Are chi squred tests always 1-tailed or 2 tailed?

[FluffyPuff]

Suppose we wanted to test whether 2 categorical variables are related as folllows:
H0: The two variables are not related
HA: The two variables ARE related

using the standard Chi-squared test for a cross-tabulation, in which we calculate the test statistic using the sum of (Oi - Ei)^2/Ei where Oi is the observed value in a particular cell and Ei is the expected value.

The examples in the text list examples of this kind as being 1-tailed. But it seems to me that this is a 2-tailed test since the hypothesis could be rejected on both sides. To me, a one tailed test would have a null hypothesis that would state the DIRECTION in which the two variables are related.

Any help would be appreciated.

[squareandrare]

For this test, only large values of the test statistic indicate significant differences. Small test statistics will have high p-values.

I seem to remember using a 2-sided Chi-square test once, but I honestly don't remember what I was testing. Just about every test you will ever encounter using Chi-square will be one-sided.

[Link]

I seem to remember using a 2-sided Chi-square test once, but I honestly don't remember what I was testing.

Please let me know if it comes to you. I can't think of any situation right now where a two sided test would be applicable. It'd be nice to know of some though for future referece.

[squareandrare]

I think it was in a paper I was reading last year about Geographically Weighted Regression. I think it had to do with testing whether allowing a single coefficient to vary spatially significantly helped the model as opposed to traditional OLS (where the coefficient is constant). It appeared that you rejected for both large and small values of a Chi-square test statistic, but I may have just been misunderstanding the test.

I didn't get far because we quickly decided to "delay" (read: abandon) the project and work on something else.

[ichbin]

Whether you should reject your hypothesis when you get a chi-square value deep in the left tail depends on the exact formulation of your hypothesis. This much is certainly true: that too-small value is telling you something important about your data. Basically, it's telling you that your variance is not as large as you had assumed it to be.

Imagine this concerte scenario: I tell you that heights in a population are normally distributed with mean 167 cm and standard deviation 15 cm. I then give you a sample of 20 measured heights and ask you whether my sample is consistent with the population. When you look at the data, you see that it is a list of 20 values, all of which are within 1 cm of 167 cm. Now, any of your usual z- or t- type tests (the two-sided forms of which are basically equivilent to a one-sided chi-squared test) is going to say your sample mean is compatible with the population mean. But if there aren't alarm bells going off in your head, you aren't a very good statistician.

As a physical scientist, I usually see chi-squares as a goodness-of-fit test in the context of model fits. I tell my students to be almost as wary of too-small chi-squared (indicating that their error bars are off) as they are of too-large chi-squared (indicating that the model does not fit the data).