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.     

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.     

The Joint Modeling of Drug and Positive Control Effects to Estimate Sample Size and Power in Crossover “Thorough” QT/QTc Studies

We present analytical results for computing the power and sample size in a thorough QT/QTc study with a four-period crossover design in which the treatments are placebo, positive control, supratherapeutic dose of investigational drug, and therapeutic dose of investigational drug. An assessment of noninferiority of the supratherapeutic dose to placebo is performed by the intersection–union test and assay sensitivity is tested (union–intersection test) at prespecified time points using positive control within the framework of a linear mixed-effects analysis. The power and sample size estimates are obtained using the joint distribution of statistics to test noninferiority of the supratherapeutic dose to placebo and to test assay sensitivity using positive control.     

Adaptive Statistical Analysis Following Sample Size Modification Based on Interim Review of Effect Size

ABSTRACT

In designing a comparative clinical trial, the required sample size is a function of the effect size, the value of which is unknown and at best may be estimated from historical data. Insufficiency in sample size as a result of overestimating the effect size can be destructive to the success of the clinical trial. Sample size re-estimation may need to be properly considered as a part of clinical trial planning. This paper is intended to give the motivations for the sample size re-estimation based partly on the effect size observed at an interim analysis and for a resulting simple adaptive test strategy. The performance of this adaptive design strategy is assessed by comparing it with a fixed maximum sample size design that is properly adjusted in anticipation of the possible sample size adjustment.     

Estimating Dose Equivalence for New Routes of Drug Administration

Abstract

For patient's convenience, dose administration of insulin via oral inhalation is often considered as an alternative to subcutaneous administration. An important statistical problem is to estimate dose equivalence, which is the amount of drug needed to be delivered by inhalation to generate an equivalent pharmacokinetic (PK) response produced by a therapeutic dose of subcutaneous insulin. Because of high intersubject variability, a crossover design clinical trial is typically used where data from both routes of administration are obtained from the same subject. A linear mixed effects model is proposed to describe the relationship between AK response and insulin dose for the two routes of administration. Estimation of dose equivalence in this setting has not been discussed in the statistical literature. Several competing methods for estimating dose equivalence are proposed and contrasted. A formula for calculating an approximate sample size necessary to estimate dose equivalence with a desired precision for the new route of administration is also provided.     

Determination of the Efficiency of First Time-In-Man Designs in Healthy Volunteers

The study objective was to develop an approach for evaluating the efficiency of FTIM study designs. Mild adverse event occurrence, dose proportionality, pharmacokinetic parameter estimation efficiency, study duration, and sample size were investigated. There were 15 designs broadly categorized as fixed sequence, crossover, alternating crossover, and parallel dose designs. An efficiency – cost metric was developed to judge design performance.

Dose proportionality was well characterized, irrespective of study design. Three designs were top tier designs with the first two – fixed sequence designs with a combination of doubling and modified Fibonacci dosing schemes differing only panel overlap – preferred.     

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.     

A NOTE ON SAMPLE SIZE CALCULATION FOR MEAN COMPARISONS BASED ON NONCENTRAL t-STATISTICS

One-sample and two-sample t-tests are commonly used in analyzing data from clinical trials in comparing mean responses from two drug products. During the planning stage of a clinical study, a crucial step is the sample size calculation, i.e., the determination of the number of subjects (patients) needed to achieve a desired power (e.g., 80%) for detecting a clinically meaningful difference in the mean drug responses. Based on noncentral t-distributions, we derive some sample size calculation formulas for testing equality, testing therapeutic noninferiority/superiority, and testing therapeutic equivalence, under the popular one-sample design, two-sample parallel design, and two-sample crossover design. Useful tables are constructed and some examples are given for illustration.     

A Bayesian Framework Allowing Incorporation of Retrospective Information in Prospective Diagnostic Biomarker-Validation Designs

Abstract

The sample size of a prospective clinical study aimed at validation of a diagnostic biomarker-based test may be prohibitively large. We present a Bayesian framework that allows incorporating available development-study information about the performance of the test. As a result, the framework allows reducing the sample size required in the validation study, which may render the latter study feasible. The validation is based on the Bayesian testing of a hypothesis regarding possible values of AUC. Toward this end, first, available information is translated into a prior distribution. Next, this prior distribution is used in a Bayesian design to evaluate the performance of the diagnostic-test. We perform a simulation study to compare the power of the proposed Bayesian design to the approach ignoring development-study information. For each scenario, 1000 studies of sample size 100, 400, and 800 are simulated. Overall, the proposed Bayesian design leads to a substantially higher power than the flat-prior design. In some of the considered simulation settings, the Bayesian design requires as little as 50% of the flat-prior traditional design’s sample size to reach approximately the same power. Moreover, a simulation-based application strategy is proposed and presented with respect to a case-study involving the development of a biomarker-based diagnostic-test for Alzheimer’s disease.     

ON SAMPLE SIZE CALCULATION BASED ON ODDS RATIO IN CLINICAL TRIALS

Sample size calculation formulas for testing equality, noninferiority, superiority, and equivalence based on odds ratio were derived under both parallel and one-arm crossover designs. An example concerning the study of odds ratio between a test compound (treatment) and a standard therapy (control) for prevention of relapse in subjects with schizophrenia and schizoaffective disorder is presented to illustrate the derived formulas for sample size calculation for various hypotheses under both a parallel design and a crossover design. Simulations were performed to assess the adequacy of the sample size calculation formulas. Simulation results were given at the end of the paper.     

On sample size calculation based on odds ratio in clinical trials

Sample size calculation formulas for testing equality, noninferiority, superiority, and equivalence based on odds ratio were derived under both parallel and one-arm crossover designs. An example concerning the study of odds ratio between a test compound (treatment) and a standard therapy (control) for prevention of relapse in subjects with schizophrenia and schizoaffective disorder is presented to illustrate the derived formulas for sample size calculation for various hypotheses under both a parallel design and a crossover design. Simulations were performed to assess the adequacy of the sample size calculation formulas. Simulation results were given at the end of the paper.     

Considering Over-Dispersion in the Sample Size Calculation for Clinical Trials With Repeated Count Measurements

Abstract

Over-dispersed count variables are frequently encountered in biomedical research. Despite extensive research in analytical methods, addressing over-dispersion in the design of clinical trials has received much less attention. In this study, we propose to directly incorporate over-dispersion into sample size calculation for clinical trials where a count outcome is repeatedly measured on each subject. The proposed method is applicable to the comparison of slopes as well as time-averaged responses. It is easy to compute and flexible enough to account for unbalanced randomization, arbitrary missing patterns, and different correlation structures. We show that sample size requirement is proportional to over-dispersion, which highlights the danger of ignoring over-dispersion in experimental design. Simulation results demonstrate that the proposed sample size calculation methods maintain the nominal levels of power and Type I error over a wide range of scenarios. Application example to an epileptic trial is presented. Supplementary materials for this article are available online.     

On Sample Size Calculation in Bioequivalence Trials

Sample size calculation plays an important role in bioequivalence trials. In practice, a bioequivalence study is usually conducted under a crossover design or a parallel design with raw data or log-transformed data. In this paper, we discuss the differences in sample size calculation between a crossover design and a parallel design with raw data or log-transformed data. Formulas for sample size calculation under a crossover design and a parallel design with raw data or log-transformed data are derived. A brief discussion for the relationship among these formulas is given.     

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.     

Adaptive clinical endpoint bioequivalence studies with sample size re-estimation based on a nuisance parameter

ABSTRACT

A clinical endpoint bioequivalence (BE) study is often used to establish bioequivalence (BE) between a locally acting generic drug (T) and an innovator drug (R), which is a double-blind, randomized three-arm (T, R and placebo: P) parallel clinical trial. BE is established if two superiority tests (T vs. P, R vs. P) and one equivalence test (T vs. R) all pass. An accurate estimate of the nuisance parameter (e.g. variance) is vital in determining an accurate sample size to attain sufficient power. However, due to potential study design variations between NDA and Abbreviated NDA (ANDA) studies and high variability of clinical endpoints, variance may be over- or under-estimated, resulting in unnecessary extra costs or underpowered studies. Traditionally, clinical endpoint BE studies use a fixed study design. In this work, we propose four sample size re-estimation approaches based on a nuisance parameter and recommend one approach after comparing various operating characteristics by simulation.

The proposed adaptive design with sample size re-estimation provides a more accurate estimate of sample size without wasting resources or under-powering the study and controls the Type 1 error rate under a negligible level, both for the family-wise alpha and individual alpha for superiority and equivalence tests.     

临床试验Williams设计中样本含量的估算方法

在临床试验中,2组交叉设计应用已相当广泛,其样本含量估算方法也被研究者所熟悉。多组交叉试验由Williams首先提出,因此被称为Wil-liams设计。本文介绍基于Williams设计的样本含量估算方法,并提供示例分析,供研究者参考使用。     

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.     

A Note on Sample Size Determination for Bioequivalence Studies with Higher-Order Crossover Designs

Similar to Liu and Chow, approximate formulas for sample size determination are derived based on Schuirmann's two one-sided tests procedure for bioequiealence studies for the additive and the multiplicative models under various higher order crossover designs for comparing two formulations of a drug product. The higher order crossover designs under study include Balaam's design, the two-sequence dual design, and two four-period designs (with two and four sequences), which are commonly used for assessment of bioequivalence between formulations. The derived formulas are simple enough to be carried out with a pocket calculator. The number of subjects required for each of the four higher order designs are tabulated for selected powers and various parameter values.     

Adjustment for unbalanced sample size for analytical biosimilar equivalence assessment

ABSTRACT

Large sample size imbalance is not uncommon in the biosimilar development. At the beginning of a product development, sample sizes of a biosimilar and a reference product may be limited. Thus, a sample size calculation may not be feasible. During the development stage, more batches of reference products may be added at a later stage to have a more reliable estimate of the reference variability. On the other hand, we also need a sufficient number of biosimilar batches in order to have a better understanding of the product. Those challenges lead to a potential sample size imbalance. In this paper, we show that large sample size imbalance may increase the power of the equivalence test in an unfavorable way, giving higher power for less similar products when the sample size of biosimilar is much smaller than that of the reference product. Thus, it is necessary to make some sample size imbalance adjustments to motivate sufficient sample size for biosimilar as well. This paper discusses two adjustment methods for the equivalence test in analytical biosimilarity studies. Please keep in mind that sufficient sample sizes for both biosimilar and reference products (if feasible) are desired during the planning stage.     

Two-Stage Adaptive Designs for Three-Treatment Bioequivalence Studies

Abstract

Bioequivalence (BE) studies are most often conducted as crossover trials, and therefore establishing their required sample size necessitates specification of the within-person variance. Given that this specification is often difficult in practice, there has been great interest in recent years in the use of adaptive designs for BE trials. However, while numerous methods for this have now been presented, their focus has been solely on two-treatment BE studies. In some instances, it will be desired to incorporate more than a single test and reference formulation into a BE trial. It would therefore be useful to establish methodology for the design of adaptive multi-treatment BE trials, to acquire the benefits in the two-treatment setting in this more complex situation. Here, we achieve this for three-treatment studies by extending previously proposed designs for two-treatment trials. First, we discuss the additional design considerations that arise when multiple comparisons are made. Next, an extensive simulation study is employed to compare the performance of the proposed procedures. With this, we demonstrate that two-stage designs with desirable statistical operating characteristics can be readily identified for three-treatment BE trials. Supplementary materials for this article are available online.