# Chi-Squared always a one-sided test?

A published article contains these 2 sentences:

"Moreover, misreporting may be caused by the application of incorrect rules or by a lack of knowledge of the statistical test. For example, the total df in an ANOVA may be taken to be the error df in the reporting of an F test, or the researcher may divide the reported p value of a $\chi^2$ or F test by two, in order to obtain a one-sided p value, whereas the p value of a $\chi^2$ or F test is already a one-sided test." Source: http://wicherts.socsci.uva.nl/BakkerWicherts2011.pdf

Why might they have said that? Chi-Squared is a two-sided test. (I have asked one of the authors, but gotten no response.)

Am I overlooking something?

Thanks.

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The chi-squared test is essentially always a one-sided test. Here is a loose way to think about it: the chi-squared test is basically a 'goodness of fit' test. Sometimes it is explicitly referred to as such, but even when it's not, it is still often in essence a goodness of fit. For example, the chi-squared test of independence on a 2 x 2 frequency table is (sort of) a test of goodness of fit of the first row (column) to the distribution specified by the second row (column), and vice versa, simultaneously. Thus, when the realized chi-squared value is way out on the right tail of it's distribution, it indicates a poor fit, and if it is far enough, relative to some pre-specified threshold, we might conclude that it is so poor that we don't believe the data are from that reference distribution.

If we were to use the chi-squared test as a two-sided test, we would also be worried if the statistic were too far into the left side of the chi-squared distribution. This would mean that we are worried the fit might be too good. This is simply not something we are typically worried about. (As a historical side-note, this is related to the controversy of whether Mendel fudged his data. The idea was that his data were too good to be true. See here for more info if you're curious.)

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+1 for mentioning the two-sided use with Mendel's pea experiments: it's memorable and gets to the heart of the question. –  whuber Feb 6 '12 at 17:01
+1 for a good question and an excellent answer. @Joel W: I can strongly recommend Khan Academys video on the $\chi^2$ test –  Max Gordon Feb 6 '12 at 17:15
My summary of this is that the $\chi^2$ is a two-sided test for which we are usually interested in only one of the tails of the distribution, indicating more disagreement, rather than less disagreement than one expects by chance. –  Frank Harrell Feb 6 '12 at 21:50
Supporting the 2-tailed view: "The two-tail probability beyond +/- z for the standard normal distribution equals the right-tail probability above z-squared for the chi-squared distribution with df=1. For example, the two-tailed standard normal probability of .05 that falls below -1.96 and above 1.96 equals the right-tail chi-squared probability above (1.96)squared=3.84 when df=1." Agresti, 2007 (2nd ed.) page 11 –  Joel W. Feb 7 '12 at 2:30
That's right. Squaring a z-score yields a chi-squared variate. For example, a z of 2 (or, -2!) when squared equals 4, the corresponding chi-squared value. The two-tailed p-value associated with a z-score of 2 is .04550026; and the one-tailed p-value associated with a chi-squared value of 4 (df=1) is .04550026. A two-tailed z test corresponds to a one-tailed chi-squared test. Looking at the left tail of the chi-squared distribution would correspond to looking for z-scores that are closer to z=0 than you might expect by chance. –  gung Feb 7 '12 at 2:52
The chi-square test $(n-1)s^2/\sigma^2$ of the hypothesis that the variance is $\sigma^2$ can be either one- or two-tailed in exactly the same sense that the t-test $(m-\mu)\sqrt{n}/s$ of the hypothesis that the mean is $\mu$ can be either one- or two-tailed.