Methods and formulas for goodness-of-fit statistics in Analyze Factorial Design

In This Topic

S
R-sq
R-sq (adj)
R-sq (pred)
PRESS
R-sq (WP)
R-sq(SP)
Log-likelihood
AICc (Akaike's Corrected Information Criterion)
BIC (Bayesian Information Criterion)
Mallows' Cp

S

Notation

Term	Description
MSE	mean square error

R-sq

R² is also known as the coefficient of determination.

Formula

Notation

Term	Description
y_i	i ^th observed response value
	mean response
	i ^th fitted response

R-sq (adj)

While the calculations for adjusted R² can produce negative values, Minitab displays zero for these cases.

Notation

Term	Description
	i^th observed response value
	i^th fitted response
	mean response
n	number of observations
p	number of terms in the model

R-sq (pred)

While the calculations for R²(pred) can produce negative values, Minitab displays zero for these cases.

Notation

Term	Description
y_i	i ^th observed response value
	mean response
n	number of observations
e_i	i ^th residual
h_i	i ^th diagonal element of X(X'X)^–1X'
X	design matrix

PRESS

Assesses your model's predictive ability and is calculated as:

Notation

Term	Description
n	number of observations
e_i	i^th residual
h_i	i^th diagonal element of X (X' X)^-1X'

Term

Description

number of observations

e_i

i^th residual

h_i

i^th diagonal element of

X (X' X)^-1X'

R-sq (WP)

In split-plot designs, R² (WP) is the proportion of variation among whole plots that is accounted for by all of the terms in the model that involve only hard-to-change factors.

Notation

Term	Description
SS (WP_error)	Sum of squares for the whole plot error term
SS Total (WP)	Sum of sequential SS for all terms that involve the whole plot error term and only hard-to-change factors

R-sq(SP)

In split-plot designs, R² (SP) is the proportion of variation among subplots (within whole plots) that is accounted for by the model.

Notation

Term	Description
SS (SP_error)	sums of squares for the subplot error term
SS Total (SP)	SS Total – SS Total(WP)
SS Total (WP)	sum of sequential SS for all terms that involve the whole plot error term and only hard-to-change factors

Log-likelihood

For unweighted analyses, Minitab uses the following equation:

For an analysis that has weights for the observations, Minitab uses the following equation:

Observations with weights of 0 are not in the analysis.

Notation

Term	Description
n	the number of observations
R	the sum of squares for error for the model
w_i	the weight of the i^th observation

AICc (Akaike's Corrected Information Criterion)

AICc is not calculated when .

Notation

Term	Description
n	the number of observations
p	the number of coefficients in the model, including the constant

BIC (Bayesian Information Criterion)

Notation

Term	Description
p	the number of coefficients in the model, including the constant
n	the number of observations

Mallows' Cp

Notation

Term	Description
SSE_p	sum of squared errors for the model under consideration
MSE_m	mean square error for the model with all candidate terms
n	number of observations
p	number of terms in the model, including the constant