L12handout - PUBH 7430 Lecture 12 J. Wolfson Division of...

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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, two-step 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/p-values even if the assumed variance function and correlation structure are incorrectly specified....
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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.

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L12handout - PUBH 7430 Lecture 12 J. Wolfson Division of...

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