This chapter provides an overview of recent developments in the design and analysis of cross-over trials. We first consider the analysis of the trial that compares two treatments, A and B, over two periods and where the subjects are randomized to the treatment sequences AB and BA. We make the distinction between fixed and random effects models and show how these models can easily be fitted using modern software. Issues with fitting and testing for a difference in carry-over effects are described and the use of baseline measurements is discussed. Simple methods for testing for a treatment difference when the data are binary are also described. Various designs with two or more treatments but with three or four periods are then described and compared. These include the balanced and partially balanced designs for three or more treatments and designs for factorial treatment combinations. Also described are nearly balanced and nearly strongly balanced designs. Random subject-effects models for the designs with two or more treatments are described and methods for analysing non-normal data are also given. The chapter concludes with a description of the use of cross-over designs in the testing of bioequivalence.

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Design and Analysis of Cross-Over Trials

Therapeutic drug monitoring (TDM) is the quantification and interpretation of drug concentrations in blood to optimize pharmacotherapy. It considers the interindividual variability of pharmacokinetics and thus enables personalized pharmacotherapy. In psychiatry and neurology, patient populations that may particularly benefit from TDM are children and adolescents, pregnant women, elderly patients, individuals with intellectual disabilities, patients with substance abuse disorders, forensic psychiatric patients or patients with known or suspected pharmacokinetic abnormalities. Non-response at therapeutic doses, uncertain drug adherence, suboptimal tolerability, or pharmacokinetic drug-drug interactions are typical indications for TDM. However, the potential benefits of TDM to optimize pharmacotherapy can only be obtained if the method is adequately integrated in the clinical treatment process. To supply treating physicians and laboratories with valid information on TDM, the TDM task force of the Arbeitsgemeinschaft fur Neuropsychopharmakologie und Pharmakopsychiatrie (AGNP) issued their first guidelines for TDM in psychiatry in 2004. After an update in 2011, it was time for the next update. Following the new guidelines holds the potential to improve neuropsychopharmacotherapy, accelerate the recovery of many patients, and reduce health care costs.

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Consensus Guidelines for Therapeutic Drug Monitoring in Neuropsychopharmacology: Update 2017.

INTRODUCTION Regulatory Requirement Basic Considerations Procedures for Sample Size Calculation Aims and Structure of the Book CONSIDERATIONS PRIOR TO SAMPLE SIZE CALCULATION Confounding and Interaction One-Sided Test Versus Two-Sided Test Crossover Design Versus Parallel Design Subgroup/Interim Analyses Data Transformation Practical Issues COMPARING MEANS One-Sample Design Two-Sample Parallel Design Two-Sample Crossover Design Multiple-Sample One-Way ANOVA Multiple-Sample Williams Design Practical Issues LARGE SAMPLE TESTS FOR PROPORTIONS One-Sample Design Two-Sample Parallel Design Two-Sample Crossover Design One-Way Analysis of Variance Williams Design Relative Risk - Parallel Design Relative Risk - Crossover Design Practical Issues EXACT TESTS FOR PROPORTIONS Binomial Test Fisher's Exact Test Optimal Multiple-Stage Designs for Single Arm Trials Flexible Designs for Multiple-Arm Trials Remarks TESTS FOR GOODNESS-OF-FIT AND CONTINGENCY TABLES Tests for Goodness-of-Fit Test for Independence -Single Stratum Test for Independence -Multiple Strata Test for Categorical Shift Carry-Over Effect Test Practical Issues COMPARING TIME-TO-EVENT DATA Basic Concepts Exponential Model Cox's Proportional Hazards Model Weighted Log-Rank Test Practical Issues GROUP SEQUENTIAL METHODS Pocock's Test O'Brien and Fleming's Test Wang and Tsiatis' Test Inner Wedge Test Binary Variables Time-to-Event Data Alpha Spending Function Sample Size Re-Estimation Conditional Power Practical Issues COMPARING VARIABILITIES Comparing Intra-Subject Variabilities Comparing Intra-Subject CVs Comparing Inter-Subject Variabilities Comparing Total Variabilities Practical Issues BIOEQUIVALENCE TESTING Bioequivalence Criteria Average Bioequivalence Population Bioequivalence Individual Bioequivalence In Vitro Bioequivalence NONPARAMETRICS Violation of Assumptions One-Sample Location Problem Two-Sample Location Problem Test for Independence Practical Issues SAMPLE SIZE CALCULATION IN OTHER AREAS Dose Response Studies ANOVA with Repeated Measures Quality of Life Bridging Studies Vaccine Clinical Trials Appendix: Tables of Quantiles References Index

Sample Size Calculations in Clinical Research

Therapeutic drug monitoring (TDM), i. e., the quantification of serum or plasma concentrations of medications for dose optimization, has proven a valuable tool for the patient-matched psychopharmacotherapy. Uncertain drug adherence, suboptimal tolerability, non-response at therapeutic doses, or pharmacokinetic drug-drug interactions are typical situations when measurement of medication concentrations is helpful. Patient populations that may predominantly benefit from TDM in psychiatry are children, pregnant women, elderly patients, individuals with intelligence disabilities, forensic patients, patients with known or suspected genetically determined pharmacokinetic abnormalities or individuals with pharmacokinetically relevant comorbidities. However, the potential benefits of TDM for optimization of pharmacotherapy can only be obtained if the method is adequately integrated into the clinical treatment process. To promote an appropriate use of TDM, the TDM expert group of the Arbeitsgemeinschaft fur Neuropsychopharmakologie und Pharmakopsychiatrie (AGNP) issued guidelines for TDM in psychiatry in 2004. Since then, knowledge has advanced significantly, and new psychopharmacologic agents have been introduced that are also candidates for TDM. Therefore the TDM consensus guidelines were updated and extended to 128 neuropsychiatric drugs. 4 levels of recommendation for using TDM were defined ranging from “strongly recommended” to “potentially useful”. Evidence-based “therapeutic reference ranges” and “dose related reference ranges” were elaborated after an extensive literature search and a structured internal review process. A “laboratory alert level” was introduced, i. e., a plasma level at or above which the laboratory should immediately inform the treating physician. Supportive information such as cytochrome P450 substrate- and inhibitor properties of medications, normal ranges of ratios of concentrations of drug metabolite to parent drug and recommendations for the interpretative services are given. Recommendations when to combine TDM with pharmacogenetic tests are also provided. Following the guidelines will help to improve the outcomes of psychopharmacotherapy of many patients especially in case of pharmacokinetic problems. Thereby, one should never forget that TDM is an interdisciplinary task that sometimes requires the respectful discussion of apparently discrepant data so that, ultimately, the patient can profit from such a joint effort.

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AGNP Consensus Guidelines for Therapeutic Drug Monitoring in Psychiatry: Update 2011

The Hill equation was first introduced by A.V. Hill to describe the equilibrium relationship between oxygen tension and the saturation of haemoglobin. In pharmacology, the Hill equation has been extensively used to analyse quantitative drug-receptor relationships. Many pharmacokinetic-pharmacodynamic models have used the Hill equation to describe nonlinear drug dose-response relationships. Although the Hill equation is widely used, its many properties are not all well known. This article aims at reviewing the various properties of the Hill equation. The descriptive aspects of the Hill equation, in particular mathematical and graphical properties, are examined, and related to Hill's original work. The mechanistic aspect of the Hill equation, involving a strong connection with the Guldberg and Waage law of mass action, is also described. Finally, a probabilistic view of the Hill equation is examined. Here, we provide some new calculation results, such as Fisher information and Shannon entropy, and we introduce multivariate probabilistic Hill equations. The main features and potential applications of this probabilistic approach are also discussed. Thus, within the same formalism, the Hill equation has many different properties which can be of great interest for those interested in mathematical modelling in pharmacology and biosciences.

The Hill equation: a review of its capabilities in pharmacological modelling

I Introduction.- II Time for {2}-Inverses.- 2.0 Introduction.- 2.1 The Three Phase Inversion Procedure.- 2.2 Constrained Inverses.- 2.3 {2}- and {1,2}-Inverses: The Null Augmented Mappings.- 2.4 {1,5}-Inverses: The Nonnull Augmented Mappings.- 2.5 Construction of Moore-Penrose Type Generalized Inverses.- 2.6 A Geometric Representation of {2}-Inverses.- 2.7 {1,5}-Inverses and Projections.- 2.8 {1,5}-Inverses and Solutions to Linear Equations.- 2.9 Decomposition of {2}-Inverses.- 2.10 Spectral Decomposition in Terms of {2}Inverses.- 2.11 Computation of {2}-Inverses.- III {2}-Inverses, Quadratic Forms and Second Degree Polynomials.- 3.0 Introduction.- 3.1 x2 Distribution and Independence of Quadratic Forms and Second Degree Polynomials.- 3.2 Generalized Inverses and Quadratic Forms.- 3.3 {2}-Inverses and x2 Distributed Quadratic Forms.- 3.4 On The Uniqueness of the {2}-Inverse Representation of x2 Distributed Quadratic Forms.- 3.5 A Minimal Sufficient Set of Coefficient Matrices for All X2 Distributed Quadratic Forms.- 3.6 Independence of X2 Distributed Quadratic Forms.- 3.7 A Canonical Representation of Second Degree Polynomials.- 3.8 X2 Distributed Second Degree Polynomials.- 3.9 {2}-Inverses and the Distribution and Independence of Second Degree Polynomials.- IV {2}-Inverses and Least Squares Solutions.- 4.0 Introduction.- 4.1 The Least Squares Problem.- 4.1 Strategies For Obtaining Least Squares Solutions.- 4.3 Symmetric {1,2}-Inverses and Sets of Nonestimable Constraints.- 4.4 Bott-Duffin Inverses and Constrained LSS's.- 4.5 {1,5}-Inverses and LSS's.- 4.6 Relationships Among LSS's.- 4.7 Minimum Norm LSS's.- 4.8 A General Theorem on Constrained LSS's.- 4.9 Residual Sum of Squares and Their Difference.- 4.10 Computing Constrained LSS's and Residual Sum of Squares.- V {2}-Inverses in Linear Models.- 5.0 Introduction.- 5.1 The Models.- 5.2 The Distribution and Relationships Among the LSS's For the Prameters in Various Models.- 5.3 Hypothesis Testing in Linear Models.- 5.4 Equivalent Numerator Sum of Squares for a Test of Hypothesis.- 5.5 Hypotheses Invariant to Cell Sizes.- 5.6 The R Approach and SAS Type I and Type II Sums of Squares.- 5.7 The R* Approach and the SAS Type III Sum of Squares.- References.

2-inverses and their statistical application

Ordinarily a Bayesian estimation procedure uses one prior distribution to obtain a unique estimation rule (its Bayes rule). From the decision theoretical point of view, this procedure can be regarded as a convenient way to obtain admissible decision rules. However, many intuitively appealing, admissible estimation rules cannot be obtained directly in this way. We propose a new mechanism, called the Stepwise Bayesian Procedure (SBP). When the parameter space contains only finitely-many points and the loss function is strictly convex, this SBP can be used to obtain every admissible estimation rule. A relationship between SBP and the limiting Bayes rules is given.

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A Stepwise Bayesian Procedure

The extent of drug availability is often measured by the area under the concentration-time curve. In animal studies, experimental constraints can limit the number of observations available on each animal. Estimation of area under the curve and its standard error are straightforward when each animal is measured at each time point. Bailer [1] and Nedelman et al. [2], have described techniques for estimating the area under the curve and its standard error when each animal is measured once. Yeh [3] has described a technique for the hybrid case where animals are measured more than once, but not at all time points. We describe a method for estimating area under the curve and its standard error which is applicable to all three types of designs. We give formulas for testing treatment differences, including dose trends and dose proportionality, in area under the curve for designs containing an arbitrary number of treatments. A jackknife estimator is also described.

A method for estimating and testing area under the curve in serial sacrifice, batch, and complete data designs.

In December of 1997, the FDA proposed a draft guidance for future in vivo bioequivalence studies. The guidance suggested specific criteria for new drug sponsors to show individual bioequivalence (IBE). The criteria use a mixed-scaling aggregate strategy. It has been generally accepted that, under some particular situations, the proposed criteria would result in a relaxation of the current bioequivalence standard set by the average bioequivalence (ABE) criterion. Here we study the magnitude of this relaxation under three scenarios: when the conditions for an ABE investigation are met; when the drugs are highly variable, and when the experiments are poorly conducted. The magnitude of relaxation we report here may be surprisingly large to many. For example, when a drug is highly variable (with the intrasubject coefficient of variation reaching 40 per cent), the allowable limit for the ratio of the formulation means could reach 55-180 per cent in an IBE investigation. In comparison, the usual allowable limit in an ABE investigation is 80-125 per cent. Our investigation raises doubts on whether the implied standard of the new proposed IBE criteria would adequately ensure switchability in highly variable drugs.

Some statistical considerations on the FDA draft guidance for individual bioequivalence.

When a new drug is under development, a conventional dose-finding study involves learning about the dose–response curve in order to bring forward right doses of the drug to late-stage development. We propose an adaptive procedure for dose-finding in clinical trials in the presence of both efficacy and toxicity endpoints. We use the principles of optimal experimental designs for bivariate continuous endpoints.However, instead of using the traditional D-optimal design, which favors collective ethics but neglects the individual ethics, we consider the penalized D-optimal design that achieves an appropriate balance between the efficient treatment of patients in the trial (by penalizing allocation of patients to ineffective or toxic doses) and the precise estimation of the model parameters to be used in the identification of the target dose. This is compared with the traditional fixed allocation design in terms of allocation of subjects and precision of the identified dose–response curve and selection of the t...

Francis Hsuan

Papers

2-inverses and their statistical application

A Stepwise Bayesian Procedure

A method for estimating and testing area under the curve in serial sacrifice, batch, and complete data designs.

Some statistical considerations on the FDA draft guidance for individual bioequivalence.

Adaptive Penalized D-Optimal Designs for Dose Finding Based on Continuous Efficacy and Toxicity