# How to interpret intercepts for individual random effects from lme4 model

I am working through an example problem from a text that models some experimental data as a Linear Mixed Effect Model. The experiment has 3 operators measure the thickness of 10 parts. Each operator measures each part and also repeats each measurement on every part one time so there is a total of n=60 data points.

The model is set up as follows using lme4:

mod_1 <- lmer(coating_thickness ~ 1 + (1|part) + (1|operator) + (1|part:operator), data = thickness_data)


I believe I understand how to interpret the output and see how much of the total variance is due to part, operator, or interaction. However, I'm struggling to understand the meaning of the intercepts for the individual random effects. When I look at them via: coef(mod_1 ) I see that they are not very different from the Fixed effect intercept. For instance, fixed effects estimate is 0.7982 and the random effects for part and operator are:

$part (Intercept) 1 0.6302115 2 0.9706044 3 0.7828980 4 0.8333266 5 0.5209495 6 0.8809536 7 0.9243782 8 0.7913028 9 0.8949615 10 0.7520805$operator
(Intercept)
1   0.7981667
2   0.7718962
3   0.7560621



I interpret this to mean that if I wanted to make an estimate of thickness and didn't know anything about which part or operator was measured/used then I would use the overall fixed effect as my estimate --> 0.7982. If I wanted to make an estimate of thickness for part 1 and all I knew was that I was measuring part 1, I'd use the random effect for Part 1 --> 0.630.

However, what would I do if I wanted to make a estimate of Part 1 by Operator 1? I would think you would sort of start with an estimate of the overall mean fixed effect and then make some adjustment based on what you know about Part 1 and then another adjustment based on what you know about Operator 1. However, based on the look of these intercepts I clearly can't just add up the intercepts to get a final estimate/predicted value for an Operator 1 measurement of Part 1. Do I average them or something? Not quite sure what to do here.

For reference, the actual data values are not that far off from the fixed effect intercept. Here are the first 10 rows:

part
<fctr>
operator
<fctr>
coating_thickness
<dbl>
1   1   0.71
1   1   0.69
1   2   0.56
1   2   0.57
1   3   0.52
1   3   0.54
2   1   0.98
2   1   1.00
2   2   1.03
2   2   0.96


I would be extremely wary of such a model. With only 3 observations for operator, this is hopelessly inadequate for random intercepts to be estimated for it.