In the pages, Gap between Models of Statistics and Real and Regression Analysis , I write that there are many cases that Multi-Regression Analysis does not work well. This page is the case that small idea is good for this problem.
There are 3 variables, Y, X1 and X2. And relationships between 2 of them are figures below. Y is the effects. X1 and X2 are causes.
The figures tell us that,
"Y and X1 may have proportion", "Y and X2 may have inverse proportion" and "X1 and X2 has nothing".
Y = a1 * X1 + a2 * X2 + b
is used as basic formula (model). I call the model as "Addition Model" in this page.
The output of Addition Model is the figure below.
The figure is the relationship between Y and the model.
If the model is perfect, the plots are on a straight line.
But in this case, it is not straight.
Use tha fact,
"Y and X2 may have inverse proportion"
and calculate the number
X1 / X2 .
The figure below is the relationship between Y and X1 / X2. This is similar to a straight.
(I made the data in this page by the calculation X1 / X2 and varied.
So X1 / X2 is correct answer.)
In Analysis of Management , Index of Environmental Assessment and logics for Natural Environment , there are numbers made by division.
When I analyze the numbers of these types, Division Model is useul.
Division Model is more difficult than Addition Model. But by the approach of Dimension Analysis , addition for different dimension is unnatural. Division Model may be natural in this case.
"Y and X2 may have inverse proportion" also leads the model,
Y = a1 * X1 + a2 / X2 + b .
The output figure is below.
This model is better than simple Addition Model.
Even if correct model is Division, there are cases that addition model is enough because the outputs of these models are roughly same.
So Multi-Regression Analysis is one of the Robust Analysis.
I use positive numbers as the data in this page. If negative numbers are also used, the models are not "Roughly same". They could be "Roughly different".
By the changes negative to positive or positive to negative, this knowledge is useful to get correct model or rough model.
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