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When we do a chi-squared test (to test goodness of fit or the dependence of two variables), we assume that the the chi-squared statistic follows the chi-squared distribution.

  • Shouldn't we first check if the chi-squared statistic follows the chi-squared distribution in that particular case?
  • If yes, then how do we do that?
  • Or have I got it all mixed up and my question itself is wrong?
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This is actually pretty straightforward. The chi-squared distribution is a distribution of continuous values. A chi-squared test statistic may or may not be able to take any positive real value. For example, the test statistic for a likelihood ratio test can take continuous values, but the test statistic from a chi-squared test of independence for a 2x2 contingency table can only take a finite set of discrete values. The former may match the theoretical distribution just fine, but the latter will be an approximation. If your sample is large enough, the approximation isn't a problem and the Yates' correction for continuity also helps a lot, so in practice it isn't usually something that you need to worry about often. To understand this further, it may help to read my answer here: Comparing and contrasting, p-values, significance levels and type I error.

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    $\begingroup$ I gave this +1, but a small niggle worries at me -- the small-sample distribution of an LRT (even when continuous) may not be very near chi-square. I'd suggest "may match" rather than "will match" as more generally correct. $\endgroup$ – Glen_b Aug 28 '15 at 2:52

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