This function gives the generalised R-squared of Nagelkerke (1991) for a GAMLSS model.

`Rsq(object, type = c("Cox Snell","Cragg Uhler","both"))`

The `Rsq()`

produces a single value if type="Cox Snell" or "Cragg Uhler" and a list if type="both".

- object
a GAMLSS object

- type
which definition of R squared. Can be the "Cox Snell" or the Nagelkerke, "Cragg Uhler" or "both".

Mikis Stasinopoulos d.stasinopoulos@londonmet.ac.uk

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 with
$$1- L(0)^(2/n)$$

Nagelkerke, N. J. (1991). A note on a general definition of the coefficient of determination. Biometrika, 78(3), 691-692.

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/).

`GAIC`

```
data(aids)
m1 <- gamlss(y~x+qrt, data=aids, family=NBI)
Rsq(m1)
Rsq(m1, type="both")
```

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