments, and analyze data from a feeding trial to demonstrate the improved performance of the proposed method compared to standard methods for a parallel study design.     

On variance estimation in crossover designs

It is very unlikely that the errors in crossover experiments with more than two treatments are uncorrelated. The assumption of uncorrelated errors generally leads to underestimation of the variances of estimates. This paper determines bounds for the degree of underestimation for arbitrary covariance matrices, provided the experiment is planned according to a design that has the same numbers of periods and treatments and is balanced for carryover effects.     

Design,Analysis, and Reporting of Crossover Trials for Inclusion in a Meta-Analysis

Randomized crossover trials are clinical experiments in which participants are assigned randomly to a sequence of treatments and each participant serves as his/her own control in estimating treatment effect. We need a better understanding of the validity of their results to enable recommendations as to which crossover trials can be included in meta-analysis and for development of reporting guidelines.

Objective

To evaluate the characteristics of the design, analysis, and reporting of crossover trials for inclusion in a meta-analysis of treatment for primary open-angle glaucoma and to provide empirical evidence to inform the development of tools to assess the validity of the results from crossover trials and reporting guidelines.

Methods

We searched MEDLINE, EMBASE, and Cochrane’s CENTRAL register for randomized crossover trials for a systematic review and network meta-analysis we are conducting. Two individuals independently screened the search results for eligibility and abstracted data from each included report.

Results

We identified 83 crossover trials eligible for inclusion. Issues affecting the risk of bias in crossover trials, such as carryover, period effects and missing data, were often ignored. Some trials failed to accommodate the within-individual differences in the analysis. For a large proportion of the trials, the authors tabulated the results as if they arose from a parallel design. Precision estimates properly accounting for the paired nature of the design were often unavailable from the study reports; consequently, to include trial findings in a meta-analysis would require further manipulation and assumptions.

Conclusions

The high proportion of poorly reported analyses and results has the potential to affect whether crossover data should or can be included in a meta-analysis. There is pressing need for reporting guidelines for crossover trials.     

Non-Parametric Methods for the 2-Period-Cross-Over Design Under Weak Model Assumptions

A nonparametric analysis for the two period cross-over design has first been suggested by Koch (1972) and has been discussed by Hills and Armitage (1979). As known rank tests on sums or differences of the data are applied in this procedure, the results on the one hand are not invariant under monotonous transformations and on the other hand the procedure is only correct for models with additive effects. Therefore, in the present article generalized effects will first be defined in the 2-period cross-over design without the assumption of a linear model and then rank test will be presented which test tese effects without the need of sums or differences of the data. In the appendix the equivalence of the hypothesis for the generalized effects to the known hypotheses for the effects in the linear model will be shown. The application of the procedures will be demonstrated by means of an example in literature.     

Contrasting principal stratum and hypothetical strategy estimands in multi-period crossover trials with incomplete data

Complete case analyses of complete crossover designs provide an opportunity to make comparisons based on patients who can tolerate all treatments. It is argued that this provides a means of estimating a principal stratum strategy estimand, something which is difficult to do in parallel group trials. While some trial users will consider this a relevant aim, others may be interested in hypothetical strategy estimands, that is, the effect that would be found if all patients completed the trial. Whether these estimands differ importantly is a question of interest to the different users of the trial results. This paper derives the difference between principal stratum strategy and hypothetical strategy estimands, where the former is estimated by a complete-case analysis of the crossover design, and a model for the dropout process is assumed. Complete crossover designs, that is, those where all treatments appear in all sequences, and which compare t treatments over p periods with respect to a continuous outcome are considered. Numerical results are presented for Williams designs with four and six periods. Results from a trial of obstructive sleep apnoea-hypopnoea (TOMADO) are also used for illustration. The results demonstrate that the percentage difference between the estimands is modest, exceeding 5% only when the trial has been severely affected by dropouts or if the within-subject correlation is low.     

Using conditional power of network meta‐analysis (NMA) to inform the design of future clinical trials

Clinical trials are typically designed with an aim to reach sufficient power to test a hypothesis about relative effectiveness of two or more interventions. Their role in informing evidence‐based decision‐making demands, however, that they are considered in the context of the existing evidence. Consequently, their planning can be informed by characteristics of relevant systematic reviews and meta‐analyses. In the presence of multiple competing interventions the evidence base has the form of a network of trials, which provides information not only about the required sample size but also about the interventions that should be compared in a future trial. In this paper we present a methodology to evaluate the impact of new studies, their information size, the comparisons involved, and the anticipated heterogeneity on the conditional power (CP) of the updated network meta‐analysis. The methods presented are an extension of the idea of CP initially suggested for a pairwise meta‐analysis and we show how to estimate the required sample size using various combinations of direct and indirect evidence in future trials. We apply the methods to two previously published networks and we show that CP for a treatment comparison is dependent on the magnitude of heterogeneity and the ratio of direct to indirect information in existing and future trials for that comparison. Our methodology can help investigators calculate the required sample size under different assumptions about heterogeneity and make decisions about the number and design of future studies (set of treatments compared).     

Randomization of Neighbour Balanced Designs

This paper is a simulation study on the influence of interference between treatments in field trials. The considerations in the paper are based on a simple model which includes additive neighbour effects of treatments. We use uniformity data where neighbour effects are added, to demonstrate the influence that these effects have on the validity of comparisons between treatments. The simulations illustrate that the influence of the neighbour effects is reduced if a neighbour balanced or a partially neighbour balanced design is used.     

Characterisation of Certain Split-Block Designs with a Control

The paper deals with an incomplete split-block design in which one or two factors are split into two parts, the first one containing test treatments and the second one — a control treatment. The aim of the experiment carried out in such design is to compare the average of the test treatment effects with the control treatment effect and the test treatment effects with the control treatment effect individually. All those comparisons can be expressed by contrasts, elementary or basic ones. The aim of the paper is to characterise particular cases of the incomplete split-block designs with respect to general balance and to efficiency factors of the design with respect to the contrasts. In the paper we restrict our attention to the designs in which each block has two rows or/and two columns only. These designs are suitable for certain agricultural experiments.     

Empirical power and sample size calculations for cluster-randomized and cluster-randomized crossover studies

In recent years, the number of studies using a cluster-randomized design has grown dramatically. In addition, the cluster-randomized crossover design has been touted as a methodological advance that can increase efficiency of cluster-randomized studies in certain situations. While the cluster-randomized crossover trial has become a popular tool, standards of design, analysis, reporting and implementation have not been established for this emergent design. We address one particular aspect of cluster-randomized and cluster-randomized crossover trial design: estimating statistical power. We present a general framework for estimating power via simulation in cluster-randomized studies with or without one or more crossover periods. We have implemented this framework in the clusterPower software package for R, freely available online from the Comprehensive R Archive Network. Our simulation framework is easy to implement and users may customize the methods used for data analysis. We give four examples of using the software in practice. The clusterPower package could play an important role in the design of future cluster-randomized and cluster-randomized crossover studies. This work is the first to establish a universal method for calculating power for both cluster-randomized and cluster-randomized clinical trials. More research is needed to develop standardized and recommended methodology for cluster-randomized crossover studies.     

Analysis of two-period crossover design in a multicenter clinical trial

A technique is discussed for analyzing a two-period crossover design for a multicenter trial using identical study protocols. The technique is a modification of the analysis originally proposed by Grizzle (1965, Biometrics 21, 467-480; 1974, Biometrics 30, 727) for analyzing a two-period crossover design when study is not a factor. A mixed model using the first baseline as a covariate is analyzed to increase the power of the test of significance of the treatment-by-period interaction. The baseline values are also used in a preliminary test.