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 Pseudo R squared values for multiple logistic regression

R squared is a useful metric for multiple linear regression, but does not have the same meaning in logistic regression. Statisticians have come up with a variety of analogues of R squared for multiple logistic regression that they refer to collectively as “pseudo R squared”. These do not have the same interpretation, in that they are not simply the proportion of variance explained by the model.

In fact, the magnitude of any particular pseudo R squared value can't be used to compare across datasets. Instead, the primary use for these pseudo R squared values is for comparing multiple models fit to the same dataset. There is not a clear "best" pseudo R squared, and Prism offers four different ones from which to choose. More information on how these pseudo R squared values are calculated is provided.

## Tjur’s R squared

This is one of the simplest pseudo R squared values to explain. Find the predicted probability from the model for every entered value of the dependent variable. For each category of the dependent variable (0 and 1), find the average predicted probability. Then, find the absolute value of the difference between these two averages. That value is Tjur’s R squared.

This value uses the log-likelihood of the specified model and a corresponding “intercept-only” model (values that Prism can report) and determines their ratio. This ratio is then subtracted from 1 to determine the reported value. A small ratio (and thus a final value close to 1) indicates that the specified model is better than an intercept-only model.

## Cox-Snell’s R squared (“generalized” R squared)

Cox-Snell’s R squared uses the likelihood (as opposed to the log-likelihood), so some additional mathematical manipulation would be required to calculate this value. Unlike other pseudo R squared values here, the maximum of Cox-Snell’s R squared is less than 1. However, this value is commonly reported by other software, and so is an option here.

## Nagelkerke’s R squared

This pseudo R squared is very similar to Cox-Snell’s R squared. The primary difference is that it adjusts Cox-Snell’s R squared to have a maximum value of 1. This is done by dividing the Cox-Snell’s R squared by its maximum possible value.