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I am trying to understand how lme4's mixed modeling works, specifically its random effects design matrix "Z". I have the following R code:

library(lme4)
A = as.factor(c(1,1,1,1,2,2,2,2))
B = as.factor(c(1,1,2,2,1,1,2,2))
R = c(1,2,3,4,6,7,9,11) + runif(8)
remdl = lmer(R ~ (1|A/B) + (1|B))
summary(remdl)
getME(remdl, "Z")

In this mixed effect model, I believe that I am telling lmer() to treat Predictor B as both nested within A and crossed with A. When I run this code, lmer() throws a "singular fit" error, but it does fit the model.

My question is: How is this possible? How can the random effect predictor B be treated as both nested and crossed with A?

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The response of the model you're fitting ((1|A/B) + (1|B)) is equivalent to (1|A) + (1|A:B) + (1|B). This means that you are allowing the intercept to vary (1) among levels of A, (2) among levels of B, and (3) among combinations of A and B. As long as you have multiple observations for (most) combinations of A and B, this should be identifiable. (In glmmTMB, which has slightly updated formula-processing machinery you could also specify this model as (1|A*B).)

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  • $\begingroup$ I understand, thanks! If I may ask a follow up question, in my example, the random effects design matrix "Z" seems to show many co-linear columns, yet the lmer() function still works. Is this design matrix not actually co-linear, or is this simply a feature of random effects design matrices that they can have co-linear columns? $\endgroup$
    – Chris Science
    Commented Jul 21, 2023 at 17:27
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    $\begingroup$ You could ask it as a separate question if you want, but the answer is the latter: because the coefficients corresponding the Z matrix are penalized, coefficients for collinear columns will 'share' the effects (similar to collinear predictors in a ridge regression, if that analogy helps ...) $\endgroup$
    – Ben Bolker
    Commented Jul 21, 2023 at 17:34
  • $\begingroup$ I'm not familiar with ridge regression, but is it possible to think of collinear columns as being weighted by a their fraction (2 columns get 50% weight each; 3 columns get 33% weight each)? Also, is there any situation where adding collinear columns to a random effect design matrix would be desirable? This concept contradicts my intuition from fixed effect design matrices about what the purpose of a model coefficient is, namely to distinguish a set of data points from them rest. $\endgroup$
    – Chris Science
    Commented Jul 21, 2023 at 17:54
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    $\begingroup$ Approximately, yes. Whenever you have more than one RE term in a model, the Z matrix is likely to be rank-deficient/have multicollinear columns, each component is a set of mutually exclusive indicator variables (so the sum of each set of columns is equal to 1). $\endgroup$
    – Ben Bolker
    Commented Jul 21, 2023 at 20:18
  • $\begingroup$ Great, thank you! $\endgroup$
    – Chris Science
    Commented Jul 21, 2023 at 22:23

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