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A published article (pdf) 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.

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

Am I overlooking something?

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  • $\begingroup$ Look at exercise 4.14 of Davidson & Mackinnon 'Econometric Theory and Methods' 2004 edition for an (exceptional) example of when the Chi-squared is used for a two-tailed test. Edit: great explanation here: itl.nist.gov/div898/handbook/eda/section3/eda358.htm $\endgroup$
    – Max
    Commented Apr 29, 2013 at 6:32
  • $\begingroup$ see stats.stackexchange.com/questions/171074/… $\endgroup$
    – user83346
    Commented Sep 13, 2015 at 6:59
  • $\begingroup$ There's at least one case where it makes sense to talk about a one-sided chi-squared: when you have two dichotomous variables. I give more details here. $\endgroup$
    – Arnaud Mortier
    Commented Feb 12, 2020 at 10:47
  • $\begingroup$ The χ² test can be either one- or two-sided in these scenarios: (A) comparing two samples' proportions in a 2x2 contingency table; (B) comparing a sample's observed variance to a null value. In case A, the p-value is equal to either (i) half the upper χ² tail (for a one-sided H1: p_A > p_B or p_A < p_B) or (ii) all of the upper χ² tail (for a two-sided H1: p_A ≠ p_B). In case B, the p-value is equal to (i) just the upper χ² tail (for one-sided H1: σ² > σ₀²), (ii) just the lower χ² tail (for one-sided H1: σ² < σ₀²), or (iii) upper χ² tail + lower χ² tail (for two-sided H1: σ² ≠ σ₀²). $\endgroup$
    – jdcrossval
    Commented Feb 4, 2023 at 0:04
  • $\begingroup$ Is this just a matter of terminology? For example, on sphweb.bumc.bu.edu/otlt/MPH-Modules/PH717-QuantCore/…: "As with the t-distribution, the χ2 distribution is actually a series of distributions, i.e., one for each number of degrees of freedom, and the upper tail area is the probability, i.e., the p-value. Even though it evaluates the upper tail area, the chi-square test is regarded as a two-tailed test (non-directional), since it is basically just asking if the frequencies differ." $\endgroup$
    – hackerb9
    Commented Feb 8 at 1:18

7 Answers 7

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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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    $\begingroup$ +1 for mentioning the two-sided use with Mendel's pea experiments: it's memorable and gets to the heart of the question. $\endgroup$
    – whuber ♦
    Commented Feb 6, 2012 at 17:01
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    $\begingroup$ +1 for a good question and an excellent answer. @Joel W: I can strongly recommend Khan Academys video on the $\chi^2$ test $\endgroup$
    – Max Gordon
    Commented Feb 6, 2012 at 17:15
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    $\begingroup$ 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. $\endgroup$
    – Frank Harrell
    Commented Feb 6, 2012 at 21:50
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    $\begingroup$ 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 $\endgroup$
    – Joel W.
    Commented Feb 7, 2012 at 2:30
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    $\begingroup$ 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. $\endgroup$
    – gung - Reinstate Monica
    Commented Feb 7, 2012 at 2:52
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Is chi-squared always a one-sided test?

That really depends on two things:

  1. what hypothesis is being tested. If you're testing variance of normal data against a specified value, it's quite possible to be dealing with the upper or lower tails of the chi-square (one-tailed), or both tails of the distribution. We have to remember that $\frac{(O-E)^2} E$ type statistics are not the only chi-square tests in town! (Consider testing equality of two variances, for one example.)

  2. whether people are talking about the alternative hypothesis being one- or two-sided (because some people use 'two-tailed' to refer to a two-sided alternative, irrespective of what happens with the sampling distribution of the statistic. This can sometimes be confusing. So for example, if we're looking at a two-sample proportions test, someone might in the null write that the two proportions are equal and in the alternative write that $\pi_1 \neq \pi_2$ and then speak of it as 'two-tailed', but test it using a chi-square rather than a z-test, and so only look at the upper tail of the distribution of the test statistic (so it's two tailed in terms of the distribution of the difference in sample proportions, but one tailed in terms of the distribution of the chi-square statistic obtained from that -- in much the same way that if you make your t-test statistc $|T|$, you're only looking at one tail in the distribution of $|T|$).

Which is to say, we have to be very careful about what we mean to cover by the use of 'chi-square test' and precise about what we mean when we say 'one-tailed' vs 'two-tailed'.

In some circumstances (two I mentioned; there may be more), it may make perfect sense to call it two-tailed, or it may be reasonable to call it two-tailed if you accept some looseness of the use of terminology.

It may be a reasonable statement to say it's only ever one-tailed if you restrict discussion to particular kinds of chi-square tests.

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  • $\begingroup$ what about this one? stats.stackexchange.com/questions/223560/… $\endgroup$
    – An old man in the sea.
    Commented Jul 16, 2016 at 9:45
  • $\begingroup$ Thank you very much for mentionning the variance test. That is actually a quite interesting use of the test, and also the reason why I ended up on this page ^^ $\endgroup$
    – Tobbey
    Commented Sep 9, 2019 at 14:04
  • $\begingroup$ @Glen_b can you please elaborate on this statement " it's two tailed in terms of the distribution of the difference in sample proportions, but one tailed in terms of the distribution of the chi-square statistic obtained from that" ? $\endgroup$
    – KKW
    Commented Apr 18, 2023 at 13:32
  • $\begingroup$ If you look at $\hat{p}_1-\hat{p}_2$ and you want to reject whether that's large and positive or small (i.e. large and negative), you're interested in both tails of that difference in proportions. If you did a z-test for example, you'd do a two tailed test. Both of those tail-cases result in large chi-squared values, so if you're doing the chi-squared test you reject for large values. In that example, small values of the chi-squared statistic occur when the sample proportions are almost identical (in the center of the null distribution of $\hat{p}_1-\hat{p}_2$); you don't want to reject those. $\endgroup$
    – Glen_b
    Commented Apr 18, 2023 at 22:22
  • $\begingroup$ By contrast, if you're doing a variance test, you'd want to reject whether $s^2/\sigma^2$ was large or small (it's values in the middle of that distribution -- near $1$ -- that are consistent with $H_0$; both tails of the that ratio are telling you the variance differs from the variance in the null. This translates directly to the chi-squared statistic being close to its d.f. when $H_0$ is true and either higher or lower than it when $H_0$ is false. In that case you want to look into both tails of the chi-squared distribution. (So we must be clear about what we're talking about the tails of) $\endgroup$
    – Glen_b
    Commented Apr 18, 2023 at 22:32
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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.

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I also have had some problems to come to grips with this question as well, but after some experimentation it seemed as if my problem was simply in how the tests are named.

In SPSS as an example, a 2x2 table can have an addition of a chisquare-test. There there are two columns for p-values, one for the "Pearson Chi-Sqare", "Continuity Correction" etc, and another pair of columns for Fisher's exact test where there are one column for a 2-sided test and another for a 1-sided test.

I first thought the 1- and 2-sides denoted a 1- or 2-sided version of the chisquare test, which seemed odd. It turned out however that this denotes the underlying formulation of the alternate hypothesis in the test of a difference between proportions, i e the z-test. So the often reasonable 2-sided test of proportions is achieved in SPSS with the chisquare test, where the chisquare measure is compared with a value in the (1-sided) upper tail of the distribution. Guess this is what other responses to the original question already have pointed out, but it took me some time to realize just that.

By the way, the same kind of formulation is used in openepi.com and possibly other systems as well.

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@gung's answer is correct and is the way discussion of $\chi^2$ should be read. However, confusion may arise from another reading:

It would be easy to interpret a $\chi^2$ as 'two-sided' in the sense that the test statistic is typically composed of a sum of squared differences from both sides of an original distribution.

This reading would be to confuse how the test statistic was generated with which tails of the test statistic are being looked at.

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  • $\begingroup$ Could you elaborate on what a "side of an original distribution" would be? It's not even evident what that "original distribution" refers to nor how it is related to the chi-squared statistic as computed from data. $\endgroup$
    – whuber ♦
    Commented Jun 15, 2015 at 16:29
  • $\begingroup$ For example, a sum of $n$ independent normals squared is $\chi^2$. The normals are the 'original' distribution. The $\chi^2$ stat incorporates information from both tails of the underlying normal distribution. $\endgroup$
    – conjectures
    Commented Jun 15, 2015 at 16:32
  • $\begingroup$ OK, but I still cannot figure out what you are contrasting that with. Could you provide an example of a non-two-sided test statistic that could be used in ANOVA and show how it is connected with the tails of some distribution? $\endgroup$
    – whuber ♦
    Commented Jun 15, 2015 at 16:39
  • $\begingroup$ I'm not contrasting it with anything. I'm pointing out a reason why people might get confused about the one-sided/two-sided jargon in the context of $\chi^2$. It's straightforward for experts to see that the $\chi^2$ test itself is usually a one-sided test on the calculated stat. Others may have some data and be thinking about deviations from the mean in both directions, which often get rolled up into a $\chi^2$ stat. They will have heard things along the lines of 'thinking of deviations from the mean in both directions=two-sided test'. Hence a misunderstanding. $\endgroup$
    – conjectures
    Commented Jun 15, 2015 at 16:52
  • $\begingroup$ I'm asking for a contrast only to help understand what you are trying to describe. I haven't been able to determine what that is yet. $\endgroup$
    – whuber ♦
    Commented Jun 15, 2015 at 17:08
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$\chi^2$ test of variance can be one or two sided: The test statistic is $(n-1)\frac{s^2}{\sigma^2}$, and the null hypothesis is: s (sample deviation)= $\sigma$ (a reference value). The alternative hypothesis could be: (a) $ s> \sigma$, (b) $s < \sigma$, (c) $s \neq \sigma$. p-value caculation involves the asymmetry of the distribution.

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The $\chi^2$ and F tests are one sided tests because we never have negative values of $\chi^2$ and F. For $\chi^2$, the sum of the difference of observed and expected squared is divided by the expected ( a proportion), thus chi-square is always a positive number or it may be close to zero on the right side when there is no difference. Thus, this test is always a right sided one-sided test. The explanation for F test is similar.

For the F test, we compare between group variance to sum of within group variances ( mean square error to $\frac{SSw}{dfw}$. If the between and within mean sum of squares are equal we get an F value of 1.

Since it is essentially the ratio of sum of squares, the value never becomes a negative number. Thus, we don't have a left sided test and F test is always a right sided one sided test. Check the figures of $\chi^2$ and F distributions, they are always positive.For both tests, you are looking at whether the calculated statistic lies to the right of the critical value.

Chi-square and F Distributions

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    $\begingroup$ A test statistic doesn't need to take negative values for us to consider both tails. Consider an F test for the ratio of two variances, for example. $\endgroup$
    – Glen_b
    Commented Mar 3, 2017 at 6:25
  • $\begingroup$ F test is one sided test Glen_b. $\endgroup$
    – Daniel
    Commented Mar 3, 2017 at 6:49
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    $\begingroup$ The F test for equality of variances, which has a statistic that's the ratio of the two variance estimates is NOT one sided; there's an approximation to it which places the larger of the two sample variances on the numerator, but it's only really right if the df are the same. But if you don't like that there's any number of other examples. The statistic for the rank sum test cannot be negative but the test is two tailed.I can supply other examples if needed. $\endgroup$
    – Glen_b
    Commented Mar 3, 2017 at 7:19
  • $\begingroup$ @Ferdi Unfortunately there's something clearly wrong with the example there -- it says it's two-sided but then implies it only rejects for large values of the statistic. If $\sigma_1^2$ was less than $\sigma_2^2$ we'd be almost never observing a large value for the ratio, so the statistic would only tend to reject when $\sigma_1^2>\sigma_2^2$ making it one-sided. $\endgroup$
    – Glen_b
    Commented Mar 3, 2017 at 11:27

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