Methods and formulas for fits and residuals in General MANOVA

The standard error of the fitted value in a regression model with one predictor is:

The standard error of the fitted value in a regression model with more than one predictor is:

For weighted regression, include the weight matrix in the equation:

When the data have a test data set or K-fold cross validation, the formulas are the same. The value of s² is from the training data. The design matrix and the weight matrix are also from the training data.

Notation

Term	Description
s2	mean square error
n	number of observations
x0	new value of the predictor
	mean of the predictor
xi	ith predictor value
x0	vector of values that produce the fitted values, one for each column in the design matrix, beginning with a 1 for the constant term
x'0	transpose of the new vector of predictor values
X	design matrix
W	weight matrix

Also called the externally Studentized residuals. The formula is:

Another presentation of this formula is:

The model that estimates the i^th observation omits the i^th observation from the data set. Therefore, the i^th observation cannot influence the estimate. Each deleted residual has a student's t-distribution with degrees of freedom.