Random intercept model Flashcards
Fixed versus random effects: Central difference
- RE intends to quantify and explain Level-2 variance (i.e. keeping school differences)
- FE controls for Level-2 variance (i.e. getting rid of school differences, only focussing to within school difference)
The random effects approach
- Think of cluster effects as latent variables (all the reasons/hidden forces why schools differ) –> when we introduce cluster effects, we try to uncover this group variability
- Random draw/sample from unknown variance (i.e. some schools will be extremely low performing, others extremely high performing)
- Estimate variance components
- Explain variance with X-variables (RI/RS)
RE assumption (MLM)
X-variables are uncorrelated with RE (uj)
(i.e. school level variables should be uncorrelated with x-variables otherwise there might be biased)
Why to use a Random Intercept model?
- Captures the group-specific variation in the baseline or average level of the outcome variable
- Visualizing models
- Residual diagnostics
How does the Empirical Bayes estimation work?
NOT ALL CLUSTERS ARE EQUAL:
- there may be inherent differences between clusters that can contribute to the observed outcomes
- can account for the within-cluster similarities and between-cluster differences -> provides more accurate estimates of the treatment effects or other parameters of interest
- then u fit your model based on the data you have and after having the model, you assign values to those schools by using a shrinkage factor
What is shrinkage and why do it?
- statistical technique used to reduce the variability or magnitude of estimated parameters towards a common value, typically the overall mean (pull to the mean)
- commonly employed in situations where the estimated values of certain parameters may be unstable, unreliable, or biased due to limited sample sizes, high levels of noise, or extreme observations
- main purpose of shrinkage is to improve the accuracy and precision of parameter estimates. By shrinking estimates towards a common value, shrinkage helps to strike a balance between individual-level estimates and the overall estimate, leading to more stable and robust results
Shrinkage example: Which school samples do we not trust?
- small school sample
- high within-school variation → the more clustered, the more we trust
shrinkage factor: k
how to get it / how it changes
gets larger under two conditions:
- if within school variance gets larger -> why is there so much spread?
- sample size of cluster gets smaller -> very unreliable
purpose:
downplays the importance of clusters with little information
Emprical Bayes (II)
- BLUP helps to ensure that the average prediction error is zero or very close to zero. This contributes to reducing the systematic bias in the predictions.
BLUP helps to reduce the variability or spread of the prediction errors around their mean. This results in more precise predictions with the smallest possible variance.
Random Intercept model
Fixed vs Random part
What are b0 and b1?
What are the 4 estimators estimators that give different weight to between-variance and within-variance?
- Total variance: ignore that there are different schools (POLS = pooled OLS) - no between/within variance
- Fixed estimator (FE): Look only at within variance (within each school) - averages the within effect - no between variance
- Between estimator (BE): only between schools, no within variance whatsoever (radically different assessment)
- Random effects estimator (RE): uses a bit of both (optimal use of data in terms of precision, efficiency, minimizing SE - only valid if RE assumption holds) → this is the estimator you get, when using mixed (Stata)
Interpret the coefficient of ESCS
Interpret the variance change
Random Intercept model with level-1 & level-2 variables
a) Interpret the coefficient of Private school in M3
b) What happened to variance from model 1 to model 3?
c) Explain the change in coefficient from 3 to 4.
d) Are private schools better at teaching maths?
