1. Working Independence versus modeling correlation Longitudinal Example
Generate data in clusters (i.e., a person)
5 observations per cluster
Response is a linear function of time
Yit = 0 + 1t + eit
The residuals are first-order autoregressive, AR(1)
eit = ei(t-1) + uit (the u’s are independent)
corr(ei(t+s) , eit) = s
Estimate the slope by
OLS: assumes independent residuals
Maximum likelihood: models the autocorrelation
Bio753: Adv. Methods III

2. Comparisons Compare the following reported Var(1)
That reported by OLS (it’s incorrect)
That reported by a robustly estimated SE for the OLS slope (It’s correct for the OLS slope)
That reported by the MLE model
It’s correct if the MLE model is correct
You can use any working correlation model,
but need a robust SE to get valid inferences
Bio753: Adv. Methods III

3. Variance of OLS & MLE Estimates of b versus , the first-lag Correlation
MLE reported variance
OLS reported variance
True variance of OLS
Bio753: Adv. Methods III

4. Benefits & Drawbacks of working non-independence
Efficient estimates
Valid standard errors and sampling distributions
Protection from some missing data processes
The MLM/RE approach allows estimating conditional-level parameters, estimating latent effects and improving estimates
Drawbacks
Working non-independence imposes more strict validity requirements on the fixed effects model (the Xs)
Can get valid SEs via working independence with robust standard errors
At a sacrifice in efficiency
Bio753: Adv. Methods III

5. There is no free lunch! Working independence models
(coupled with robust SEs!!!)
are sturdy, but inefficient
Fancy models are potentially
efficient, but can be fragile
Bio753: Adv. Methods III