Methods and formulas for coefficients in Analyze Mixture Design

Select the method or formula of your choice.

Coefficient (Coef)

Minitab uses least squares estimation to calculate the coefficients.

In matrix terms, the least squares estimates of the coefficients are:

b = (X'X)^-1X'y

For more information on coefficients of higher order models, see Cornell¹.

Term	Description
X	design matrix
y	response column

J.A. Cornell (1990). Experiments With Mixtures: Designs, Models, and the Analysis of Mixture Data, John Wiley & Sons.

For simple linear regression, the standard error of the coefficient is:

The standard errors of the coefficients for multiple regression are the square roots of the diagonal elements of this matrix:

Term	Description
x_i	i^th predictor value
	mean of the predictor
X	design matrix
X'	transpose of the design matrix
s²	mean square error

Term	Description
	test statistic for the coefficient
	estimated coefficient
	standard error of the estimated coefficient

The two-sided p-value for the null hypothesis that a regression coefficient equals 0 is:

The degrees of freedom are the degrees of freedom for error, as follows:

n – p

Term	Description
	The cumulative distribution function of the t distribution with degrees of freedom equal to the degrees of freedom for error.
t_j	The t statistic for the j^th coefficient.
n	The number of observations in the data set.
p	The sum of the degrees of freedom for the terms.

The VIF can be obtained by regressing each predictor on the remaining predictors and noting the R²value.

For predictor x_j, the VIF is:

Term	Description
R²( x_j)	coefficient of determination with x_j as the response variable and the other terms in the model as the predictors