Interpreting ANOVA results

Question

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 

Answer 1

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)

Answer 2

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.