# Timeline for Repeated measures analysis: why nest experimental factors within subject factor?

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Jul 16 '20 at 19:18 history edited
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Jul 15 '20 at 12:24 comment @amoeba no worries. I didn't realise this was an old question until I answered it and got a revival badge :D
Jul 15 '20 at 12:19 comment Thanks for pinging me @RobertLong! I have not been using SE much lately, so am not following these threads as closely as I was a few years ago.
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Jul 15 '20 at 8:32 history edited
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Jul 14 '20 at 18:22 history edited
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Jul 14 '20 at 18:17 answer timeline score: 11
Jul 14 '20 at 11:18 comment @amoeba Aaaaaahhhh this problem again :) I really want to understand this and by coincidence I started looking at it again a few days ago. I was led down the path of "split plot" designs and that's where I am currently at !
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Jul 10 '17 at 15:22 comment Finally, the only possible reason I found why some sources recommend to calculate RM ANOVA with (1|id/A/B/C) is that they want to use nlme, not lme4, and that it is very difficult in nlme to set up a model like I described it in the first paragraph in this comment. (Of course, I'm sure that's nothing new for you, but I wanted to have written it down, maybe get some feedback if the general ideas are right!)
Jul 10 '17 at 15:21 comment However, as far as I understood it, it is actually not that sensible to model it like described above, with y~ABC + (1|id/A/B/C). The last term translates to (1|id) + (1|id:A) + (1|id:A:B) + (1|id:A:B:C), which is quite different to the other one. While this can give similar results to the RM ANOVA, the results also depend on the order in which you “nest” the factors – so (1|A/B/C) will give you a different result than (1|C/B/A) – which they should not, when you want to replicate RM ANOVA.
Jul 10 '17 at 15:21 comment As far as I understood it, this is to model the factor-specific error terms, in which real error terms and the interactions between subject and within-subject factors are confounded. (And, I believe, the random intercept for the interaction of subject with the highest order interaction cannot be modelled because it is not discernible from the residual).
Jul 10 '17 at 15:20 comment Thanks! I did some research myself in the meantime, and found that with lme4, you can nearly perfectly replicate the RM ANOVA (with three factors A,B,C, like in my example) with the model 'y~ABC + (1|id) + (1|id:A) + (1|id:B) + (1|id:C) + (1|id:A:B) + (1|id:A:C) + (1|id:B:C)'. Or, put more simply, a random intercept for subject, plus a random intercept for every interaction of subject and every main effect and interaction of the within-subject factors, except for the highest order interaction.
Jun 15 '17 at 21:52 history tweeted
Jun 15 '17 at 12:03 history edited
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Jun 14 '17 at 21:23 comment Still, some partial answers. (i) You are right, nothing is really "nested" here. We use the "nesting operator" because we want to have (1|id) + (1|id:A) + (1|id:B) + (1|id:A:B) + ... etc. structure, but it would not be correct to say that A,B,C are nested in subjects. They are not. (ii) We want (1|id) + (1|id:A) + ... structure because it mimics the approach of repeated measures ANOVA (RM-ANOVA). I can't properly explain why RM-ANOVA does that though. (iii) Yes, you can have random slopes (A*B*C - A:B:C | id) instead. But it's very different model when A,B,C have many levels.
Jun 14 '17 at 21:19 comment +1. This is an excellent question. It's very closely related to something I asked here some time ago: stats.stackexchange.com/questions/232109 - but unfortunately there is no satisfactory answer in that thread (existing answers might only add to the confusion, so be careful). I am planning to write an answer there myself at some point, but I still need to do some investigations beforehand...
Jun 13 '17 at 9:11 comment This is a purely theoretical question - I don't have a specific real situation. What I am interested in is why this is recommended as a standard way to analyze repeated measures data (at least by some authors), and what it means conceptually and mathematically to have a random intercept on each of these levels of "nested" experimental factors.
Jun 13 '17 at 1:35 comment Try to write models mathematically, then comparing them with your real situation to see which one is more reasonable.
Jun 12 '17 at 22:36 history asked