Change in prediction (second prediction - first prediction)
Change in prediction between exams (updating)

Multilevel model Summary
## Linear mixed model fit by REML. t-tests use Satterthwaite's method [
## lmerModLmerTest]
## Formula: pred_delta ~ PE_lag1 + (1 + PE_lag1 | id)
## Data: df
##
## REML criterion at convergence: 11480.4
##
## Scaled residuals:
## Min 1Q Median 3Q Max
## -5.3640 -0.4760 -0.0148 0.5107 5.2881
##
## Random effects:
## Groups Name Variance Std.Dev. Corr
## id (Intercept) 0.1942 0.4407
## PE_lag1 0.0467 0.2161 -1.00
## Residual 194.4667 13.9451
## Number of obs: 1407, groups: id, 519
##
## Fixed effects:
## Estimate Std. Error df t value Pr(>|t|)
## (Intercept) 1.348e+00 3.847e-01 1.365e+03 3.503 0.000474 ***
## PE_lag1 3.626e-01 2.784e-02 3.992e+02 13.027 < 2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Correlation of Fixed Effects:
## (Intr)
## PE_lag1 -0.096
## convergence code: 0
## boundary (singular) fit: see ?isSingular
Raw data with regression line

Plotting random effects
