Semiparametric Regression for Periodic Longitudinal Hormone Data from Multiple Menstrual Cycles

We consider semiparametric regression for periodic longitudinal data. Parametric fixed effects are used to model the covariate effects and a periodic nonparametric smooth function is used to model the time effect. The within-subject correlation is modeled using subject-specific random effects and a random stochastic process with a periodic variance function. We use maximum penalized likelihood to estimate the regression coefficients and the periodic nonparametric time function, whose estimator is shown to be a periodic cubic smoothing spline. We use restricted maximum likelihood to simultaneously estimate the smoothing parameter and the variance components. We show that all model parameters can be easily obtained by fitting a linear mixed model. A common problem in the analysis of longitudinal data is to compare the time profiles of two groups, e.g., between treatment and placebo. We develop a scaled chi-squared test for the equality of two nonparametric time functions. The proposed model and the test are illustrated by analyzing hormone data collected during two consecutive menstrual cycles and their performance is evaluated through simulations.

[1]  J. Rice,et al.  Smoothing spline models for the analysis of nested and crossed samples of curves , 1998 .

[2]  B. Silverman,et al.  Nonparametric regression and generalized linear models , 1994 .

[3]  B. Yandell.  Spline smoothing and nonparametric regression , 1989 .

[4]  Steven G. Gilmour,et al.  The analysis of designed experiments and longitudinal data by using smoothing splines - Discussion , 1999 .

[5]  Peter Hall,et al.  Bootstrap test for difference between means in nonparametric regression , 1990 .

[6]  F. E. Satterthwaite.  An approximate distribution of estimates of variance components. , 1946, Biometrics.

[7]  G. Wahba.  Automatic Smoothing of the Log Periodogram , 1980 .

[8]  Naomi Altman,et al.  Kernel Smoothing of Data with Correlated Errors , 1990 .

[9]  P. Green.  Penalized Likelihood for General Semi-Parametric Regression Models. , 1987 .

[10]  J. Randolph,et al.  Urinary Ovarian and Gonadotropin Hormone Levels in Premenopausal Women with Low Bone Mass , 1998, Journal of bone and mineral research : the official journal of the American Society for Bone and Mineral Research.

[11]  K. B. Kulasekera.  Comparison of Regression Curves Using Quasi-Residuals , 1995 .

[12]  D. Harville.  Maximum Likelihood Approaches to Variance Component Estimation and to Related Problems , 1977 .

[13]  M. C. Jones,et al.  Spline Smoothing and Nonparametric Regression. , 1989 .

[14]  Yuedong Wang.  Mixed effects smoothing spline analysis of variance , 1998 .

[15]  H. Müller,et al.  Kernel estimation of regression functions , 1979 .

[16]  M. B. Brown,et al.  A flexible model for human circadian rhythms. , 1996, Biometrics.

[17]  Miss A.O. Penney.  (b) , 1974, The New Yale Book of Quotations.

[18]  J. Ware,et al.  Prediction and creation of smooth curves for temporally correlated longitudinal data , 1995 .

[19]  B. Silverman,et al.  Spline Smoothing: The Equivalent Variable Kernel Method , 1984 .

[20]  Yuedong Wang.  Smoothing Spline Models with Correlated Random Errors , 1998 .

[21]  B. Silverman,et al.  Estimating the mean and covariance structure nonparametrically when the data are curves , 1991 .

[22]  James Stephen Marron,et al.  Semiparametric Comparison of Regression Curves , 1990 .

[23]  Satterthwaite Fe.  An approximate distribution of estimates of variance components. , 1946 .

[24]  Adrian Bowman,et al.  NON-PARAMETRIC ANALYSIS OF COVARIANCE , 1995 .

[25]  P. Diggle,et al.  Semiparametric models for longitudinal data with application to CD4 cell numbers in HIV seroconverters. , 1994, Biometrics.

[26]  J. Raz,et al.  Semiparametric Stochastic Mixed Models for Longitudinal Data , 1998 .

[27]  M. Kenward,et al.  The Analysis of Designed Experiments and Longitudinal Data by Using Smoothing Splines , 1999 .