Syntax differences between aov and lmer for two-way repeated measures design

Question

I'm working with the following data frame using R. It consists of measurements obtained from 7 subjects with two independent variables (IV1 and IV2) with two levels each (OFF/ON, ALT/ISO, respectively):

>myData
Subject      DV         IV1     IV2
        1   2.567839      OFF      ALT         
        1  58.708027       ON      ALT          
        1  44.504265      OFF      ISO         
        1 109.555701       ON      ISO          
        2  99.043735      OFF      ALT         
        2  75.958737       ON      ALT          
        2 182.727396      OFF      ISO         
        2 364.725795       ON      ISO          
        3  45.788988      OFF      ALT         
        3  52.941263       ON      ALT          
        3  54.719013      OFF      ISO         
        3  41.909909       ON      ISO          
        4 116.145279      OFF      ALT         
        4 162.927971       ON      ALT          
        4  34.162077      OFF      ISO         
        4  74.029748       ON      ISO          
        5 114.412913      OFF      ALT          
        5 121.127983       ON      ALT          
        5 192.379708      OFF      ISO         
        5 229.192453       ON      ISO          
        6 213.421076      OFF      ALT         
        6 526.739206       ON      ALT          
        6 150.596812      OFF      ISO         
        6 217.931951       ON      ISO          
        7 117.931273      OFF      ALT         
        7 102.467813       ON      ALT           
        7  57.823062      OFF      ISO         
        7  85.181033       ON      ISO

(1) Is this a repeated measures (RM) design? Some folks have mentioned that it is not since it isn't a longitudinal study, but I thought that as long as there are measurements from each experimental unit for every single level of a factor, one can say this as a RM design. What is correct? Also, is an RM design synonymous with having a within-subject factor?

(2) I'm interested in both the main and the interaction effects of IV1 and IV2, but due to having measurements from each subject for all level combinations, I think I have to include Subject as a random effect. I have looked at aov and lmer but I'm confused about the difference in syntax: This cheat sheet recommends:

m1 <- aov(DV ~ IV1 * IV2 + Error(Subject / (IV1 * IV2)), myData)

However it's not clear to me whether Error(x / (y * z)) means x is a random effect and y and z are nested in x. Is this interpretation correct? If so, would m1 be inappropriate for my data since my data isn't nested, but fully crossed? And if so, would

m2 <- aov(DV ~ IV1 * IV2 + Error(Subject), myData)

be the correct syntax? I have also been told that in m2 the Error term should be dropped - is this correct?

(3) In a previous question I was told the linear mixed effects model

m3 <- lmer(DV ~ IV1 * IV2 + (1 | Subject), myData)

was appropriate more my data. Just to better understand lmer syntax: if I had n subjects and for each subject measurements were obtained for both levels of IV2 but half of the subjects were OFF and the other half ON, would the model be

m4 <- lmer(DV ~ IV1 * IV2 + (1 | Subject / IV1), data = myData)

? And if there was only one measurement per IV1*IV2 combination, would that mean this is no longer a repeated-measures design and therefore the model is just

m5 <- lmer(DV ~ IV1 * IV2, data = myData)

? In which case lm would probably suffice.

Related: https://stats.stackexchange.com/questions/13784/repeated-measures-anova-with-lme-lmer-in-r-for-two-within-subject-factors. — amoeba, Apr 12 '19 at 19:47
(1) Yes, this is RM design. You are right. (2) `Error(Subject / (IV1 * IV2))` is a correct `aov` syntax for two-way RM-ANOVA. In the two-way RM-ANOVA, nothing is nested: the two factors (IV1 and IV2) are crossed, as you correctly say. Nevertheless, this is the correct syntax. Your `m2` model does *not* correspond to the classical two-way RM-ANOVA. (3) `m3` recommended by @RobertLong is a reasonable model for this design. However, it's not the most flexible model possible. [cont.] — amoeba, Apr 12 '19 at 21:47
[cont.] The most flexible model would have `(IV1+IV2 | Subject)` random term or even `(IV1*IV2 | Subject)` if you have multiple measurements per IV1/IV2 combination. This is a rather complicated model with 4*4 random covariance matrix. So there are several ways to simplify/restrict it. `m3` corresponds to the maximal simplification. One intermediate specification is `(1|subject)+(1|subject:IV1)+(1|subject:IV2)`. This corresponds to RM-ANOVA. This has the same specification *as if* IV1 and IV2 were random and nested within `subject`, but this is just a coincidence. Here they are fixed. [cont.] — amoeba, Apr 12 '19 at 21:51
[cont.] Your `m4` and `m5` are completely off and show that you don't understand the logic of mixed models very well. I'd recommend to read some more systematic introduction to mixed models if you plan to use them. — amoeba, Apr 12 '19 at 21:52
By the way, I have been confused by the same thing and asked this Q some time ago: https://stats.stackexchange.com/questions/232109. Unfortunately it does not contain a fully satisfactory answer. I will try to find time to answer it myself, now that I have a better understanding of these issues... — amoeba, Apr 12 '19 at 21:57
First of all, thanks for clarifications and for pointing me to those threads, reLLY helpful. I do realise I need to focus on understanding the theory first, the references here seem good: https://stats.idre.ucla.edu/other/mult-pkg/introduction-to-linear-mixed-models/. Is GLM theory a prerequisite to LMM? — user3050269, Apr 12 '19 at 22:04
@amoeba Your comments are very useful - why not turn them into an answer ? I would certainly upvote it ! — Robert Long, Apr 13 '19 at 13:23
@Robert I was hoping you would write a better and more detailed answer, maybe showing some model summaries using the OP's data etc... Feel free to use my comments if you decide to compose an answer... — amoeba, Apr 13 '19 at 13:32
@amoeba ahh - OK - in that case I will try to have a go at it. This brings back memories of [this](https://stats.stackexchange.com/questions/232109/does-it-make-sense-for-a-fixed-effect-to-be-nested-within-a-random-one-or-how-t) question of yours, from some time ago :) — Robert Long, Apr 13 '19 at 13:36
@robert Yes, I linked it above already :) We should get back to it once! Either you extend/edit your answer, or I should write my own one... — amoeba, Apr 13 '19 at 13:38
@Robert That's a good reference by the way http://dwoll.de/rexrepos/posts/anovaMixed.html — amoeba, Apr 13 '19 at 13:50

Syntax differences between aov and lmer for two-way repeated measures design

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