GAIC and Generalised (Pseudo) R-squared for GAMLSS Models

Description

Functions to compute the GAIC and the generalised R-squared of Nagelkerke (1991) for GAMLSS models.

Usage

## Information criteria.
GAIC(object, ...,
  k = 2, corrected = FALSE)

## R-squared.
Rsq(object, ...,
  type = c("Cox Snell", "Cragg Uhler", "both", "simple"),
  newdata = NULL)

Arguments

object	A fitted model object
…	Optionally more fitted model objects.
k	Numeric, the penalty to be used. The default k = 2 corresponds to the classical AIC.
corrected	Logical, whether the corrected AIC should be used? Note that it applies only when k = 2.
type	Which definition of R squared. Can be the “Cox Snell” or the Nagelkerke, “Cragg Uhler” or “both”, or “simple”, which computes the R-squared based on the median. In this case also newdata may be supplied.
newdata	Only for type = “simple”, the R-squared can be evaluated using newdata.

Details

The Rsq() function uses the definition for R-squared:

\(R^2=1- \left(\frac{L(0)}{L(\hat{\theta})}\right)^{2/n}\)

where \(L(0)\) is the null model (only a constant is fitted to all parameters) and \(L(\hat{\theta})\) is the current fitted model. This definition sometimes is referred to as the Cox & Snell R-squared. The Nagelkerke /Cragg & Uhler’s definition divides the above by

\(1 - L(0)^{2/n}\)

Note that GAIC() function is fully represented by the AIC and BIC functions. We only included GAIC() because of the previous package so users who relied on this function would still have the option.

Value

Numeric vector or data frame, depending on the number of fitted model objects.

References

Nagelkerke NJD (1991). “A Note on a General Definition of the Coefficient of Determination.” Biometrika, 78(3), 691–692. doi:10.1093/biomet/78.3.691

See Also

gamlss2

Examples

library("gamlss2")


## load the aids data set
data("aids", package = "gamlss.data")

## estimate negative binomial count models
b1 <- gamlss2(y ~ x + qrt, data = aids, family = NBI)

GAMLSS-RS iteration  1: Global Deviance = 492.7033 eps = 0.148555     
GAMLSS-RS iteration  2: Global Deviance = 492.6374 eps = 0.000133     
GAMLSS-RS iteration  3: Global Deviance = 492.6373 eps = 0.000000     

b2 <- gamlss2(y ~ s(x) + s(qrt, bs = "re"), data = aids, family = NBI)

GAMLSS-RS iteration  1: Global Deviance = 408.2411 eps = 0.294515     
GAMLSS-RS iteration  2: Global Deviance = 379.8718 eps = 0.069491     
GAMLSS-RS iteration  3: Global Deviance = 375.4344 eps = 0.011681     
GAMLSS-RS iteration  4: Global Deviance = 374.4315 eps = 0.002671     
GAMLSS-RS iteration  5: Global Deviance = 374.1784 eps = 0.000676     
GAMLSS-RS iteration  6: Global Deviance = 374.1352 eps = 0.000115     
GAMLSS-RS iteration  7: Global Deviance = 374.1317 eps = 0.000009     

## compare models
Rsq(b1)

[1] 0.8095853

Rsq(b1, type = "both")

$CoxSnell
[1] 0.8095853

$CraggUhler
[1] 0.809588

Rsq(b1, b2)

       b1        b2 
0.8095853 0.9863226 

GAIC(b1, b2)

        AIC      df
b2 398.7665 12.3174
b1 504.6373  6.0000

AIC(b1, b2)

        AIC      df
b2 398.7665 12.3174
b1 504.6373  6.0000

BIC(b1, b2)

        AIC      df
b2 421.0199 12.3174
b1 515.4773  6.0000