Pseudo-R-squared

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.

Pseudo-R-squared values are used when the outcome variable is nominal or ordinal such that the coefficient of determination R² cannot be applied as a measure for goodness of fit and when a likelihood function is used to fit a model.

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. ^[1] In logistic regression analysis, there is no agreed upon analogous measure, but there are several competing measures each with limitations. ^{[1] [2]}

Four of the most commonly used indices and one less commonly used one are examined in this article:

• Likelihood ratio R²_L
• Cox and Snell R²_CS
• Nagelkerke R²_N
• McFadden R²_McF
• Tjur R²_T

R²_L by Cohen

R²_L is given by Cohen: ^[1]

${\displaystyle R_{\text{L}}^{2}={\frac {D_{\text{null}}-D_{\text{fitted}}}{D_{\text{null}}}}.}$

This is the most analogous index to the squared multiple correlations in linear regression. ^[3] 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. ^[3] One limitation of the likelihood ratio R² is that it is not monotonically related to the odds ratio, ^[1] meaning that it does not necessarily increase as the odds ratio increases and does not necessarily decrease as the odds ratio decreases.

R²_CS by Cox and Snell

R²_CS is an alternative index of goodness of fit related to the R² value from linear regression. ^[2] It is given by:

${\displaystyle {\begin{aligned}R_{\text{CS}}^{2}&=1-\left({\frac {L_{0}}{L_{M}}}\right)^{2/n}\\[5pt]&=1-e^{2(\ln(L_{0})-\ln(L_{M}))/n}\end{aligned}}}$

where L_M and L₀ are the likelihoods for the model being fitted and the null model, respectively. The Cox and Snell index is problematic as its maximum value is ${\displaystyle 1-L_{0}^{2/n}}$. The highest this upper bound can be is 0.75, but it can easily be as low as 0.48 when the marginal proportion of cases is small. ^[2]

R²_N by Nagelkerke

R²_N, proposed by Nico Nagelkerke in a highly cited Biometrika paper, 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. ^[1] 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 R² is often preferred to the alternatives as it is most analogous to R² in linear regression, is independent of the base rate (both Cox and Snell and Nagelkerke R²s increase as the proportion of cases increase from 0 to 0.5) and varies between 0 and 1.

R²_McF by McFadden

The pseudo R² by McFadden (sometimes called likelihood ratio index ^[4] ) is defined as

${\displaystyle R_{\text{McF}}^{2}=1-{\frac {\ln(L_{M})}{\ln(L_{0})}},}$

and is preferred over R²_CS by Allison. ^[2] The two expressions R²_McF and R²_CS are then related respectively by,

${\displaystyle {\begin{matrix}R_{\text{CS}}^{2}=1-\left({\dfrac {1}{L_{0}}}\right)^{\frac {2(R_{\text{McF}}^{2})}{n}}\\[1.5em]R_{\text{McF}}^{2}=-{\dfrac {n}{2}}\cdot {\dfrac {\ln(1-R_{\text{CS}}^{2})}{\ln L_{0}}}\end{matrix}}}$

R²_T by Tjur

Allison ^[2] prefers R²_T which is a relatively new measure developed by Tjur. ^[5] It can be calculated in two steps:

1. For each level of the dependent variable, find the mean of the predicted probabilities of an event.
2. Take the absolute value of the difference between these means

Interpretation

A word of caution is in order when interpreting pseudo-R² statistics. The reason these indices of fit are referred to as pseudoR² is that they do not represent the proportionate reduction in error as the R² in linear regression does. ^[1] 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 R² as a proportionate reduction in error in a universal sense in logistic regression. ^[1]

References

1. Cohen, Jacob; Cohen, Patricia; West, Steven G.; Aiken, Leona S. (2002). Applied Multiple Regression/Correlation Analysis for the Behavioral Sciences (3rd ed.). Routledge. p. 502. ISBN   978-0-8058-2223-6.
2. Allison, Paul D. "Measures of fit for logistic regression" (PDF). Statistical Horizons LLC and the University of Pennsylvania.
3. Menard, Scott W. (2002). Applied Logistic Regression (2nd ed.). SAGE. ISBN   978-0-7619-2208-7.^{[ page needed ]}
4. Hardin, J. W., Hilbe, J. M. (2007). Generalized linear models and extensions. USA: Taylor & Francis. Page 60, Google Books
5. Tjur, Tue (2009). "Coefficients of determination in logistic regression models". American Statistician. 63 (4): 366–372. doi:10.1198/tast.2009.08210. S2CID   121927418.