DROPOUTS IN LONGITUDINAL STUDIES: DEFINITIONS AND MODELS

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 2×2mreplicated 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 2×4 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 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.     

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.     

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.     

Sample size determination for testing equality in a cluster randomized trial with noncompliance

For administrative convenience or cost efficiency, we may often employ a cluster randomized trial (CRT), in which randomized units are clusters of patients rather than individual patients. Furthermore, because of ethical reasons or patient's decision, it is not uncommon to encounter data in which there are patients not complying with their assigned treatments. Thus, the development of a sample size calculation procedure for a CRT with noncompliance is important and useful in practice. Under the exclusion restriction model, we have developed an asymptotic test procedure using a tanh(-1)(x) transformation for testing equality between two treatments among compliers for a CRT with noncompliance. We have further derived a sample size formula accounting for both noncompliance and the intraclass correlation for a desired power 1 - β at a nominal α level. We have employed Monte Carlo simulation to evaluate the finite-sample performance of the proposed test procedure with respect to type I error and the accuracy of the derived sample size calculation formula with respect to power in a variety of situations. Finally, we use the data taken from a CRT studying vitamin A supplementation to reduce mortality among preschool children to illustrate the use of sample size calculation 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.     

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.     

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.     

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.     

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.     

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.     

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.     

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.     

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.     

Nonparametric Sample Size Estimation for Sensitivity and Specificity with Multiple Observations per Subject

We propose a sample size calculation approach for the estimation of sensitivity and specificity of diagnostic tests with multiple observations per subjects. Many diagnostic tests such as diagnostic imaging or periodontal tests are characterized by the presence of multiple observations for each subject. The number of observations frequently varies among subjects in diagnostic imaging experiments or periodontal studies. Nonparametric statistical methods for the analysis of clustered binary data have been recently developed by various authors. In this paper, we derive a sample size formula for sensitivity and specificity of diagnostic tests using the sign test while accounting for multiple observations per subjects. Application of the sample size formula for the design of a diagnostic test is discussed. Since the sample size formula is based on large sample theory, simulation studies are conducted to evaluate the finite sample performance of the proposed method. We compare the performance of the proposed sample size formula with that of the parametric sample size formula that assigns equal weight to each observation. Simulation studies show that the proposed sample size formula generally yields empirical powers closer to the nominal level than the parametric method. Simulation studies also show that the number of subjects required increases as the variability in the number of observations per subject increases and the intracluster correlation increases.     

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.     

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.     

Bioequivalence review for drug interchangeability.

To monitor the performance of the approved generic copies of a brand-name drug, we propose some methods in assessing bioequivalence among generic copies and the brand-name drug, and among generic copies themselves, using data from several bioequivalence studies adopting the standard 2 x 2 crossover design without carryover effects. We propose a meta-analysis method that increases statistical power when the between-subject variability is not large. A nonmeta-analysis is also considered. A numerical example of applying both methods is presented for illustration.     

Test non-inferiority and sample size determination based on the odds ratio under a cluster randomized trial with noncompliance

Because the odds ratio (OR) possesses certain desirable statistical properties, the OR has been recommended elsewhere to measure the relative treatment effect in establishing non-inferiority. For cost efficiency, we may often employ a cluster randomized trial (CRT), in which randomized units are clusters of patients. Furthermore, it is not uncommon to encounter data in which there are patients not complying with their assigned treatment. Under the Dirichlet multinomial model, we have developed a test statistic for assessing non-inferiority based on the OR between two treatments under a CRT with noncompliance. We have further derived a sample size formula accounting for both noncompliance and the intraclass correlation for a desired power 1 - β of detecting non-inferiority with respect to the OR at a nominal α level. Using Monte Carlo simulation, we have evaluated the performance of the proposed test statistic and sample size formula. Finally, we use the CRT studying the effect of vitamin A supplementation on mortality among preschool children to illustrate the use of the sample size formula given here.