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When I start with the maximal random effects structure for a repeated measures design with three within-subject factors m_1 <- lmer(y ~ a*b*c + (1 + a*b*c|subject), data) and my model doesn't converge, how do I reduce it?

I think a model that converges most of the time should be the following random intercept model

m_interc <- lmer(y ~ a*b*c + (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), data)

But I don't really know how to get from m_1 to m_interc. E.g. when the random slope variance for the a * b * c interaction is zero do I follow up with m_2or m_3 

  m_2 <- lmer(y ~ a*b*c + (1 + a*b*c - a:b:c|subject) + (1|a:b:c:subject), data)
  m_3 <- lmer(y ~ a*b*c + (1 + a*b*c - a:b:c|subject), data)

Or, given the random slope variance for factor a is zero, do I substitute the random slope for a with the term (1|a:subject) or just exclude it?

Also, Bates states (slide 91) that if the variance-covariance matrix for the factor by subject slope, e.g. in a model with (0 + a|subject), has the form of compound symmetry it is equivalent to a model with random intercepts for subject (1|subject) and for the factor:subject interaction (1|a:subject). Why is this the case?

statmerkur
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  • Terms like `(1|a:subject)` are not really in the spirit of mixed models. This is approach used in ANOVA and one usually uses such terms to replicate the ANOVA results with lmer. If you don't care about classical ANOVA then you usually want to have your factors `a`, `b`, etc. only on the left side of `|`. So if `(1 + a*b*c|subject)` does not converge, try `(1 + a*b*c - a:b:c|subject)`. – amoeba Sep 20 '17 at 08:29
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    Bates states that there are two ways to define an interaction between factors and and groups (s. my edited post). So, I think this could be an option although I don't understand the exact difference. – statmerkur Sep 21 '17 at 16:00
  • It's a good discussion, thanks for the link. You might know that compound symmetry is an assumption of repeated-measures ANOVA, which is related to why I said above that terms like `(1|a:subject)` correspond to what ANOVA does. The main practical difference is discussed by Bates: when `a` has lots of levels, `(a|subject)` is a vastly more complicated model because it estimates the full covariance matrix between levels. – amoeba Sep 21 '17 at 18:44
  • Can you explain in an intuitive way why - in case of compound symmetry - `(0 + a|subject)` is equivalent to `(1|subject) + (1|a:subject)`? Or do you know any resources that discuss this issue? – statmerkur Sep 21 '17 at 19:27
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    I think this would be an excellent separate question and I suggest you ask it as such. (I am not sure I can answer it though.) – amoeba Sep 21 '17 at 19:36
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    I asked it [here](https://stats.stackexchange.com/q/304374/136579) – statmerkur Sep 21 '17 at 20:13

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