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I'm having problems understanding how to interpret correlations in mixed models

sub <- c(1,1,1,1,2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,5,6,6,6,6,7,7,7,7,8,8,8,8,9,9,9,9,10,10,10,10)
f1 <- c(13,13,17,22,29,21,12,34,22,26,28,29,11,11,14,12,18,12,9,33,33,38,22,37,12,16,22,21,11,10,10,10,22,23,35,21,2,3,5,6)
f2 <- c(22,25,33,13,21,33,12,2,26,22,33,31,21,26,27,26,21,11,14,17,45,37,34,35,31,27,29,39,10,12,12,14,17,12,13,14,10,10,10,10)
y <-  c(12,14,21,19,25,32,21,22,33,23,28,32,14,15,18,14,18,12,11,21,33,43,32,38,9,9,21,19,16,14,14,14,21,11,44,41,14,11,11,10)

dat <- data.frame(sub=sub, f1=f1, f2=f2, y=y)

m <- lmer(y ~ f1 + f2 + (1|sub) ,data=dat)

Fixed effects:
            Estimate Std. Error      df t value Pr(>|t|)    
(Intercept)   4.7549     3.2024 12.1903   1.485    0.163    
f1            0.6502     0.1256 27.5841   5.178 1.78e-05 ***
f2            0.1813     0.1214 25.0848   1.493    0.148

My questions are:

1) how can I extract the individual correlations between f1 and y, and between f2 and y from m? Do I need to create separate models for each predictor?

2) how can I interpret the significant Estimate of f1? Is it accurate to say that a unit increase in f1 changes the y by 0.65?

3) the summary function tells me that the correlation between predictors is -0.31 but cor.test(dat$f1,dat$f2) gives me 0.42. I know I'm overlooking something but I wonder what.

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A couple of points:

  • The correlation between the outcome variable y and a predictor f1 is related to the coefficient of the predictor. When you put two predictors in the model, like in your case, then you have partial correlations.
  • In the case of simple linear regression, you can easily transform the coefficients to correlations. But in the case of clustered/grouped data, things become more complex.
  • To my view, coefficients are easier to interpret than correlations. And indeed the interpretation is as you wrote, i.e., that a unit increase in f1 changes the average y by 0.65. Note that the interpretation of the coefficients is independent of their statistical significance.
  • The correlation between f1 and f2 you obtain in the output of the summary() function is the correlation between the estimated fixed effects coefficients - not the correlation between the predictors f1 and f2.
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  • $\begingroup$ Thank you Dimitris, that is very helpful. Regarding the first point, I was not sure adding two predictors would amount to doing partial correlations. For example the answers in this post seem to suggest that lm does not use partial correlations, or am I interpreting the answers wrong? $\endgroup$ – locus Nov 16 '18 at 0:19
  • $\begingroup$ Regarding the second point, I tried to find a way to transform coefficients into correlations here in CV but only found a similar question with no answer. Have you ever attempted doing this yourself, or are you aware of any package that can help me with that? $\endgroup$ – locus Nov 16 '18 at 0:22

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