Specifying group covariates and cross level interactions in lmer

Question

I'm new to R, and despite trying to read as much as I can about how lmer works in R, I still don't feel like I know how to correctly specify more complex models using the lmer syntax.

For example, at the moment I want to use lmer for a two-level multilevel model, where the first level is features of a specific course taken by a specific student in a specific semester (with covariates like kind of course, teaching method, etc) and the second level is the student, which also has a number of covariates (e.g. ethnicity, gender, age, gpa at the beginning of the study, and score on a specific instrument). I also want to assess interactions with the teaching method, for example by including cross-level interactions between the second-level covariates ethnicity, gender, age, gpa, and the instrument score and the first-level covariate teaching method.

For the sake of readability, let's limit the equation to teaching method, course type, and score. It seems to me that I have seen three different ways of doing something like what I want to do in R - they are clearly all specifying something somewhat different, but I can't figure out what the underlying math is supposed to be for each case. So, for example, in different online references and the books on MLM that I have on hand, it seems that one of these three models is recommended for what I want to do:

course_outcome ~ course_type*teaching_method + score*teaching_method + (1|student)
course_outcome ~ course_type*score + (teaching_method|student)
course_outcome ~ course_type*teaching_method + score*teaching_method + (teaching_method|student)

I'm a little confused about the differences between how lmer interprets each of these three codings - is there any chance that someone who really understands R better could possibly translate this into the basic regression equation(s) structure that would be calculated in each of these three examples? Or direct me to a reference that might more clearly explain the difference by giving the regression equations for each case?

Thanks for your time!

Answer 1

The transformations from lmer codes to regression equations are listed below, where $u$ denotes the random effect, $i$ and $j$ indicate student and course respectively. Suppose the course features would not change with the students.

1. course_outcome ~ course_type*teaching_method + score*teaching_method + (1|student)

   $$\mathrm{course\_outcome}_{ij}= \beta_0 +\beta_1\mathrm{course\_type}_j +\beta_2\mathrm{teaching\_method}_j +\beta_3\mathrm{score}_i +\beta_4\mathrm{course\_type}_j*\mathrm{teaching\_method}_j +\beta_5\mathrm{score}_i*\mathrm{teaching\_method}_j +u_i+e_{ij}$$

2. course_outcome ~ course_type*score + (teaching_method|student)

   $$\mathrm{course\_outcome}_{ij}= \beta_0 +\beta_1\mathrm{course\_type}_j +\beta_2\mathrm{score}_i +\beta_3\mathrm{course\_type}_j*\mathrm{score}_i +u_{0i} +u_{1i}\mathrm{teaching\_method}_j +e_{ij}$$

3. course_outcome ~ course_type*teaching_method + score*teaching_method + (teaching_method|student)

   $$\mathrm{course\_outcome}_{ij}= \beta_0 +\beta_1\mathrm{course\_type}_j +\beta_2\mathrm{teaching\_method}_j +\beta_3\mathrm{score}_i +\beta_4\mathrm{course\_type}_j*\mathrm{teaching\_method}_j +\beta_5\mathrm{score}_i*\mathrm{teaching\_method}_j +u_{0i}+u_{1i}\mathrm{teaching\_method}_j +e_{ij}$$

It may help you form some general ideas to interpret the codes. Note that $\beta$s across the equations have different meanings. It seems that course_type and teaching_method may be factors. If so, they would have several coefficients corresponding to the number of levels.