Select the method or formula of your choice.

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

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

Binomial | Logit | |

Binomial | Normit | |

Binomial | Gompit | |

Poisson | Natural log | |

Poisson | Square root | |

Poisson | Identity |

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. The following is the general formula for a 100(1 − *α*)% two-sided confidence interval:

The following table gives specific formulas for the different model types and link functions:

Type | Link | Standard error of the fit |
---|---|---|

Binary logistic | Logit | |

Binary logistic | Normit | |

Binary logistic | Gompit | |

Poisson | Log | |

Poisson | Square root | |

Poisson | Identity |

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 |
---|---|

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 and Poisson models | |

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 cumulative distribution function of the standard normal distribution |