I am confused by the results of two-way ANOVA, and I'd greatly appreciate any help, please!
For instance, I am collecting light intensity data for a 2-factor design: Season (summer, winter) and Site (site 1 and site 2).
The two-way ANOVA output indicates a significance in both Season and Site, but no significance in Season * Site (p = 0.991). However, I found that there is indeed a significant difference in light intensities of Summer Site 1 and Winter Site 1, when I ran a one-way ANOVA with Season and Site combined.
Why is this so? Should all two-way ANOVAs be followed by a one-way ANOVA of combinations of the factors?
The answer to your last question is no. Your 2 x 2 ANOVA should answer all relevant questions because it contains the main effects of both factors and the potential interaction. No follow-up one-way ANOVAs are needed or useful.
The fact that the interaction effect Season*Site is non-significant means that the summer/winter differences in light intensity do not affect Site 1 and Site 2 differently (i.e., the summer/winter differences in intensity are similar for Site 1 and Site 2).
I would plot the main and interaction effects using a line graph of the 4 cell means (e.g., summer/winter on the X axis and Sites as separate lines). This should clarify things. The lines should be about parallel.
Adding to Christian Geiser's good explanation, it might help you to think about the F-test for the interaction as a test for a difference in differences. I.e., the null hypothesis for the interaction could be stated as either of the following:
- The seasonal difference (if any) does not depend on site
- The site difference (if any) does not depend on season
Which one you choose depends on which of the two variables is your focal variable, and which you think of as a possible effect modifier.
In addition to the perfect answers given by Christian and Bruce I'd like to stress that the ANOVA tests for differences (and that the interaction term is a term adressing the differences of differences, as Bruce explained). Hence, you are analyzing differences in light intensity (e.g. between sites). I am not enough an expert to judge if this reasonable, but my gut feeling tells me that it might be more resonable to analyze ratios rather than differences. Imagine in summer you have a high intensity, and an a large difference between the sites. For instance, say you get 10 units for site 1 and 6 for site 2. So the difference is 4. In Winter, the intensity is much smaller, say only 2 units for site 1. It now is impossible to get a difference of 4, as this would mean a negative intensity for site 2. If you get an intensity of only 1.2 units on site 2 in winter, the difference would only be 0.8, and the difference of difference (summer-winter) would be 4 - 0.8 = 3.2, which is some interaction that may well be significant. But if you look at the ratios, in summer ratio (size 2/1) 6/10 = 0.6, and in winter it is 1.2/2, what is also 0.6. So despite the very same values, now there is no interaction (in terms of relative changes).
Take-home message: when interpreting interactions, it makes a relevant difference if you look at absolute or relative changes!
PS: to analyze relative changes in light intensity you can simply analyse the absolute changes in log(light intensity), because log(A/B) = log(A)-log(B) (in words: the log of a ratio equals the difference of the logs; so by analyzing the difference of the logs you actually analyze the log of the ratio, and since the log is a monotone transformation, larger values of log(A/B) mean larger values of A/B as well).
Great technical explanations above. I always find simple real life scenarios with graphs helpful in understanding core statistical concepts so I will burrow one from a favorite book that I read (Hypothesis Testing by Jim Frost). Imagine you did a survey of people enjoying hot dogs and ice cream, obviously you will get an average response that is very high for enjoyment. Then you decided to add condiments such as chocolate sauce and mustard on hot dogs and ice cream. Hot dog with chocolate sauce might sound tasty to some, but we find out in our street testing survey it actually wasn't because people enjoyed hot dogs with mustard more. Same with ice cream, people tend to enjoy ice cream with chocolate sauce more than they enjoy mustard. I am posting an interaction graph that Christian was alluding to (pg. 250 of 394 in Hypothesis Testing by Jim Frost). In the scenario I mention you should see a shape like X in the graph when you plot the average enjoyment. So in this scenario we shouldn't be surprised to get in the 2-Way ANOVA output table if food * condiment gets a high f-value with a low p-value because hot dogs with chocolate sauce turned out to be a revolting combination of condiments as well as ice cream with mustard! But as you can see in the picture, the reverse is true, mustard is great with hot dogs but not with ice cream and that chocolate sauce is great with ice creams but not hot dogs. I'm sure Jochen enjoys his hot dogs with chocolate sauce.
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