# Interpreting ANOVA results [duplicate]

I am having difficulty interpreting the results of a colleague. I don't know much about ANOVA other than it is a regression from classifications to floating point numbers.

From reading Gotelli's Ecological Statistics book, I have figured out more or less that the F-ratio is a step in getting the P value. The P value indicates the chance of obtaining this data if the null hypothesis were true.

I am stuck on matching that explanation with the data he provided me, shown below. I assume the Pr(>F) is the P value, but what is the hypothesis tested? Is it clear from just this?

fit <- with(divergence_slopes, aov(slopes~init_diversity + seed_population + init_diversity*seed_population))

summary(fit)
Df   Sum Sq   Mean Sq F value   Pr(>F)
init_diversity                   2 6.29e-14 3.146e-14   8.802 0.000228 ***
seed_population                  1 7.93e-14 7.925e-14  22.171 5.08e-06 ***
init_diversity:seed_population   2 8.34e-14 4.171e-14  11.667 1.76e-05 ***
Residuals                      174 6.22e-13 3.570e-15
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

• As y ~ x1 * x2 expands to y ~ x1 +x2 + x1:x2 the model could at least be simplified... – Rasmus Bååth Apr 2 '13 at 11:59
• "The P value indicates the chance of obtaining this data if the null hypothesis were true." --- no, it doesn't. It's the probability of obtaining a result at least as extreme as this one if the null were true. – Glen_b Apr 2 '13 at 13:19
• Many very basic questions like this have been asked on the site before - this one was asked: Anova from R output interpretation and probably in several other places. Those who ask (and answer) such questions do the site a service by checking on that before posting, rather than allowing us to compile several answers to the same question in various places. – Macro Apr 2 '13 at 19:25

## 2 Answers

The null hypothesis being tested by the full model is that slopes are not related to init_diversity. seed_population, or the interaction. But that doesn't seem to be in what you've shown. You have individual F-values (and associated p-values) for each of three null hypotheses: 1. That there is no relation between slopes and init_diversity, after controlling for seed_population and the interaction

2, 3, similar, but for seed_pop and the interaction.

All three null hypotheses are rejected (p < 0.0001)

what is the hypothesis tested? Is it clear from just this?

There are three hypotheses being tested.

[Each p-value is effectively for that effect "fitted last", though this looks like it's probably a balanced ANOVA in which case the order makes no difference to the sum of squares (and hence to the p-values).]

Let's take some simpler cases first.

Imagine that only seed_population was considered to have any impact on the response; then the model would have been slopes ~ seed_population and the null hypothesis on seed_population (which I assume is a factor) would be that the population means for the two groups were equal.

Now, consider if you had a model like slopes~init_diversity (I realize this is probably not a very interesting model, but bear with me). This has 2 d.f., so it corresponds to three levels. The null hypothesis would be that the population mean of the 'slopes' variable for all three levels would be equal.

Now if you had say a main-effects model: slopes~init_diversity + seed+population
you'd be testing both those hypotheses I mentioned - each would get its own row of the ANOVA table and its own p-value.

Now to deal with your actual model. To those two terms we now add an interaction term between the two factors. There's now the additional null hypothesis that the population means for the interaction terms are both zero (which also corresponds to a particular linear combination of cell population means being zero). This hypothesis gets its own line (and its own p-value) in the table.

--

It's not common to formally test the main effects on finding a significant interaction.

Usually, you'd only concern yourself with the the interaction when it's significant - that alone is enough to tell you both variables are important in explaining variation in $y$ and that they interact.

• that's a good point; more generally. I notice someone downvoted my answer as well. But neither of us can tell why. – Peter Flom Apr 2 '13 at 23:56
• @PeterFlom Likely the same person who downvoted the original question. I thought your answer was good and voted it up before. I don't mind downvotes - that's part of how we arrive at sorting out reputations after all - but I do want to understand what the problem is and try to fix it; downvotes are an opportunity to give a better answer, but it only works if we know what the issue is. – Glen_b Apr 3 '13 at 0:16
• I thought it was obvious that I downvoted. The main reason was stated in my comment to the main question. It does no one any good to have the same introductory questions asked, and answered, again and again on this site. You two are veteran members of this site and know that answering such questions is a waste of time and undermines site policy against question duplication, yet you insist on doing it (for points I guess?). I didn't place a comment here before because it seemed unconstructive - I don't expect you two to stop answering duplicates as a result. Don't expect me to stop downvoting. – Macro Apr 3 '13 at 1:15
• Based purely on their merits as answers, I would downvote both regardless of anything else - glen_b, this answer is a comment, if anything, to the main question. I don't see how this answers the OP's question, in any way. Regarding @Peter's answer - "That there is no relation between slopes and init_diversity, after controlling for seed_population and the interaction" - no, that is not the null hypothesis that is being tested. – Macro Apr 3 '13 at 1:18
• @Macro Thanks for the explanation. My reasons for responding were simply that the question had been asked; if I'd seen a suitably similar one in the first few results returned I'd have pointed to it. Your complaint that my answer is more suitable as a comment ... yes, that's a reasonable criticism of it as an answer. I appreciate you taking the time. – Glen_b Apr 3 '13 at 1:36