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I conducted a full 3x2x2 (W1,W2,W3) within design with every participant experiencing the same 12 conditions. Within each condition, there were several measurements of my outcome variable(outcome). Additionally, I want to include a covariate (covariate) in my model that was measured as on the same level as my outcome, i.e. several times within each condition.

I set up the model using nlme(following Field, Miles, & Field, 2012; chapter 13):

Model_lme <-lme(outcome ~ W1 + W2 + W3 + W1:W2 + W1:W3 + W2:W3 + W1:W2:W3 + covariate , 
random = ~covariate|Subject/W1/W2/W3, data = data,control = list(opt = "optim"), method = "ML")

And using lme4

Model_lme4 <- lmer(outcome ~ W1*W2*W3 + covariate +  (1 + covariate|Subject/W1:W2:W3), 
data=data, REML=FALSE)

I would be very grateful if someone could answer the following questions:

1) Are these model really equivalent? I'm not sure about the Lme4 approach right now.

2) When comparing the models without the covariate, I find a significant effect of W2. After introducing the covariate, this main effect is no longer present. Is it then fine to state that the main effect of W2 was explained by the significant covariate?

3) Finally, I am also not sure how to report my model. As I understand, the models describe my data as follows: W1, W2, and W3 are nested within participants. These factors have random intercepts (meaning that there is an intercept for each condition within each participant), but no random slopes. The covariate is a factor with random intercept and slope.
Is that the way how to describe in a paper? It rather seems like papers only state whether something was handled a fixed or random effect (e.g. Jainta, Blythe, Nikolova, Jones,& Liversedge, 2015, p. 121)

I know these are many questions, but currently I am still really puzzled about mixed models, even though I read and try to understand every literature I can get regarding this topic.

Thanks in advance for your answers!

Genscher
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  • Your W1,W2,W3 are *not* "nested within participants. To say this will be incorrect and misleading. You have a repeated measures design, but subjects are crossed with W1/2/3. Nothing is nested here. The so called "maximal" model would have `(W1*W2*W3 + covariate | Subject)` random term. You can try but this is a lot of random effects to fit, and you might not have enough data to fit this reliably. So you can simplify, e.g. `(W1 + W2 + W3 + covariate | Subject)`, or with `||`. What you have is an even further simplification `(1 + covariate | Subject) + (1 + covariate | Subject:W1:W2:W3)`. – amoeba Mar 14 '19 at 10:26
  • This last simplification assumes so called "compound symmetry". See https://stats.stackexchange.com/questions/304374/ https://stats.stackexchange.com/questions/302951 and further linked threads. – amoeba Mar 14 '19 at 10:30
  • Thanks @amoeba for the very useful clarification and linked threads. As i feel not very confident with the conduction and interpretation of mixed models, I want to build up a model as close to the approach of a repeated measures ANOVA, but with the integrated covariate with random intercept and slope. Would my reduced model with the assumption of "compound symmetry" meet this criteria? – Genscher Mar 14 '19 at 12:42
  • Yes. RM-ANOVA corresponds to compound symmetry. – amoeba Mar 14 '19 at 12:51
  • Well, actually not quite - I think RM-ANOVA with three factors will have more random terms, see here: https://stats.stackexchange.com/questions/117660. Smth like `(1|subject) + (1|a:subject) + (1|b:subject) + (1|c:subject) + (1|a:b:subject) + (1|a:c:subject) + (1|b:c:subject) + (1|a:b:c:subject)`... It's tricky to reproduce this kind of anova with lmer exactly, it might be easier to adopt the mixed model style of thinking ant put w1/w2/w3 to the left of the vertical bar. – amoeba Mar 14 '19 at 12:54
  • Thank you again very much for your help. Now I think there is no other way for me than leaving the restricted ANOVA based thinking behind and form a real understanding of mixed models. – Genscher Mar 14 '19 at 13:33

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