Crossed or nested random effects? I've conducted an experiment in which 20 pairs of talkers are conversing in their first and second language (L1 and L2, respectively) both in quiet and in noise in a fully crossed design:


*

*L1 in quiet 

*L1 in noise

*L2 in quiet

*L2 in noise


Each pair is replicating the experiment three times.
Therefore, I have three fixed effects: background (levels = quiet, noise), language (levels = L1, L2), and replicate (levels = 1, 2, 3). Furthermore, I have a random effect of pair, talker and gender. This is where I need some help to understand the nested vs crossed structure of the random effects. I have 20 pairs making up a total of 40 talkers (labeled talker 1:40). Each of those talkers has a gender. I'm assuming there's a correlation between the outcome measures within the pairs, so that talker 1 & 2 show correlated behavior that's uncorrelated with talker 3 & 4.
Which one of these models would be correct (if either)?


*

*y ~ background * language * replicate + (1 + background + language +
replicate  | pair) + (1 + background + language + replicate  |
person) + (1 + background + language + replicate  | gender)

*y ~ background * language * replicate + (1 + background + language + replicate  | pair/person/gender)


The latter model fails to converge. I have 480 data points in total.
 A: A couple of points:


*

*Very often, even if you have a complex design with multiple levels of nesting or crossing, the correlations in the data are not strong enough to support including all random effects the design would dictate. Because the variances of random effects are more challenging to be estimated than the fixed effects coefficient from an algorithmic viewpoint (i.e., leading to convergence problems), it is often preferable to start with a simpler model and build up the random-effects structure performing likelihood ratio tests to see if you need the extra random effects. If you run into convergence problems when you include an additional random effect that persist (e.g., even if you change optimizers and/or initial values), then this is again an indication that perhaps you do need this extra random term. Hence, in your specific case, I'd first start with a simple random intercept for each pair, then try to include the nested random effect for person, and following also try random slopes.

*I would not include a random effect for gender. Random effects are used to model correlations in the measurements within each level of your grouping variable. By including a random effect for gender, you assume that measurements of your outcome variable y within the same gender are correlated (even if they come from different persons). I don't know all the details of your experiment, but this sounds like something that you wouldn't want to assume.

