Deviance measures the discrepancy between the current model and the full model. The full model is the model that has *n* parameters, one parameter per observation. The full model maximizes the log-likelihood function. The full model provides a point of comparison for models with fewer than *n* parameters. Comparisons to the full model use the scaled deviance.

The following equation gives the contribution to the scaled deviance for the binomial model:

The degrees of freedom for the test depend on the sample size and the number of terms in the model:

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

L _{f} | the log-likelihood for the full model |

L_{c} | the log-likelihood of the model with a subset of terms from the full model |

y _{i} | the number of events for the i^{th} row in the data |

the estimated mean response for the i^{th} row in the data | |

m_{i} | the number of trials for the i^{th} row in the data |

n | the number of rows in the data |

p | the regression degrees of freedom |

The generalized Pearson chi-square statistic assesses the relative difference between the observed and fitted values.

The degrees of freedom for the test depend on the sample size and the number of terms in the model. The Pearson statistic has an exact chi-square distribution for normal data. For non-normal data, like the binomial distribution and the Poisson distribution, the statistic approaches the distribution asymptotically.

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

n | the number of rows in the data |

p | the regression degrees of freedom |

y_{i} | the response value for the i^{th} factor/covariate pattern |

the estimated mean response of the i^{th} row | |

V(·) | the variance function for the model, defined below |

The following equation gives the variance function for a binomial model:

A goodness-of-fit test for models with binary responses based on grouping data based on the estimated probabilities. It is the chi-square statistic from a 2 × *g* table of observed and estimated expected frequencies, where *g* is the number of groups. The degrees of freedom for the test is *g* − 2.

The formula is:

To form the groups, Minitab orders the estimated probabilities and then attempts to create 10 groups of equal size.

The expected number of events in a group is:

expected events =

The expected value for the number of nonevents is:

expected nonevents =

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

The number of trials in the k^{th} group | |

o _{k} | The number of events among the factor/covariate patterns |

The average estimated probability for each group | |

π _{i} | The fitted probabilities for the factor/covariate patterns in a group |