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Revista Colombiana de Estadística

Print version ISSN 0120-1751

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

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.


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


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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Abeyasekera, S. & Curnow, R. N. (1984), `The desirability of adjusting for residual effects in a crossover design.', Biometrics 40, 1071-1078. [ Links ]

Berg, M., Welty, T. E., Gidal, B. E., Diaz, F. J., Krebill, R., Szaarski, J. P., Dworetzky, B. A., Pollard, J. R., Elder Jr, E. J., Jiang, W., Jiang, X., Switzer, R. D. & Privitera, M. D. (2017), `Bioequivalence between generic and branded lamotrigine in people with epilepsy: the EQUIGEN randomized clinical trial.', JAMA Neurology 74, 919-926. [ Links ]

Botts, S., Diaz, F., Santoro, V., Spina, E., Muscatello, M. R., Cogollo, M., Castro, F. E. & de Leon, J. (2008), `Estimating the eects of co-medications on plasma olanzapine concentrations by using a mixed model.', Progress in Neuro-Psychopharmacology & Biological Psychiatry, 32, 1453-1458. [ Links ]

Breslow, N. E. & Clayton, D. G. (1993), `Approximate inference in generalized linear mixed models', Journal of the American Statistical Association 88, 9- 25. [ Links ]

Bronson, R. (1989), Matrix Operations, 1st Edition. Schaum's Outline Series, New York. McGraw-Hill. [ Links ]

Center for Drug Evaluation and Research (2001), US Food and Drug Administration. Guidance for Industry: Statistical Approaches to Establishing Bioequivalence. Accessed February 22, 2018. [ Links ]

Center for Drug Evaluation and Research (2003), US Food and Drug Administration. Guidance for Industry: Bioavailability and Bioequivalence Studies for Orally Administered Drug Products - General Considerations. BioEquiv.pdf. Accessed February 22, 2018. [ Links ]

Christensen, R. (2011), Plane Answers to Complex Questions: The Theory of Linear Models, New York: Springer. [ Links ]

Diaz, F. J. (2016), `Measuring the Individual Benet of a Medical or Behavioral Treatment Using Generalized Linear Mixed-Eects Models.', Statistics in Medicine 35, 4077-4092. [ Links ]

Diaz, F. J. (in press), `Estimating individual benets of medical or behavioral treatments in severely ill patients', Statistical Methods in Medical Research DOI: 10.1177/0962280217739033. [ Links ]

Diaz, F. J., Berg, M. J., Krebill, R., Welty, T., Gidal, B. E., Alloway, R. & Privitera, M. (2013), `Random-effects linear modeling and sample size tables for two special cross-over designs of average bioequivalence studies: the 4-period, 2-sequence, 2-formulation and 6-period, 3-sequence, 3-formulation designs', Clinical Pharmacokinetics 52, 1033-1043. [ Links ]

Ebbes, P., Bockenholt, U. & Wedel, M. (2004), `Regressor and random-effects dependencies in multilevel models', Statistica Neerlandica 58, 161-178. [ Links ]

Fava, M., Rush, A. J., Trivedi, M. H., Nierenberg, A. A., Thase, M. E., Sackeim, H. A., Quitkin, F. M., Wisniewski, S., Lavori, P. W., Rosenbaum, J. F. & Kupfer, D. J. (2003), `Background and rationale for the sequenced treatment alternatives to relieve depression (STAR*D) study', Psychiatric clinics of North America 26, 457-494. [ Links ]

Fitzmaurice, G. & Molenberghs, G. (2004), Advances in longitudinal data analysis: An historical perspective. In Longitudinal Data Analysis, Fitzmaurice, G., Davidian, M., Verbeke, G., Molenberghs, G., Editors. London: Chapman & Hall/CRC. London: Chapman & Hall/CRC. [ Links ]

Fleiss, J. L. (1989), `A critique of recent research on the two-treatment Crossover design', Controlled Clinical Trials 10, 237-43. [ Links ]

Frees, E. W. (2001), `Omitted variables in longitudinal data models', The Canadian Journal of Statistics 29, 573-595. [ Links ]

Frees, E. W. (2004), Longitudinal and Panel Data, Cambridge: Cambridge University Press. [ Links ]

Grajales, L. F. & Lopez, L. A. (2006), `Data imputation in switchback designs using a mixed model with correlated errors', Revista colombiana de estadística 29, 221-238. [ Links ]

Hausman, J. A. (1978), `Specication tests in econometrics', Econometrica 46, 1251-1271. [ Links ]

Henderson, C. R. (1953), `Estimation of variance and covariance components', Biometrics 9, 226-252. [ Links ]

Hofmann, M., Wrobel, N., Kessner, S. & Bingel, U. (2014), `Minimizing carryover effects after treatment failure and maximizing therapeutic outcome. Can changing the route of administration mitigate the influence of treatment history?', Zeitschrift fur Psychologie 222, 171-178. [ Links ]

Hooks, T., Marx, D., Kachman, S. & Pedersen, J. (2009), `Optimality criteria for models with random effects', Revista Colombiana de Estadística 32, 17-31. [ Links ]

Huber, P. J. (1967), `The behavior of maximum likelihood estimates under nonstandard conditions', In Vol. 1 of Proceedings of the Fifth Berkeley Symposium on Mathematical Statistics and Probability, pp. 221-233. [ Links ]

Jones, B. & Kenward, M. G. (2015), Design and Analysis of Cross-over Trials, Boca Raton: CRC Press. [ Links ]

Kim, J.-S. & Frees, E. W. (2006), `Omitted variables in multilevel models', Psychometrika 71, 659-690. [ Links ]

Kim, J.-S. & Frees, E. W. (2007), `Multilevel modeling with correlated effects', Psychometrika 72, 505-533. [ Links ]

Laird, N. M. & Ware, J. H. (1982), `Random eects models for longitudinal data', Biometrics 38, 963-974. [ Links ]

Long, D. L., Preisser, J. S., Herring, A. H. & Golin, C. E. (2015), `A marginalized zero-inflated Poisson regression model with random effects', Journal of the Royal Statistical Society: Series C 64, 815-830. [ Links ]

Mundlak, Y. (1978), `On the pooling of time series and cross-section data', Econometrica 46, 69-85. [ Links ]

Nelder, J. A. & Wedderburn, R. W. M. (1972), `Generalized linear models', Journal of the Royal Statistical Society, Series A 135, 370-384. [ Links ]

Privitera, M. D., Welty, T. E., Gidal, B. E., Diaz, F. J., Krebill, R., Szaflarski, J. P., Dworetzky, B. A., Pollard, J. R., Elder, E. J., Jiang, W., Jiang, X. & Berg, M. (2016), `Generic-to-Generic Lamotrigine Switches in People with Epilepsy: A Randomized Controlled Trial', Lancet Neurology 15, 365-372. [ Links ]

Rabe-Hesketh, S. & Skrondal, A. (2009), Generalized linear mixed-eects models. In Longitudinal Data Analysis, Fitzmaurice, G., Davidian, M., Verbeke, G., Molenberghs, G., Editors. Chapman & Hall/CRC, London. [ Links ]

Rabe-Hesketh, S., Skrondal, A. & Pickles, A. (2005), `Maximum likelihood estimation of limited and discrete dependent variable models with nested random effects', Journal of Econometrics 128, 301-323. [ Links ]

Searle, S. R. (1966), Matrix Algebra for the Biological Sciences, Wiley, New York. [ Links ]

Senn, S. (2002), Cross-over Trials in Clinical Research, 2nd Edition, Wiley, Hoboken. [ Links ]

Senn, S., D'Angelo, G. & Potvin, D. (2004), `Carry-over in cross-over trials in bioequivalence: theoretical concerns and empirical evidence', Pharmaceutical Statistics 3, 133-142. [ Links ]

Senn, S. & Lambrou, D. (1998), `Robust and realistic approaches to carry-over', Statistics in Medicine 17, 2849-2864. [ Links ]

Shao, J. (2003), Mathematical Statistics, 2nd Edition, Springer, New York. [ Links ]

Vermunt, J. K. (2005), `Mixed-effects logistic regression models for indirectly observed discrete outcome variables', Multivariate Behavioral Research 40, 281- 301. [ Links ]

White, H. (1982), `Maximum likelihood estimation of misspecified models', Econometrica 50, 1-25 [ Links ]

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