# Approximation to Log-likelihood of Nonlinear Mixed-effects Model

##### Posted on Nov 26, 2023

Gaussian quadrature is used to approximate integrals of functions with respect to a given kernel by a weighted average of the integrand evaluated at predetermined abscissas.

Particularly, for the kernel $\exp(-x^2)$, the quadrature is called Gauss-Hermite quadrature.

For the nonlinear mixed-effect model, the $j$-th observation on the $i$-th subject is modeled as

$y_{ij} = f(\phi_{ij}, x_{ij}) + \varepsilon_{ij}, i=1,\ldots, M, j=1,\ldots, n_i$

where $f$ is a nonlinear function of a subject-specific parameter vector $\phi_{ij}$ and the predictor vector $x_{ij}$.

The subject-specific parameter vector is modeled as

$\phi_{ij} = A_{ij}\beta + B_{ij} b_i, b_i\sim N(0, \sigma^2D)$

MLE is based on the marginal density of $y$

$p(y\mid \beta, D, \sigma^2) = \int p(y\mid b, \beta, D, \sigma^2) p(b)db$

In general this integral does not have a closed-form expression when the model function $f$ is nonlinear in $b_i$, so different approximations have been proposed for estimating it.

the paper consider four different approximations

• take the first-order Taylor expansion of the model function $f$ around the conditional (on $D$) modes of the random effects. Lindstrom and Bates’s (1990)’s LME method
• a modified Laplacian approximation
• importance sampling