Data analysis of incomplete repeated measures using a multivariate extension of the Brown-Forsythe procedure / Análisis de datos de medidas repetidas incompletas usando una extensión multivariante del enfoque de Brown-Forsythe

BACKGROUND: A multivariate extension of the Brown-Forsythe (MBF) procedure can be used for the analysis of partially repeated measure designs (PRMD) when the covariance matrices are arbitrary. However, the MBF procedure requires complete data over time for each subject, which is a significant limitation of this procedure. This article provides the rules for pooling the results obtained after applying the same MBF analysis to each of the imputed datasets of a PRMD. METHOD: Montecarlo methods are used to evaluate the proposed solution (MI-MBF), in terms of control of Type I and Type II errors. For comparative purposes, the MBF analysis based on the complete original dataset (OD-MBF) and the covariance pattern model based on an unstructured matrix (CPM-UN) were studied. RESULTS: Robustness and power results showed that the MI-MBF method performed slightly worse than tests based on CPM-UN when the homogeneity assumption was met, but slightly better when that assumption was not met. We also note that without assuming equality of covariance matrices, little power was sacrificed by using the MI-MBF method in place of the OD-MBF method. CONCLUSIONS: The results of this study suggest that the MI-MBF method performs well and could be of practical use

ANTECEDENTES: para analizar diseños de medidas parcialmente repetidas (DMPR) con matrices de covarianza arbitrarias se puede usar una extensión multivariante del enfoque de Brown-Forsythe (MBF). Una importante limitación de este enfoque es que requiere datos completos para cada sujeto. Este artículo proporciona las reglas para agrupar los resultados obtenidos tras aplicar el análisis MBF a los diferentes conjuntos de datos imputados de un DMPR. MÉTODO: se aplican técnicas de Montecarlo para evaluar la solución propuesta (IM-MBF), en términos de control de los errores Tipo I y Tipo II. Con fines comparativos, también se evalúan los resultados obtenidos con el enfoque MBF basado en los datos originales (DO-MBF), así como con el modelo de patrones de covarianza basado en asumir una matriz no estructurada (MPC-NE). RESULTADOS: cuando se cumple el supuesto de homogeneidad, el desempeño de la prueba IM-MBF es ligeramente inferior al obtenido con la prueba MPC-NE, mientras que sucede lo contrario cuando se incumple dicho supuesto. También encontramos que se pierde poca potencia usando el enfoque MI-MBF, en lugar del enfoque DO-MBF, cuando las matrices de covarianza son heterogéneas. CONCLUSIONES: los resultados sugieren que el enfoque MI-MBF funciona bien y podría ser de uso práctico

Multilevel bootstrap analysis with assumptions violated / Análisis bootstrap multinivel con supuestos incumplidos

Background: Likelihood-based methods can work poorly when the residuals are not normally distributed and the variances across clusters are heterogeneous. Method: The performance of two estimation methods, the non-parametric residual bootstrap (RB) and the restricted maximum likelihood (REML) for fitting multilevel models are compared through simulation studies in terms of bias, coverage, and precision. Results: We find that (a) both methods produce unbiased estimates of the fixed parameters, but biased estimates of the random parameters, although the REML was more prone to give biased estimates for the variance components; (b) the RB method yields substantial reductions in the difference between nominal and actual confidence interval coverage, compared with the REML method; and (c) for the square root of the mean squared error (RMSE) of the fixed effects, the RB method performed slightly better than the REML method. For the variance components, however, the RB method did not offer a systematic improvement over the REML method in terms of RMSE. Conclusions: It can be stated that the RB method is, in general, superior to the REML method with violated assumptions (AU)

Antecedentes: los métodos basados en la verosimilitud pueden trabajar con dificultad cuando los errores no se distribuyen normalmente y las varianzas a través de los grupos son heterogéneas. Método: el desempeño de dos métodos de estimación, el bootstrap residual (BR) no paramétrico y el de la máxima verosimilitud restringida (MVR), para ajustar modelos multinivel es comparado mediante estudios de simulación en términos de sesgo, cobertura y precisión. Resultados: encontramos que: (a) ambos métodos proporcionan estimaciones no sesgadas de los efectos fijos, pero sesgadas de los efectos aleatorios, aunque el método MVR es más propenso a generar estimaciones sesgadas para los componentes de la varianza; (b) el método BR depara diferencias más pequeñas entre las tasas de cobertura real y nominal de los intervalos de confianza que el método MVR; y (c) los valores de la raíz del error cuadrático medio (RECM) para los efectos fijos son algo más pequeños bajo el método BR que bajo el método REML. Sin embargo, en lo referido a los componentes de la varianza, el método de BR no ofrece una mejora sistemática sobre el método MVR en términos de RECM. Conclusiones: en general, se puede afirmar que el método BR resulta superior al método MVR con supuestos incumplidos (AU)

Multilevel bootstrap analysis with assumptions violated.

BACKGROUND: Likelihood-based methods can work poorly when the residuals are not normally distributed and the variances across clusters are heterogeneous. METHOD: The performance of two estimation methods, the non-parametric residual bootstrap (RB) and the restricted maximum likelihood (REML) for fitting multilevel models are compared through simulation studies in terms of bias, coverage, and precision. RESULTS: We find that (a) both methods produce unbiased estimates of the fixed parameters, but biased estimates of the random parameters, although the REML was more prone to give biased estimates for the variance components; (b) the RB method yields substantial reductions in the difference between nominal and actual confidence interval coverage, compared with the REML method; and (c) for the square root of the mean squared error (RMSE) of the fixed effects, the RB method performed slightly better than the REML method. For the variance components, however, the RB method did not offer a systematic improvement over the REML method in terms of RMSE. CONCLUSIONS: It can be stated that the RB method is, in general, superior to the REML method with violated assumptions.

[Generalization of the Brown-Forsythe approach to factorial designs]. / Generalización del enfoque Brown-Forsythe a diseños factoriales.

The current paper proposes a solution that generalizes ideas of Brown and Forsythe to the problem of comparing hypotheses in two-way classification designs with heteroscedastic error structure. Unlike the standard analysis of variance, the proposed approach does not require the homogeneity assumption. A comprehensive simulation study, in which sample size of the cells, relationship between the cell sizes and unequal variance, degree of variance heterogeneity, and population distribution shape were systematically manipulated, shows that the proposed approximation was generally robust when normality and heterogeneity were jointly violated.

Generalización del enfoque Brown-Forsythe a diseños factoriales / Generalization of the Brown-Forsythe approach to factorial designs

El presente trabajo propone una solución basada en generalizar las ideas de Brown y Forsythe al problema de contrastar hipótesis en diseños factoriales carentes de homogeneidad. A diferencia del tradicional modelo de análisis de la varianza, el enfoque propuesto no requiere satisfacer el supuesto de homogeneidad de las varianzas. Un comprensivo estudio de simulación, en el cual se manipuló sistemáticamente el tamaño de muestra de las celdas, la relación entre el tamaño de las celdas y el tamaño de las varianzas, el grado de heterogeneidad y la forma de la distribución de la población, soporta la robustez de la aproximación propuesta para contrastar los efectos del diseño factorial en ausencia de heterogeneidad y también bajo no normalidad

The current paper proposes a solution that generalizes ideas of Brown and Forsythe to the problem of comparing hypotheses in two-way classification designs with heteroscedastic error structure. Unlike the standard analysis of variance, the proposed approach does not require the homogeneity assumption. A comprehensive simulation study, in which sample size of the cells, relationship between the cell sizes and unequal variance, degree of variance heterogeneity, and population distribution shape were systematically manipulated, shows that the proposed approximation was generally robust when normality and heterogeneity were jointly violated