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My question is triggered by this question. I can't see that it has been asked here before, even though it looks like a natural enough question.

Suppose I have hierarchical data. The Wikipedia article uses as an example classes and pupils together with some response, so let's consider that. I'm old-school, so I would set up a regression model using these predictor variables: pupils, indicator columns for the different classes, and interaction terms between pupils and the class columns. How is this different from a bi-level model? Are there any differences in the assumptions, particularly the equal variance assumption? Are the resulting coefficients different? (Why?) Are the hierarchical model folks just blowing smoke, or am I missing something fundamental?

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  • $\begingroup$ Could you please explain what you mean by a "bi-level linear model"? $\endgroup$
    – whuber ♦
    Aug 13, 2015 at 14:08
  • $\begingroup$ I gave a link to the wikipedia article, which essentially sums up my knowledge (as well as my frustration). It gives a number of similar terms, which are not specifically flagged as synonyms, such as nested models, random effects models, mixed models, etc.) The prototypical bi-level model seems to be a sample of students from different classes, where one is interested, as far as I can tell. in isolating a "student effect" and a "class effect". The article states that the class effect is affect is modeled as $\beta_{0,j}$ and $\beta_{1j}$ which are intercepts and slopes for each class. (cont'd) $\endgroup$
    – user3697176
    Aug 13, 2015 at 14:39
  • $\begingroup$ (cont'd): the level-2 attaches errors to the estimations of the betas and then uses these level-2 estimates to compute the error for the individual students. The article has this to say about the assumptions: "Multilevel models have the same assumptions as other major general linear models (e.g., ANOVA, regression), but some of the assumptions are modified for the hierarchical nature of the design". So, is the bi-level design the same as a regression of whatever response I have on whatever "first-level" predictors I have (income, time spent studying, etc.) plus class dummies and interactions? $\endgroup$
    – user3697176
    Aug 13, 2015 at 14:43
  • $\begingroup$ For "a regression model using these predictor variables: pupils, indicator columns for the different classes, and interaction terms between pupils and the class columns", not sure what the pupils variable is. Pupil indicators? I guess you are asking estimating the effects for each class using class indicators? If so, then it's essentially a question about the difference between random-effects model in biostatistics and fixed-effects model in econometrics. $\endgroup$
    – Randel
    Aug 19, 2015 at 3:26

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