Pseudo-R-squared

In linear regression, the squared multiple correlation, R2 is used to assess goodness of fit as it represents the proportion of variance in the criterion that is explained by the predictors.

The Cox and Snell index corresponds to the standard R2 in case of a linear model with normal error.

[2] R2N, proposed by Nico Nagelkerke in a highly cited Biometrika paper,[4] provides a correction to the Cox and Snell R2 so that the maximum value is equal to 1.

Nevertheless, the Cox and Snell and likelihood ratio R2s show greater agreement with each other than either does with the Nagelkerke R2.

The pseudo R2 by McFadden (sometimes called likelihood ratio index[5]) is defined as and is preferred over R2CS by Allison.

The reason these indices of fit are referred to as pseudo R2 is that they do not represent the proportionate reduction in error as the R2 in linear regression does.

Logistic regression will always be heteroscedastic – the error variances differ for each value of the predicted score.

This displays R output from calculating pseudo-r-squared values using the "pscl" package by Simon Jackman. The pseudo-R-squared by McFadden is clearly labelled “McFadden”, which is equal to the pseudo-R-squared by Cohen. 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.