# Methods and formulas for event predictions in Fit Binary Logistic Model

Select the method or formula of your choice.

## Fitted and predicted values

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 ### Notation

TermDescription
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

## Standard error of fitted values and predictions

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.

### Notation

TermDescription 1, for the binomial and Poisson models
xithe vector of a design point the transpose of xi
Xthe design matrix
Wthe 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

## Confidence limits for fits and predictions

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.

### Notation

TermDescription 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 αthe significance level  Xthe design matrix
Wthe 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
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