Page 379 - Six Sigma Demystified

Part 3 S i x S i g m a To o l S 359

than 2.0 for any observation (such as observation 12 in the example), they
are often considered outliers that should be investigated and removed.
Similarly, Cook’s distance (such as provided in Minitab) provides a mea-
sure of the squared distance between the β estimate with and without the
observation.
• Residuals versus independent variable (x). We shouldn’t see any pattern or
any significant correlation when we analyze the residuals (on the y axis)
and the independent variable (on the x axis) using a scatter diagram. If
patterns exist, they would imply that the model changes as a function of
the x variable, meaning that a separate model for particular values of x (or
higher-order terms in x) should be considered.
• Residuals versus fitted response. A pattern here would suggest that the
model does not fit well over the range of the response. When we have
repeated runs at each experimental condition, we also can look for non-
constant variance of the response (shown in Figure F.39). If nonconstant
variance of the response is suspected, then Bartlett’s equality-of-variance
test can be used to test for the condition. Reasons for nonconstant vari-
ance include an increase in measurement error for larger responses (such
as when error is a fixed percentage of the measured value) and the distri-
butional properties of the response (such as for skewed distributions
where the variance is a function of the mean).
If the data can be ordered in a time-based manner (meaning that for
each datum you have a unit of time at which it occurred), then we can do
some additional analysis of the model.

• Control chart of residuals. A control chart of the residuals would indi-
cate whether the er ror changes over the time in which the data oc-
curred. An out-of-control condition (or run-test violation) on this
chart implies that another factor (that changes over time) is influenc-
ing the response.
• Test for autocorrelation of residuals. Significant autocorrelation of the re-
siduals (which may be detected using the autocorrelation function) im-
plies that the error at one period in time is influenced by earlier error. In
this case, it’s likely we are missing a factor that varies over time. We can
use the autocorrelation function or the Durbin-Watson test (shown in the
“Regression Analysis” example) for independence of consecutive residuals.
The Durbin-Watson statistic tests for first-order (lag one) serial correla-