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When I fit any model in lmer(), summary identifies a correlation between fixed effect(s) and the (fixed) intercept. How should I interpret this, and why is it not $0$? In my way of thinking any correlation between the intercept and a fixed effect should be zero, since, in the design matrix $X$, the intercept term is just a column of $1$s, and therefore constant. There is a related post on this topic (lmer interpretation of correlation), but it doesn't help me much. Below is model summary output, so you can see what I mean. See the block labeled Correlation of Fixed Effects:, at the very bottom.

>summary(exp2modFull)
Linear mixed model fit by REML ['lmerMod']
Formula: RT_log ~ Condition + (Condition | Subject) + (Condition | Item)
   Data: exp2

REML criterion at convergence: -1978.6

Scaled residuals: 
    Min      1Q  Median      3Q     Max 
-3.7080 -0.5775 -0.0634  0.4801  7.8186 

Random effects:
 Groups   Name            Variance  Std.Dev. Corr       
 Subject  (Intercept)     0.0261472 0.16170             
          Conditionoddman 0.0028821 0.05369  -0.21      
          Conditionhetero 0.0037356 0.06112  -0.46  0.80
 Item     (Intercept)     0.0018914 0.04349             
          Conditionoddman 0.0002885 0.01699  -0.97      
          Conditionhetero 0.0010140 0.03184  -0.65  0.81
 Residual                 0.0320147 0.17893             
Number of obs: 3600, groups:  Subject, 20; Item, 12

Fixed effects:
                 Estimate Std. Error t value
(Intercept)      6.583310   0.038622  170.46
Conditionoddman -0.009109   0.014883   -0.61
Conditionhetero  0.021487   0.018018    1.19

Correlation of Fixed Effects:
            (Intr) Cndtnd
Conditnddmn -0.309       
Conditinhtr -0.472  0.722
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This is a general issue of parameter estimates and not limited to mixed models (compare How to interpret parameter estimates correlated with the intercept parameter estimate?).


For your case with the $1$-vector and one covariate $x$ (both with length $n$), let's define the $n \times 2$ matrix $X = \begin{bmatrix}1 & x\end{bmatrix}$. The covariance for the fixed effects is $\sigma \cdot (X'X)^{-1}$ (see for example eq. 34 in https://web.stanford.edu/~mrosenfe/soc_meth_proj3/matrix_OLS_NYU_notes.pdf). In your case this would be $\sigma \cdot \begin{bmatrix} \sum x^2 & -\sum x \\ -\sum x & n\end{bmatrix}$. Here you see the clear correlation between the estimates for the intercept and slope parameter (and also that it becomes $0$ if you center $x$ around $0$).

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