# AICvlm: Akaike's Information Criterion In VGAM: Vector Generalized Linear and Additive Models

## Description

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

## Usage

 ```1 2 3 4 5``` ``` AICvlm(object, ..., corrected = FALSE, k = 2) AICvgam(object, ..., k = 2) AICrrvglm(object, ..., k = 2) AICqrrvglm(object, ..., k = 2) AICrrvgam(object, ..., k = 2) ```

## Arguments

 `object` Some VGAM object, for example, having class `vglm-class`. `...` Other possible arguments fed into `logLik` in order to compute the log-likelihood. `corrected` Logical, perform the finite sample correction? `k` Numeric, the penalty per parameter to be used; the default is the classical AIC.

## Details

The following formula is used for VGLMs: -2*log-likelihood + k*npar, where npar represents the number of parameters in the fitted model, and k = 2 for the usual AIC. One could assign k = \log(n) (n the number of observations) for the so-called BIC or SBC (Schwarz's Bayesian criterion). This is the function `AICvlm()`.

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

Any estimated scale parameter (in GLM parlance) is used as one parameter.

For VGAMs and CAO the nonlinear effective degrees of freedom for each smoothed component is used. This formula is heuristic. These are the functions `AICvgam()` and `AICcao()`.

The finite sample correction is usually recommended when the sample size is small or when the number of parameters is large. When the sample size is large their difference tends to be negligible. The correction is described in Hurvich and Tsai (1989), and is based on a (univariate) linear model with normally distributed errors.

## Value

Returns a numeric value with the corresponding AIC (or BIC, or ..., depending on `k`).

## Warning

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

## Note

AIC has not been defined for QRR-VGLMs, yet.

Using AIC to compare `posbinomial` models with, e.g., `posbernoulli.tb` models, requires `posbinomial(omit.constant = TRUE)`. See `posbinomial` for an example. A warning is given if it suspects a wrong `omit.constant` value was used.

Where defined, `AICc(...)` is the same as `AIC(..., corrected = TRUE)`.

T. W. Yee.

## References

Hurvich, C. M. and Tsai, C.-L. (1989). Regression and time series model selection in small samples, Biometrika, 76, 297–307.

VGLMs are described in `vglm-class`; VGAMs are described in `vgam-class`; RR-VGLMs are described in `rrvglm-class`; `AIC`, `BICvlm`, `TICvlm`, `drop1.vglm`, `extractAIC.vglm`.

## Examples

 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13``` ```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) AIC(fit1) AICc(fit1) # Quick way AIC(fit1, corrected = TRUE) # Slow way (fit2 <- vglm(cbind(normal, mild, severe) ~ let, cumulative(parallel = FALSE, reverse = TRUE), data = pneumo)) coef(fit2, matrix = TRUE) AIC(fit2) AICc(fit2) AIC(fit2, corrected = TRUE) ```

### Example output

```Loading required package: stats4

Call:
vglm(formula = cbind(normal, mild, severe) ~ let, family = cumulative(parallel = TRUE,
reverse = TRUE), data = pneumo)

Coefficients:
(Intercept):1 (Intercept):2           let
-9.676093    -10.581725      2.596807

Degrees of Freedom: 16 Total; 13 Residual
Residual deviance: 5.026826
Log-likelihood: -25.09026
(Intercept)          -9.676093         -10.581725
let                   2.596807           2.596807
[1] 56.18052
[1] 62.18052
[1] 62.18052

Call:
vglm(formula = cbind(normal, mild, severe) ~ let, family = cumulative(parallel = FALSE,
reverse = TRUE), data = pneumo)

Coefficients:
(Intercept):1 (Intercept):2         let:1         let:2
-9.593308    -11.104791      2.571300      2.743550

Degrees of Freedom: 16 Total; 12 Residual
Residual deviance: 4.884404
Log-likelihood: -25.01905