Just to make sure I understand your question:

your outcome variable is position, it may take 5*3*6=90 distinct values (categories)?

This sounds quite a lot, particularly as you have only 21 measurements per session.

My first feeling is that the number of outcome categories should be reduced. Preferably in a way that makes sense from the biological point of view, alternatively using some dimensionality reduction method (PCA, MDS, ...) that substitutes the 21 locations with a few numbers that describe location patterns.

Dear Gabor,

Thanks, your idea sounds great. But actually, analyszing 3D position as a whole was out of my aim. sorry I was not clear. I want to test habitat and predator effect on position, dimension by dimension separately. I have hypotheses related especially to Z (i.e. the deepth) and also to X (because I have a mixed habitat treatement for testing habitat choice). I expect no pattern along Z, a signal would be only artefact. Hence, I will have from 3 to 6 catogories in my response variable. Is it still too large (e.g. 6 categories for 21 observations, 3.5 obs/cat as null model).

About log-linear model, I was thinking: Because I have for each combination of Pred and Hab the same number of observations, the simple term Pred and Hab as well the Hab:Pred are useless no?

I planned to fit these models using glm, family quasipoisson, and to test difference between model by using AIC.

Full: Freq~Pos + Pos:Hab + Pos:Pred + Pos:Hab:Pred

m0: Freq~ Pos:Hab + Pos:Pred + Pos:Hab:Pred

m1: Freq~Pos + Pos:Pred + Pos:Hab:Pred

m2: Freq~Pos + Pos:Hab + Pos:Hab:Pred

m3: Freq~Pos + Pos:Hab + Pos:Pred

Full vs m0, testing Pos ; Is there an overhole trend in the frequence of occupency of each position? (e.g. I expect that in average prey fish occupy more the bottom than the water column)

Full vs m1, testing Pos:Hab ; Does Habitat type affect the position?

Full vs m2, testing Pos:Pred ; Does Predator affect the position?

Full vs m3, testing Pos:Hab ; Does the effect of predator on the position depend on the habitat?

What do you think of this approach? Is it well parametred as regard to my question?

Tank in advance,

chears, Pierre

The log-linear model looks as a reasonable approximation to me.

Do fish behave in a symmetric way? If yes and if the habitat is also symmetric, you could reduce the variable

- X to X2 with 3 categories (1 or 5, 2 or 4, 3)

- Y to Y2 with 2 categories (1 or 3, 2).

You could compare models with X and X2, Y and Y2 to decide.

You mention that you have 4 individuals, so these must be the prey fish. Were the predators always the same 2 individuals (the P and the A)?

[I have to re-think the rest before I can answer.]

Dear Gabor,

Do you mean for X2, I pool on one hand 1 & 5 and on the other 2 & 4. The problem is that the remaining third will be twice smaller, and therefore expected frequency by chance would be the same. I would keep my originnal variables for that and also because an asymmetri could indicate an artefact (tankiness, e.g. a spot-light in the room attracted fish).

The two levels of the factor Predator are Absent and Present. For each combination of the levels of Hab and Pred, i.e. 8 combinations, I had 4 new individuals (4 sessions) of prey fish. This lead to a total of 8*4 differents individuals.

It was the same for predator fish, of course only for the treatment Predator Present.

Chears,

Pierre

Dear Pierre,

With your full vs m0 comparison you are investigating whether prey fish give preference to any location by an axis. So you are not excluding the possibility that they like sides of the tank. If so, why couldn't they especially prefer edges or corners? Or the near the spotlight? These preferences can only be detected by treating the 3 dimensions together. I don't think analysing the 3 dimensions separately is wrong, but you may bypass something characteristic.

In your above models you are neglecting the subsession effect and the effect of time within subsessions, let's assume that this is OK, this way a unit of investigation is a session (with 3*21 location data points). I don't know fish, but I could imagine that there are differences among the individuals, I would explore if the location preferences and the responses to the predator are similar enough across prey fish to neglect or not that similar. Let's assume this is negligible, too. Then the units of investigation are independent, so using an ANOVA type linear predictor is correct.

To be continued.

Gabor

Pierre,

to my opinion, your experiment can be analyzed in different ways.

Firstly, you should decide how many Y variables should be considered. I think that you can use one Y variable, viz. XYZ-position. Alternatively, as you probably wish, you can use three Y variables, viz. X-position, Y-position and Z-position, each analyzed separately.

Secondly, Position Recordings (21 times every 30 sec) should be a repeated measure, given that Fishes (4) are the subjects (random effect).

One possible parameterization could be:

Y=Position (categorical)

X1=Pred (2 levels; categorical; fixed, crossed effect)

X2=Hab (4 levels; categorical; fixed, crossed effect)

Session=Ses (4 levels; categorical; random effect)

Subsession=SubSes (3 levels; categorical; random effect nested within Ses)

Fish=Fish (4 level; subjects; random effect)

Position Recordings=PosRec (21 levels; repeated measure)

In this framework:

- multinomial models would be adequate, however I know that they require some assumptions, which I ignore, thus you should eventually document on this.

- Log-linear models would be adequate, as you mentioned, in the case the Y variable is a count; this means that your data should include a number (count) per each XYZ-position

- ANOVA would be inadequate and incorrect.

- Permutation test for ANOVA would be inadequate, because suitable for continuous data violating the assumptions for ANOVA (I’m not sure about this, because it’s not a familiar test for me)

- Non-parametric analysis of variance-type (AVT) statistic would be adequate, correct, and easy-to-apply. I recommend this method. To learn this method see ‘Shah, D. A., and Madden, L. V. 2004. Nonparametric analysis of ordinal data in designed factorial experiments. Phytopathology 94:33-43.’. Read the article to learn the theory, while I can advise on the SAS programming, if you want to gain familiarity with it.

Thanks to both of you.

I had a quick look to Shah and Madden 2004. That sounds great. I have many other things on fire at this time, so I will soon dig in this way and look for the R package and programming. I will let you know.

Cheers,

Pierre