Methods and formulas for event predictions in Analyze Binary Response for Definitive Screening Design

Fitted and predicted values

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

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:

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