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
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
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
Term
Description
1, for the binomial and Poisson models
xi
the vector of a design point
the transpose of xi
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
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
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
is from the training data only when there is a test data set for validation.
is from the training data only when there is a test data set for validation.
