# 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

### Notation

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

## Standard error of the coefficient (SE Coef)

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:

### Notation

TermDescription
xiith predictor value mean of the predictor
Xdesign matrix
X'transpose of the design matrix
s2mean square error

## T-value

### Notation

TermDescription test statistic for the coefficient  estimated coefficient standard error of the estimated coefficient

## P-value – Coefficients table

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:

np

### Notation

TermDescription The cumulative distribution function of the t distribution with degrees of freedom equal to the degrees of freedom for error.
tjThe t statistic for the jth coefficient.
nThe number of observations in the data set.
pThe sum of the degrees of freedom for the terms.

## Variance inflation factor (VIF)

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

### Formula

For predictor xj, the VIF is:

### Notation

TermDescription
R2( xj)coefficient of determination with xj as the response variable and the other terms in the model as the predictors
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