Methods and formulas for the ANOVA table for Stability Study for fixed batches

Select the method or formula of your choice.

Sum of squares (SS)

In matrix terms, these are the formulas for the different sums of squares:

Minitab breaks down the SS Regression or SS Treatments component into the amount of variation explained by each term using both the sequential sum of squares and adjusted sum of squares.

Notation

TermDescription
bvector of coefficients
Xdesign matrix
Yvector of response values
nnumber of observations
Jn by n matrix of 1s

The formula for the Mean Square (MS) of the regression is:

Notation

TermDescription
mean response
ith fitted response
pnumber of terms in the model

The Mean Square of the error (also abbreviated as MS Error or MSE, and denoted as s2) is the variance around the fitted regression line. The formula is:

Notation

TermDescription
yiith observed response value
ith fitted response
nnumber of observations
pnumber of coefficients in the model, not counting the constant

F

If all the factors in the model are fixed, then the calculation of the F-statistic depends on what the hypothesis test is about, as follows:

F(Term)
F(Lack-of-fit)

If there are random factors in the model, F is constructed using the expected mean square information for each term. For more information, see Neter et al.1.

Notation

TermDescription
Adj MS TermA measure of the amount of variation that a term explains after accounting for the other terms in the model.
MS ErrorA measure of the variation that the model does not explain.
MS Lack-of-fitA measure of variation in the response that could be modeled by adding more terms to the model.
MS Pure errorA measure of the variation in replicated response data.
1. J. Neter, W. Wasserman and M.H. Kutner (1985). Applied Linear Statistical Models, Second Edition. Irwin, Inc.

p-value (P)

Used in hypothesis tests to help you decide whether to reject or fail to reject a null hypothesis. The p-value is the probability of obtaining a test statistic that is at least as extreme as the actual calculated value, if the null hypothesis is true. A commonly used cut-off value for the p-value is 0.05. For example, if the calculated p-value of a test statistic is less than 0.05, you reject the null hypothesis.