Methods and formulas for goodness-of-fit statistics in Analyze Variability

S

Notation

TermDescription
MSEmean square error

R-sq

R2 is also known as the coefficient of determination.

Formula

Notation

TermDescription
yi 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

TermDescription
ith observed response value
ith fitted response
mean response
nnumber of observations
pnumber 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

TermDescription
yi i th observed response value
mean response
n number of observations
ei i th residual
hi i th diagonal element of X(X'X)–1X'
X design matrix

PRESS

Assesses your model's predictive ability.

Formula

Notation

TermDescription
n number of observations
ei i th residual
hi i th diagonal element of X(X'X)–1X'

Log-likelihood

The calculation of the log-likelihood depends on the estimation method for the analysis.

Least squares estimation

Minitab uses the following equation for the log-likelihood:
The weights for the analysis are either the specified weights or given by the following equation:

Maximum likelihood estimation

This calculation of the log-likelihood assumes that the standard deviations are from normal distributions and that the natural log of the standard deviation follows a linear model.

Notation

TermDescription
nthe number of rows with no missing data
Rthe sum of squares for error for the model
the trigamma function
vithe degrees of freedom for the ith standard deviation
the gamma function
Sithe ith sample standard deviation
the row of the design matrix associated with the ith standard deviation
the maximum likelihood estimates of the model coefficients

AICc (Akaike's Corrected Information Criterion)

AICc is not calculated when .

Notation

TermDescription
pthe number of coefficients in the model, including the constant
nthe number of rows in the data with no missing data

BIC (Bayesian Information Criterion)

Notation

TermDescription
pthe number of coefficients in the model, not counting the constant
nthe number of rows in the data with no missing data

Mallows' Cp

Notation

TermDescription
SSEpsum of squared errors for the model under consideration
MSEmmean square error for the model with all candidate terms
nnumber of observations
pnumber of terms in the model, including the constant
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