There are two excellent CV posts on specifying crossed effects models (post 1, post 2).
The issue I'm trying to wrestle with pertains to part of the answer to post 2, in particular how to nest crossed random effects.
In my study, I have:
- About 20 individuals per site
- About 10 sites
- Within each site, there were about 20 samples
The outcome in the example is participant's "interest" (the study is about out-of-school programs).
Because there are dependencies by both participant and sample, I think there are two crossed random effects, one for observations associated with each individual, and one for observations associated with each sample. The hard part for me is that these random effects are nested in one of the 10 programs.
The samples were at the same time for all of the individuals within the site, but at different times at different sites, so that sample 1 in site A was not necessarily at the same time in any sense (not the same date / time nor at the same interval from the "start" of the site's activities). Therefore, to create the variable identifying the time of the sample, I combined the
site variable, the date that the sample was collected, and another variable specifying whether the sample was the 1st, 2nd, 3rd, or 4th sample collected for that date. It's a factor.
The data (in
R) are as follows:
# A tibble: 2,970 × 4 interest participant_ID site sample <dbl+lbl> <dbl> <chr> <fctr> 1 2 1001 1 1-2015-07-14-1 2 2 1001 1 1-2015-07-14-2 3 4 1001 1 1-2015-07-15-1 4 3 1001 1 1-2015-07-15-2 5 3 1001 1 1-2015-07-21-1 6 1 1001 1 1-2015-07-21-2 7 3 1001 1 1-2015-07-21-4 8 3 1001 1 1-2015-07-22-1 9 4 1001 1 1-2015-07-22-4 10 3 1001 1 1-2015-07-28-1 # ... with 2,960 more rows
In the answer to post 2, the author of the selected answer wrote:
Because you do not have unique values of the tow variable (i.e. because as you say below tows are specified as 1, 2, 3 at every station), you do need to specify the nesting, as (1|station:tow:day). If you did have the tows specified uniquely, you could use either (1|tow:day) or (1|station:tow:day) (they should give equivalent answers).
In mapping this to my example, I do have unique values of the sample (tow variable), I do not need to specify the nesting. I'm having trouble specifying this model mathematically, and, thus, in terms of model syntax. (I am using
But, here seem to be the options:
Not nesting the crossed random effects within the site because the sample variable includes a site identifier:
lmer(interest ~ 1 + (1|participant_ID) + (1|sample), data = df)
Creating the sample variable without a site identifier but in a way so that samples within each site were still identified uniquely and nesting the crossed random effects within the site:
lmer(interest ~ 1 + (1|site/participant_ID) + (1|site/sample), data = df)
Other examples interact the crossed random effects, via adding a term such as
Does either of these seem like they would account for dependencies by both participant and sample? Or, are there other options or better ways to model this?
(1|site)into there. Your option #2 is also fine, and you don't need to change anything about how you code your sample variable;
(1|site/sample)is equivalent to
(1|site)+(1|site:sample)and if your sample is coded like it is then this is further equivalent to
(1|site)+(1|sample). The same goes for the participant term. So option #2 will be equivalent to #1 if you add the site term to #1 as I suggested above. $\endgroup$
(1|participant_ID:sample)is another issue. I would try fitting models with and without this term and see what comes out. $\endgroup$
(1|site) + (1|participant_ID) + (1|sample)and if the
siteterm comes out with zero variance then so be it; but you cannot know in advance, so it makes sense to put it in the model anyway. $\endgroup$