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I'm trying to check whether a variable is a confounder or not. Specifically, for a randomized trial where I want to investigate the effects of a reduction in class size on student performance, would free lunch status be a confounder? According to this answer, a confounder must satisfy the following conditions:

  1. Be independently associated with the outcome;
  2. Be associated with the exposure
  3. Must not lie on the causal pathway between exposure and outcome.

I have verified 1 and 3 but am confused about 2 because I perform a chi-square test and find that the Pearson chi2(2) = 1.6e+03 and its Pr = 0.000, which would indicate that it is independent of the exposure and thus lunch status is not a confounder. Intuitively, this doesn't make sense to me since in the study I'm trying to replicate, they condition on free lunch status, which would mean it is a confounder. Am I just being stubborn in believing that free lunch status must be a confounder that needs to be adjusted for or am I just wrong?

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You have exactly misinterpreted what the chi square result means. A high chi square and low p value indicates that the two variables are related, not that they are not.

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    $\begingroup$ Ah - so that is the source of my mistake and confusion. Thank you! $\endgroup$ – kpz Jan 26 '16 at 19:23
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    $\begingroup$ Check the marital "satisfaction" example taken from Agresti here - $H_0: Independence$. You reject $H_0$. $\endgroup$ – Antoni Parellada Jan 26 '16 at 19:27
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    $\begingroup$ Free lunch status is also an excellent indicator for socioeconomic status -- specifically, poverty. It's probably the single best index for that, actually. $\endgroup$ – Mike Hunter Jan 26 '16 at 19:29
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Unfortunately, the definition you cite is wrong. Section 6.3 of Judea Pearl's book "Causality" has more on this topic.

Section 6.2 of that book is called "Why There Is No Statistical Test For Confounding", which is exactly true. One needs to either randomize or make substantive assumptions about causal mechanisms in order to rule out confounding. You can do that by drawing a graph of what determines treatment and exposure and then, for example, checking whether you can work with the back-door criterion. See Pearl, section 3.3.

In your example, you state that you are analyzing data from a randomized experiment, so treatment and exposure can only be confounded because of limited observations and bad luck. I'm sure free-lunch status of a student is associated with performance because socio-economic status drives both, but I don't see immediately how this could be related to class size.

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