Multilevel model clustering university classes

Question

I'm having a hard time figuring out whether a MLM would buy me anything with an analysis I'm doing

Questionnaires were administered to university classes and lets say I'm interested in variables X1 and X2 on that questionnaire. I'm interested in whether a students class grade can be predicted from those variables: grade ~ b1X1 + b2X2. I predict that both b1 and b2 will be significant and positive.

So far thats just straightforward multiple regression. The reason I started thinking about MLM is because class grades at this university are scaled and professors will scale differently. That means for one class, the mean grade might be 80, whereas for another class mean grade might be scaled down to 60. This might turn the model into something more like grade ~ b1X1 + b2X2 + b3Class

I only just started learning about MLM so likely misunderstanding some of the concepts here. I initially thought that due to scaling, the regression surface for each class is going to have a different intercept so maybe I should run a random intercepts model. Further thinking changed my mind on that because I'm going in fully expecting mean grades for each class to be different precisely because of scaling, so seeing significant variance in grade due to class isnt going to tell me anything I dont already know/expect

Then I wondered whether a random slopes model would actually be more appropriate because while I expect the intercepts to be different for each class, I dont know if the slopes will differ, so maybe a different slope should be modelled for each class (even though I have no theoretical reason to suspect that slopes will differ between classes)

My question:

1. Is grade scaling enough to warrant sacrificing power and running an MLM?
2. Under what conditions might MLM be useful here?
3. Is my logic correct in thinking that random intercepts should not be tested here?

OR, do I have this completely backwards, and should instead be testing whether classes differ on X1 or X2 instead of grade?

EDIT:

I have about 800 samples in total

Split into 7 different classes

Each class has a different number of students. Most have around 130 students, one class has 80, and another one has about 50

Answer 1

Here are my answers:

1. I ask for sample size and number of groups in comments to point out a possible problem. I think you might experience some issues because of the number of classes. There are some discussions about the sufficient sample size for multilevel analysis, please check this, this, and this for some examples. Having only 7 classes might lead to problems (e.g.,unreliable estimates), but not necessarily. See Andrew Gelman's post.

2. Still, I would be careful about fitting a multilevel model. Let's assume there is no problem about sample size (and number of groups) and you have theoretical reasons to use multilevel modeling. Then, you should do it. In the end, we have enough computational power. At least, you can fit a null-model and see whether significant variation exist among classes (also intraclass correlation coefficient indicates how much variance in grades is between classes). But as you see, this is an empirical question; I think you shouldn't simply assume difference between mean grades, and skip the analysis. You should test, and...

3. You should fit a random-intercept model here (following the null-model). As far as I understand random-slopes models without random-intercepts are quite rare. In fact, Gelman and Hill (2006) argues that "almost always, when a slope is allowed to vary, it makes sense for the intercept to vary also" (p. 283). They talk about an example of random-slopes without random-intercepts, but I think in your case, you should have random-intercepts anyway as (you argue) the classes are not identical in their mean grades. Here is a visualization of random-intercept, random-slopes, and random-intercept and random-slopes models (Gelman and Hill, 2006, p.238):

If you realize that fitting multilevel model is not appropriate, you can fit multiple regression model with class as a (categorical) predictor, and maybe, you will add some interaction terms between class and other variables. I hope the answer helps.

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Gelman, A., & Hill, J. (2006). Data Analysis Using Regression and Multilevel Models. Cambridge: Cambridge University Press.

See also:

Luke, D. A. (2004). Multilevel Modeling. London: Sage Publications.

Raudenbush, S. W., & Bryk, A. S. (2002). Hierarchical Linear Models. Thousand Oaks London New Delhi: Sage Publications.