Abstract
This research examines the Type I error rates obtained when using the mixed model with the Kenward-Roger correction and compares them with the Between–Within and Satterthwaite approaches in split-plot designs. A simulated study was conducted to generate repeated measures data with small samples under normal distribution conditions. The data were obtained via three covariance matrices (unstructured, heterogeneous first-order auto-regressive, and random coefficients), the one with the best fit being selected according to the Akaike criterion. The results of the simulation study showed the Kenward-Roger test to be more robust, particularly when the population covariance matrices were unstructured or heterogeneous first-order auto-regressive. The main contribution of this study lies in showing that the Kenward–Roger method corrects the liberal Type I error rates obtained with the Between–Within and Satterthwaite approaches, especially with positive pairings between group sizes and covariance matrices.
Acknowledgments
This research was supported by Grants MEC-SEJ2005-01923 and MEC-SEJ2005-01883 from Spain's Ministry of Education and Science.
Notes
Note: UN=unstructured model; ARH=heterogeneous first-order autroregressive model; RC=random coefficients model; ϵ = sphericity index.
Note: J: groups, K: number of repeated measurements, ϵ: sphericity, N: total sample size, n 1, n 2, and n 3: group sizes, Δ n j : variance coefficient of the group size, =/ ≠ : homogeneity/heterogeneity of covariance matrices between groups, null/+/−: null/positive/negative pairing of group sizes and covariance matrices.
Note: The structure with the highest selection percentage is in bold.
Note: The structure with the highest selection percentage is in bold.
Note: The structure with the highest selection percentage is in bold.
Note: BW, SW, and KR=methods for calculating the degrees of freedom (Between–Within, Satterthwaite, and Kenward–Roger). In bold=liberal.
Note: BW, SW, and KR=methods for calculating the degrees of freedom (Between–Within, Satterthwaite, and Kenward–Roger). In bold=liberal.
Note: BW, SW, and KR=methods for calculating the degrees of freedom (Between–Within, Satterthwaite, and Kenward–Roger). In bold=liberal; in italics=conservative.
Note: BW, SW, and KR=methods for calculating the degrees of freedom (Between–Within, Satterthwaite, and Kenward–Roger). In bold=liberal.
Note: BW, SW, and KR=methods for calculating the degrees of freedom (Between–Within, Satterthwaite, and Kenward–Roger). In bold=liberal; in italics=conservative.
Note: BW, SW, and KR=methods for calculating the degrees of freedom (Between–Within, Satterthwaite, and Kenward–Roger). In bold=liberal; in italics=conservative.