Takeuchi's Information Criterion
Calculates the Takeuchi information criterion for a fitted model object for which a log-likelihood value has been obtained.
TIC(object, ...)
TICvlm(object, ...)object |
A VGAM object having
class |
... |
Other possible arguments fed into
|
The following formula is used for VGLMs:
-2*log-likelihood
+ 2 * trace(V K),
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 / √{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.
Returns a numeric TIC value.
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.
TIC has not been defined for RR-VGLMs, QRR-VGLMs, etc., yet.
See AICvlm about models
such as posbernoulli.tb
that require posbinomial(omit.constant = TRUE).
T. W. Yee.
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.
VGLMs are described in vglm-class;
AIC,
AICvlm.
BICvlm.
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)Please choose more modern alternatives, such as Google Chrome or Mozilla Firefox.