The use of patterned covariance structures in the parametric analysis of longitudinal data is both elegant and efficient. However, this strategy has not been well studied for semi-parametric models for analysing such data. We propose to estimate the covariance matrix in the semi-parametric model by rearranging the non-parametric component as a profiled linear function of the data and using a local smoothing technique. This results in a parametric regression formulation that enables us to construct likelihood functions and use various information criteria to select the best fitting covariance matrix. We apply our method to reanalyse data from a two-armed clinical trial for Scleroderma patients and show our method is more efficient for estimating the parametric components in the semi-parametric model.
Get full access to this article
View all access options for this article.
References
1.
Fan J., Zhang J-T.Two-step estimation of functional linear models with applications to longitudinal data. Journal of Royal Statistical Society, Series B (Methodological)2000; 62: 303-22.
2.
Fan J., Li R.New estimation and model selection procedures for semi-parametric modeling in longitudinal data analysis. Journal of the American Statistical Association2004; 99: 710-23.
3.
Fan J., Huang T., Li R.Analysis of longitudinal data with semi-parametric estimation of covariance function. Journal of the American Statistical Association2007; 102: 632-41.
4.
Wu H., Zhang J-T.In John Wiley & Sons, Inc. Nonparametric regression methods for longitudinal data analysis: mixed-effects modeling approaches. Hoboken , New Jersey; 2006.
5.
Fitzmaurice GM , Laird NM, Ware JHIn John Wiley & Sons, Inc. Applied longitudinal analysis. Hoboken, New Jersey; 2004.
Burnham KP, Anderson DRModel selection and inference: a practical information-theoretic approach . Springer, New york; 1988.
8.
Postlethwaite AE, Wong WK, Clements P., Chatterjee S., Fessler BJ, Kang AH, Korn J., Mays M., Merkel PA, Molitor JA, Moreland L., Rothfield N., Simms RW, Smith EA, Spiera R., Steen V., Warrington K., White B., Wigley F., Daniel EFA multicenter, randomized, double-blind, placebo-controlled trial of oral type I collagen treatment in patients with diffuse cutaneous systemic sclerosis: I. Oral type I collagen does not improve skin in all patients, but may improve skin in late-phase disease. Arthritis & Rheumatism2008; 58: 1810-22.
9.
Li J., Wong WKA semi-parametric analysis for identifying scleroderma patients responsive to an anti-fibrotic agent. Contemporary Clinical Trials2009; 30: 105-13.
10.
Fan J., Gijbels I.In Chapman and Hall. Local polynomial modeling and its applications . London; 1996.
11.
Eubank RLIn Marcel Dekker, Inc. Spline smoothing and nonparametric regression. New York; 1988.
12.
Fan J., Huang T.Profile likelihood inferences on semi-parametric varying-coefficient partially linear models. Bernoulli2005; 11: 1031-57.
13.
Li J., Palta M.Bandwidth selection through cross validation for semi-parametric varying-coefficient partially linear models. Journal of Statistical Computation and Simulation2009; 79: 1277-86.
14.
Smyth GK, Verbyla APA conditional likelihood approach to REML in generalized linear models. Journal of the Royal Statistical Society. Series B (Methodological)1996; 58: 565-72.
15.
Pinheiro JC, Bates DMMixed-effects models in S and splus. Springer VerlagNew York, LLC; 2000.
16.
Lindstrom MJ , Bates DMNewton-Raphson and EM algorithms for linear mixed-effects models for repeated-measures data. Journal of the American Statistical Association1988; 83: 1014-22.
Stram DO, Lee, JWVariance components testing in the longitudinal mixed-effects models. Biometrics1994; 50: 1171-77.
19.
Akaike H.Information theory and an extension of the maximum likelihood principle. In B. N. Petrov and F. Csaki, eds. Second International Symposium on Information Theory, Academiai Kiado, Budapest , 1973: 267-281
20.
Schwarz G.Estimating the dimension of a model. Annals of Statistics1978; 6: 461-64.
21.
Burnham KP, Anderson DRMultimodel inference, understanding AIC and BIC in model selection. Sociological Methods and Research2004; 33: 261-304.
22.
Atkinson ACA note on the generalized information criterion for choice of model. Biometrika1988; 67: 413-18.
23.
Tobias A., Campbell MJAkaike’s informaiton criterion and Schwarz’s criterion. Stata Technical Bulletin, 45, 23-25. In Stata Techincal Bulletin Reprints, Vol. 8; 1998: 174-177. College Station, Stata Press, TX.
24.
Long JS, Freese J.Regression models for categorical dependent variables using STATA (2nd edn). Stata Press, College Station, TX; 2006.
25.
Raftery A.Bayesian model selection in social research. In Marsden, PV. ed. Sociological methodology, Vol. 25, Basil Blackwell, Oxford, 1996: 111-63.
26.
Palta M.In John Wiley & Sons. Quantitative Methods in Population Health: Extensions of Ordinary Regression. Hoboken , New Jersey; 2003.
27.
Reschenhofer E.Prediction with vague prior knowledge. Communication in Statistics, Theory and Methods1996; 25: 601-08.
28.
Wolfinger RDHeterogeneous variance: covariance structures for repeated measures. Journal of Agricultural, Biological and Environmental Statistics1996; 1: 205-30.
29.
Xia Y., Zhang W., Tong H.Efficient estimation for semivarying-coefficient models. Biometrika2004; 91: 661-81.
30.
Zhang CM, Li J., Meng J.On Stein’s lemma, dependent covariates and functional monotonicity in multi-dimensional modeling. Journal of Multivariate Analysis2008; 99: 2285-303.
31.
Noda K., Miyaoka E., Itoh M.On bias correction of the Akaike information criterion in linear models. Communications in Statistics, Theory and Methods1996; 25: 1845-57.
32.
Neath AA, Cavanaugh JERegression and time series model selection using variants of the Schwarz information criterion. Communication in Statistics, Theory and Methods1997; 26: 559-80.
33.
Cavanaugh JE , Neath AAGeneralizing the derivation of the Schwarz information criterion. Communication in Statistics, Theory and Methods1999; 28: 49-66.
34.
Harville DABayesian inference for variance components using only error contrasts. Biometrika1974; 61: 383-385.
