Methods and formulas for the diagnostic measures in Binary Fitted Line Plot

Select the method or formula of your choice.

Deviance residuals

Deviance residuals are based on the model deviance and are useful in identifying ill-fitted factor/covariate patterns. The model deviance is a goodness-of-fit statistic based on the log-likelihood function. The deviance residual defined for the ith factor/covariate pattern is:

Notation

TermDescription
yi the response value for the ith factor/covariate pattern
the fitted value for the ith factor/covariate pattern
the deviance for the ith factor/covariate pattern

Standardized deviance residual

The standardized deviance residual is helpful in the identification of outliers. The formula is:

Notation

TermDescription
rD,iThe deviance residual for the ith factor/covariate pattern
hiThe leverage for the ith factor/covariate pattern

Deleted deviance residual

The deleted deviance residual measures the change in the deviance due to the omission of the ith case from the data. Deleted deviance residuals are also called likelihood ratio deviance residuals. For the deleted deviance residual, Minitab calculates a one-step approximation based on the Pregibon one-step approximation method1. The formula is as follows:

Notation

TermDescription
yithe response value at the ith factor/covariate pattern
the fitted value for the ith factor covariate pattern
hithe leverage for the ith factor/covariate pattern
r'D,ithe standardized deviance residual for the ith factor/covariate pattern
r'P,ithe standardized Pearson residual for the ith factor/covariate pattern

1. Pregibon, D. (1981). "Logistic Regression Diagnostics." The Annals of Statistics , Vol. 9, No. 4 pp. 705–724.

Variance inflation factor (VIF)

To calculate a VIF, perform a weighted regression on the predictor with the remaining predictors. The weight matrix is that given in McCullagh and Nelder1 for the estimation of the coefficients. In this case, the VIF formula is equivalent to the formula for a linear regression. For example, for predictor xj the formula for the VIF is:

Notation

TermDescription
R2( xj)coefficient of determination with xj as the response variable and the other terms in the model as the predictors

1. P. McCullagh and J. A. Nelder (1989). Generalized Linear Models, 2nd Edition, Chapman & Hall/CRC, London.

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