Methods and formulas for fits and residuals in Analyze Mixture Design

Mixture designs do not include a constant.

Notation

Term	Description
	fitted value
xk	kth term. Each term can be a single predictor, a polynomial term, or an interaction term.
bk	estimate of kth regression coefficient

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:

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
x'0	transpose of the new vector of predictor values
X	design matrix

The residual is the difference between an observed value and the corresponding fitted value. This part of the observation is not explained by the model. The residual of an observation is:

Notation

Term	Description
yi	ith observed response value
	ith fitted value for the response

Standardized residuals are also called "internally Studentized residuals."

Formula

Notation

Term	Description
ei	i th residual
hi	i th diagonal element of X(X'X)–1X'
s2	mean square error
X	design matrix
X'	transpose of the design 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.

Notation

Term	Description
ei	ith residual
s(i)2	mean square error calculated without the ith observation
hi	i th diagonal element of X(X'X)–1X'
n	number of observations
p	number of terms
SSE	sum of squares for error