Tests for inter-subject and total variabilities under crossover designs

In this paper, we consider statistical tests for inter-subject and total variabilities between treatments under crossover designs. Since estimators of variance components for inter-subject variability and total variability in crossover design are not independent, the usual F-test cannot be applied. Alternatively, we propose a test based on the concept of the extension of the modified large sample method to compare inter-subject variability and total variability between treatments under a 2 x 2 m replicated crossover design. An asymptotic power of the proposed test is derived. A sensitivity analysis is performed based on the asymptotic power to determine how the power changes with respect to various parameters such as inter-subject correlation and intra-class correlation. Also the two methods for sample size calculation for testing total variability under 2 x 4 crossover design are discussed. The method based on the Fisher-Cornish inversion shows better performance than the method based on the normal approximation. Several simulation studies were conducted to investigate the finite sample performance of the proposed test. Our simulation results show that the proposed test can control type I error satisfactorily.     

ON STATISTICAL POWER FOR AVERAGE BIOEQUIVALENCE TESTING UNDER REPLICATED CROSSOVER DESIGNS

In its recent guidance on bioequivalence, the U.S. Food and Drug Administration (FDA) recommends a two-sequence, four-period (2×4)replicated crossover design be used for assessment of population and individual bioequivalence [FDA. Guidance for Industry on Statistical Approaches to Establishing Bioequivalence; Center for Drug Evaluation and Research, Food and Drug Administration: Rockville, MD, 2001]. The recommended replicated crossover design not only allows estimates of both the inter-subject and the intra-subject variabilities and the variability due to subject-by-formulation interaction, but also provides an assessment of average bioequivalence (ABE). In this article, power function for assessment of ABE under a general replicated crossover design (i.e., a 2×2mreplicated crossover design) based on the traditional analysis of variance model and the mixed effects model as suggested by the FDA are studied. It is found that the power of a 2×2mreplicated crossover design depends upon the variability due to subject-by-formulation interaction and the number of replicates. Based on the derived power function, formula for sample size calculation for assessment of ABE under a 2×2mreplicated crossover design is also provided.     

On statistical power for average bioequivalence testing under replicated crossover designs

In its recent guidance on bioequivalence, the U.S. Food and Drug Administration (FDA) recommends a two-sequence, four-period (2 x 4) replicated crossover design be used for assessment of population and individual bioequivalence [FDA. Guidance for Industry on Statistical Approaches to Establishing Bioequivalence; Center for Drug Evaluation and Research, Food and Drug Administration: Rockville, MD, 2001]. The recommended replicated crossover design not only allows estimates of both the inter-subject and the intra-subject variabilities and the variability due to subject-by-formulation interaction, but also provides an assessment of average bioequivalence (ABE). In this article, power function for assessment of ABE under a general replicated crossover design (i.e., a 2 x 2m replicated crossover design) based on the traditional analysis of variance model and the mixed effects model as suggested by the FDA are studied. It is found that the power of a 2 x 2m replicated crossover design depends upon the variability due to subject-by-formulation interaction and the number of replicates. Based on the derived power function, formula for sample size calculation for assessment of ABE under a 2 x 2m replicated crossover design is also provided.     

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.     

Statistical assessment of QT/QTc prolongation based on maximum of correlated normal random variables

To establish noninferiority in QT/QTc prolongation of a test drug with respect to either a placebo or an active control, a thorough QT/QTc study is recommended by ICH (ICH E14, ICH 2005) which concerns statistical inference on the maximal time-matched drug effect. The existing statistical methods for assessing such effects suffer either power loss or parameter restriction. In this paper, we propose a new asymptotic test with small sample correction based on distribution of maximum of correlated random variables under both a parallel-group design and a crossover design. Simulations indicate that our proposed test has adequate powers.     

Application of Clinical Trial Simulation to Compare Proof-of-Concept Study Designs for Drugs with a Slow Onset of Effect; An Example in Alzheimer's Disease

Objective Clinical trial simulation (CTS) was used to select a robust design to test the hypothesis that a new treatment was effective for Alzheimer's disease (AD). Typically, a parallel group, placebo controlled, 12-week trial in 200–400 AD patients would be used to establish drug effect relative to placebo (i.e., Ho: Drug Effect = 0). We evaluated if a crossover design would allow smaller and shorter duration trials.Materials and Methods A family of plausible drug and disease models describing the time course of the AD assessment scale (ADAS-Cog) was developed based on Phase I data and literature reports of other treatments for AD. The models included pharmacokinetic, pharmacodynamic, disease progression, and placebo components. Eight alternative trial designs were explored via simulation. One hundred replicates of each combination of drug and disease model and trial design were simulated. A ‘positive trial’ reflecting drug activity was declared considering both a dose trend test (p < 0.05) and pair-wise comparisons to placebo (p < 0.025).Results A 4 × 4 Latin Square design was predicted to have at least 80% power to detect activity across a range of drug and disease models. The trial design was subsequently implemented and the trial was completed. Based on the results of the actual trial, a conclusive decision about further development was taken. The crossover design provided enhanced power over a parallel group design due to the lower residual variability.Conclusion CTS aided the decision to use a more efficient proof of concept trial design, leading to savings of up to US$4M in direct costs and a firm decision 8–12 months earlier than a 12-week parallel group trial.     

Mean group tests for stationarity in heterogeneous panels

Summary This paper proposes a panel‐based mean group test for the null of stationarity against the alternative of unit roots in the presence of both heterogeneity across cross‐section units and serial correlation across time periods. Using both sequential and joint asymptotic analyses the proposed test statistic is shown to be distributed as standard normal under the null for large N (number of groups) and large T (number of time periods). Monte Carlo results support the use of joint asymptotic limits (under the further condition that N/T→ 0) as a guide to finite sample performance, but also clearly indicate that the power of our suggested panel‐based test is substantially higher than that of the single time‐series‐based test.     

Crossover Designs With Two Binary Endpoints

ABSTRACT

We propose a crossover design for simultaneous significance testing of two binary endpoints, in which the AB/BA crossover design is carried out for each endpoint. An asymptotic α-level test is obtained by applying the intersection-union principle to the marginal Mainland–Gart tests. Power approximations and sample size calculation are derived and implemented in R programs. An adaptive design with sample size reestimation is also presented. We demonstrate the numerical accuracy of the proposed design through an extensive simulation study. Supplementary materials for this article are available online.     

Evaluation of parametric and nonparametric two one-sided tests procedures for assessing bioequivalence of average bioavailability

A simulation study was conducted to compare the levels of significance and power between Schuirmann's and nonparametric two one-sided tests procedures for a 2 ×2 crossover design under different combinations of sample sizes, intrasubject variabilities, and underlying distributions. Empirical results suggest that Schuirmann's two one-sided tests procedure is robust to minor departure from the assumption of normality.     

Cost-efficient higher-order crossover designs in comparative bioavailability studies

BACKGROUND AND OBJECTIVE: Cost is an extremely important factor to consider when planning drug clinical trials. Higher-order crossover designs have recently drawn considerable attention in comparative bioavailability studies because of their desirable statistical properties. In this paper, we compared the cost efficiency of five commonly used higher-order crossover designs under certain cost function for comparative bioavailability studies. METHODS: Multivariate normal data were simulated under scenarios of a wide range of variability and correlations (coefficient of variation = 10-40%; correlation coefficient rho = 0.2-0.8). Monte Carlo simulations and mixed-effects models were carried out to obtain empirical sample sizes for each design using Schuirmann's two-one sided test procedure, under an 80% power and a 5% significance level, based on the US FDA bioequivalence criteria (80-125%). The five crossover designs studied were the two-period four-sequence (D2 x 4), the three-period two-sequence (D3 x 2), the three-period four-sequence (D3 x 4), the four-period two-sequence (D4 x 2), and the four-period four-sequence (D4 x 4). Costs for each design were then determined by a cost function, which takes into account costs for recruiting and screening, costs associated with period, and the overheads incurred for multiple sequences. Comparison of the costs for the above-mentioned designs was made under different scenarios. RESULTS: There was no single design uniformly dominating the others in terms of cost efficiency for comparative bioavailability studies. The designs D3 x 2 and D4 x 4 (especially the former) have the best overall performance in terms of cost efficiency for comparative bioavailability studies. They dominated the other designs under most of the scenarios. The design D2 x 4 showed the worst performance among the five crossover designs. CONCLUSIONS: A D3 x 2, and D4 x 2 crossover designs are recommended to achieve cost efficiency with a given power. The D2 x 4 crossover design is not recommended in general for comparative bioavailability studies.     

Specification testing in nonstationary time series models

In this paper, we consider a specification testing problem in nonlinear time series models with nonstationary regressors, and we propose using a nonparametric kernel‐based test statistic. The null asymptotics for the proposed nonparametric test statistic have been well developed in the existing literature. In this paper, we study the local asymptotics of the test statistic (i.e. the asymptotic properties of the test statistic under a sequence of general nonparametric local alternatives) and show that the asymptotic distribution depends on the asymptotic behaviour of the distance function, which is the local deviation from the parametrically specified model in the null hypothesis. In order to implement the proposed test in practice, we introduce a bootstrap procedure to approximate the critical values of the test statistic and establish a new Edgeworth expansion, which is used to justify the use of such an approximation. Based on the approximate critical values, we develop a bandwidth selection method, which chooses the optimal bandwidth that maximizes the local power of the test while its size is controlled at a given significance level. The local power is defined as the power of the proposed test for a given sequence of local alternatives. Such a bandwidth selection is made feasible by an approximate expression for the local power of the test as a function of the bandwidth. A Monte Carlo simulation study is provided to illustrate the finite sample performance of the proposed test.     

Sample Size Calculations for Population Pharmacodynamic Experiments Involving Repeated Dichotomous Observations

Analysis of repeated binary measurements presents a challenge in terms of the correlation between measurements within an individual and a mixed-effects modelling approach has been used for the analysis of such data. Sample size calculation is an important part of clinical trial design and it is often based on the method of analysis. We present a method for calculating the sample size for repeated binary pharmacodynamic measurements based on analysis by mixed-effects modelling and using a logit transformation. Wald test is used for hypothesis testing. The method can be used to calculate the sample size required for detecting parameter differences between subpopulations. Extensions to account for unequal allocation of subjects across groups and unbalanced sampling designs between and within groups were also derived. The proposed method has been assessed via simulation of a linear model and estimation using NONMEM. The results showed good agreement between nominal power and power estimated from the NONMEM simulations. The results also showed that sample size increases with increased variability at a rate that depends on the difference in parameter estimates between groups, and designs that involve sampling based on an optimal design can help to reduce cost.     

Sample size calculations for population pharmacodynamic experiments involving repeated dichotomous observations

Analysis of repeated binary measurements presents a challenge in terms of the correlation between measurements within an individual and a mixed-effects modelling approach has been used for the analysis of such data. Sample size calculation is an important part of clinical trial design and it is often based on the method of analysis. We present a method for calculating the sample size for repeated binary pharmacodynamic measurements based on analysis by mixed-effects modelling and using a logit transformation. Wald test is used for hypothesis testing. The method can be used to calculate the sample size required for detecting parameter differences between subpopulations. Extensions to account for unequal allocation of subjects across groups and unbalanced sampling designs between and within groups were also derived. The proposed method has been assessed via simulation of a linear model and estimation using NONMEM. The results showed good agreement between nominal power and power estimated from the NONMEM simulations. The results also showed that sample size increases with increased variability at a rate that depends on the difference in parameter estimates between groups, and designs that involve sampling based on an optimal design can help to reduce cost.     

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.     

On hybrid parallel–crossover designs for assessing drug interchangeability of biosimilar products

ABSTRACT

In recent years, a specific hybrid parallel–crossover design that consists of two sequences of treatments, namely R–R–R–R and R–T–R–T, where T and R is a proposed biosimilar product and an innovative biological product, respectively, have been proposed and received much attention for assessing drug interchangeability between T and R, where R could be either a US-licensed product or an EU-reference product. In practice, there are three types of hybrid parallel–crossover designs that are commonly employed in assessing drug interchangeability of biosimilar products. These three types of parallel–crossover hybrid designs include (1) a parallel + 2 × 2 crossover design, (2) a parallel + 2 × 3 crossover design, and (3) a parallel + 2 × 4 crossover design. This article provides a comprehensive review of these study designs including a complete N-of-1 randomized trial design. A specific hybrid parallel–crossover design, that is, (RRRR, RTRT) for addressing drug interchangeability in terms of switching and the relative risk between with/without alternation is discussed.     

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.     

Tests for equivalence based on odds ratio for matched-pair design

Currently, methods for evaluation of equivalence under a matched-pair design use either difference in proportions or relative risk as measures of risk association. However, these measures of association are only for cross-sectional studies or prospective investigations, such as clinical trials and they cannot be applied to retrospective research such as case-control studies. As a result, under a matched-pair design, we propose the use of the conditional odds ratio for assessment of equivalence in both prospective and retrospective research. We suggest the use of the asymptotic confidence interval of the conditional odds ratio for evaluation of equivalence. In addition, a score test based on the restricted maximum likelihood estimator (RMLE) is derived to test the hypothesis of equivalence under a matched-pair design. On the other hand, a sample size formula is also provided. A simulation study was conducted to empirically investigate the size and power of the proposed procedures. Simulation results show that the score test not only adequately controls the Type I error but it can also provide sufficient power. A numerical example illustrates the proposed methods.     

Tests for Equivalence Based on Odds Ratio for Matched-Pair Design

Currently, methods for evaluation of equivalence under a matched-pair design use either difference in proportions or relative risk as measures of risk association. However, these measures of association are only for cross-sectional studies or prospective investigations, such as clinical trials and they cannot be applied to retrospective research such as case-control studies. As a result, under a matched-pair design, we propose the use of the conditional odds ratio for assessment of equivalence in both prospective and retrospective research. We suggest the use of the asymptotic confidence interval of the conditional odds ratio for evaluation of equivalence. In addition, a score test based on the restricted maximum likelihood estimator (RMLE) is derived to test the hypothesis of equivalence under a matched-pair design. On the other hand, a sample size formula is also provided. A simulation study was conducted to empirically investigate the size and power of the proposed procedures. Simulation results show that the score test not only adequately controls the Type I error but it can also provide sufficient power. A numerical example illustrates the proposed methods.     

Sample Size Calculations for Combination Drugs of Two Monotherapies With One Approved Dose Level

In this article, we consider sample size calculations for combination drugs of two monotherapies that each has only one approved dose level. We modify the method of Laska and Meiner by employing the asymptotic joint distribution of test statistics to derive the power function and using unequal allocation to minimize the total sample sizes. Two cases are investigated. In the first case, each monotherapy has the same indication. A heuristic method, the method of Laska and Meiner, and the proposed method are compared in terms of the total sample sizes. We show that the proposed method produces the smallest total sample sizes. In the second case, each monotherapy has a different indication. While the method of Laska and Meiner cannot be applied in this case, we show that the proposed method can be employed and that it produces smaller total sample sizes than a heuristic method.     

On power and sample size calculation for QT studies with recording replicates at given time point

The problem of the impact on power and sample size calculation for routine QT studies with ECG recording replicates under a parallel-group design and a crossover design is examined. Replicate ECGs are defined as single ECG recorded within several minutes of a nominal time (PhRMA, 2003). Formulas for sample size calculations with and without adjustment for covariates such as some pharmacokinetic responses (e.g., AUC or C(max)), which are known to be correlated to the QT intervals, were derived under both the parallel-group design and the crossover design. The results indicate that the approach of replicates may require a smaller sample size for achieving the same power when the correlation coefficient between the recording replicates (or repeated measures) is close to 0 (i.e., these replicate ECGs are almost independent). On the other hand, if the correlation coefficient is close to 1, then there is not much gain regardless of whether replicate ECGs are considered. In this paper, an approach to identifying optimal allocation between the number of subjects and the number of replicates per subject is proposed for achieving the maximum power under a fixed budget constraint. The proposed approach can also be applied to minimize the cost for a given power.