# Correlation with repeated measures

For each of my 20 subjects, I have:

• physiological parameter: brain activity (y1) at time T1;
• physiological parameter: brain activity (y2) at time T2;
• behavioral parameter: reaction time (X);

I want to know if brain activity and reaction times are correlated, using all the data I have.

1. If Y is the concatenation of y1 and y2, is it correct to use a linear regression model, in which I use Y as the response and X as the predictor, repeating twice the elements in X?
2. Is it more appropriate to estimate two correlations separately (y1 with X, and y2 with X)? Should I compare the effect sizes, then, only if both correlations are significant? • Perhaps you should start with a free-form examination of the relationships using a 3d plot of all of the values. It's surprising how effectively you can discern relationships visually. There are lots of ways to do it, but you could look here for how to use R: genomearchitecture.com/2014/03/3d-animations-with-r Nov 24, 2016 at 20:31
• I added the graph. It suggest a correlation, especially between reaction time and brain activity in T1. Still, my question remains. Nov 25, 2016 at 10:00

The answer depends on the hypothesis/-es you want to test. If you simply want to know if activity at either of these time points correlates with reaction time, then you should compute two separate correlations. You could then make statements like "reaction time is significantly correlated with y1, but not with y2". To additionally say something about whether one correlation is significantly different from the other, you'd have to do a test on the difference between them.

With regards to testing this difference, it doesn't matter in this case if one of the correlations is not significantly different from 0. In fact this is a common mistake that people make when interpreting patterns of results, where they will assert that a is significantly different from b, based on the fact that a is significantly different from 0 while b is not. To actually be able to claim this, you have to test a against b directly, and it is perfectly possible for that test to fail to reject the null hypothesis under those circumstances.

I don't see for what reason concatenating y1 and y2 would make sense. If you want to know if y1 and y2 explain independent variance in X, then the thing to do would be to run a multiple regression analysis with y1 and y2 as predictors (and an intercept as the third predictor) in the regression model and X as the outcome variable, and compare that model to a model with only y1 or y2 (and again including an intercept).

If you want to know whether there is any relationship between y1 and y2, and X, you could test the model with both predictors against a model with only an intercept. Opinions vary (I believe) on whether it is better to start with a full model and then work your way down the various nested hypotheses, or start with the simple hypotheses and then build to more complex models.

To compare regression models, you would use an F-test. In general, I would advise you to read up on regression analyses. Note that a correlation is simply a normalized version of a regression, where the regression slope is expressed in terms of the variance shared between the predictor and the outcome variable. Multiple regression in turn is equivalent to a partial correlation analysis (where you assess whether each predictor variable explains unique variance after the effects of the other predictors have been removed).

• Thanks. This was useful. Effectively, maybe I should just compare correlation coefficients, using a measure of effect size (like Cohen's q, or maybe bootstrapping the correlation coefficients before). Nov 25, 2016 at 13:29
• In your question you said 'using all the data I have'. I think Ruben's suggestion in his last three paragraphs makes use of all your data in a way in which your suggestion in this comment does not. Nov 25, 2016 at 13:43
• Actually it is a matter of 'all the data I have at once' or 'not at once'. Anyway, I'm considering both ways. Any suggestion is appreciated! Nov 25, 2016 at 15:13