TY - JOUR
T1 - Regularized reduced rank growth curve models
AU - Takane, Yoshio
AU - Jung, Kwanghee
AU - Hwang, Heungsun
N1 - Funding Information:
The work reported in this paper is supported by Grants 10603 and 290439 from the Natural Sciences and Engineering Research Council of Canada to the first and the third author, respectively. Correspondence regarding this article should be sent to Yoshio Takane, Department of Psychology, McGill University, 1205 Dr. Penfield Avenue, Montreal, QC H3A 1B1 Canada.
PY - 2011/2/1
Y1 - 2011/2/1
N2 - The growth curve model (GCM), also known as GMANOVA, is a useful technique for investigating patterns of change in repeated measurement data over time and examining the effects of predictor variables on temporal trajectories. The reduced rank feature had been introduced previously to GCM for capturing redundant information in the criterion variables in a parsimonious way. In this paper, a ridge type of regularization was incorporated to obtain better estimates of parameters. Separate ridge parameters were allowed in column and row regressions, and the generalized singular value decomposition (GSVD) was applied for rank reduction. It was shown that the regularized estimates of parameters could be obtained in closed form for fixed values of ridge parameters. Permutation tests were used to identify the best dimensionality in the solution, and the K-fold cross validation method was used to choose optimal values of the ridge parameters. A bootstrap method was used to assess the reliability of parameter estimates. The proposed model was further extended to a mixture of GMANOVA and MANOVA. Illustrative examples were given to demonstrate the usefulness of the proposed method.
AB - The growth curve model (GCM), also known as GMANOVA, is a useful technique for investigating patterns of change in repeated measurement data over time and examining the effects of predictor variables on temporal trajectories. The reduced rank feature had been introduced previously to GCM for capturing redundant information in the criterion variables in a parsimonious way. In this paper, a ridge type of regularization was incorporated to obtain better estimates of parameters. Separate ridge parameters were allowed in column and row regressions, and the generalized singular value decomposition (GSVD) was applied for rank reduction. It was shown that the regularized estimates of parameters could be obtained in closed form for fixed values of ridge parameters. Permutation tests were used to identify the best dimensionality in the solution, and the K-fold cross validation method was used to choose optimal values of the ridge parameters. A bootstrap method was used to assess the reliability of parameter estimates. The proposed model was further extended to a mixture of GMANOVA and MANOVA. Illustrative examples were given to demonstrate the usefulness of the proposed method.
KW - A mixture of GMANOVA and MANOVA
KW - Generalized singular value decomposition (GSVD)
KW - K-fold cross validation
KW - Permutation tests
KW - Reduced rank approximation
KW - Ridge-type regularization
KW - The bootstrap method
KW - The growth curve model (or GMANOVA)
UR - http://www.scopus.com/inward/record.url?scp=78049237659&partnerID=8YFLogxK
U2 - 10.1016/j.csda.2010.08.012
DO - 10.1016/j.csda.2010.08.012
M3 - Article
AN - SCOPUS:78049237659
VL - 55
SP - 1041
EP - 1052
JO - Computational Statistics and Data Analysis
JF - Computational Statistics and Data Analysis
SN - 0167-9473
IS - 2
ER -