A B-Spline Based Semiparametric Nonlinear Mixed Effects Model

The Semiparametric Nonlinear Mixed Effects Model (SNMM) (Ke and Wang 2001) provides a flexible framework for longitudinal comparisons of curve shapes between groups. In this article, we develop an alternative method for fitting the SNMM by reformulating Ke and Wang's smoothing spline based model in terms of B-splines. The existing algorithm is based on a backfitting procedure that iterates between two mixed models whose corresponding likelihoods are not equivalent to the likelihood of all model parameters. The consequence is a lack of reliable convergence and statistical inference. Using B-splines, however, overcomes these disadvantages by simplifying the likelihood computations without sacrificing model flexibility. Therefore, the algorithm can be expressed in terms of existing, accurate techniques based on Adaptive Gaussian Quadrature. The model is applied to labor curves, cervical dilation measured longitudinally, from women attempting a vaginal birth after cesarean. Only partial curves were measured on cases of uterine rupture given the need for emergency c-section while controls completed delivery naturally. The model allowed us to estimate and compare the average labor curve shape between cases and controls and also determine the earliest time at which clinicians could distinguish between the average labor curves in different groups. Supplemental materials are available online.