Why do we need Multi-Level Models when we have Logistic Regression? I am an MBA student taking courses in Statistics.
I attended a seminar hosted by the Statistics faculty in my university in which students present the research they are doing.
The students had data (e.g. age, gender, race, number of children they have, which university they studied at, what they studied in university, neighborhood they live in, city they live in, province they live in , etc.) on different people from across the country - and they are building models to understand the effect of these different variables on "university graduation rate" (e.g. in the data they have, the student either "graduated" or "dropped out").
In my very naïve understanding of statistics, this kind of problem can be solved using a Logistic Regression Model. Provided that the model is fit properly and the results of the model are statistically significant (e.g. p-values), a Logistic Regression Model should be able to tell you the effect of different combinations of dependent variables on "university graduation rate". As I understand, this is closely related to the Odds Ratio - how much more likely are "students with children living in city A" to graduate university compared to "students without children in city B".
However, the students were instead presenting a type of model called a "Multilevel Regression Model" (I think this also called a "Hierarchical Model").
When I asked the students why they chose a "Multilevel Regression Model" instead of a "Logistic Regression Model", they gave me the following answer:

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*It is a reasonable assumption to believe that students in the same faculty within the same university have more similar graduation rates compared to students in the same university but from different faculties. On the other hand, it is also a reasonable assumption to believe that students in the same neighborhood have similar graduation rates to one another compared to students in different neighborhoods. Apparently a Multilevel Regression Model can potentially address these "within group correlations" whereas a Logistic Regression Model can not. (Some further comments were mentioned about "Fixed Effects", "Mixed Effects" and "Random Effects" - but I could not fully understand these concepts nor understand the relevance of these concepts when comparing the aptness of Logistic Regression vs. Multilevel Logistic Regression.)

While this sounds like a reasonable justification as to why one would prefer a Multilevel Regression  Model compared to a Logistic Regression Model - due to my lack of knowledge in Statistics, I have no choice but to blindly accept this justification.
In my opinion, a Logistic Regression Model can provide estimates and help understand the effects of different independent variables on the dependent variable - for example, all students in the same neighborhood should all be subject to the same effect of this neighborhood on the university graduation rate. Therefore, in this regard, it seems to me that both the Logistic Regression Model and the Multilevel Regression Model can take into consideration similarities within groups when estimating the effect of independent variables on the dependent variable.
Thus, why might the students from the seminar chosen a Multilevel Regression Model compared to a Logistic Regression Model - given that both models can take into consideration the effects of similar groups when estimating the effect of independent variables on the dependent variable?
Some Final Notes:

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*Even though a  Logistic Regression Model can estimate the effect of independent variables on the dependent variable, a Logistic Regression Model assumes that all observations are independent from one another (I.I.D.), whereas a Multilevel Regression model does not make this assumption (i.e. cluster/group correlation)


*If this is really the case - why not just fit a whole bunch of individual models for each combination of factors within the independent variables? I understand that this might result in fitting a large number of models, but aren't modern computers strong enough to do this?

Multilevel models provide us with a few benefits when compared to logistic regression. Here are a few I can think of without being too pedantic:
Shrinkage
Suppose you are estimating the mean/proportion of a response variable per group. Sometimes a group can have a mean too high just because that group has fewer samples on it (just think of thumbs-up reviews on a product that is not very popular on an e-commerce website). You know you can't trust the review number on those instances, so instead of removing them from the analysis, you "shrink" the estimates towards the gran mean/proportion in an "if I don't trust your estimate because of your sample size I consider you close to the gran estimate" rationale.
This shrinking effect will happen for all of the parameters you estimate, ie: per group intercepts and slopes.
Variance components
Whenever we try to fit models to test a statistical hypothesis, the goal is to have as little variance around the estimates of parameters of interest so we can test them, have p-values, etc. Adding covariates, for example, if they help explain part of the variability of the response variable together with the parameters of interest will usually aid in reducing the variability around the estimates of the parameters of interest.
In multilevel models, we get "extra" variance parameters, each for the groups of random slopes/effects and intercepts, since those have distributions tied to them as well. This helps us distribute explainable variance of the response variable toward those new components in a way that allows for that type of justification of "grouped observations are usually similar" meaning they have their own variance component.
Usually less prediction error
Now, this is due to the shrinkage and variance components we have in those models. Usually, since we make the parameters closer to the grand mean, we have less quadratic error overall. The read more about this type of relationship between shrinkage and prediction error, search for Stein's Paradox.
Allow for finding interesting between patterns in groups
Now again due to the shrinkage and individual groups parameters, we can have some discussions like: "even though variable X effect is overall tied to such and such changes in the response variable, we can see that in group A the effect is zero (or even opposite) to the overall trend".
Some further reading
I think you will be interested in reading a bit more about those modeling techniques. And from your question, I think you will be able to follow those texts with no trouble at all. Gelman et al's text has some really nice examples of the type of conclusions you can get from the multilevel models and does a superb job of explaining the math around those. Singer et al's text has some nice intro chapters explaining the why's and how to perform exploratory/descriptive analysis around those Multilevel models.
