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?

As another comment notes - multilevel regression and logistic regression are not mutually exclusive, and in fact it might be that the best model to run is a mutlilevel logit model! In this case, the key issue is whether or not you want to consider "school" as a variable or a unit of analysis.
But first, this question conflates two separate questions about regression modeling that don't have much to do with one another.
The first question is - what "flavor" of regression model should I use, based on the way my dependent variable is measured. Normal "OLS" regression assumes that the dependent variable is continuous and has a more or less normal-ish distribution. But if your dependent variable is binary - for example "did you graduate or not?" then an OLS model isn't really appropriate. Instead you want to use a logit (or logistic) regression model. This type of model is specifically set up to analyze binary dependent variables - it tells you the extent to which a one unit increase in a particular independent variable changes the probability of getting a "1" on the binary dependent variable, holding all other variables constant.
Totally sperate from that entire discussion, we have the question of what to do when you have observations that are "nested" in other groups, and where some variables are at a different "level" than others. So in this case you have students who are nested in schools, and some variables (like gender, or graduation) refer the characteristics of students while other variables (like school size, public/private) refer to characteristics of the school, so that every student who went to school #345 will have identical values for those variables. As you allude to this messes with the assumption that observations are independent which is common to "single level" version of ALL types of regression analysis, including logit, OLS, tobit etc.
Making whatever type of model you are running a "multilevel" model is one particular way to deal with this issue, although there are actually a bunch of different kinds of multilevel models. In this case what the student was probably proposing is a model that allows the intercept term to vary randomly (that is, according to a normal curve) at the school level - what's called a random intercept model. This sort of approach can work on an OLS or a logit model, although the implementation is a bit different in each case. However, there is one feature of this approach that may be problematic, and might be the source of your confusion. In a multilevel model you don't actually ANALYZE the effect of being in one "group" (school in this case) vs the other. The model assumes that, in general, school level intercepts vary randomly around an overall intercept and just estimates the variance of that normal curve using the data available. So it won't actually tell you if students at school A are more likely to graduate at school B, because it's not thinking about school as a variable but as another unit of analysis (like people). You can, however, use a method called empirical bayes estimation to try and figure out which schools do better or worse, based on the model you just ran.
An alternative to a multilevel model would be a "fixed effects" model that just includes a dummy variable for each school. This model treats school like a variable and it will tell you how each school does in comparison to some reference school (although for various reasons the empirical bayes approach I discussed above might be better for this question). But a fixed effects model might end up with big standard errors, and it won't allow you to include school level variables (like school size) that might tell you WHY some schools do better than others.
So to answer the question "what type of model should I use" you need to answer two separate questions:

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*How is the dependent variable measured? If it's binary, use a logit model. If it's continuous use OLS. If it's something else...use another type of model (ologit, tobit, negative binomial etc)


*How are you going to deal with the fact that students are nested in schools? if you want to treat "school" like a variable, you can use a fixed effects model. If you want to include school level variables in the model, then you might explore a random intercept model.
