spaMM (version 3.11.3)

AIC: Extractors for information criteria such as AIC

Description

get_any_IC computes model selection/information criteria such as AIC. See Details for more information about these criteria. The other extractors AIC and extractAIC are methods for HLfit objects of generic functions defined in other packages: AIC is equivalent to get_any_IC, and extractAIC returns the marginal AIC and the number of degrees of freedom for the fixed effects.

Usage

get_any_IC(object, nsim=0L, ..., verbose=interactive(),
           also_cAIC=TRUE, short.names=NULL)
# S3 method for HLfit
AIC(object, nsim=0L, ..., k, verbose=interactive(),
                    also_cAIC=TRUE, short.names=NULL)
# S3 method for HLfit
extractAIC(fit, scale, k, ..., verbose=FALSE)

Arguments

object, fit

A object of class HLfit, as returned by the fitting functions in spaMM.

scale, k

Currently ignored, but are required in the definitions for consistency with the generic.

verbose

Whether to print the model selection criteria or not.

also_cAIC

Whether to include the plug-in estimate of conditional AIC in the result (its computation may be slow).

nsim

Controls whether to include the bootstrap estimate of conditional AIC (see Details) in the result. If positive, nsim gives the number of bootstrap replicates.

short.names

NULL, or boolean; controls whether the return value uses short names (mAIC, etc., as shown by screen output if verbose is TRUE), or the descriptive names (" marginal AIC:", etc.) also shown in the screen output. Short names are more appropriate for programming but descriptive names may be needed for back-compatibility. The default (NULL) ensures back-compatibility by using descriptive names unless the bootstrap estimate of conditional AIC is reported.

…

Other arguments that may be needed by some method. If nsim is positive, a seed argument may be passed to simulate, and the other “…” will be used in a call to dopar to control the optional parallel execution of the bootstrap computations.

Value

For AIC and get_any_IC, a numeric vector whose possible elements are described in the Details, and whose names are controlled by the short.names argument. Note that the bootstrap computation actually makes sense and works also for fixed-effect models (although it is not clear how useful it is in that case). The return value will still refer to its results as conditional AIC.

For extractAIC, a numeric vector of length 2, with first and second elements giving

edf

the degree of freedom of the fixed-effect terms of the model for the fitted model fit.

AIC

the (marginal) Akaike Information Criterion for fit.

This output aims to be equivalent (except for the explicit names) to the one from stats::extractAIC, despite the obscurities of the latter's documentation, and is indeed equivalent in value for GLMs (see Examples).

Details

The AIC is a measure (by Kullback-Leibler directed distance, up to an additive constant) of quality of prediction of new data by a fitted model. Comparing information criteria may be viewed as a fast alternative to a comparison of the predictive accuracy of different models by cross-validation. Further procedures for model choice may also be useful (e.g. Williams, 1970; Lewis et al. 2010).

The conditional AIC (Vaida and Blanchard 2005) applies the AIC concept to new realizations of a mixed model, conditional on the realized values of the random effects. Lee et al. (2006) and Ha et al (2007) defined a corrected AIC [i.e., AIC(D*) in their eq. 7] which is here interpreted as the conditional AIC.

Such Kullback-Leibler relative distances cannot generally be evaluated exactly and various estimates have been discussed. get_any_IC computes, optionally prints, and returns invisibly one or more of the following quantities: (1) Akaike's classical AIC (marginal AIC, mAIC); (2) a plug-in estimate (cAIC) and/or a bootstrap estimate (b_cAIC) of the conditional AIC; and (3) a focussed AIC for dispersion parameters (dispersion AIC, dAIC).

For the conditional AIC, Vaida and Blanchard's plug-in estimator involves the conditional likelihood, and degrees of freedom for (i) estimated residual error parameters and (ii) the overall linear predictor characterized by the Effective degrees of freedom already discussed by previous authors including Lee and Nelder (1996), which gave a plug-in estimator (\(p_D\)) for it in HGLMs. By default, the plug-in estimate of both the conditional AIC and of \(n-p_D\) (GoFdf, where \(n\) is the length of the response vector) are returned by get_any_IC. But these are biased estimates of conditional AIC and effective df, and an alternative procedure is available if a non-default positive nsim value is used. In that case, the conditional AIC is estimated by a bootstrap version of Saefken et al. (2014)'s equation 2.5; this involves refitting the model to each bootstrap samples, so it may take time, and a full cross-validation procedure might as well be considered for model selection.

The dispersion AIC has been defined from restricted likelihood by Ha et al (2007; eq.10). The present implementation will use restricted likelihood only if made available by an REML fit, otherwise marginal likelihood is used.

References

Ha, I. D., Lee, Y. and MacKenzie, G. (2007) Model selection for multi-component frailty models. Statistics in Medicine 26: 4790-4807.

Lee Y. and Nelder. J. A. 1996. Hierarchical generalized linear models (with discussion). J. R. Statist. Soc. B, 58: 619-678.

Lewis, F., Butler, A. and Gilbert, L. (2011), A unified approach to model selection using the likelihood ratio test. Methods in Ecology and Evolution, 2: 155-162. 10.1111/j.2041-210X.2010.00063.x

Saefken B., Kneib T., van Waveren C.-S., Greven S. (2014) A unifying approach to the estimation of the conditional Akaike information in generalized linear mixed models. Electron. J. Statist. 8, 201-225.

Vaida, F., and Blanchard, S. (2005) Conditional Akaike information for mixed-effects models. Biometrika 92, 351-370.

Williams D.A. (1970) Discrimination between regression models to determine the pattern of enzyme synthesis in synchronous cell cultures. Biometrics 26: 23-32.

Examples

# NOT RUN {
data("wafers")
m1 <- fitme(y ~ X1+X2+X3+X1*X3+X2*X3+I(X2^2)+(1|batch), data=wafers, 
            family=Gamma(log))

get_any_IC(m1) 
# => The plug-in estimate is stored in the 'm1' object 
#    as a result of the previous computation, and is now returned even by: 
get_any_IC(m1, also_cAIC=FALSE)

if (spaMM.getOption("example_maxtime")>4) {
 get_any_IC(m1, nsim=100L, seed=123) # provides bootstrap estimate of cAIC.
 # (parallelisation options could be used, e.g. nb_cores=detectCores()-1L)
}

extractAIC(m1)

# }
# NOT RUN {
# Checking consistency with glm example from help("stats::extractAIC"):
utils::example(glm)
dataf <- data.frame(counts=counts,outcome=outcome, treatment=treatment)
extractAIC(glm.D93) 
extractAIC(fitme(counts ~ outcome + treatment, family = poisson(), data=dataf))
lm.D93 <- glm(counts ~ outcome + treatment)
extractAIC(lm.D93)
extractAIC(fitme(counts ~ outcome + treatment, data=dataf))
# }
# NOT RUN {
# }