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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:

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 |

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

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 |

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 method^{1}. The formula is as follows:

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.

To calculate a VIF, perform a weighted regression on the predictor with the remaining predictors. The weight matrix is that given in McCullagh and Nelder^{1} 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:

Term | Description |
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R^{2}( x)_{j} | coefficient of determination with x as the response variable and the other terms in the model as the predictors_{j} |

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