Interaction as main effect in lme4 vs. nlme

Question

Let's say I am trying to see if there are speed differences among different types of cars (e.g., jeep, SUV, truck, sports car). I also want to see whether the car store (A, B, C, D, E) has an effect on this. I don't have a specific hypothesis about the car stores, but I assume that certain stores tend to sell certain types of cars (e.g., SUVs at Store A and trucks at Store B).

So, I tried to fit a linear mixed effect model with lme4 and nlme. I have a question about the difference between these two models:

nlme:

m1 <- lme(speed ~ car_type + car_store + car_type:car_store, 
          data = data1, random = ~ 1 | car_owner, method = "ML")

lme4 equivalent is:

m2 <- lmer(speed ~ car_type * car_store + (1 | car_owner), 
           data = data1, REML = FALSE)

Now, my question is, can I have interaction as the main effect in nlme like this:

m3 <- lme(speed ~ car_type + car_type:car_store, data = data1, 
          random = ~ 1 | car_owner, method = "ML")

So far, I can't find the equivalent of m3 in lme4. Can anyone help me understand why?

Also, ideally, I think I can achieve my purpose with m3. But is there anything else that I should consider? like perhaps it's better to use lme4 and car_store as main/fixed effect as well?

Answer 1

There should be no difference between lme4 and nlme regarding the specification of the fixed effects. As mentioned by Dimitris Rizopoulos in a comment to the question, there is nothing specific to mixed models here.

I agree that your models m1 and m2 indeed have the same fixed effects.

Now, my question is, can I have interaction as the main effect in nlme like this:

m3 <- lme(speed ~ car_type + car_type:car_store, data = data1, random = ~ 1 | car_owner, method = "ML")

Yes, you do that, but it is not (usually) a good idea. Models that include an interaction, but exclude one or both main effects, have been discussed here many times, for example:

Including the interaction but not the main effects in a model

Unless you have a very good reason, I suggest your model's fixed effects should be:

car_type + car_store + car_type:car_store

or just:

car_type * car_store

which is shorthand for the same thing.