C:\OSU\Stat 600-601\2C. Exponential Explosion Factor\the exponential explosion factor.doc
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The Exponential Explosion Factor
Consider the logistical problem of building statistical models involving K predictor
variables and R response variables. How many different models are possible?
Consider the case of creating a simple regression model involving the K predictor
variables and restrict the variables in the models to first-order terms and all possible
interactions (up to and including the one kth order interaction. That is, consider only
terms of the form
1,0jXx....xXxX i
j
K
j
2
j
1
K
21 =
as candidates for inclusion in the model.
In this case, there is a total of
KK
K
0j
2)11(
j
K
M=+=
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
=∑
=
first-order terms and cross products that could be chosen for inclusion in the model. How
many models can be made from these 2K terms? Recognizing that the term 00
2
0
1... K
XXX
represents the intercept term in the model, and if we ignore the model Y ≡ 0 + ε as a
legitimate model, then there are
12)11(
222
2
0
−=+=
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
=∑
=
KK
K
j
K
j
N
possible models for each of the R response variables. Given R response variables, then
there is a total of NxRN =
* total models to consider.
Example (K=2 and R = 3)
Suppose there are two predictor variables X1 and X2 and three response variables.
The 22 = 4 possible terms in the model are:
• 0
2
0
1XX
• 0
2
1
1XX
• 1
2
0
1XX
• 1
2
1
1XX
