# Group-level correlation effect

I am measuring two continuous variables (X,Y) for 21 subjects. X and Y have 216 data points each (per subject). I would like to see if X and Y are correlated at the group level. I can think of 3 options:

a) Concatenate all the subjects together and compute the correlation. I believe this method will inflate my Type I error rate quite a bit and seems like a bad idea.

b) Run a separate correlation for each subject, and then run a 2nd level analysis to see if the t-values from each correlation are significantly different from 0 (single sample t-test). This seems to be quite prominent in analyzing fMRI data at the group level.

c) Construct a mixed-effect linear model, treating subject as a random variable. I've done this in R using lmer(y ~ x + (1|sub)) and got the same result as b, albeit a different p-value. However, I use R infrequently and am somewhat suspect when pvals.fnc reports a p value of "0" (which I interpret as less than .0001).

What is the proper way to run this analysis, and more specifically what are the differences between (b) and (c)?

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I don't understand the use of the term concatenate in 1. Do you simply mean pooling all the subjects into one large sample of x-y pairs? Why are you saying that it will inflate your type I error? What are you testing? Is it that the correlation is different from 0? If that is the case and the individual subjects have varying correlations running both positive and negative than the type I error would be increased. I don't think inflated would be the right way to characterize that though. If that is the case b) would be more reasonable. –  Michael Chernick Aug 8 '12 at 16:49
The difference between b and c is that in b you are running several independent tests on the subject correlations. With 21 tests some form of multiplicity adjustment would seem to be in order. Option c models the subject effect and the p-value tells you that the relationship differs across subjects. It does not specifically address whether or not some of the correlations are 0 (or not statistically significantly different from 0). –  Michael Chernick Aug 8 '12 at 16:54
By concatenate, I mean treat X and Y as 4536 (216*21) observations from a single subject. My belief is that X causes Y, but the strength of the relationship between the two variables does not necessarily have to be the same between subjects. I am under the impression that the p-value of the random effect tells me 'that the relationship differs across subjects', but the p-value of the fixed effect should tell me whether there is a linear relationship between X and Y adjusted for subject variability, no? –  Jeff Aug 8 '12 at 17:06
Then concatenating is pooling all subjects together. Okay your interpretation of the model sounds right. Then you are interpeting a significant regression parameter as the same as a significant correlation which should be okay since it is just between X and Y. –  Michael Chernick Aug 8 '12 at 17:35
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