Explained variation for logistic regression.

Different measures of the proportion of variation in a dependent variable explained by covariates are reported by different standard programs for logistic regression. We review twelve measures that have been suggested or might be useful to measure explained variation in logistic regression models. The definitions and properties of these measures are discussed and their performance is compared in an empirical study. Two of the measures (squared Pearson correlation between the binary outcome and the predictor, and the proportional reduction of squared Pearson residuals by the use of covariates) give almost identical results, agree very well with the multiple R2 of the general linear model, have an intuitively clear interpretation and perform satisfactorily in our study. For all measures the explained variation for the given sample and also the one expected in future samples can be obtained easily. For small samples an adjustment analogous to Radj2 in the general linear model is suggested. We discuss some aspects of application and recommend the routine use of a suitable measure of explained variation for logistic models.

[1]  Taylor Francis Online,et al.  The American statistician , 1947 .

[2]  N. Wermuth,et al.  A Comment on the Coefficient of Determination for Binary Responses , 1992 .

[3]  H. Theil.  On the Estimation of Relationships Involving Qualitative Variables , 1970, American Journal of Sociology.

[4]  David M. Allen,et al.  The Relationship Between Variable Selection and Data Agumentation and a Method for Prediction , 1974 .

[5]  Alan Agresti.  Applying R 2 -Type measures to ordered categorical data , 1986 .

[6]  L. Magee,et al.  R 2 Measures Based on Wald and Likelihood Ratio Joint Significance Tests , 1990 .

[7]  S. le Cessie,et al.  Predictive value of statistical models. , 1990, Statistics in medicine.

[8]  A. C. Rencher,et al.  Inflation of R2 in Best Subset Regression , 1980 .

[9]  M. Schemper,et al.  The relative importance of prognostic factors in studies of survival. , 1991, Statistics in medicine.

[10]  R. H. Myers.  Classical and modern regression with applications , 1986 .

[11]  P. Schmidt,et al.  Limited-Dependent and Qualitative Variables in Econometrics. , 1984 .

[12]  R Simon,et al.  Measures of explained variation for survival data. , 1990, Statistics in medicine.

[13]  Douglas C. Crocker.  Some Interpretations of the Multiple Correlation Coefficient , 1972 .

[14]  W. P. Dixon,et al.  BMPD statistical software manual , 1988 .

[15]  Richard Simon,et al.  Explained Residual Variation, Explained Risk, and Goodness of Fit , 1991 .

[16]  D. McFadden.  MEASUREMENT OF URBAN TRAVEL DEMAND , 1974 .

[17]  Barry H. Margolin,et al.  An Analysis of Variance for Categorical Data, II: Small Sample Comparisons with Chi Square and other Competitors , 1974 .

[18]  T. O. Kvålseth.  Cautionary Note about R 2 , 1985 .

[19]  J. G. Cragg,et al.  The Demand for Automobiles , 1970 .

[20]  M. Kendall,et al.  Rank Correlation Methods , 1949 .

[21]  David W. Hosmer,et al.  Applied Logistic Regression , 1991 .

[22]  S. Cessie,et al.  Logistic Regression, a review , 1988 .

[23]  B. Efron.  Regression and ANOVA with Zero-One Data: Measures of Residual Variation , 1978 .

[24]  D. Hosmer,et al.  Applied Logistic Regression , 1991 .

[25]  D. Freedman.  A Note on Screening Regression Equations , 1983 .

[26]  J. Clarke,et al.  Medicine , 1907, Bristol medico-chirurgical journal.

[27]  J. C. van Houwelingen,et al.  Predictive value of statistical models , 1990 .

[28]  N. Nagelkerke,et al.  A note on a general definition of the coefficient of determination , 1991 .

[29]  S. Haberman.  Analysis of Dispersion of Multinomial Responses , 1982 .

[30]  R. Forthofer,et al.  Rank Correlation Methods , 1981 .

[31]  L. A. Goodman,et al.  Measures of association for cross classifications , 1979 .

[32]  Robert H. Somers,et al.  A new asymmetric measure of association for ordinal variables. , 1962 .

[33]  M. Stone.  An Asymptotic Equivalence of Choice of Model by Cross‐Validation and Akaike's Criterion , 1977 .