# Methods and formulas for the regression model for Stability Study for fixed batches

Select the method or formula of your choice.

## GLM model

In matrix terms, this is the formula for the general linear regression model:

### Notation

TermDescription
Yvector of responses
Xdesign matrix
βvector of parameters
εvector of independent normal random variables

## Design matrix

The design matrix contains the predictors in a matrix (X) with n rows, where n is the number of observations. There is a column for each linearly independent coefficient in the model. Because one coefficient is linearly dependent on the others, the number of columns to represent the batch term is one less than the number of batches. Batches use −1, 0, 1 coding.

To calculate the columns for an interaction term, multiply all of the corresponding values for the predictors in the interaction. For example, suppose the first observation has a value of 4 for predictor A and a value of 2 for predictor B. In the design matrix, the interaction between A and B is represented as 8 (4 x 2).

## x'x inverse

A p x p matrix, where p is the number of coefficients in the model. Multiplying x'x inverse by MSE produces the variance-covariance matrix of the coefficients. Minitab also uses the x'x inverse to calculate the regression coefficients and the hat matrix.
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