Page 378 - Six Sigma Demystified

358 Six SigMa DemystifieD

Minitab provides each of these stepwise regression techniques in the Stat\
Regression\Stepwise menu. To further test the fitted model, we should conduct
residuals analysis.

Residuals Analysis

Residuals analysis refers to a collection of techniques used to check the regres-
sion model for unnatural patterns, which may be indicative of an error in the
model. A residual is calculated as the difference between the observed value of
a response (the dependent variable) and the prediction for that response based
on the regression model and the actual value of the dependent variables.

e = y – ˆ y i
i
i
Minitab and Excel will calculate a standardized residual. The effect of stan-
dardizing the residuals is to scale the error to achieve a variance of 1, which
helps to make outliers more prominent.
Standardized residual:

e / s 2
i

There are a number of tools (discussed elsewhere in this part) that are useful
for finding patterns in residuals.

• Normality test for residuals. If the error between the model and the data is
truly random, then the residuals should be normally distributed with a
mean of zero. A normal probability plot or goodness-of-fit test will allow

us to see departures from normality, indicating a poor fit of the model to
particular data. The normal probability plot for the residuals in the pre-
ceding “Regression Analysis” topic example provides a K-S test value of
0.991, indicating that the standardized residuals approximately fit a nor-
mal distribution, supporting the premise that the regression model fits the
data. The residual associated with observation 12 (marked in red in the
software) was significantly beyond the variation expected of a normal
distribution with mean of 0 and standard deviation of 1. Data far removed
from the “middle” of the data may have a large influence on the model
parameters (the β coefficients, the adjusted correlation coefficient, and
the mean square error terms). When standardized residuals are greater