mvMORPH (version 1.2.1)

EIC: Extended Information Criterion (EIC) to compare models fit with mvgls (or mvols) by Maximum Likelihood (ML) or Penalized Likelihood (PL)

Description

The EIC (Ishiguro et al. 1997, Kitagawa & Konishi 2010), uses bootstrap to estimate the bias term of the Extended Information Criterion. This criterion allows comparing models fit by Maximum Likelihood (ML) or Penalized Likelihood (PL).

Usage

EIC(object, nboot=100L, nbcores=1L, ...)

Value

a list with the following components

LogLikelihood

the log-likelihood estimated for the model with estimated parameters

EIC

the EIC criterion

se

the standard error of the bias term estimated by bootstrap

bias

the values of the bias term estimated from the boostrapped replicates to compute the EIC

Arguments

object

An object of class 'mvgls'. See ?mvgls or ?mvols

nboot

The number of boostrap replicates used for estimating the EIC.

nbcores

The number of cores used to speed-up the computations (uses the 'parallel' package)

...

Options to be passed through.

Author

J. Clavel

Details

The Extended Information Criterion (EIC) allows comparing the fit of various models estimated by Penalized Likelihood or Maximum Likelihood (see ?mvgls). Similar to the GIC or the more common AIC, the EIC has the form:

$$EIC = -2*(Likelihood) + 2*bias$$

Where Likelihood corresponds to either the full or the restricted likelihood (see the note below), and the bias term is estimated by (semi-parametric) bootstrap simulations rather than by using analytical or approximate solutions (see for instance ?GIC). The smaller the EIC, the better is the model. With small sample sizes, the variability around the bootstrap estimate is expected to be high, and one must increase the number of bootstrap replicates. Parallel computation (argument nbcores) allows to speed-up the computations.

Note: for models estimated by REML, it is generally not possible to compare the restricted likelihoods when the models fit have different fixed effects. However, it is possible to compare models with different fixed effects by using the full likelihood evaluated at the REML estimates (see e.g. Yafune et al. 2006, Verbyla 2019). Both options - evaluating the restricted likelihood or the full likelihood with parameters estimated by REML - are available through the REML argument in the EIC function. The default has been set to REML=FALSE to allow the comparison of models with different fixed effects using the full likelihood evaluated with the REML estimates (see Verbyla 2019).

References

Clavel J., Aristide L., Morlon H., 2019. A Penalized Likelihood framework for high-dimensional phylogenetic comparative methods and an application to new-world monkeys brain evolution. Syst. Biol. 68:93-116.

Ishiguro M., Sakamoto Y., Kitagawa G., 1997. Bootstrapping log likelihood and EIC, an extension of AIC. Ann. Inst. Statist. Math. 49:411-434.

Kitagawa G., Konishi S., 2010. Bias and variance reduction techniques for bootstrap information criterion. Ann. Inst. Stat. Math. 62:209-234.

Konishi S., Kitagawa G., 1996. Generalised information criteria in model selection. Biometrika. 83:875-890.

Verbyla A. P., 2019. A note on model selection using information criteria for general linear models estimated using REML. Aust. N. Z. J. Stat. 61:39-50.

Yafune A., Funatogawa T., Ishiguro M., 2005. Extended information criterion (EIC) approach for linear mixed effects models under restricted maximum likelihood (REML) estimation. Statist. Med. 24:3417-3429.

See Also

GIC mvgls mvols manova.gls

Examples

# \donttest{

set.seed(1)
n <- 32 # number of species
p <- 50 # number of traits

tree <- pbtree(n=n) # phylogenetic tree
R <- crossprod(matrix(runif(p*p), ncol=p)) # a random symmetric matrix (covariance)
# simulate a dataset
Y <- mvSIM(tree, model="BM1", nsim=1, param=list(sigma=R))

fit1 <- mvgls(Y~1, tree=tree, model="BM", method="H&L")
fit2 <- mvgls(Y~1, tree=tree, model="OU", method="H&L")


EIC(fit1); EIC(fit2)

# We can improve accuracy by increasing the number of bootstrap samples
# EIC(fit1, nboot=5000, nbcores=8L)
# EIC(fit2, nboot=5000, nbcores=8L)
# }