A Semiparametric Nonlinear Mixed Effects Model with Penalized Splines Using Automatic Differentiation

Abstract

We present an estimation procedure for nonlinear mixed-effects models in which the population trajectory is represented by penalized splines and adapted to individuals via subject-specific transformation parameters. By exploiting the mixed model representation of penalized splines, the level of smoothness can be estimated jointly with other variance components. The integration over random effects needed to obtain the marginal likelihood is carried out using the Laplace approximation. Exact derivatives for evaluation and maximization of the resulting likelihood are obtained via automatic differentiation implemented through Template Model Builder. In simulation studies, the method produces improved inferential performance and reduced computational burden when compared to the existing procedure. The approach is further illustrated through a case study on infant height growth in the first two years of life.

Figures (5)

• Figure 1: Results of the sine curve simulation on the population curve. The first four settings correspond to $(n, m, \sigma, \boldsymbol{D})$ equal to $(10,10,1,\text{high})$, $(10,10,1,\text{low})$, $(10,10,0.4,\text{high})$, and $(10,10,0.4,\text{low})$. The remaining groups of 4 follow the same order but with sample sizes $(10,20)$, $(20,10)$, and $(20,20)$. The top panel displays the average simultaneous coverage of the population curve across 300 replications; bars indicate binomial confidence intervals for the coverage probability. The bottom panel shows boxplots of the average confidence-band width; for visual clarity, some results from assist fall outside the plotting range.
• Figure 2: Results of the sine curve simulation on the subject curves. The parameter settings follow those of Figure \ref{['fig:simulation1_assistcomparison']}. The top panel displays the average simultaneous coverage of the subject curves across 300 replications; bars indicate binomial confidence intervals for the coverage probability. The bottom panel shows boxplots of the average confidence-band width.
• Figure 3: Results of the bell curve simulation on the population curve. The first four settings correspond to $(n, m, \sigma)$ equal to $(10,10,0.2)$, $(20,10,0.2)$, $(10,20,0.2)$, and $(20,20,0.2)$. The remaining groups of 4 follow the same order but with error variance $0.4$. The top panel displays the average simultaneous coverage of the population curve across 300 replications; bars indicate binomial confidence intervals for the coverage probability. The bottom panel shows boxplots of the average confidence-band width.
• Figure 4: Height observations and model-based fits for four individuals. Green points denote observed height measurements. The solid black line shows the estimated mean growth trajectory, with the blue and red shaded areas representing pointwise and simultaneous confidence bands respectively. Sex and gestational age are (i) Female, 40 weeks; (ii) Male, 36 weeks; (iii) Female, 41 weeks; (iv) Male, 38 weeks.
• Figure 5: Parametric bootstrap assessment of bias and standard errors. The left panel shows the original parameter estimates against their bootstrap averages. The right panel shows the empirical standard deviation of the bootstrap estimates versus the average asymptotic standard errors. Results from $200$ simulations. The standard error for $\beta_1$ and $\beta_2$ deviate most from their bootstrap average and are labeled.