# Why do we use a one-tailed test [F-test] in analysis of variance (ANOVA)?

Can you give the reason for using a one tailed test in the analysis of variance test?

Why do we use a one-tail test - the F-test - in ANOVA?

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Some questions to guide your thinking... What does a very negative t statistic mean? Is a negative F statistic possible? What does a very low F statistic mean? What does a high F statistic mean? –  rpierce Aug 16 '13 at 6:41
Why are you under the impression that a one-tailed test has to be an F-Test? To answer your question: The F-Test allows to test a hypothesis with more than one linear combination of parameters. –  IMA Aug 16 '13 at 7:17
Do you want to know why one would use a one-tailed instead of a two-tailed test? –  Jens Kouros Aug 16 '13 at 7:53
@tree what constitutes a credible or official source for your purposes? –  Glen_b Oct 27 '13 at 23:24
@tree Incidentally, an explanation of why is already given in the last paragraph of this answer. I would expand on that answer in detail but I doubt that meets your criteria of 'credible or official', whatever that means. –  Glen_b Oct 27 '13 at 23:26

F tests are most commonly used for two purposes:

1. in ANOVA, for testing equality of means (and various similar analyses); and

2. in testing equality of variances

Let's consider each in turn:

1) F tests in ANOVA (and similarly, the usual kinds of chi-square tests for count data) are constructed so that the more the data are consistent with the alternative hypothesis, the larger the test statistic tends to be, while arrangements of sample data that looks most consistent with the null corresponds to the smallest values of the test statistic.

Consider three samples (of size 10, with equal sample variance), and arrange them to have equal sample means, and then move their means around in different patterns. As the variation in the sample means increases from zero, the F statistic becomes larger:

The black lines ($|$) are the data values. The heavy red lines are the group means.

If the null hypothesis (equality of population means) were true, you'd expect some variation in sample means, and would typically expect to see F ratios roughly around 1. Smaller F statistics result from samples that are closer together than you'd typically expect ... so you aren't going to conclude the population means differ.

That is, for ANOVA, you'll reject the hypothesis of equality of means when you get unusually large F-values and you won't reject the hypothesis of equality of means when you get unusually small values (it may indicate something, but not that the population means differ).

Here's an illustration that might help you see that we only want to reject when F is in its upper tail:

2) F tests for equality of variance*. Here, the ratio of two sample variance estimates will be large if the numerator sample variance is much larger than the variance in the denominator, and the ratio will be small if the denominator sample variance is much larger than variance in the numerator.

That is, for testing whether the ratio of population variances differs from 1, you'll want to reject the null for both large and small values of F.

* (Leaving aside the issue of the high sensitivity to the distributional assumption of this test (there are better alternatives) and also the issue that if you're interested in suitability of ANOVA equal-variance assumptions, your best strategy probably isn't a formal test.)

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Good answer. Curious what test you'd consider a better alternative for testing equal variances (in general). –  TLJ Oct 29 '13 at 4:08
@TaylerJones Levene's test is somewhat more robust. Browne-Forsythe is more robust (but loses a little power near the normal). Fligner-Killeen more so again. In several decades, I've used Levene or Browne-Forsythe no more than twice each. (If it came up again, likely something like Browne-Forsythe would suit me fine but I don't generally have situations where it makes any sense to test several groups variances for equality.) –  Glen_b Oct 29 '13 at 7:34
I appology. I still haven't understood why do we use one-tail in ANOVA . More specifically, from your discussion i understood that under null hypothesis i would not have any treatment effect and consequently $F=\frac{MS_{TREATMENT}}{MS_{ERROR}}$ will be close to $1$ , while if alternative hypothesis is true the $F$-Ratio will be larger. But how does it imply "that is the reason for using one-tail test in ANOVA?" –  tree Oct 29 '13 at 12:18
@tree it sounds like you don't understand something about hypothesis testing more generally, but it's hard to be certain exactly where. You say you understand that if you get a large F you want to reject and if you get a small F you don't want to reject. The large values of F are those values in the upper tail while small values of F are those values in the lower tail. You only want to reject when the values are large ... i.e. in the upper tail, but not the lower tail. How can you not see that's one tailed? I'll include another plot that might help. –  Glen_b Oct 29 '13 at 22:03
Now it is clear. This is really an excellent explanation. Also your last comment. –  tree Oct 30 '13 at 0:35

It must be understood that the objective of ANOVA is to check whether there is inequality of means...which implies that we are concerned with large variations between samples (& thus means as variations are computed from the means) as compared with variations within samples (again computed from individual sample mean). When the variations between samples is small (resulting in F value being on the left side) it does not matter as this difference is insignificant. The variations between samples matters if it is significantly higher than the within variations & in such case the F value would be greater than 1, & therefore in the right tail.

The only question remains is why put the entire level of significance in the right tail & the answer is again similar. THe rejection happens only when the F ratio is on the right side & never when the F ratio is on the left side. The level of significance is the measure of error due to statistical limitations. As the rejection happens only on the right the entire level of significance (error risk of misconclusion) is kept in the right. `

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