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Revista Colombiana de Estadística

Print version ISSN 0120-1751

Rev.Colomb.Estad. vol.41 no.2 Bogotá July/Dec. 2018

http://dx.doi.org/10.15446/rce.v41n2.63332 

Artículos originales de investigación

Construction of the Design Matrix for Generalized Linear Mixed-Effects Models in the Context of Clinical Trials of Treatment Sequences

Construcción de la matriz de diseño en modelos lineales de efectos mixtos generalizados en un contexto de ensayos clínicos de secuencias de tratamientos

Francisco J. Diaz1  a 

1 Department of Biostatistics, The University of Kansas Medical Center, Kansas City, Kansas, United States.

Abstract

The estimation of carry-over effects is a di-cult problem in the design and analysis of clinical trials of treatment sequences including cross-over trials. Except for simple designs, carry-over effects are usually unidentifiable and therefore nonestimable. Solutions such as imposing parameter constraints are often unjustified and produce differing carry-over estimates depending on the constraint imposed. Generalized inverses or treatment-balancing often allow estimating main treatment eects, but the problem of estimating the carry-over contribution of a treatment sequence remains open in these approaches. Moreover, washout periods are not always feasible or ethical. A common feature of designs with unidentifiable parameters is that they do not have design matrices of full rank. Thus, we propose approaches to the construction of design matrices of full rank, without imposing artificial constraints on the carry-over effects. Our approaches are applicable within the framework of generalized linear mixed-effects models. We present a new model for the design and analysis of clinical trials of treatment sequences, called Antichronic System, and introduce some special sequences called Skip Sequences. We show that carry-over effects are identifiable only if appropriate Skip Sequences are used in the design and/or data analysis of the clinical trial. We explain how Skip Sequences can be implemented in practice, and present a method of computing the appropriate Skip Sequences. We show applications to the design of a cross-over study with 3 treatments and 3 periods, and to the data analysis of the STAR*D study of sequences of treatments for depression.

Key words: Augmented regression; carry-over effects; cross-over design; design matrix; estimability; generalized inverses; generalized least squares; identifiability; maximum likelihood; placebo; quasi-likelihood; random effects linear models; robust fixed-effects estimators

Resumen

La estimación de los efectos de arrastre es un problema difícil en el diseño y análisis de ensayos clínicos de secuencias de tratamientos, incluyendo ensayos cruzados. Excepto por diseños simples, estos efectos son usualmente no identificables y, por lo tanto, no estimables. La imposición de restricciones a los parámetros es a menudo no justificada y produce diferentes estimativos de los efectos de arrastre dependiendo de la restricción impuesta. Las inversas generalizadas o el balance de tratamientos a menudo permiten estimar los efectos principales de tratamiento, pero no resuelven el problema de estimar la contribución de los efectos de arrastre de una sequencia de tratamiento. Además, los períodos de lavado no siempre son factibles o éticos. Los diseños con parámetros no identificables comúnmente tienen matrices de diseño que no son de rango completo. Por lo tanto, proponemos métodos para la construcción de matrices de rango completo, sin imponer restricciones artificiales en los efectos de arrastre. Nuestros métodos son aplicables en un contexto de modelos lineales mixtos generalizados. Presentamos un nuevo modelo para el diseño y análisis de ensayos clínicos de secuencias de tratamientos, llamado Sistema Anticrónico, e introducimos secuencias de tratamiento especiales llamadas Secuencias de Salto. Demostramos que los efectos de arrastre son identificables sólo si se usan Secuencias de Salto apropiadas. Explicamos como implementar en la práctica estas secuencias, y presentamos un método para calcular las secuencias apropiadas. Presentamos aplicaciones al diseño de un estudio cruzado con 3 tratamientos y 3 períodos, y al análisis del estudio STAR*D de secuencias de tratamientos para la depresión.

Palabras-clave: Cuasi-verosimilitud; diseño cruzado; efectos de arrastre; estimabilidad; estimadores robustos de efectos fijos; identificabilidad; inversas generalizadas; matriz de diseño; máxima verosimilitud; mínimos cuadrados generalizados; modelos lineales de efectos aleatorios; placebo

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a E-mail: fdiaz@kumc.edu

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