Does the “Sig.” column in the SPSS output for ANOVA

The "Sig." column in SPSS output for t-test is a two-tailed p-value, i.e. if one want to decide whether to reject a null hypothesis, they need to compare the predetermined significant level with the "Sig." value divided by 2 instead of the value itself.

What about the SPSS output for F-test? It seems that it is a one-tailed p-value (i.e. one can directly compare it with the significance level of a one-tailed test). It would also not make sense if it was two-tailed as F-distribution is not symmetric. But I am still unable to find the confirmation.

• As an aside, the asymmetry of the F distribution does not mean that one cannot form two-sided null hypotheses with it. One can certainly calculate $p= P(F_{1} \le F_{df_{n},df_{d}} \le F_{2})$, for two critical values of $F$. The issue is more of why one would want to pose such a question... the assumptions underlying such a test, and the kinds of conclusions one would draw from it. – Alexis Jan 30 '18 at 17:01
• I was not talking about the null hypothesis formed regarding F-test; I was talking about why the SPSS output of F-test unlikely displays 2-tailed p-value. – Aqqqq Jan 30 '18 at 17:07
• Yes. Which is why I made a comment as an aside, and not an answer. :) (Welcome to CV, by the way!) – Alexis Jan 30 '18 at 17:18
• I see. Sorry for the misunderstanding. – Aqqqq Jan 30 '18 at 17:52
• This is nearly the same as the question at stats.stackexchange.com/questions/325354/…, whose answers might be informative. – whuber Jan 30 '18 at 20:31

Conceptually, this is a two-tailed $p$-value. The right tail of the $F$ distribution reflects more variability than expectation and the left tail less variability than expectation. If we consider the restricted case with 1 degree of freedom in the numerator we have an $F$ that is simply $t^2$. The two-tailed $p$-value for this $t$ will be the same as $p$-value reported in the SPSS ANOVA table for the right tail of the $F$ distribution. Once you have 2 or 3 degrees-of-freedom in the numerator (i.e., when you are comparing 3 or more means), the directionality of the mean differences is no longer meaningful. Hence, the right tail (one-tailed) is for larger differences in either direction between the means then you would expect given the standard errors.

• So if one want to decide whether to reject a null hypothesis, they would need to compare the predetermined significant level with the "Sig." value divided by 2 instead of the value itself? – Aqqqq Jan 30 '18 at 17:53
• Not unless the F has a single df in the numerator. Otherwise, what you are trying to do doesn't make sense. That is, with 3 or more means, there isn't a one-tailed test in the sense you are getting at. The F reflects the squared differences of each mean around the grand mean. There is no directionality. – dbwilson Jan 30 '18 at 18:09
• Thank you for your remark. So when the numerator df is 1 for the critical value, I should compare the predetermined significant level with the "Sig." value divided by 2? (I am a bit confused as my lecturer did not talk about the dividing by 2 at all when we talked about the relevant questions. (he is unreachable now.)) – Aqqqq Jan 30 '18 at 19:00
• If the numerator of the F has 1 df, then you can divide by 2 to get a one-tailed p-value. This assumes, that you specified a one-tailed test, that is, that you specified a specific direction for the mean difference and will ignore the p-value if the mean difference is in the opposite direction. The F-value can only be positive and as such does not indicate the direction of the mean difference. You must examine the means to determine this. Essentially, this is using the F-test to do what would typically be done with an independent t-test (which would be preferred). – dbwilson Jan 30 '18 at 22:01

It is one tailed. In the ANOVA context, the statistic is calculated as between groups variance / within groups variance. Formulated as such, you only care if the ratio is positive; values <= 1.0 can never be significant, and so it is pointless to consider. Thus, you by definition have a directional hypothesis, and a one-tailed test.

You could reformulate the question nondirectionally, as in the context of a variance ratio test, but this is less common, and often times one simply puts the larger variance in the numerator anyway, leading again to a one-tailed test.

• So if one want to decide whether to reject a null hypothesis, they only need to compare the predetermined significant level with the "Sig." value itself (without dividing it by 2)? – Aqqqq Jan 31 '18 at 7:51
• Just evaluate whether it is less than your desired alpha level (e.g. 0.05) – HEITZ Jan 31 '18 at 14:01