Published

2018-07-01

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

DOI:

https://doi.org/10.15446/rce.v41n2.63332

Keywords:

Augmented regression, robust fixed-effects estimators, generalized least squares, maximum likelihood, quasi-likelihood, random effects linear models (en)
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 (es)

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The problem of constructing a design matrix of full rank for generalized linear mixed-effects models (GLMMs) has not been addressed in statistical literature in the context of clinical trials of treatment sequences. Solving this problem is important because the most popular estimation methods for GLMMs assume a design matrix of full rank, and GLMMs are useful tools in statistical practice. We propose new developments in GLMMs that address this problem. We present a new model for the design and analysis of clinical trials of treatment sequences, which utilizes some special sequences called skip sequences. We present a theorem showing that estimators computed through quasi-likelihood, maximum likelihood or generalized least squares, or through robust approaches, exist only if appropriate skip sequences are used. We prove theorems that establish methods for implementing skip sequences in practice. In particular, one of these theorems computes the necessary skip sequences explicitly. Our new approach allows building design matrices of full rank and facilitates the implementation of regression models in the experimental design and data analysis of clinical trials of treatment sequences. We also explain why the standard approach to constructing dummy variables is inappropriate in studies of treatment sequences. The methods are illustrated with a data analysis of the STAR*D study of sequences of treatments for depression.

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 secuencia 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 cómo 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.

References

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

Berg, M., Welty, T. E., Gidal, B. E., Diaz, F. J., Krebill, R., Sza-arski, 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.

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

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

Bronson, R. (1989), Matrix Operations, Schaum's Outline Series, 1 edn, McGraw-Hill, New York.

Center for Drug Evaluation and Research (2001), Us food and drug administration. guidance for industry: Statistical approaches to establishing bioequivalence. https://www.fda.gov/ downloads/drugs/guidances/ucm070244.pdf, Accessed February 22, 2018.

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. https://www.fda.gov/ohrms/dockets/ac/03/brie-ng/3995B1-07-GFIBioAvail-BioEquiv.pdf, Accessed February 22, 2018.

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

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

Diaz, F. J. (2017), `Estimating individual benefits of medical or behavioral treatments in severely ill patients', Statistical Methods in Medical Research (DOI: 10.1177/0962280217739033).

Diaz, F. J., Berg, M. J., Krebill, R., Welty, T., Gidal, B. E., Alloway, R. & Privitera, M. (2013), `Random-e-ects 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.

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

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.

Fitzmaurice, G. & Molenberghs, G. (2004), Advances in longitudinal data analysis, in G. Fitzmaurice, G. Molenberghs, M. Davidian & G. Verbeke, eds, `Longitudinal data analysis', Chapman and Hall/CRC, London.

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

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

Frees, E. W. (2004), Longitudinal and Panel Data, Cambridge, Cambridge University Press.

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.

Hausman, J. A. (1978), `Specification tests in econometrics', Econometrica 46, 1251-1271.

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

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.

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

Huber, P. J. et al. (1967), The behavior of maximum likelihood estimates under nonstandard conditions, in `Proceedings of the fifth Berkeley symposium on mathematical statistics and probability', Vol. 1, University of California Press, pp. 221-233.

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

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

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

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

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.

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

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

Privitera, M. D., Welty, T. E., Gidal, B. E., Diaz, F. J., Krebill, R., Sza-arski, 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.

Rabe-Hesketh, S. & Skrondal, A. (2009), Generalized linear mixed-effects models, in `Longitudinal Data Analysis', Chapman and Hall/CRC, London, pp. 93-120.

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.

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

Senn, S. (2002), Cross-over Trials in Clinical Research, 2 edn, Wiley, Hoboken.

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.

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

Shao, J. (2003), Mathematical Statistics, 2 edn, Springer, New York.

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

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

How to Cite

APA

Diaz, F. J. (2018). Construction of the Design Matrix for Generalized Linear Mixed-Effects Models in the Context of Clinical Trials of Treatment Sequences. Revista Colombiana de Estadística, 41(2), 191–233. https://doi.org/10.15446/rce.v41n2.63332

ACM

[1]
Diaz, F.J. 2018. Construction of the Design Matrix for Generalized Linear Mixed-Effects Models in the Context of Clinical Trials of Treatment Sequences. Revista Colombiana de Estadística. 41, 2 (Jul. 2018), 191–233. DOI:https://doi.org/10.15446/rce.v41n2.63332.

ACS

(1)
Diaz, F. J. Construction of the Design Matrix for Generalized Linear Mixed-Effects Models in the Context of Clinical Trials of Treatment Sequences. Rev. colomb. estad. 2018, 41, 191-233.

ABNT

DIAZ, F. J. Construction of the Design Matrix for Generalized Linear Mixed-Effects Models in the Context of Clinical Trials of Treatment Sequences. Revista Colombiana de Estadística, [S. l.], v. 41, n. 2, p. 191–233, 2018. DOI: 10.15446/rce.v41n2.63332. Disponível em: https://revistas.unal.edu.co/index.php/estad/article/view/63332. Acesso em: 28 mar. 2025.

Chicago

Diaz, Francisco J. 2018. “Construction of the Design Matrix for Generalized Linear Mixed-Effects Models in the Context of Clinical Trials of Treatment Sequences”. Revista Colombiana De Estadística 41 (2):191-233. https://doi.org/10.15446/rce.v41n2.63332.

Harvard

Diaz, F. J. (2018) “Construction of the Design Matrix for Generalized Linear Mixed-Effects Models in the Context of Clinical Trials of Treatment Sequences”, Revista Colombiana de Estadística, 41(2), pp. 191–233. doi: 10.15446/rce.v41n2.63332.

IEEE

[1]
F. J. Diaz, “Construction of the Design Matrix for Generalized Linear Mixed-Effects Models in the Context of Clinical Trials of Treatment Sequences”, Rev. colomb. estad., vol. 41, no. 2, pp. 191–233, Jul. 2018.

MLA

Diaz, F. J. “Construction of the Design Matrix for Generalized Linear Mixed-Effects Models in the Context of Clinical Trials of Treatment Sequences”. Revista Colombiana de Estadística, vol. 41, no. 2, July 2018, pp. 191-33, doi:10.15446/rce.v41n2.63332.

Turabian

Diaz, Francisco J. “Construction of the Design Matrix for Generalized Linear Mixed-Effects Models in the Context of Clinical Trials of Treatment Sequences”. Revista Colombiana de Estadística 41, no. 2 (July 1, 2018): 191–233. Accessed March 28, 2025. https://revistas.unal.edu.co/index.php/estad/article/view/63332.

Vancouver

1.
Diaz FJ. Construction of the Design Matrix for Generalized Linear Mixed-Effects Models in the Context of Clinical Trials of Treatment Sequences. Rev. colomb. estad. [Internet]. 2018 Jul. 1 [cited 2025 Mar. 28];41(2):191-233. Available from: https://revistas.unal.edu.co/index.php/estad/article/view/63332

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CrossRef citations4

1. Francisco J. Diaz. (2021). Using population crossover trials to improve the decision process regarding treatment individualization in N‐of‐1 trials. Statistics in Medicine, 40(20), p.4345. https://doi.org/10.1002/sim.9030.

2. Francisco J. Diaz, Xuan Zhang, Nikos Pantazis, Jose De Leon. (2022). Measuring Individual Benefits of Medical Treatments Using Longitudinal Hospital Data with Non-Ignorable Missing Responses Caused by Patient Discharge: Application to the Study of Benefits of Pain Management Post Spinal Fusion. Revista Colombiana de Estadística, 45(2), p.275. https://doi.org/10.15446/rce.v45n2.101597.

3. Zhiwen Wang, Francisco J. Diaz. (2020). A graphical approach to assess the goodness-of-fit of random-effects linear models when the goal is to measure individual benefits of medical treatments in severely ill patients. BMC Medical Research Methodology, 20(1) https://doi.org/10.1186/s12874-020-01054-3.

4. Xuan Zhang, Jose de Leon, Benedicto Crespo-Facorro, Francisco J. Diaz. (2020). Measuring individual benefits of psychiatric treatment using longitudinal binary outcomes: Application to antipsychotic benefits in non-cannabis and cannabis users. Journal of Biopharmaceutical Statistics, 30(5), p.916. https://doi.org/10.1080/10543406.2020.1765371.

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