Methods and formulas for the analysis of variance in Analyze Definitive Screening Design

In This Topic

Adjusted sum of squares
Sum of squares (SS)
Sequential sum of squares
Degrees of freedom (DF)
Adj MS – Model
Adj MS – Term
Adj MS – Error
F
P-value – Analysis of variance table
P-value – Lack-of-fit test

Adjusted sum of squares

The adjusted sums of squares does not depend on the order that the terms enter the model. The adjusted sum of squares is the amount of variation explained by a term, given all other terms in the model, regardless of the order that the terms enter the model.

For example, if you have a model with three factors, X1, X2, and X3, the adjusted sum of squares for X2 shows how much of the remaining variation the term for X2 explains, given that the terms for X1 and X3 are also in the model.

The calculations for the adjusted sums of squares for three factors are:

SSR(X3 | X1, X2) = SSE (X1, X2) - SSE (X1, X2, X3) or
SSR(X3 | X1, X2) = SSR (X1, X2, X3) - SSR (X1, X2)

where SSR(X3 | X1, X2) is the adjusted sum of squares for X3, given that X1 and X2 are in the model.

SSR(X2, X3 | X1) = SSE (X1) - SSE (X1, X2, X3) or
SSR(X2, X3 | X1) = SSR (X1, X2, X3) - SSR (X1)

where SSR(X2, X3 | X1) is the adjusted sum of squares for X2 and X3, given that X1 is in the model.

You can extend these formulas if you have more than 3 factors in your model¹.

J. Neter, W. Wasserman and M.H. Kutner (1985). Applied Linear Statistical Models, Second Edition. Irwin, Inc.

Sum of squares (SS)

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

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

Notation

Term	Description
b	vector of coefficients
X	design matrix
Y	vector of response values
n	number of observations
J	n by n matrix of 1s

Sequential sum of squares

Minitab breaks down the SS Model component of variance into sequential sums of squares for each factor term or set of factor terms. The sequential sums of squares depend on the order that the factors or predictors enter the model. The sequential sum of squares is the unique portion of the SS Model that each term explains, given any terms that previously entered.

For example, if you have a model with three factors, X1, X2, and X3, the sequential sum of squares for X2 shows how much of the remaining variation X2 explains, given that X1 is already in the model. To obtain a different sequence of terms, repeat the analysis and enter the terms in a different order.

Degrees of freedom (DF)

Different sums of squares have different degrees of freedom.

DF for a numeric factor = 1

DF for a categorical factor = b − 1

DF for a quadratic term = 1

DF for blocks = c − 1

DF for error = n − p

DF for pure error =

DF for lack of fit = m − p

DF total = n − 1

For interactions among factors, multiply the degrees of freedom for the terms in the factor. For example, if the factors are A and B then the interaction AB has these degrees of freedom:

To find the degrees of freedom for a type of term, sum the degrees of freedom for the terms. For example, if the factors are A and B then the main effects in the model have this many degrees of freedom:

Note

Categorical factors in screening designs in Minitab have 2 levels. Thus, the degrees of freedom for a categorical factor are 2 – 1 = 1. By extension, interactions between factors also have 1 degree of freedom.

Notation

Term	Description
b	The number of levels in the factor
c	The number of blocks
n	The total number of observations
n_i	The number of observations for i^th factor level combination
m	The number of factor level combinations
p	The number of coefficients

Adj MS – Model

Notation

Term	Description
	The mean of the response variable
	The i^th fitted value of the response
p	The number of terms in the model, not including the constant term

Adj MS – Term

The calculation for the Mean Square (MS) for the model term is:

Adj MS – Error

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

Term	Description
y_i	i^th observed response value
	i^th fitted response
n	number of observations
p	number of coefficients in the model, not counting the constant

F

The calculation of the F-statistic depends on the hypothesis test, as follows:

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

Notation

Term	Description
Adj MS Term	A measure of the amount of variation that a term explains after accounting for the other terms in the model.
MS Error	A measure of the variation that the model does not explain.
MS Lack-of-fit	A measure of variation in the response that could be modeled by adding more terms to the model.
MS Pure error	A measure of the variation in replicated response data.

J. Neter, W. Wasserman and M.H. Kutner (1985). Applied Linear Statistical Models, Second Edition. Irwin, Inc.

P-value – Analysis of variance table

The p-value is a probability that is calculated from an F-distribution with the degrees of freedom (DF) as follows:

Numerator DF: sum of the degrees of freedom for the term or the terms in the test
Denominator DF: degrees of freedom for error

Formula

1 − P(F ≤ f_j)

Notation

Term	Description
P(F ≤ f)	cumulative distribution function for the F-distribution
f	f-statistic for the test

P-value – Lack-of-fit test

This p-value is for the test of the null hypothesis that the coefficients are 0 for any terms that are possible to estimate from these data that are not in the model. The p-value is the probability from an F distribution with degrees of freedom (DF) as follows:

Numerator DF: degrees of freedom for lack-of-fit
Denominator DF: degrees of freedom for pure error

Formula

1 − P(F ≤ f_j)

Notation

Term	Description
P(F ≤ f_j)	cumulative distribution function for the F-distribution
f_j	f-statistic for the test