Pseudo-R-squared

The pseudo 2 credited to McFadden (sometimes called likelihood ratio index or log-likelihood ratio ::R^2_\text{L} = 1 - \frac{\ln(L_M)}{\ln(L_0)}, or expressed using deviance It represents the proportional reduction in the deviance wherein the deviance is treated as a measure of variation analogous but not identical to the variance in linear regression analysis. ::R^2_\text{L} = 1 - \frac{\ln(L_M) - K}{\ln(L_0)}, == 2M by Cox and Snell ==

2M is an alternative index of goodness of fit described by Maddala in 1983 attributed to Cox and Snell The Cox and Snell index corresponds to the standard 2 in case of a linear model with normal error. In certain situations, 2M may be problematic as its maximum value is 1 - L_0^{2/n} which means it never takes the value of one. For example, for logistic regression, the upper bound is R^2_\text{M}\leq0.75 for a symmetric marginal distribution of events and decreases further for an asymmetric distribution of events. provides a correction to the Cox and Snell 2 so that the maximum value is equal to 1. This correction is done by dividing M by its upper bound. :: \begin{align} R^2_\text{N} & = \frac{\left[1 - \left(\frac{L_0}{L_M}\right)^{2/n}\right]}{1 - L_0^{2/n}} \end{align} Nevertheless, the Cox and Snell and likelihood ratio 2s show greater agreement with each other than either does with the adjusted Nagelkerke 2. Of course, this might not be the case for values exceeding 0.75 as the Cox and Snell index is capped at this value. The likelihood ratio 2 is often preferred to the alternatives as it is most analogous to 2 in linear regression, is independent of the base rate (both Cox and Snell and Nagelkerke 2s increase as the proportion of cases increase from 0 to 0.5) and varies between 0 and 1. == 2 by Tjur==

Tjur proposed an alternative measure of 2, which is in two steps: • For each level of the dependent variable, find the mean of the predicted probabilities of an event. • Take the absolute value of the difference between these means This quantity is not an R^2 per se, however Tjur shows how it is related to usual R^2 metrics. == Example ==

This displays R output from calculating pseudo-r-squared values using the "pscl" package by Simon Jackman. The pseudo-R-squared calculated using the formule named for McFadden is labelled “McFadden”. Next to this, the pseudo-r-squared by Cox and Snell is labelled “r2ML” and this type of pseudo-R-squared By Cox and Snell is sometimes simply called “ML”. The last value listed, labelled “r2CU” is the pseudo-r-squared by Nagelkerke and is the same as the pseudo-r-squared by Cragg and Uhler. == Interpretation ==

A word of caution is in order when interpreting pseudo-2 statistics. The reason these indices of fit are referred to as pseudo 2 is that they do not represent the proportionate reduction in error as the 2 in linear regression does. Linear regression assumes homoscedasticity, that the error variance is the same for all values of the criterion. Logistic regression will always be heteroscedastic – the error variances differ for each value of the predicted score. For each value of the predicted score there would be a different value of the proportionate reduction in error. Therefore, it is inappropriate to think of 2 as a proportionate reduction in error in a universal sense in logistic regression. == See also ==