Timeline for Estimated vs. true expected value in $\chi^2$ test of independence of two categorical variables
Current License: CC BY-SA 4.0
9 events
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| Jan 10, 2022 at 7:45 | comment | added | Sextus Empiricus | @RichardHardy I remember that the first time that I came across the $\chi^2$ test I was wondering why it isn't (O-E)^2/O instead of (O-E)^2/E. It was only when I read Pearson's 1900 article that I came to view the $\chi^2$ test as an equivalent to a linear regression where we can have a geometrical view of the distribution of the observations as a n-dimensional cloud that partitions into two spaces with dimensions n-p and p. | |
| Jan 10, 2022 at 7:26 | comment | added | Richard Hardy | So my original concern did not go as far as to question whether $\frac{(\text{observed}_i-\text{expected}_i)^2}{\text{expected}_i}$ is actually $\chi^2$ distributed, it stopped earlier. In the meantime, your answer addresses a fuller point, I suppose. | |
| Jan 10, 2022 at 7:22 | comment | added | Richard Hardy | You got deeper into the matter than I intended (and that is good). My original confusion concerned how presentations of $\chi^2$ test get away by saying $E_i$ is the expected value when it is actually only an estimate thereof, and then basing the intuition on how $O_i$ should not be far from the expected value under $H_0$. I remember being confused the first time I heard the argument, but I never got to think about it in any depth. When I remembered the issue 15 years later, it dawned on me that I now had some intuition as to why it works in spite of the discrepancy. (Why? Because it shrinks). | |
| Jan 10, 2022 at 7:15 | comment | added | Sextus Empiricus | I see mainly 2 points of which the 2nd relates to yours, but it is only in the denominator where it counts. So we have 1 The influence of $E_i$ is very small in the numerator term $E_i-O_i$ because of the (approximate) independence. This is independence is not because $E_i$ approaches a constant (law of large numbers) but instead because the observations approach a multivariate normal distribution (where the parameters like $O_1 - O_2$ and $O_1+ O_2$ are independent). 2 There is some influence of the parameter $E_i$ in the denominator. In that place you get that your point plays a role | |
| Jan 10, 2022 at 7:10 | comment | added | Richard Hardy | Thank you, that is helpful. I wonder if your two points are sufficient by themselves, or is my point also needed to show that "this is not a problem" (w.r.t. the OP's question). My intuition was that my point was of first-order importance, but perhaps it is not? In other words, if the variance of the distribution of the difference between the true and the estimated $E_i$ did not shrink with the sample size and stayed as large as with $n=1$, would this be a problem? | |
| Jan 9, 2022 at 22:11 | history | edited | Sextus Empiricus | CC BY-SA 4.0 |
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| Jan 9, 2022 at 21:54 | history | edited | Sextus Empiricus | CC BY-SA 4.0 |
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| Jan 9, 2022 at 21:47 | history | edited | Sextus Empiricus | CC BY-SA 4.0 |
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| Jan 9, 2022 at 21:33 | history | answered | Sextus Empiricus | CC BY-SA 4.0 |