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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.

locus
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1 Answers1

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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.
Dimitris Rizopoulos
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  • 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](https://stats.stackexchange.com/questions/37785/does-lm-use-partial-correlation-r-squared-change) seem to suggest that `lm` does not use partial correlations, or am I interpreting the answers wrong? – locus Nov 16 '18 at 00:19
  • 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? – locus Nov 16 '18 at 00:22