Solved – How to use the Laplace approximation to fit a spatial GLM model

generalized linear modelmodelingr

I would like to fit a spatial generalized linear mixed model, for example, to model spatial count responses by using the Laplace approximation in available statistical software.

I am particularly interested in implementations in R.

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Nested Laplace approximations can be used to fit various spatial models. Papers, tutorials, and lots of examples can be found here

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Solved – Reference for generalized linear mixed models using Laplace approximation

Here are some papers:

Breslow and Clayton, 1993: Approximate Inference in Generalized Linear Mixed Models.

Tierney and Kadane, 1986: Accurate Approximations for Posterior Moments and Marginal Densities.

Breslow, 2003: Whither PQL?.

Tuerlinckx, et al. 2006: Statistical Inference in generalized linear mixed models: A review.

Wolfinger, 1993: Laplace's approximation for nonlinear mixed models.

Vonesh, 1996: A Note on the Use of Laplace's Approximation for Nonlinear Mixed-Effects Models.

Solved – How to interpret coefficients of a multivariate mixed model in lme4 without overall intercept

Your idea is good, but in your example, you forgot to model different intercept and different random variances for each trait, so your output is not interpretable as is. A correct model would be:

m1 <- lmer(value ~ -1 + variable + variable:gear + variable:carb + (0 + variable | factor(carb))

In that case, you would get the estimates of fixed effects on each variable (for example, variabledrat:gear is the effect of predictor gear on response drat), but you would also get the intercepts for each variable (for example variabledrat for the intercept of response drat) and the random variance of each variable and the correlations between variables:

Groups       Name         Std.Dev. Corr     
 factor(carb) variabledrat 23.80             
              variablempg  24.27    0.20     
              variablehp   23.80    0.00 0.00
 Residual                  23.80

A more detailed description of these methods has been written down by Ben Bolker, as well as the use of MCMCglmmin a Bayesian framework. Another new package, mcglm can also handle multivariate models, even with non-normal responses, but you have to code your random design matrices. A tutorial should be available soon (see the R help page).

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Solved – Reference for generalized linear mixed models using Laplace approximation

Solved – How to interpret coefficients of a multivariate mixed model in lme4 without overall intercept

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