# state reason for using one tail in the analysis of variance test ?

#### Cynderella

##### New Member
I think in this way that
since in anova we partition the total variability into various independent components ,
and since the Mean square of this components follow chi-square distribution,
and since the ratio of two chi-square distribution follows F- distribution,
and since F- distribution is a one tailed test,
we use one tail in the analysis of variance test .

Does it make sense ?

#### Miner

##### TS Contributor
I think you over analyzed it. Look at it from your null and alternate hypotheses. Your null hypothesis is that the variance due to the factor is <= to the variance due to the random error. Your alternate hypothesis is the variance due to the factor is > that due to random error. This means a one tailed test.

A two-tailed test means that you care only that they are different. Why would you care whether the variance due to the factor was < than that due to random error?

#### Dason

##### Ambassador to the humans
and since F- distribution is a one tailed test,
The F-distribution is a distribution - not a test. You can most certainly use a two-tailed test for test statistics that follow the F distribution - but most of the time this isn't something that makes sense to do.

#### trinker

##### ggplot2orBust
Can you even have a directional hypothesis with an ANOVA that uses the F distribution (at least with more than 2 groups)?

#### Cynderella

##### New Member
Look at it from your null and alternate hypotheses. Your null hypothesis is that the variance due to the factor is <= to the variance due to the random error. Your alternate hypothesis is the variance due to the factor is > that due to random error.
My null hypothesis is $$\mu_1=\mu_2=...=\mu_a$$ and alternative hypothesis is $$\mu_i\neq\mu_j$$ for some $$i\neq j$$. And I did ANOVA test for testing the hypothesis.

Isn't the hypothesis two-tailed test ?

#### rogojel

##### TS Contributor
My null hypothesis is $$\mu_1=\mu_2=...=\mu_a$$ and alternative hypothesis is $$\mu_i\neq\mu_j$$ for some $$i\neq j$$. And I did ANOVA test for testing the hypothesis.

Isn't the hypothesis two-tailed test ?
As far as I know with ANOVA you are actually testing the consequence of this hypothesis on the variance structure. If the assumption that at leastbsome group means are different is a good explanation of the total variance in the data then the original null is rejected.

So, we really test whether the model with different means has a much smaller residual variance ( contribution to the sum of squares really) then the model with the same means. The test for this is one sided.

regards
rogojel

#### Miner

##### TS Contributor
My null hypothesis is $$\mu_1=\mu_2=...=\mu_a$$ and alternative hypothesis is $$\mu_i\neq\mu_j$$ for some $$i\neq j$$. And I did ANOVA test for testing the hypothesis.

Isn't the hypothesis two-tailed test ?
Technically, you are correct about the hypotheses. However, you must look at the way that ANOVA tests these hypotheses. The F-test is actually a ratio of the s^2 due to the effect of a factor divided by the s^2 due to random variation. Therefore, in effect the null hypothesis becomes s^2 effect <= s^2 random, and the alternate, in effect, becomes s^2 effect > s^2 random, which is a 1-tailed test using the F distribution.