Correlation/Regression using Aggregrated Data

EH11

New Member
#1
Hi,

This is something that I've been wondering about for some time now, but which I've never managed to find a formal reference for. Specifically, I've been wondering about the difference between using correlations or regression on individual level data (e.g. each data point is a single individual) versus using correlations or regression on aggregated data (e.g. each data point is based on the responseS of 1000 individuals).

I'm of the opinion that the effect size estimate should be associated with a smaller standard error in the latter than in the former case, but I've never found any information either way, so please see below an example of what I mean - I've created some dummy data from 5 imaginary classes in order to demonstrate that when you average out the correlations across all 5 classes (of variables that are genuinely correlated), you would usually end up with an average correlation that is lower from the one you would obtain if you correlated the same two variables using aggregated data (each data point is based on 10 pupils from a different Class). This is presumably because the level of error in each aggregated data point (which represents an entire class) is much lower than that of each individual score.

Could anyone attempt to clarify if and how one should treat correlation / regression based on aggregated data in a different way? E.g. how would you work out the Standard Error of pearson's r when the correlation is based on aggregated data as in the second example below?


CLASS 1
SAT EXAM
student 1 55 83
student 2 55 91
student 3 86 66
student 4 47 39
student 5 63 79
student 6 86 87
student 7 55 39
student 8 52 72
student 9 57 57
student 10 67 90

pearson's r 0.38


CLASS 2
SAT EXAM
student 1 66 86
student 2 56 86
student 3 67 65
student 4 76 93
student 5 46 98
student 6 46 68
student 7 45 46
student 8 54 65
student 9 66 82
student 10 56 87

pearson's r 0.37


CLASS 3
SAT EXAM
student 1 55 77
student 2 64 73
student 3 34 66
student 4 47 88
student 5 63 79
student 6 64 34
student 7 46 39
student 8 34 23
student 9 54 57
student 10 54 90

pearson's r 0.30


CLASS 4
SAT EXAM
student 1 67 81
student 2 64 54
student 3 45 66
student 4 47 39
student 5 42 79
student 6 32 46
student 7 37 39
student 8 52 45
student 9 43 57
student 10 46 90

pearson's r 0.30


CLASS 5
SAT EXAM
student 1 52 83
student 2 45 91
student 3 34 66
student 4 44 39
student 5 34 79
student 6 88 87
student 7 65 59
student 8 23 24
student 9 34 25
student 10 22 29

pearson's r 0.60


AVERAGE Pearson's r (at pupil level) across 5 CLASSES = 0.39


SAT EXAM
CLASS 1 62.3 70.3
CLASS 2 57.8 77.6
CLASS 3 51.5 62.6
CLASS 4 47.5 59.6
CLASS 5 44.1 58.2

Pearson's r (at CLASS level) based on average SAT and EXAM scores of 10 pupils from each CLASS = 0.85