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

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In This Topic

Deviance residuals
Standardized deviance residual
Deleted deviance residual
Variance inflation factor (VIF)

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 i^th factor/covariate pattern is:

Notation

Term	Description
y_i	the response value for the i^th factor/covariate pattern
	the fitted value for the i^th factor/covariate pattern
	the deviance for the i^th factor/covariate pattern

Standardized deviance residual

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

Notation

Term	Description
r_D,i	The deviance residual for the i^th factor/covariate pattern
h_i	The leverage for the i^th factor/covariate pattern

Deleted deviance residual

The deleted deviance residual measures the change in the deviance due to the omission of the i^th 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

Term	Description
y_i	the response value at the i^th factor/covariate pattern
	the fitted value for the i^th factor covariate pattern
h_i	the leverage for the i^th factor/covariate pattern
r'_D,i	the standardized deviance residual for the i^th factor/covariate pattern
r'_P,i	the standardized Pearson residual for the i^th 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 x_j the formula for the VIF is:

Notation

Term	Description
	coefficient of determination with x_j 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, 2^nd Edition, Chapman & Hall/CRC, London.