VGAM (version 1.1-8)

TIC: Takeuchi's Information Criterion

Description

Calculates the Takeuchi information criterion for a fitted model object for which a log-likelihood value has been obtained.

Usage

TIC(object, ...)
    TICvlm(object, ...)

Value

Returns a numeric TIC value.

Arguments

object

A VGAM object having class vglm-class.

...

Other possible arguments fed into logLik in order to compute the log-likelihood.

Author

T. W. Yee.

Warning

This code has not been double-checked. The general applicability of TIC for the VGLM/VGAM classes has not been developed fully. In particular, TIC should not be run on some VGAM family functions because of violation of certain regularity conditions, etc.

Some authors note that quite large sample sizes are needed for this IC to work reasonably well.

Details

The following formula is used for VGLMs: $-2\text{log-likelihood}+2trace(VK)$, where $V$ is the inverse of the EIM from the fitted model, and $K$ is the outer product of the score vectors. Both $V$ and $K$ are order-$p.VLM$ matrices. One has $V$ equal to vcov(object), and $K$ is computed by taking the outer product of the output from the deriv slot multiplied by the large VLM matrix and then taking their sum. Hence for the huge majority of models, the penalty is computed at the MLE and is empirical in nature. Theoretically, if the fitted model is the true model then AIC equals TIC.

When there are prior weights the score vectors are divided by the square root of these, because $(a_i{U}_i/\sqrt{a_i}{)}^2=a_i{U}_i^2$.

This code relies on the log-likelihood being defined, and computed, for the object. When comparing fitted objects, the smaller the TIC, the better the fit. The log-likelihood and hence the TIC is only defined up to an additive constant.

Currently any estimated scale parameter (in GLM parlance) is ignored by treating its value as unity. Also, currently this function is written only for vglm objects and not vgam or rrvglm, etc., objects.

References

Takeuchi, K. (1976). Distribution of informational statistics and a criterion of model fitting. (In Japanese). Suri-Kagaku (Mathematic Sciences), 153, 12--18.

Burnham, K. P. and Anderson, D. R. (2002). Model Selection and Multi-Model Inference: A Practical Information-Theoretic Approach, 2nd ed. New York, USA: Springer.

See Also

VGLMs are described in vglm-class; AIC, AICvlm. BICvlm.

Examples

pneumo <- transform(pneumo, let = log(exposure.time))
(fit1 <- vglm(cbind(normal, mild, severe) ~ let,
              cumulative(parallel = TRUE, reverse = TRUE), data = pneumo))
coef(fit1, matrix = TRUE)
TIC(fit1)
(fit2 <- vglm(cbind(normal, mild, severe) ~ let,
              cumulative(parallel = FALSE, reverse = TRUE), data = pneumo))
coef(fit2, matrix = TRUE)
TIC(fit2)