Methods and formulas for the diagnostic measures in Analyze Variability

Leverages (Hi)

Leverages are obtained from the hat matrix (H), which is an n x n projection matrix:

The leverage of the ith observation is the ith diagonal element, hi of H. If hi is large, the ith observation has unusual predictors (X1i, X2i, ..., Xpi). That is, the predictor values are far from the mean vector , using Mahalanobis distance.

Leverage values fall between 0 and 1. Minitab identifies observations with leverages over 3p/n or .99, whichever is smaller, with an X in the table of unusual observations. Usually, you examine values with large leverages.

Notation

TermDescription
Xdesign matrix
hiith diagonal element of the hat matrix
pnumber of terms in the model, including the constant
nnumber of observations

Cook's distance

Overall measure, D, of the combined impact across all of the estimated regression coefficients on an observation. Minitab calculates D using leverage values and standardized residuals, and considers whether an observation is unusual with respect to both x- and y-values. Observations with large D values may be outliers.

Formula

Cook's distance is the distance between the coefficients calculated with and without the i th observation. Minitab calculates Cook's distance without fitting a new regression equation each time an observation is omitted. This calculation is:

Notation

TermDescription
ei i th residual
hi i th diagonal element of
p number of model parameters, including the constant
s 2 mean square error
b coefficient vector
b(i) coefficient vector calculated after deleting the i th observation
X design matrix

DFITS

Combines leverage and studentized residual (deleted t residuals) values into one overall measure of how unusual an observation is. DFITS measures the influence of each observation on the fitted values in a regression and ANOVA model. Observations with large DFITS values may be outliers.

DFITS represents roughly the number of standard deviations that the fitted value changes when each observation is removed from the data set and the model is refit. Minitab can calculate DFITS without fitting a new regression equation each time an observation is omitted.

Formula

Notation

TermDescription
ei i th residual
hi i th diagonal element of
X design matrix
i th fitted response
fitted value calculated without the i th observation
MSE (i) mean square error calculated without the i th observation
n number of observations
p number of model parameters
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