# How to interpret F- and p-value in ANOVA?

I am new to statistics and I currently deal with ANOVA. I carry out an ANOVA test in R using

aov(dependendVar ~ IndependendVar)


I get – among others – an F-value and a p-value.

My null hypothesis ($H_0$) is that all group means are equal.

There is a lot of information available on how F is calculated, but I don't know how to read an F-statistic and how F and p are connected.

So, my questions are:

1. How do I determine the critical F-value for rejecting $H_0$?
2. Does each F have a corresponding p-value, so they both mean basically the same? (e.g., if $p<0.05$, then $H_0$ is rejected)
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Have you tried the commands summary(aov(dependendVar ~ IndependendVar))) summary(lm(dependendVar ~ IndependendVar)) Do you mean all the group means are equal to each other and equal to 0 or just to each other? –  RyanB Jun 27 '11 at 15:46
yes, I did try the summary(aov... thanks for the lm., did not know about this :-) I don't get what you mean by equal to 0. If that's short for my 0-Hypothesis than the Hypothesis would need a value, and I did not test on specific one, so in this case: just to each other! –  JanD Jun 27 '11 at 19:57

1. You find the critical F value from an F distribution (here's a table). See an example. You have to be careful about one-way versus two-way, degrees of freedom of numerator and denominator.

2. Yes.

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It is not meaningful to talk about one- or two-way comparisons in an omnibus test such as the F-test. –  Marcus Morrisey Apr 16 at 20:19

The F statistic is a ratio of 2 different measure of variance for the data. If the null hypothesis is true then these are both estimates of the same thing and the ratio will be around 1.

The numerator is computed by measuring the variance of the means and if the true means of the groups are identical then this is a function of the overall variance of the data. But if the null hypothesis is false and the means are not all equal, then this measure of variance will be larger.

The denominator is an average of the sample variances for each group, which is an estimate of the overall population variance (assuming all groups have equal variances).

So when the null of all means equal is true then the 2 measures (with some extra terms for degrees of freedom) will be similar and the ratio will be close to 1. If the null is false, then the numerator will be large relative to the denominator and the ratio will be greater than 1. Looking up this ratio on the F-table (or computing it with a function like pf in R) will give the p-value.

If you would rather use a rejection region than a p-value, then you can use the F table or the qf function in R (or other software). The F distribution has 2 types of degrees of freedom. The numerator degrees of freedom are based on the number of groups that you are comparing (for 1-way it is the number of groups minus 1) and the denominator degrees of freedom are based on the number of observations within the groups (for 1-way it is the number of observations minus the number of groups). For more complicated models the degrees of freedom get more complicated, but follow similar ideas.

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Thanks for the explanation! I assume that if I can look up the F value on a table to see the p-value, than the the p and F are just two ways to express the likelyhood that a result like the one analysed can occur if the H0 is right? –  JanD Jun 27 '11 at 19:49
In all parametric statistics there is a direct functional link between the test statistic (F in this case) and the p-value. These have been put into table for convenience, but can also be computed directly. You can either use alpha to find the cut-off for a critical region to compare the test statistic to (which I think is more intuitive) or use the computed test statistic to find the p-value to compare to alpha. In either case we start with an alpha level and a test statistic formula that follows a given distribution when the null is true. –  Greg Snow Jun 28 '11 at 18:00