# Estimating individual-level outcome probabilities given from group-level training data

Given multi-level data, I'm interested in estimating member-level outcomes when only group-level outcomes, along with both group- and member-level predictors, are known. I'm lacking in statistical terminology, so I'll explain with an example:

We'd like to estimate the likelihood that a given high school student will be granted college admission. In this case, "groups" will be schools and "members" will be students. We have:

1. Aggregated, group-level data on the dependent variable (total number of accepted/rejected applications from each high school)
2. Aggregated, group-level independent variables for each school (number of students who have taken at least three advanced/honors courses, distribution of GPA, number of students in the top 10% of standardized test scores, etc.)
3. The same independent variables at the member/individual level (for each student, whether he/she has taken at least three advanced/honors courses, actual GPA, whether they scored in the top 10% on a standardized test, etc.)

What we're lacking, however, is member-level outcome data on college admission (for each student, whether he/she was admitted to college).

Intuitively, I get the sense that some estimation may be made for group members even in absence of this individual-level outcome data: we may see, for instance, that schools with high proportions of students taking at least three advanced/honors courses have correspondingly high acceptance rates. It makes sense that, knowing this, a student's likelihood of being granted college admission should be positively influenced by whether he/she has taken at least three advanced/honors courses (possibly along with other, group-level effects).

Is it possible, then, to model group-level outcomes and apply those models to individual cases to estimate individual-level probabilities? If so, what terminology describes this type of model? Any pointers to literature on this topic would be appreciated as well.

• Your response variable (college acceptance) is non-normal (specifically, Bernoulli). When the response is normal, the population mean & the average of individuals are the same, but this is not typically true w/ non-normal responses. To understand this issue, it may help to read my answer here: Difference between generalized linear models & generalized linear mixed models. – gung Feb 17 '13 at 20:48
• If you have multiple accounts, Dan, you can merge them by filling out the form at stats.stackexchange.com/help/user-merge. – whuber Feb 17 '13 at 22:07
• Thanks gung, very enlightening. This distinction makes sense, as individual-level responses in my case are binomial while group-level aggregates are not, but I'm wondering if these group-level outcomes (which may, in fact, be normally-distributed) can inform estimates of individual-level probabilities? A rudimentary estimation could be made by assigning those member-level probabilities as equal to their group-level means, but given the additional information we have about each member (test scores, grade, etc.), my intuition tells me better estimates for individuals should be possible. – danpelota Feb 20 '13 at 23:32