Why having a 3rd level nested design, requires the 2nd level random-effects to be shown as a combination 2nd:3rd levels (data provided) Why when we have a 3-level nested design, the 2nd level is shown as a combination or interaction with the 3rd level i.e., 2nd:3rd?
For example, below I have 10 schools (schoolid) within which there are different number of students (childid). On average, each student has been measured 6 times. If we plot each student's growth trajectory over time (year), we get a plot like shown.
Using this plot, what is the meaning of childid:schoolid which is the random variation in intercepts among children nested in schools? Can we possibly see childid:schoolid or describe childid:schoolid in the plot?
library(lme4)
dd <- read.csv('https://raw.githubusercontent.com/rnorouzian/e/master/3.csv')
m31 <- lmer(math~year+(1|schoolid/childid), data = dd)



 A: schoolid
Can you think of schoolid as a placeholder for all the school-level factors that can potentially affect a child's math score on a given year? (These factors can be either known to you or unknown; if they are known, they can be measured in your study or not measured; if they are measured, they are nevertheless omitted from your model.)
Note that these school-level factors are believed to affect the math score of that child in the same way over time - in other words, they are time-invariant.
For example, one such factor might be type of school (private versus public); a child who studies in a private school might have a consistently higher math score each year compared to one who studies in a public school.
Another such factor might be school size (e.g., < 1000 students versus 1000 or more students);  a child who studies in a small school with fewer than 1000 students might have consistently higher math scores than one who studies in a large school with 1000 students or less.
Not only are these school-level factors time-invariant - they are assumed to affect each child in a particular school in the same way.
childid:schoolid
For each child within a school, there could also be child-level factors (not included in your model) that can affect that child's math score on each year. Examples of such factors could be the student's age, their gender, etc.  Again, these are time-invariant factors, so they would be expected to influence the child's score in the same way each year.  Because the child attends a specific school, how these factors affect a child's yearly math scores in this school could conceivably be different from how they affect the yearly math scores of a child with the same child-level factor values in another school (hence the "interaction" you mentioned).  In other words, the child operates within a context defined by his/her school and this could affect how these child-specific factors work to affect his/her yearly math score.
