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

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

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

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

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