1. Stat 414 – Day 19

2. Last Time – Random coefficients
JMP and R output no longer agreeing
Example 5.1
Is variation between slopes large?
Interpretation of intercept
Interpretation of negative correlation between slopes and intercepts
Classes with a higher performance for a pupil of average intelligence (higher intercept) have a lower within-class effect of intelligence (lower slope). So their high intercept coming from high performance of lower IQ students and less by higher performance of higher IQ students.

3. Case Study - Musicians
Looking at “na” – negative affect (anxiety) before each performance
Null model
na ~ 1 + (1 | id)
18% of variation in anxiety is at the
musician level (correlation of responses by same individual)
With random effect Without random effect

4. Case Study – Musicians Random intercepts & slopes
Subjects with greater intercepts tend to have steeper negative slopes
Random intercepts & slopes
na ~ large + (large | id)
16.7=  mean performance anxiety level before solos and small perfs.
-1.7= the mean decrease in performance anxiety before large perfs.
21.8= perf-to-perf variation within musician.
5.5 (6.3)= the between-person variance in performance anxiety scores with small performances
0.7 = the between-person variance in increases (or decreases) in scores before large ensembles.
subjects with higher levels of performance anxiety before solos and small ensembles tend to have smaller increases (or greater decreases) in performance anxiety before large ensembles; those with higher levels of performance anxiety before solos and small ensembles have more opportunity for decreases before large ensembles.

5. Multiple random slopes
Consideration of whether or not want the random effects to be associated

6. Level 2 Covariate Composite model
Orch, Large, Orch*Large, random intercepts and random slopes for Large
na ~ orch + large + orch:large + (large | id)
15.9. estimated mean performance anxiety for small ensembles (Large=0) for keyboard players and vocalists (Orch=0)
0.9. Estimated mean decrease in anxiety for keyboard players and vocalists (Orch = 0) when playing in large instead of small
1.7. Orchestral instrumentalists have an estimated mean performance anxiety for small ensembles (Large = 0) 1.7 points higher than keyboard players and vocalists.
-1.4: Estimated mean decrease in anxiety for orch when playing in large instead of small is -.9 – 1.4 = -2.3

7. Level 2 Covariate Composite model
Orch, Large, Orch*Large, random intercepts and random slopes for Large
na ~ orch + large + orch:large + (large | id)
5.7: Estimated variance of na for small ensembles, after controlling for instrument played.
0.4: The estimated variance of differences in performance anxiety levels between large ensembles and small, after controlling for instrument
21.8: Estimated variation in residuals for the individual regression models

8. Level 2 Covariate Composite model
Orch, Large, Orch*Large, random intercepts and random slopes for Large
na ~ orch + large + orch:large + (large | id)
Negative covariation between intercepts and slopes: After controlling for instrument played, those subjects with higher performance anxiety scores for solos and small ensembles tend to have greater decreases in performance anxiety for large ensemble performances.

9. Example 5.2 IQ and mean IQ in text Coefficient of IQ_verb in a school:
meanIQ + random slope effect
A negative mean IQ means the effect of individual IQ on language test is much larger

10. Interactions
Random slopes – interaction with level 2 grouping variable
Interactions
Within level
Cross level – including Level 2 covariates to explain variation in intercepts, slopes

11. To Do
HW 4
Next project report