Ch. 16. Selecting the Best Regression Equation
(Model Development)
I. Situation
A. Selecting meaningful (important) X variables to
establish the best set of X variables to predict Y.
B. If we have a model which includes all possible
interactions and power functions of the X variables as
well as all the original X variables, we will have the
maximum model.
C. The maximum model usually includes some trivial X
variables which add no meaningful contribution to the
prediction of Y.
D. Therefore, we want to select the most efficient line
according to the rule of parsimony.
II. Maximum Model
A. Includes all possible function of all possible X
variables of interest.
B. Assume k is the number of all possible X components.
C. If k = n - 1, then we have a perfect regression
(R²=1).
D. The "Rule of Thumb" requires a minimum of n=5k, n=10k,
n=20k or n=k+40. However, typically we have a minimum
of 100 for n and we feel comfortable if n>200.
III. Criteria for Model Development
A. R²p (p: # of X variables for the best model).
1. Goodness-of-fit index.
2. It is almost always overestimated --->
misleading.
3. Adding any predictors will increase R².
4. Should not be the sole criterion for the model.
B. Fp - Value
1. Index of the significance of the model.
2. The ratio of reduced variance to the error
variance of the maximum model (MSRp/MSEk).
3. The larger, the better.
C. MSEp
1. The error (residual) variance for the best model.
2. The smaller, the better.
D. Cp
1. Mallow's Cp.
2. Cp = p+1 when MSEp is almost equal to MSEk.
3. Sometimes it gives us a ridiculous number.
E. SAS gives us all of them.
IV. Methods of Model Development.
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