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Consider a scenario where a two-way contingency table is analyzed by a chi-squared test of independence and a significant result is found. Now, it turns out that this table is an aggregation of data from two subgroups which are heterogenous and one much bigger than the other. When analysed at the sub-group level, both groups give a non-significant result. How is this best explained ?

Edit: I managed to find the paper where I read about this. They explain it as Simpson's Paradox. http://www.amstat.org/publications/jse/secure/v7n3/datasets.morrell.cfm

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  • $\begingroup$ Is this a homework assignment? If so, it should have the homework tag. $\endgroup$
    – Peter Flom
    Dec 8, 2011 at 10:53
  • $\begingroup$ Nope, it's not homework. I came across it in a paper recently, which, frustratingly, I can't find now, but I'm going to have a good look again and try to post a link. The paper said it was Simpson's Paradox, but I always though of SP as a reversal of association which isn't quite what we have here, though I can see perhaps it can be interpreted that way.... $\endgroup$
    – LeelaSella
    Dec 8, 2011 at 11:29

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Simpson's paradox is an extreme form of confounding where the apparent sign of correlation is reversed; you haven't said this is the position here.

I can see at least three possibilities here: the heterogenity between the subgroups, the reduction in sample sizes in each, and poor definition of the subgroups which presuppose the results. Ignoring the third, both of the first two can have an impact: from past experience it is often the small sample size which lead to non-significance in the smaller subgroup and heterogenity which causes the whole group to produce a significant result wile the large subgroup does not.

That was an over-generalisation - each case will have its own issues.

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    $\begingroup$ Thanks for the reply. I've updated the question with a link to the paper where I saw this, which explains it as Simpson's Paradox. $\endgroup$
    – LeelaSella
    Dec 8, 2011 at 11:32
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    $\begingroup$ In your specific example, South African race is clearly a confounding variable, affecting both whether mothers had medical aid (insurance) at birth and whether children were traced for this longitudinal study. It is an example of Simpson's paradox (hence the article), but easily could not have been if one fewer White child who had medical aid had been traced. $\endgroup$
    – Henry
    Dec 8, 2011 at 12:08
  • $\begingroup$ Thanks again. I also thought it was confounding (hence the question) - though the word "confounding" is not mentioned in the article at all which struck me as odd, while Simpson's Paradox is usually known (by me anyway) as a situation where a variable reverses an association, so it wasn't obvious to me that it was Simpson's Paradox at all. $\endgroup$
    – LeelaSella
    Dec 8, 2011 at 12:37

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