P values. One-tail or  two-tail ?

When comparing two groups, you must distinguish between one- and two-tail P values. Some books refer to one-sided and two-sided P values, which mean the same thing.

What does one-sided mean?

It is easiest to understand the distinction in context. So let’s imagine that you are comparing the mean of two groups (with an unpaired t test). Both one- and two-tail P values are based on the same null hypothesis, that two populations really are the same and that an observed discrepancy between sample means is due to chance.

A two-tailed P value answers this question:

To interpret a one-tail P value, you must predict which group will have the larger mean before collecting any data. The one-tail P value answers this question:

If the observed difference went in the direction predicted by the experimental hypothesis, the one-tailed P value is half the two-tailed P value (with most, but not quite all, statistical tests).

When is it appropriate to use a one-sided P value?

A one-tailed test is appropriate when previous data, physical limitations, or common sense tells you that the difference, if any, can only go in one direction. You should only choose a one-tail P value when both of the following are true.

• You predicted which group will have the larger mean (or proportion) before you collected any data.
• If the other group had ended up with the larger mean – even if it is quite a bit larger – you would have attributed that difference to chance and called the difference 'not statistically significant'.

Here is an example in which you might appropriately choose a one-tailed P value: You are testing whether a new antibiotic impairs renal function, as measured by serum creatinine. Many antibiotics poison kidney cells, resulting in reduced glomerular filtration and increased serum creatinine. As far as I know, no antibiotic is known to decrease serum creatinine, and it is hard to imagine a mechanism by which an antibiotic would increase the glomerular filtration rate. Before collecting any data, you can state that there are two possibilities: Either the drug will not change the mean serum creatinine of the population, or it will increase the mean serum creatinine in the population. You consider it impossible that the drug will truly decrease mean serum creatinine of the population and plan to attribute any observed decrease to random sampling. Accordingly, it makes sense to calculate a one-tailed P value. In this example, a two-tailed P value tests the null hypothesis that the drug does not alter the creatinine level; a one-tailed P value tests the null hypothesis that the drug does not increase the creatinine level.

The issue in choosing between one- and two-tailed P values is not whether or not you expect a difference to exist. If you already knew whether or not there was a difference, there is no reason to collect the data. Rather, the issue is whether the direction of a difference (if there is one) can only go one way. You should only use a one-tailed P value when you can state with certainty (and before collecting any data) that in the overall populations there either is no difference or there is a difference in a specified direction. If your data end up showing a difference in the “wrong” direction, you should be willing to attribute that difference to random sampling without even considering the notion that the measured difference might reflect a true difference in the overall populations. If a difference in the “wrong” direction would intrigue you (even a little), you should calculate a two-tailed P value.

Recommendation
I recommend using only two-tailed P values for the following reasons:

• The relationship between P values and confidence intervals is more straightforward with two-tailed P values.
• Two-tailed P values are larger (more conservative). Since many experiments do not completely comply with all the assumptions on which the statistical calculations are based, many P values are smaller than they ought to be. Using the larger two-tailed P value partially corrects for this.
• Some tests compare three or more groups, which makes the concept of tails inappropriate (more precisely, the P value has more than two tails). A two-tailed P value is more consistent with P values reported by these tests.
• Choosing one-tailed P values can put you in awkward situations. If you decided to calculate a one-tailed P value, what would you do if you observed a large difference in the opposite direction to the experimental hypothesis? To be honest, you should state that the P value is large and you found “no significant difference.” But most people would find this hard. Instead, they’d be tempted to switch to a two-tailed P value, or stick with a one-tailed P value, but change the direction of the hypothesis. You avoid this temptation by choosing two-tailed P values in the first place.

When interpreting published P values, note whether they are calculated for one or two tails. If the author didn’t say, the result is somewhat ambiguous.

How to convert between one- and two-tail P values

If the actual difference (effect) went in the predicted direction:

• The one-tail p-value is half the two-tail p-value. So if the two-tailed p-value is 0.1, the one-tailed p-value is 0.05.
• The two-tail p-value is twice the one-tail p-value.

If the actual difference (effect) went opposite to the predicted direction:

• The one-tail  p-value equals one minus half the two-tailed value. So if the two-tailed p-value is 0.1, the one-tailed p-value is 0.95
• The two-tail p-value is twice the one-tail p-value.