# Methods and formulas for event predictions in Binary Fitted Line Plot

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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