The function `GAIC()`

calculates the Generalised Akaike information criterion (GAIC) for a given penalty `k`

for a fitted GAMLSS object.

The function `AIC.gamlss()`

is the method associated with a GAMLSS object of the generic function `AIC()`

. Note that `GAIC()`

is a synonymous of the function `AIC.gamlss`

.

The function `IC()`

is an old version of `GAIC()`

.

The function `GAIC.table()`

produces a table with different models as rows and different penalties, `k`

, as columns.

The function `GAIC.scaled()`

produces, [for a given set of different fitted models or for a table produced by `chooseDist()`

], the scaled Akaike values (see Burnham and Anderson (2002) section 2.9 for a similar concept the GAIC weights. The scaled Akaike should not be interpreted as posterior probabilities of models given the data but for model selection purpose they produce a scaled ranking of the model using their relative importance i.e. from the worst to the best model.

The function `extractAIC`

is a the method associated a GAMLSS object of the generic function `extractAIC`

and it is
mainly used in the `stepAIC`

function.

The function `Rsq`

compute a generalisation of the R-squared for not normal models.

```
IC(object, k = 2)
# S3 method for gamlss
AIC(object, ..., k = 2, c = FALSE)
GAIC(object, ..., k = 2, c = FALSE )
GAIC.table(object, ..., k = c(2, 3.84, round(log(length(object$y)), 2)),
text.to.show=NULL)
GAIC.scaled(object,..., k = 2, c = FALSE, plot = TRUE,
text.cex = 0.7, which = 1, diff.dev = 1000,
text.to.show = NULL, col = NULL, horiz = FALSE)
# S3 method for gamlss
extractAIC(fit, scale, k = 2, c = FALSE, ...)
```

The function `IC()`

returns the GAIC for given penalty k of the GAMLSS object.
The function `AIC()`

returns a matrix contains the df's and the GAIC's for given penalty k.
The function `GAIC()`

returns identical results to `AIC`

.
The function `GAIC.table()`

returns a table which its rows showing different models and its columns different `k`

's.
The function `extractAIC()`

returns vector of length two with the degrees of freedom and the AIC criterion.

- object
an gamlss fitted model(s) [or for

`GAIC.scaled()`

a table produced by`chooseDist()`

].- fit
an gamlss fitted model

- ...
allows several GAMLSS object to be compared using a GAIC

- k
the penalty with default

`k=2.5`

- c
whether the corrected AIC, i.e. AICc, should be used, note that it applies only when

`k=2`

- scale
this argument is not used in gamlss

- plot
whether to plot the ranking in

`GAIC.scaled()`

.- text.cex
the size of the models/families in the text of the plot of

`GAIC.scaled()`

.- diff.dev
this argument prevents models with a difference in deviance greater than

`diff.dev`

from the `best' model to be considered (or plotted).- which
which column of GAIC scaled to plot in

`GAIC.scaled()`

.- text.to.show
if NULL,

`GAIC.scaled()`

shows the model names otherwise the character in this list- col
The colour of the bars in

`GAIC.scaled()`

- horiz
whether to plot the bars vertically (default) or horizontally

Mikis Stasinopoulos d.stasinopoulos@londonmet.ac.uk

Burnham K. P. and Anderson D. R (2002). *Model Selection and Multi model Inference
A Practical Information-Theoretic Approach*, Second Edition, Springer-Verlag New York, Inc.

Rigby, R. A. and Stasinopoulos D. M. (2005). Generalized additive models for location, scale and shape,(with discussion),
*Appl. Statist.*, **54**, part 3, pp 507-554.

Rigby, R. A., Stasinopoulos, D. M., Heller, G. Z., and De Bastiani, F. (2019)
*Distributions for modeling location, scale, and shape: Using GAMLSS in R*, Chapman and Hall/CRC. An older version can be found in https://www.gamlss.com/.

Stasinopoulos D. M. Rigby R.A. (2007) Generalized additive models for location scale and shape (GAMLSS) in R.
*Journal of Statistical Software*, Vol. **23**, Issue 7, Dec 2007, https://www.jstatsoft.org/v23/i07/.

Stasinopoulos D. M., Rigby R.A., Heller G., Voudouris V., and De Bastiani F., (2017)
*Flexible Regression and Smoothing: Using GAMLSS in R*, Chapman and Hall/CRC.

(see also https://www.gamlss.com/).

`gamlss`

```
data(abdom)
m1 <- gamlss(y~x, family=NO, data=abdom)
IC(m1)
extractAIC(m1,k=2)
m2 <- gamlss(y~x, sigma.fo=~x, family=NO, data=abdom)
m3 <- gamlss(y~pb(x), sigma.fo=~x, family=NO, data=abdom)
m4 <- gamlss(y~pb(x), sigma.fo=~pb(x), family=NO, data=abdom)
AIC(m1,m2, m3, m4)
AIC(m1,m2, m3, m4, c=TRUE)
AIC(m1,m2, m3, m4, k=3)
GAIC.table(m1,m2, m3, m4)
GAIC.scaled(m1,m2, m3, m4)
if (FALSE) {
MT <- chooseDist(m3)
GAIC.scaled(MT)
GAIC.scaled(MT, which=2)}
```

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