This preview shows pages 1–8. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: PUBH 7430 Lecture 12 J. Wolfson Division of Biostatistics University of Minnesota School of Public Health October 13, 2011 Estimation, briefly • In the GLM, we model the mean E ( Y i  x i ) = μ i = X i β • To assess covariate effects, we need to: • Estimate the coefficient vector β = [ β , β 1 , . . . , β p ] • Estimate the uncertainty in the coefficient estimates via Var ( ˆ β ) ≡ Σ ˆ β • NOTE: Since Σ ˆ β is a function of the variance matrix Var ( Y i  X i ) ≡ V i , we can focus on estimation of V i Estimation, briefly Estimating ˆ β and ˆ V i involves solving score equations * of the form K X i =1 ∂ μ i ∂ β V i 1 ( y i μ i ) = 0 • ∑ K i =1 : Each cluster contributes something • ∂ μ i ∂β j : How quickly does mean react to changes in β ? • y i μ i : Residual vector which decreases in magnitude when (estimated) mean is closer to observed outcome • V i : Variance matrix which scales residual vector * details about how to derive them in more advanced classes. Estimation, briefly K X i =1 ∂ μ i ∂ β V i 1 ( y i μ i ) = 0 Solving these equations is generally an iterative, twostep procedure: 1 Using current “guess” ˆ V i of V i , obtain estimate ˆ β 2 Using current “guess” ˆ β , obtain estimate ˆ V i of V i : 3 Repeat steps 1 and 2 until convergence Relaxing assumptions • Correctness of ˆ β and ˆ V depends on correctness of: • Assumed mean function • Assumed variance function and correlation structure • Hard to be confident that variance function and correlation structure are specified correctly, potentially serious consequences if they are not. • Would be nice to have method which gives approximately correct inference, even if we guess wrong about the structure of V The sandwich variance estimator Idea • “Fix up” the estimate of V using the estimated score function (details unimportant for this class) • Resulting estimator is a “matrix sandwich”: ˆ V sand = ˆ V 1 ˆ M ˆ V 1 which we call the sandwich variance estimator . Note: Can derive ˆ Σ sand ˆ β (sandwich variance estimate for variance matrix of ˆ β ) from ˆ V sand The sandwich variance estimator The sandwich variance estimator: • Yields approximately correct confidence intervals/pvalues even if the assumed variance function and correlation structure are incorrectly specified....
View
Full
Document
This note was uploaded on 11/21/2011 for the course PUBH 7430 taught by Professor Prof.eberly during the Fall '04 term at Minnesota.
 Fall '04
 Prof.Eberly

Click to edit the document details