According to the principle of marginality, when you have interactions in the model, you should not try to interpret the individual main effects of the predictors that constitute the interactions, because it will be misleading. Just a report of your results for these predictors is more than enough, but the primary focus of your results should be the significant interactions and their interpretation.

Moreover in both lm3 and lm4 you specify a three-way interaction that includes all lower order interactions and main effects:

Group*divedeph*divedur = Group:divedeph:divedur + Group:divedeph + Group:divedur + divedeph:divedur + Group + divedeph + divedur

Thus, specifying Group again when calling lmer is redundant.

You may think that you could specify:

Group:divedeph:divedur + Group:divedeph + Group:divedur + divedeph:divedur + divedeph + divedur

and just omit Group, so that you can compare the two models with anova. This is not a good idea, as a) when you specify an interaction, you should specify all lower order interactions and predictors as well (another implication of the principle of marginality) and b) the likelihood ratio is not recommended to test the significance of fixed effects in LMMs (Pinheiro & Bates: Mixed Effects Models in S and S-Plus, 2000).

Nevertheless, if your higher order interaction (Group:divedeph:divedur) is significant, just keep it in the model along with all its constituents. Explain the interaction, but not the constituents. You may plan your strategy on model building along the lines described in Zuur et al: Mixed Effects Models and Extensions in Ecology with R, 2009. You might even decide to include a random slope in your model as well.

Log-transforming your data has implications on the interpretation of your results. Specifically, although the significance (p-values) have the same interpretation as with untransformed data, the coefficients (and the underlying equation) are no longer linear, where they work in an additive way:

Eijk = (β0 +b0i) + (β1 + b1i) x1j + β2 x2j +…+ εijk,

but instead they work in a multiplicative way:

Eijk = eβ0 + b0i (eβ1 + b1i)x1j (eβ2)x2j … eεijk.

So, you should treat them with caution (especially when you compute confidence intervals).

It would greatly help your interpretation, if you used the emmeans package (or something similar), which can compute estimated marginal means of the outcome for each level of your variables. It can even back-transform your outcome, if need be.

Thank you very much Abraham This has been very helpful.

Indeed I forgot to mention that I'm not doing the comparison between models using likelihood ratio for the fixed effects. So I think is enough then to work and focus the explanation in the significance of the interaction instead of only the fixed variable GROUP. This lead me to test the following:

lm3= lmer (log (BoutDur)~Group*divedeph*divedur+ BCI + (1IFemID), data=DIVING, REML=FALSE)

lm4= lmer(log (BoutDur)~ 1 + BCI+ (1IFemID), data=DIVING, REML=FALSE)

ANOVA(lm3,lm4)

A&und out also that BCI is also interacting with group. lactating animals will be more fat than non lactating animals and therefore the Body condition index (Mass&

We recently also found out that BCI is not an additional fixed component but also interact with GROUP (lactating and not lactating animals) since the Body condition index (BCI, Mass/total length) is affected by whether animals are lactating or not. should I add this interaction along the long chain of interactions or should I presented as another interaction? ( GROUP*divedeph*divedur*BCI or GROUP*BCI). my confusion arise because biologically, divedeph and divedur affect BOUTdur in a different way than the interaction between BCI*group but when it comes to creating the model, maybe is the same?

Again, this was a great answer, thank you very much for your help!

Cheers

R

Actually, calling the anova:

ANOVA(lm3, lm4)

does exactly a comparison between the model with and without the interaction (and all the effects below) with an LR test. Apart from the advice from Pinheiro and Bates to avoid comparisons between fixed structures with the LR test, you also compare all seven components of the Group*divedeph*divedur term with null (BCI and the random structure remain the same). You can estimate the fixed effects with three tools: F-statistic, t-test, and LR-test. The former two are better estimated with ML, the latter can only be used with ML (although REML is better for random effects).

Zuur suggests that in a linear model you should first find the maximal fixed structure (stuff it with as many effects and interactions as you can) with a linear (not mixed effects) model. Then you should step up adding random effects in LMMs until you find the optimal random structure using LR-tests. Once you are there, you optimize your fixed structure removing one fixed effect at a time (interaction or main effect), obviously starting from the highest ranking interaction and going downwards, until no more insignificant higher order terms remain. Some researchers suggest that you should insist on a predefined fixed structure from the beginning through the end, according to information from your research field, in order to avoid invalid and unsustainable conclusions. Given that automated ill-defined analyses of past decades have declined in favor, this is quite justifiable, although the Zuur strategy remains very useful in the LMM labyrinth.

If you consider to include BCI to the interaction, well, is not contraindicated and R can do it very efficiently, given the adequate sample size... However, consider how could you interpret such a four-way interaction. Even the three-way interaction may be quite cumbersome to understand and interpret.

Anyway, you are free to try anything to see what conclusions you can get out of it, given the background information from your field. It will only cost you some time. In the end, just keep in mind that you have to produce a model that better reflects your measurements and it is meaningful.