Chad d’Entremont writes,
I have been examining racial sorting in New Jersey
charter schools. Much of the literature on this
subject runs linear regressions with the percent
African-American students as the dependent variable.
However, most studies include student-level,
school-level, and district-level data all in the same
equation.For example, one study runs the equation
Sorting = aX + BY + cM + year
Where X is a vector of student characteristics, Y is a
vector of school characteristics, and M is a vector of
district characteristics. (The study is longitudinal
and the variable time identifies the year a given
child enrolled in a charter school).My understanding is that such a regression is
improper. But, it is also common in the literature.
Is there an acceptable time and place for such an
analysis? Or, was this simply a commonly used
technique before multilevel modeling became
widespread?In my own analysis, I have both school-level and
district level data (no student-level data). I
envision running an equation that includes two
district-level parameters (income and location). A
simple equation would be the following:Race = a + b1(income) + b2(achievement) + b3(school
type)…where, a ~ N(u0 + u1(income) + u2(location), sigma^2)
Does this equation make sense, or is this type of
multilevel analysis unnecessary? I just want to make
sure that it is appropriate to break from the norms
found in the published literature.
My response: It’s not clear to me what you mean by “racial sorting.” Is it a school-level charactistic (for example, %African-American in the school being different from %African-American in the district)? Or is it a district-level characteristic, something about segregation of different groups in different schools? I’d think the appropriate analysis would depend on what is being studied, and what causal questions are being asked (explicitly or implicitly). But the short answer is, yes, it’s ok to have predictors at different levels as long as you allow for unexplained variation at each level also.