The degree of freedom for a Person's X^2 test is (I-1)*(J-1) and it is obtained by (IJ-1) -([I-1]+[J-1]) =(I-1)(J-1).

I am just wondering why it is (IJ-1) -([I-1]+[J-1])??

What is (IJ-1) referred to and what is [I-1]+[J-1] referred to and why do we use the difference of those two terms?

I go through some online material, but most of them just directly state that the df is (I-1)(J-1) without process.

My personal thought is (IJ-1) is refers to the situation when the X Y variable is independent and (I-1)(J-1)is referred to the situation when they are not independent.

I am just quite curious about the process and I am not very familiar with X^2 distribution and degree of freedom.

Thank you. : )


The heuristic answer is that there are $IJ$ numbers in the table, but since they add to the sample size, only $IJ-1$ degrees of freedom. The test estimates the row and column margins in order to compute expected proportions, and this uses up $I-1$ and $J-1$ degrees of freedom respectively, so there are $(IJ-1)-(I-1)-(J-1)$ left.

This is actually kind of a bad answer, because the claim that estimating the row and column proportions uses up degrees of freedom is so non-obvious that Fisher and Pearson had a feud about it. The answer is correct and it's correct for basically the stated reasons, but if you don't believe it you are in good company (wrong, but in good company).

The real answer is the same, but done with linear algebra and multivariate Normal approximations, where the degrees of freedom are a well-defined computation.


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