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I was wondering if anyone could help explain to me why someone would use a two tailed hypothesis test? Let's say I am doing a t-test, presumably I know which mean was greater than the other and I am testing if that difference is statistically significant (i.e. a one tailed test). Given the formula for computing the t-statistic, whether the value is positive or negative just depends on which group I label as group 1 or 2, no? Are there applications for a two tailed test with a different hypothesis testing method? Or am I missing something?

Edit: To clarify, I am really wondering why you wouldn't just identify which group gives greater results, and use a one-tailed test. Because if there is a difference, that difference is going to go one way or the other, and then you can test accordingly.

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    $\begingroup$ You should never design your statistical analysis after the data is collected. With this in mind it is clear that in the majority of cases you don't know in advance the relative order of statistics of group 1 and 2. This is why we have two tailed tests. $\endgroup$ – Mur1lo Apr 17 at 2:11
  • $\begingroup$ When you say “a different type of test” do you mean other than a one- or two-tailed test? $\endgroup$ – Thomas Bilach Apr 17 at 2:27
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Actually, it tends to be the other way around --- it is almost never appropriate to use a one-sided test. The reason for this is that we need to specify our objects of inference (e.g., hypotheis tests, confidence intervals, etc.) prior to seeing the data, or we will induce bias in these objects. When seeking to make inference about two unknown quantities, it is generally best not to assume that the direction of interest is known a priori, and so it is usually best to test for a difference rather than a directional difference. Others will argue that it is legitimate to use a one-sided test when you have specified a direction of interest a priori, but I am sceptical even in this case. I would counsel that you should either avoid classical hypothesis testing altogether (e.g., using a confidence interval instead) or use a two-sided hypothesis test, even if you are interested in a relationship with a specified direction.

In regard to this issue, it is worth noting that classical hypothesis tests have some unusual (and not very helpful) properties when you compare across different tests. One of their properties is that, for a symmetric test, the p-value of the one-sided test is twice as high as the p-value for the two-sided test when data is in the relevant tail. This means that if you do a one-sided test for a disparity in the direction of the data, the p-value will be half the size of a two-sided test. So, if you correctly guess the direction of the trend a priori, the result of using the one-sided test is that you see evidence that looks twice as strong for the more specific hypothesis! This property of classical hypothesis tests gives good reason to avoid one-sided tests.

In any case, whether or not you agree with my view here, what you are proposing is definitely a bad idea. If you identify the direction of the test from the observed data, and then perform a one-sided test in the identified direction, you will bias your test towards rejection of the null hypothesis.

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  • $\begingroup$ You mentioned 'classical hypothesis tests,' is there a circumstance with different tests where this problem doesn't exist when doing a one tailed test? Is this problem just for tests where the test statistical is distributed symmetrically? And if this is the case, doesn't that make symmetrical tests less likely to give you statistically significant results inherantly, making a one tailed test closer to that of an assymetrical test? $\endgroup$ – ajax2112 Apr 17 at 10:11
  • $\begingroup$ The same basic problem exists with non-symmetric tests in the classical framework (as opposed to say, the Bayesian framework), but the p-value is not exactly halved. $\endgroup$ – Ben - Reinstate Monica Apr 17 at 10:19
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A one-tailed test is appropriate if you only want to test if there is a difference between your groups in a specific direction. You would use a two-tailed test if you want to determine if there is any difference between the two groups you're comparing.

As user Mur1lo says in their comment - you should never design your analysis after the data is collected. Therefore a two-tailed test is often more appropriate. A one-tailed test can only be justified if you have made a prediction prior to data collection about the direction of the difference, and you are completely uninterested in the possibility that the opposite outcome could be true.

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