Note: I posted this question on the Stats.Stackexchange site a week ago, but I think it might be better here. It's about proving a theorem I came up with about experiment data set structures, specifically nested predictors...
QUESTION
I am studying the structure of experiment data sets, and I want to propose a rule that I call the "transitive property of nested predictors".
The general idea is that…
- if there are three predictors: A, B, and C
- and if B is nested within A
- and if C is nested within B
Then…
- C is nested within A
My question is:
How can I prove that this rule is always true, using either mathematics, symbolic logic, set theory, diagrams, or something else? (See Example 1, below)
or
How can I prove that it is impossible to violate this rule? (See Example 2, below)
Example 1
An example data set that agrees with the proposed rule given above is shown here.
The dataset is:
A text description of this data set is:
And a graphical diagram of this data set is:
Example 2
An example experiment diagram that disagrees with the proposed rule is shown here, but I claim that such a data set does not exist.
EDIT: DEFINITION OF NESTED PREDICTOR
I define "nested" only in terms of the structure of the data set itself, not from any real-world situations. This is because the real world situations are irrelevant, as long as the levels of the nested predictors are coded properly, i.e. with unique level IDs.
A categorical predictor is nested when each of its levels has a measurement in only one level of another categorical predictor, and there are at least two of its levels measured in each level of the other predictor.
This can be determined algorithmically or visually. Take the example data set shown here:
Algorithmically
Visually
