r.squaredLR: Likelihood-ratio based pseudo-R-squared

Description

Calculate a coefficient of determination based on the likelihood-ratio test (latex{$R_{LR}^{2}$}{R_LR-squared}).

Usage

r.squaredLR(x, null = NULL, null.RE = FALSE)
null.fit(x, evaluate = FALSE, RE.keep = FALSE, envir = NULL)

Arguments

a fitted model object.

null

a fitted null model. If not provided, null.fit will be used to construct it. null.fit's capabilities are limited to only a few model classes, for others the null model has to be specified manually.

null.RE

logical, should the null model contain random factors? Only used if no null model is given, otherwise omitted, with a warning.

evaluate

if TRUE evaluate the fitted model object else return the call.

RE.keep

if TRUE, the random effects of the original model are included.

envir

the environment in which the null model is to be evaluated, defaults to the environment of the original model's formula.

Value

r.squaredLR returns a value of $R_{LR}^{2}$, and the attribute "adj.r.squared" gives the Nagelkerke's modified statistic. Note that this is not the same as nor equivalent to the classical adjusted R squared.
null.fit returns the fitted null model object (if evaluate = TRUE) or an unevaluated call to fit a null model.

Details

This statistic is is one of the several proposed pseudo-R-squared's for nonlinear regression models. It is based on an improvement from null (intercept only) model to the fitted model, and calculated as $$R_{LR}^{2}=1-\exp(-\frac{2}{n}(\log\mathit{Lik}(x)-\log\mathit{Lik}(0)))$$ where logLik(x) and logLik(0) are the log-likelihoods of the fitted and the null model respectively. ML estimates are used for this purpose in when models have been fitted by REstricted ML (by calling logLik with argument REML = FALSE). Note that the null model can include the random factors of the original model, in which case the statistic represents the variance explained by fixed effects.

For OLS models the value is consistent with classical $R^{2}$. In some cases (e.g. in logistic regression), the maximum $R_{LR}^{2}$ is less than one. The modification proposed by Nagelkerke (1991) adjusts the $R_{LR}^{2}$ to achieve 1 at its maximum: $\bar{R}^{2} = R_{LR}^{2} / \max(R_{LR}^{2})$ where $max(R_{LR}^{2}) = 1 - \exp(\frac{2}{n}\log\mathit{Lik}(\textrm{0}))$.

null.fit tries to guess the null model call, given the provided fitted model object. This would be usually a glm. The function will give an error for an unrecognized class.

References

Cox, D. R. and Snell, E. J. (1989) The analysis of binary data, 2nd ed. London, Chapman and Hall

Magee, L. (1990) R$^{2}$ measures based on Wald and likelihood ratio joint significance tests. Amer. Stat. 44: 250-253