To calculate the prediction, invert the link function for the model. The inverse functions are in this table.

Link Function | Formula for Prediction |
---|---|

Logit | |

Normit | |

Gompit |

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

exp(·) | the exponential function |

Φ(·) | the cumulative distribution function of the normal distribution |

X' | the transpose of the vector of points to predict for |

the vector of estimated coefficients |

In general, the standard error of the fit has the following form:

The following formulas give the standard error of the fit for different link functions:

- Logit
- Normit
- Gompit

Note the following relationship that applies to the formulas in the table:

where is from the training data only when there is a test data set for validation.

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

1, for the binomial and Poisson models | |

x_{i} | the vector of a design point |

the transpose of x_{i} | |

X | the design matrix |

W | the weight matrix |

the first derivative of the link function evaluated at | |

the predicted mean response | |

the predicted probability for the design point in a binary logistic model | |

the inverse cumulative distribution function of the standard normal distribution for the predicted probability in a binary logistic model | |

the probability density function of the standard normal distribution |

The confidence limits use the Wald approximation method. This is the formula for a 100(1 − *α*)% two-sided confidence interval:

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

the inverse of the link function evaluated at x | |

the transpose of the vector of the predictors | |

the vector of estimated coefficients | |

the value of the inverse cumulative distribution function for the normal distribution evaluated at | |

α | the significance level |

X | the design matrix |

W | the weight matrix |

1, for binomial models |