Mixed effect nested anova

Question

I try to find a good model to my data but I get stuck all the time as I'm not so good at statistics.

I have data on approximately 100 fishes and each fish has been incubated in one of four possible tanks. Two of the tanks contained cold (control) water temperature and two contained warm water. The fishes come from four different crosses (AA, AR, RR, RA). I want to investigate if incubation temperature influence different parameters, e.g. total swim time.

I have thought about a two way anova with crosses and incubation temperature as independent fixed factors, tank as random factor or both tank and crosses as random factors. But it seems to me that tank is nested within incubation temperature and even fish individual is nested within tank. So should I use a mixed effect nested anova instead?

Answer 1

In reading your question, one possible set up that comes to mind is that you can treat tank as a random grouping factor. Then, for each tank, you have multiple values of your response variable - total swim time - coming from the multiple fish in that tank:

Level 2: Tank 1 $......$ Tank 2 $.....$ Tank 3 $...$ Tank 4

Level 1: 🐟🐟🐟 $...$ 🐟🐟🐟🐟 $...$ 🐟🐟🐟 $...$ 🐟🐟

In the above diagram, your response obsevation level is Level 1 and your random grouping factor level is Level 2.

Given the above, temperature is a Level 2 predictor variable (i.e., a tank-level predictor variable). Indeed, its values change from tank to tank, but not from fish-to-fish within the same tank.

It is not clear from the information you provided if you included fish from all 4 crosses in any particular tank. If you did, then cross can be considered a Level-1 predictor (i.e., a fish-level predictor). If the 4 crosses are all the crosses you care about in your study, it makes sense to consider cross as a predictor in the fixed effects portion of your model.

Of course, if cross has more than the 4 levels you were able to include in your study, you would be able to consider cross as a random grouping factor. But if these 4 levels are all you care about, then see the previous paragraph.

The tricky thing here is that you are interested in the effect of a Level 2 predictor: temperature. Assuming your model only includes a single grouping factor, I wonder if your best bet would be to actually consider a GEE-type model rather than a mixed effects model to conduct inference on this Level 2 predictor.