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)`

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

- 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.

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

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".

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

```
r.glm <- glm(Survived ~ ., data=Untable(Titanic), family=binomial)
PseudoR2(r.glm)
PseudoR2(r.glm, c("McFadden", "Nagel"))
```

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