Page 368 - Six Sigma Demystified

348 Six SigMa DemystifieD

Multiple Regression First-Order Model
Multiple regression is used when there is more than one factor that influences
the response. For example, the cycle time for a sales process may be affected by
the number of cashiers, the number of floor clerks, and the time of day. In this
case, there are three independent variables: (1) number of cashiers, (2) number
of floor clerks, and (3) time of day.
Multiple regression requires additional terms in the model to estimate each
of the other factors. If we have enough data of the right conditions, such as in
an analysis of a designed experiment, we also can estimate the interaction
between these factors. For example, perhaps the effect of time of day varies
depending on the number of cashiers such that when only a few floor clerks are
working, the time of day has a big effect on cycle time variation, yet when many
floor clerks are working, time of day has little effect on cycle time variation.
The first-order model for this example is of the form
Y = β + β X + β X + β X + β X + β X + β X + error
1
2
12
0
12
1
2
23
3
3
13
13
23
where X is the number of cashiers
1
X is the number of floor clerks
2
X is the time of day
3
X is the interaction between the number of cashiers and the number of
12
floor clerks
X is the interaction between the number of cashiers and the time of day
13
X is the interaction between the number of floor clerks and the time of day
23

The error is assumed to have zero mean and a common variance (for all
runs). The errors are independent and normally distributed. These assumptions
will be tested in the residuals analysis of the model (the next tool).
The first-order model produces a plane when viewed in three dimensions of
two significant factors and the response. Interaction effects between the two
factors cause a twisting, or flexing, of the plane.
First-order models work well for a great number of cases. They are often used
in Six Sigma projects because so much can be learned about reducing process
variation with few data. We only need to identify significant factors and two-
factor interactions and may have little use for precisely defining the model or a
higher-order curvilinear response surface. Over limited regions, it is not uncom-
mon for linear effects to dominate, so higher-order models are not necessary.