On Graphically Checking Goodness-of-fit of Binary Logistic Regression Models

Summary

Objectives: This paper is concerned with checking goodness-of-fit of binary logistic regression models. For the practitioners of data analysis, the broad classes of procedures for checking goodness-of-fit available in the literature are described. The challenges of model checking in the context of binary logistic regression are reviewed. As a viable solution, a simple graphical procedure for checking goodness-of-fit is proposed.

Methods: The graphical procedure proposed relies on pieces of information available from any logistic analysis; the focus is on combining and presenting these in an informative way.

Results: The information gained using this approach is presented with three examples. In the discussion, the proposed method is put into context and compared with other graphical procedures for checking goodness-of-fit of binary logistic models available in the literature.

Conclusion: A simple graphical method can significantly improve the understanding of any logistic regression analysis and help to prevent faulty conclusions.

Keywords

Nonlinear models - residuals

• References

• 1 Strasak AM. et al. The Use of Statistics in Medical Research: A Comparison of The New England Journal of Medicine and Nature Medicine. The American Statistician 2007; 61: 47-55.
• 2 Davison AC. Statistical models. Cambridge: Cambridge University Press; 2003. Section 8.6.
• 3 Draper N, Smith H. Applied Regression Analysis. New York: Wiley & Sons; 1998
• 4 Weisberg S. Applied Linear Regression. New York: Wiley & Sons; 1985
• 5 Daniel C, Wood F, Gorman JW. Fitting Equations to Data: Computer Analysis of Multifactor Data. New York: Wiley & Sons; 1980
• 6 Hosmer J, Lemeshow S. Applied Logistic Regression. New York: Wiley & Sons; 1989
• 7 Brown CC. On a goodness-of-fit test for the logistic model based on score statistics. Communications in Statistics 1982; 11 (10) 1087-1105.
• 8 Stukel TA. Generalized logistic models. Journal of the American Statistical Association 1988; 83: 426-431.
• 9 Le Cessie S, van Houwelingen JC. A Goodness-offit Test for Binary Regression Models Based on Smoothing Methods. Biometrics 1991; 47: 1267-1282.
• 10 Hosmer D, Hosmer T, le Cessie S, and Lemeshow S. A Comparison of Goodness-of-Fit Tests for the Logistic Regression Model. Statistics in Medicine 1997; 16: 965-980.
• 11 Fienberg SE, Gong GD. Comment following the paper of Landwehr, Pregibon and Shoemaker. Journal of the American Statistical Association 1984; 79: 72-77.
• 12 Copas JB. Binary Regression Models for Contaminated Data (with discussion). Journal of the Royal Statistical Society B 1988; 50: 225-265.
• 13 Copas JB. Plotting p against x. Journal of the Royal Statistical Society C. (Applied Statistics) 1983; 32: 25-31.
• 14 Copas JB, Marshall P. The offender group reconviction scale: A statistical reconviction score for use by probation officers. Journal of the Royal Statistical Society, Series C. (Applied Statistics) 1998; 47: 159-171.
• 15 Stuck AE, Minder CE, Peter-Wuest I, Gillmann G. et al. A Randomized Trial of In-Home Visits for Disability Prevention in Community-Dwelling Older People at Low and High Risk for Nursing Home Admission. Archives of Internal Medicine 2000; 160: 977-986.
• 16 Spiegelhalter DJ. Probabilistic prediction in patient management and clinical trials. Statistics in Medicine 1986; 5: 421-433.
• 17 Harrell FE. et al. Tutorial in Biostatistics Development of a clinical prediction model for an ordinal outcome. Statistics in Medicine 1998; 17: 909-944.
• 18 Steyerberg EW, Vergouwe Y, Keizer HJ, Habbema JDF. Residual mass histology in testicular cancer: development and validation of a clinical prediction rule. Statistics in Medicine 2001; 20: 3847-3859.
• 19 Cleveland WS. Robust Locally Weighted Regression and Smoothing Scatterplots. Journal of the American Statistical Association 1979; 74: 829-836.
• 20 Fowlkes E. Some Diagnostics for Binary Logistic Regression via Smothing Methods. Biometrika 1987; 74: 503-515.
• 21 Cook RD, Weisberg S. Graphics for Assessing the Adequacy of Regression Models. Journal of the American Statistical Association 1997; 92: 490-499.
• 22 Pardoe I, Cook RD. A Graphical Method for Assessing the Fit of a Logistic Regression Model. The American Statistician 2002; 56: 263-272.
• 23 Landwehr JM, Pregibon D, Shoemaker AC. Graphical Methods for Assessing Logistic Regression Models. Journal of the American Statistical Association 1984; 79: 61-71.
• 24 Cox DR, Snell EJ. A General Definition of Residuals. Journal of the Royal Statistical Society B 1968; 30: 248-275.
• 25 Pregibon D. Logistic Regression Diagnostics. The Annals of Statistics 1981; 9 (04) 705-724.