Pseudo R2 Statistics

Although there's no commonly accepted agreement on how to assess the fit of a logistic regression, there are some approaches. The goodness of fit of the logistic regression model can be expressed by some variants of pseudo R squared statistics, most of which being based on the deviance of the model.

PseudoR2(x, which = NULL)

Arguments

x: the glm, polr or multinom model object to be evaluated.
which: character, one out of "McFadden", "McFaddenAdj", "CoxSnell", "Nagelkerke", "AldrichNelson", "VeallZimmermann", "Efron", "McKelveyZavoina", "Tjur", "all". Partial matching is supported.

Details

Cox and Snell's \(R^2\) is based on the log likelihood for the model compared to the log likelihood for a baseline model. However, with categorical outcomes, it has a theoretical maximum value of less than 1, even for a "perfect" model.

Nagelkerke's \(R^2\) (also sometimes called Cragg-Uhler) is an adjusted version of the Cox and Snell's \(R^2\) that adjusts the scale of the statistic to cover the full range from 0 to 1.

McFadden's \(R^2\) is another version, based on the log-likelihood kernels for the intercept-only model and the full estimated model.

Veall and Zimmermann concluded that from a set of six widely used measures the measure suggested by McKelvey and Zavoina had the closest correspondance to ordinary least square R2. The Aldrich-Nelson pseudo-R2 with the Veall-Zimmermann correction is the best approximation of the McKelvey-Zavoina pseudo-R2. Efron, Aldrich-Nelson, McFadden and Nagelkerke approaches severely underestimate the "true R2".

Value

the value of the specific statistic. AIC, LogLik, LogLikNull and G2 will only be reported with option "all".

McFadden: McFadden pseudo-\(R^2\)
McFaddenAdj: McFadden adjusted pseudo-\(R^2\)
CoxSnell: Cox and Snell pseudo-\(R^2\) (also known as ML pseudo-\(R^2\))
Nagelkerke: Nagelkerke pseudo\(R^2\) (also known as CraggUhler \(R^2\))
AldrichNelson: AldrichNelson pseudo-\(R^2\)
VeallZimmermann: VeallZimmermann pseudo-\(R^2\)
McKelveyZavoina: McKelvey and Zavoina pseudo-\(R^2\)
Efron: Efron pseudo-\(R^2\)
Tjur: Tjur's pseudo-\(R^2\)
AIC: Akaike's information criterion
LogLik: log-Likelihood for the fitted model (by maximum likelihood)
LogLikNull: log-Likelihood for the null model. The null model will include the offset, and an intercept if there is one in the model.
G2: differenz of the null deviance - model deviance

References

Aldrich, J. H. and Nelson, F. D. (1984): Linear Probability, Logit, and probit Models, Sage University Press, Beverly Hills.

Cox D R & Snell E J (1989) The Analysis of Binary Data 2nd ed. London: Chapman and Hall.

Efron, B. (1978). Regression and ANOVA with zero-one data: Measures of residual variation. Journal of the American Statistical Association, 73(361), 113–121.

Hosmer, D. W., & Lemeshow, S. (2000). Applied logistic regression (2nd ed.). Hoboke, NJ: Wiley.

McFadden D (1979). Quantitative methods for analysing travel behavior of individuals: Some recent developments. In D. A. Hensher & P. R. Stopher (Eds.), Behavioural travel modelling (pp. 279-318). London: Croom Helm.

McKelvey, R. D., & Zavoina, W. (1975). A statistical model for the analysis of ordinal level dependent variables. The Journal of Mathematical Sociology, 4(1), 103–120

Nagelkerke, N. J. D. (1991). A note on a general definition of the coefficient of determination. Biometrika, 78(3), 691–692.

Tjur, T. (2009) Coefficients of determination in logistic regression models - a new proposal: The coefficient of discrimination. The American Statistician, 63(4): 366-372

Veall, M.R., & Zimmermann, K.F. (1992) Evalutating Pseudo-R2's fpr binary probit models. Quality&Quantity, 28, pp. 151-164

Author

Andri Signorell <andri@signorell.net> with contributions of Ben Mainwaring <benjamin.mainwaring@yougov.com> and Daniel Wollschlaeger

Examples

r.glm <- glm(Survived ~ ., data=Untable(Titanic), family=binomial)
PseudoR2(r.glm)
#>  McFadden 
#> 0.2019875 

PseudoR2(r.glm, c("McFadden", "Nagel"))
#>   McFadden Nagelkerke 
#>  0.2019875  0.3135111