Select the method or formula of your choice.

- Pearson residuals
- Standardized and deleted Pearson residuals
- Standardized Pearson residuals with validation
- Deviance residuals
- Standardized deviance residual
- Standardized deviance residual with validation
- Deleted deviance residual
- Delta chi-square
- Delta deviance
- Delta beta (standardized)
- Delta beta
- Leverages
- Leverages with validation
- Cook's distance
- DFITS
- Variance inflation factor (VIF)

Elements of the Pearson chi-square that can be used to detect ill-fitted factor/covariate patterns. Minitab stores the Pearson residual for the *i*^{th} factor/covariate pattern. The formula 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 | |

V | the variance function for the model at |

The variance function depends on the model:

Model |
Variance function |

Binomial | |

Poisson |

Used to detect ill-fitted factor/covariate patterns. Minitab stores the standardized Pearson residual for the *i*^{th} factor/covariate pattern. Deleted Pearson residuals are also called likelihood ratio Pearson residuals. For the deleted Pearson residual, Minitab calculates the one-step approximation described in Pregibon.^{1} This approximation is equal to the standardized Pearson residual. The formula is:

Term | Description |
---|---|

the Pearson residual for the i^{th} factor/covariate pattern | |

1, for the binomial and Poisson models | |

the leverage for the i^{th} factor/covariate pattern |

For validation data, the denominator of the formula for the
standardized Pearson residual adds the leverage instead of subtracting the
leverage.

Term | Description |
---|---|

the Pearson residual for the
i^{th} validation row | |

1, for the binomial and Poisson models | |

the leverage for the
i^{th} validation row |

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 |

For validation data, the denominator of the formula for the
standardized deviance residual adds the leverage instead of subtracting the
leverage.

Term | Description |
---|---|

r_{D,i} | The deviance residual for the
i^{th} validation row |

h_{i} | The leverage for the
i^{th} validation row |

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.

Minitab calculates the change in the Pearson chi-square due to deleting all the observations with the *j*^{th} factor/covariate pattern. Minitab stores one delta chi-square value for each distinct factor/covariate pattern in the data. You can use delta chi-square to detect ill-fitted factor/covariate patterns. The formula for the delta chi-square is:

Term | Description |
---|---|

h _{j} | leverage |

r _{j} | Pearson residuals |

Minitab calculates the change in the deviance statistic by deleting all the observations with the j^{th} factor/covariate pattern. Minitab stores one value for each distinct factor/covariate pattern in the data. You can use delta deviance to detect ill-fitted factor/covariate patterns. The change in the deviance statistic is:

Term | Description |
---|---|

h_{j} | leverage |

r_{j} | Pearson residuals |

d_{j} | deviance residuals |

Minitab calculates the change by deleting all observations with the *j*^{th} factor/covariate pattern. One value is stored for each distinct factor/covariate pattern in the data. You can use standardized delta *β* to detect factor/covariate patterns that have a strong influence on the estimates of the coefficients. This value is based on the standardized Pearson residual.

Term | Description |
---|---|

h _{j} | leverage |

rs _{j} | standardized Pearson residuals |

Minitab calculates the change by deleting all observations with the *j*^{th} factor/covariate pattern. One value is stored for each distinct factor/covariate pattern in the data. You can use delta *β* to detect factor/covariate patterns that have a strong influence on the estimates of the coefficients. This value is based on the Pearson residual.

Term | Description |
---|---|

h _{j} | leverage |

r _{j} | Pearson residuals |

The leverages are the diagonal elements of the generalized hat matrix. The leverages are useful in detecting factor/covariate patterns that may have a significant influence on the results.

Term | Description |
---|---|

w _{j} | the j^{th} diagonal element of the weight matrix from fitting the coefficients |

x_{j} | the j^{th} row of the design matrix |

X | the design matrix |

X' | the transpose of X |

W | the weight matrix from the estimation of the coefficients |

Term | Description |
---|---|

w _{i} | the internal weight for the
i^{th} validation row |

x_{i} | the row of the design matrix for the
predictors in the
i^{th} validation row |

X | the design matrix for the training data set |

X' | the transpose of
X |

W | the diagonal matrix of internal weights for the training data set |

Minitab calculates an approximate Cook's distance.

Term | Description |
---|---|

h_{i} | the leverage for the i^{th} factor/covariate pattern |

the standardized Pearson residual for the i^{th} factor/covariate pattern | |

p | the regression degrees of freedom |

A measure of the influence of a single deletion on the fitted values. Observations with large DFITS values may be outliers. Minitab calculates an approximate value for DFITS.

Term | Description |
---|---|

h_{i} | The leverage for the data point |

The deleted Pearson residual for the data point |

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 |
---|---|

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.