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L6handout - PUBH 7430 Lecture 6 J Wolfson Division of...

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Unformatted text preview: PUBH 7430 Lecture 6 J. Wolfson Division of Biostatistics University of Minnesota School of Public Health September 22, 2011 Canonical correlation structures independence When looking at sample correlation and scatterplot matrices, it is useful to have in mind some “canonical” correlation structures. Independence 1 if u = 0 0 otherwise 1 0 0 ... R (A, B , . . . ) = 0 1 0 . . . ... . ... . ... . A(u ) = Canonical correlation structures independence 0.8 0.4 0.0 1 2 3 4 5 lag −2 012 −2 012 012 3 2 q qq q q q q qq q qq q qq q qq q q q qq q qq q q q q q q qq q qq qq q qq q q q q 1 q −1 0 1 −1 0 1 qq qq q qq q qq q q qqq q q qq qq q qq q q qqqq q q q q q qq qq q q q qq q q q q qq q q q qq q qq qq q q q q q q qq q q qq q q q q qq q qq q q q q qq q q q q qq q qq q q q −3 −2 Lag 5 qq −3 q q q qq q q q q qq q q qq q q q q qq q q q qqq q q qq q q qq q q q q qq q q q q q qqq q qq qq q q qq q q q q q qq q q q q q qq qq q q qqq qq qq q q q qq qq qq q qq q q qq qq q q q qq q q q q q qq q q q q q qq q q q qq q q q q qq q q qq q q q qq 2 3 Lag 4 q −1 0 q qq q q q q q q qq q q qq q qq qq q q qq q q qq qq q qq q q q q q q q qq q qq q qq q q q q qq q q qq q q q q qq q qq q q qq qq q q qq q q qqq q qq q q qq q q q qq q qqq qqqq qqq q q qq qq qq q q q q q q q q qqqq q qqq q qq q q qq qq q q qq q q qq q qq q qqq qqq q q qq q q q qq qq q q q q qq q q qq q q q qq q qq q q q q 2 3 Lag 3 −3 1 2 3 Lag 2 −1 0 q q qq qq q qq qq q q q q q q qqq q qq q qq q q qq q q q qq q q q q qq q q q q q q qqq q q q qq q qqq q qq qq q q q qq q q q q q qqqq qqq qq qq q q q qq q qq q qq q q qqq q q qq q q q qq q q qq q q qq q q qqqqq q qqq q q qq q qqqq q q q q qq q q qq q q q q qq qq q q qq q q q qq q q qq q q q qq qq q q qqqq q q q q qq q q q q qq q q qq q q q qq qq q q q q q qq qq q q q q qq q q q q qq q −3 −1 0 1 2 3 Lag 1 −3 Autocorrelation Autocorrelation function −2 012 −2 012 Canonical correlation structures exchangeable Exchangeable (compound symmetry) 1 if u = 0 ρ otherwise 1 ρ ρ ... R (A, B , . . . ) = ρ 1 ρ . . . ... . ... . ... . A(u ) = Example: FEV measured weekly from students in this class Canonical correlation structures exchangeable 0.8 0.4 0.0 1 2 3 4 5 lag −1 12 −1 12 −1 0 3 qq q q 12 2 q qq −3 q −3 −1 q q q q qq q q q qq qq q q qqq qq q qq qqq qq q q q q qqq q q qq q qq qq qq 1 1 2 qq qq qq q qq q q q qqq qq q qqqq qq q qq qq q q q qq q qqqqq q qqq qq qq q q q qqqq q qqqq q q qq q qq q q qq q q q qq q q q q qq q −1 0 3 2 q q q q q q q qq qq q qq q qq q qq qq q q qq qq q q q qq qq q q q q q qq q qq q qq qq q qq qqqqq qqqqqq q qq qq q qq qq q qqqq q qqq qq qqqq qqq q qqqq qq q qq qq q qq qq q qqq q q qqq q qq q qq q q q Lag 5 3 Lag 4 q −3 q qq q qq qqq qq qqq q q qq qqqq q qqq q q qq q qq q q q qq q q qq q q q q qqqq q q qqqqq q qqqqq qqqqq q qqqq qqqq qqq qq q qqqq q qqqq q q q qqqq qq qq qqqq qqq qqqq qq qq qq qqqqqq q qq q q qq q qq q q qq q q qq qq q q qq q q 1 3 2 q 1 Lag 3 q qq q −3 q qq q qq qq qq qqq qq q q q q qq q qq q qqqq q qqq q qq qq q q q qq q qq q q q qqq q qq q qqqqqqq qq q qq q q q qq q qq qq q q qqqq q qq q qqqq qqqq qqqq q qqq qqqqq q qq qqqq q q qqqq q q qqq q qq qq q qqqq q q qqqq qq q q q qqq qq q q q qq q qq q qqqq q qq qq qq qq q q q qq qq q q q q q −1 0 3 2 1 −1 0 Lag 2 q −1 0 Lag 1 −3 Autocorrelation Autocorrelation function −1 12 −1 12 Canonical correlation structures auto-regressive First-order auto-regressive (AR-1) A(u ) = 1 ρu if u = 0 otherwise For A, B , C , . . . equally spaced at 1 ρ R (A, B , C , . . . ) = . . . times t = 1, 2, 3, . . . , ρ ρ 2 ρ3 . . . 1 ρ ρ2 . . . ... . ... . ... . Example: Heart rate measured every minute over an hour from patients in a hospital Canonical correlation structures auto-regressive 0.8 0.4 0.0 1 2 3 4 5 lag −2 012 −2 012 −2 012 1 3 −2 012 q 2 2 q q qq q q qq qq qqqq q q q q q q q qq q q q qq q q qq q q q q q qq qqqqqq q q q q qq q qq qq q q qq q qqqq qq q q qqq qqq qq qq qq q q q q q qq q q qq q q q q qq qq qq q qq qq q q q qq qq qq q q qq q q qq q q qq q q q q qq q qq q q q qq q q −3 −1 0 q −1 0 q q qq q qqq q qq qq qq q qq qqq q q q q q q qq q qq q qq q q qqq q qq q q q qqqq qq qq q qqq qq q q q qq q q q qq qqqq qq qqq q q q q q qq q q q qqq q q q qq q q qq q qq q q q qq q q q qq q q q q qq q q q qq q q qq q q q q q qqq q qq Lag 5 1 3 Lag 4 −3 1 −1 0 q qq q qq qq q q q q qq q q q qq q q qq q q q qq qqq q q q qqqq q q q qq q qq q q qqq q q qqq q q q q qq q q q q qq q q q q q q qq q qq q q q qq q q q q q qqqq qq q q q q qqq qq q q q q q q qq qq q q q q q qqq q qq q qq q q q q q qqq q qq q q q qq q q q q qq q q q qq q q q q qq q q qq q q qq q q qq q q qq q q qq q q 2 3 Lag 3 −3 1 −1 0 q q q qq q qqq qq q q qq q qq q qq qq qq q qq qqqqq q qq q q qq qq qq q q q qq q qq q q qq q qqq qqq qq q q q qq q q q qqqq qq q q qq q q q q q qq q q q q q qq q q q q qq q q q q q q qq q q qq qqqqqqq q q q q qq q qq q qq q qq q qq q qqq q qqq q q qqqq q q qqq qq q q q q qq q q q q q q q q qq q q qq q q qqq qqq q q qq qq q q q qq qqq qq q qq qqqq q q qq q q q qq q 2 3 Lag 2 −3 −1 0 1 2 3 Lag 1 −3 Autocorrelation Autocorrelation function −2 012 Canonical correlation structures m-dependent M-dependent (banded) A(u ) = 1 ψ (u ) if u = 0 otherwise for some function ψ . For A, B , C , . . . equally spaced at times t = 1, 2, 3, . . . , 1 ψ (1) ψ (2) ψ (3) . . . ψ (1) 1 ψ (1) ψ (2) . . . R (A, B , C , . . . ) = ψ (2) ψ (1) 1 ψ (1) . . . . . . . . . . . . . . . . . . Canonical correlation structures m-dependent 0.8 0.0 0.4 A(u) = ρexpit(− 3 + 0.8u) 1 2 3 4 5 lag −2 0 2 −2 0 2 0 2 3 2 q qq q 0 −3 q −2 q q q q qq q qq q qq q q q qqq q q q q qq q qq q q qq q qq q q qq q q q qq qq qq 1 q qq q qq q q q q qqq q qq q q qq qqq q qq qqq q qqq q qq q q q qq q q q q q qq q qq q q q q qq q qq q q qqq q qqq q q q qqq qq q qq qq q q q q q q q q −1 0 2 1 −1 0 q qq q q qqq qq q qq q q qq q q q qq q q qq q q q q qq q q q qq q q qq q qqq q q qq q qq q qqq q qqqq qq q qqq q q q qqqq qq q q qq q qq q q q q q qqq q qq q q qq q q qqqqq q q q q q qqq q qq q q qq q q q q qq qq q qq q q q q qq q q q q q −2 Lag 5 3 Lag 4 q −3 1 qq qq q qq q qqq q q qq qq q qqq q q q qq q q q q q q q qq q q q qq q q q qq q qqqq q qqqq q q q q qq q q q q qq qq q q q qq qq q q qq q qq q q q q qqqqqq q q q q qq q q q q qqq q qq q q q q qq qq qq q q q q q qqqq q q qq q q qq q qq q qq q qqqq qqq q q q qqq qqq qq q qq q qq qq q q qq 2 3 Lag 3 q −1 0 q −3 1 −1 0 q q qq qq q q qq q q q qq q q qq q qq q q q q q q qqq q q q qq q qq q qq qqqq q q q qqqq qq q q q qqq q qq q q qq q q qqqqqq q q q qq qq qq q qq q q q q q qq q q q q qqqqqqq q q q q q qqqq q q q qqqq q qq qqq q qq q qqqq qq qq q q qq qq qq q q qqq qq q qq q q q qq q qqq q q q q qq q q qqq q qq q q qqq q qqq qq qq qq q q q qq q q q q q qqq q qq q qq q q qq q qq 2 3 Lag 2 q q −3 −1 0 1 2 3 Lag 1 −3 Autocorrelation Autocorrelation function 2 q −2 0 2 Canonical correlation structures unstructured Unstructured For A, B , C , . . . equally spaced at times t = 1, 2, 3, . . . , 1 ρ12 ρ13 ρ14 . . . ρ21 1 ρ23 ρ24 . . . R (A, B , C , . . . ) = ρ 1 ρ34 . . . 31 ρ32 . . . . . . . . . . . . . . . A(u ) not informative since process not stationary Autocorrelation: FEV data 0.8 0.4 0.0 2 4 6 8 10 lag 1.0 q 0.4 0.8 1.2 q q q 0.6 q qq qq qq q q Lag = 10 0.5 q q q 0.3 q qq q q qq qq q q qq q q q qq qq q qq q q q q q q q qq q qq q q q qq q q qq q q q q q qq q qq q qq q q 1.0 1.4 0.9 qq q q qq q qq q q q q q q q q q qq q qq q q q 0.1 0.4 0.2 0.8 0.4 1.2 0.8 q qq q qq q qq q qq q q q q qq qq qq q q qq q q qq qq q q q qqq q q q qq q q qq q qq qq qq qqq q q qq qq q q qqq qq q q qqq q qq q qqq q qq q q q q qq q qq q q q qq q qq qq qqqqqq q qqqq q q qqqq q q qqqqq q qqqq qqq q qq qq q q q q qq q qq q q q q Lag = 8 qq q q q q qq q q qq q q q q qqq q q qq qq qq q qq q q qq q qq q q qq q qq q q qq q qq qq q q q qq q qq qqq q q q q qqqq q qqq q q qq qq qqqq q q q qq q q q q q qq q q q qq q qq q q q q q q 0.6 Lag = 6 q 0.0 q 0.4 Lag = 4 0.4 qq q q q qq qq q q q q q qqq q qq q q q qq q qq qqq q q qq q qq q q qqq qq qq qq q qq qq qq qq qq q q qq qqq q qq q qqq q qq qq q qqqq qqqq q qqq q qq qq qq qqqq q qqqq q qq q qq qq qq q qqqq q qq qq q qq qq q qqq qqq qq q qq q qqq qq qq q q q qq q qqqq qqqq q q qq q q qq q qq q qq qq qq qqq qq q q 0.0 0.4 0.8 1.2 Lag = 2 0.0 Autocorrelation Sample autocorrelation function q 1.1 1.3 0.9 1.2 1.5 Covariates Role of covariates/predictors Often, we will want to describe the effects of covariates on a correlated outcome. Examples include: • Treatment • Time • Demographic factors • Other co-varying processes Summarizing covariate effects If covariates are time-invariant and categorical, can stratify usual summary plots/statistics by covariate levels. log(FEV) by Initial Height log(FEV) vs. Age 1.5 1.0 log(FEV) 0.5 0.0 0.0 0.5 log(FEV) 1.0 1.5 q q q qq q q q qq q q q q q q q q q q qq q q q q q q qq q qqq q q qq q q qq q q q qq q q qq q q q qq q qq qqq qq qq q q qq q qq q qq qqq q q qq q qq qq q q qq q q q qq q q q q q q q q q q qqq q q qqq q q q q qqq qq q q q q qq q q q q qq q q qq q q qq qq q q q q q q q q q q qq q qq q q q q q q qq q qq qqqq qqq qq qqqq q q qq q qq q q qqq q q q q qqq qqq q qq qq qqqqq q q q qqq qqq q q q q q q qq qqq qq q qqqq qqq qq q q qq q q qq q qq qq qqqqq q qqqqqq q qq q q qq q q q q qq q q q q qq q qq qq qq qqqqqq qqqqqq q qq qqq q q qqq qq qq q q q q q q q qqq qqqq q q q qqqqq q q qqq qq q qqqqqqqq qqqqqqqqq q qq qq q qqq qqqqqqq qqqqqqq q q q qq qqqq qq qq q q q q q q q qq q q q qqqq q qqqqq qq q q q qq qqq qqq qq q qqqqq qq q q q q qq q q q qq q qq q qq q q qq q q qqq qqqqq qqqq qqq q qq q q q q q qq q q q q q q q q qq qq qqqqqqqqq q qqqq q qqqq q q qq q q q qq qqqq q q qq q q q qqq qqq qqqqq q q q qqqq q q q q qq q q q q q q q q q q q q q q qq q q qq qqq qqq q q qqq q qq q q qq q qqq q q q q q q q q qq q q q q q q q qqq qq q qqq q qq q q q q qq q q q q q q qq q q q qq q q q q qq q qq qq q q q q q qq qqq qqq qq qqq qq q qq qq q q q qq qq q qq q q q qq q q qq qqq qq q q q qq q qq q qq q qq q q q q q q q q q q q q q qqqq q q qqqq q qq q q q q qqq q qqq qqq q q q q qqqq q qq qqqq qqq q q q qqq q q qq q q qq q qq q q q q q q qq q q qqqqqqq q q q q q q q q qq q qq q q q qqqqqq qqqqqq qq q q q q q q qq q qq qq qqq qq qq qqqqqq q qq qq q qq q qq qqq q q qq qq q qq q q q qq q qq q qq q q q q qq q q q qq qq qq qq qq q q q q qq q qqq q qq q qq qqqqqqqqq q q q qq qq qqq q q q qq q qqqq qq qq qq qq q qq q q q qq q q q q q qq q qq q q q q q q q q q qq qq qq q q q q q qq qq q qq q q qq q qq q qq q q qqqqqq q q q qqq q q qq q q q q qq qqqqqqqqq q qq q qq q qqq qq qqqq q q q q q q qq q q qq q q q qq qqq q q q q q qqqqq qq q qqq q q q qqqqqqq q q qq q qq q qqq qq q q q q qq q qq qq qq qq q q qqqq qq q q qq qq q qqq q qq q q qq q qqq q qqqqqqqqqqq qqqqq qqqqqq q q q qq qq q q q qq qq q qqqqqqqqqqq q q q q qq q qqq q q qqqqqqqqq qqqqqq qq q q q q q qq q qq q q qq qqqqqqq qqq q q q qq q qq qqqq q q q q qq q q q q qq q qq qqqq q qq qqq q q qqq q q q q qq qq q q q qq q qq q q q qqqq q q q q q q qq q q qq q qq q q qqqqq q q q q q qqq qq q q q qq q q q q q qqqqqqqqq qqq qq qq q qq qqqq q q q q qqq q q q qq q q qq qq q q qq qq q q qq q q q q q q q qqq qqq qqqq q q q q qqqq qq qqqqq q q q qq q q q q q q qqqq q q qqqq q q qq q q q q qq q qq qqq qqq qq q q q qqqq q q qq q q qq q qq q q qq q qqq q q q qq q q qq q qq q q qq qqq qqq qqq q qqq q q q q qq q q q q qq Init ht < median qqq qqq q q q qq qqqq q q qq qq qq qq qq q q q qq q q qq q qqq qqq q qq Init ht >= median q qq qq q q qq q q q q qq q qq q qq q q q q q qq q qqq q q q q q q q q q q qq q q qq q Init ht >= median Init ht < median 6 8 10 12 Age 14 16 18 Summarizing covariate effects If covariates are time-invariant and continuous, can plot outcomes at different times vs. covariate values. 1.10 1.20 1.30 Initial height 1.40 1.2 1.3 1.4 Initial height 1.5 q q q q q q q q q q qq q 0.8 q q q qq q q qq q qq q q qqq q q qq q q q q q qq q q q qq q q q qq q qqq q q q qqq q q q qq qq q qq q qq q qq q q q qq qqqqqq q q q qqq qqqq q qqq q q q qqqqqq q q q q qqqq q qqq q qq qq qq q qq qq q qq qq q q qqqqq q q qq q q qq q q q qqqqq qqqq q q qq q q qq q qq q q q qqq q q q qqqqq qq q qqqqqq qq q qq q qq q qq q q qq qq q qqqqqq qqq q qq qq q qqqqqqq q q q q q qq qqq q qqq q q q q qqq q q qq q qq q qqq q q q qqq q qqqq qq qq q q qqq q q qqqq q q q qq qq q q qqq qq q q qqq q q q q q q qqq qq q qq q qq qq q q q q qqqqqqq q q qq q q q qq q q q q qqq qq q qq q qq qq qqq qqq q qq q q q q q qq q q q qq q qq q qq q qq q qq q q qq q q qq q q q 1.3 1.5 Initial height q q q q q q q 1.1 1.2 q 1.0 q log(FEV) q qq qqq q qq qq q qq q q qqqq qqq qq q q qq q q qq q qqq q q q q qqq q q q q qqqqqqqq q q qq q q qqq q q q qqq qq q q q qq q q qqq q qqq q q qq q q qq q qq q qqq qq q qqq qqq qq q q qq qq qqqqqqqqq q q qq qqqq q q q q q q q qqqq q q q q qq q qq q q q qqq qq q q q q q q qqqq qq q qq qq q qqqqqq q q q qq qq qq qqqqqqq q qq q q qq q q qqq qq qq q q q q q q q q q qq q qq q q qq q q qq qqqq qq q q q qq q qq q q q q qq q q q q q q q qq q q q q q q qq q qq q 1.4 q q 0.6 1.2 1.0 0.8 q q qq qq qq qq q qqq q q qq q q qq q qq qqq qqq q q qq q qq qq q q q qq qqqqq q qq q qq q q qqq q q q qq q q q q q qq q q q qqq qqqqq q qq q q q q q q q q qq q qqqq q qqqqqq q qq q q q qqq q qq q qq qqq q q qq q q qq q q q qqq qqq qqq q q q qq qq qqqqqqqqqqqq q q q qqqq qqqq q q qqq q qqqqqqq qq q q q qq q qqqqqqqq q q qq q q qq q q qq q q q q q qqq q q q qq qqq qq q qq qqqqq q q q qq qq q q q q q q qq qq q qq q q q qq q q qq q q q qqq q q q q qq log(FEV) q 0.6 q 15−18 q q 0.4 0.8 log(FEV) 0.6 q 0.4 q q qq q q qq q qq q q qq q q qqqqqq qq q q qq qq q qqqqqq q qqqqq qq q qq qq q qq qqq q q qq q q qq q qq qq qqqqqq qq qqq q q q qqq qqqq q q q qq q q qq qq qq q q qqqq q qq qq q qqqq q q qq q q q q qqq q q qq q qq q q q qq qq q q qq q qq q qq qq qq q q q q qqq q q q qq q qq q q q 1.0 q 0.2 0.4 0.2 log(FEV) 0.6 q 0.0 12−14 q 1.2 q 1.4 9−11 0.8 6−8 q 1.1 1.3 1.5 Initial height 1.7 Summarizing covariate effects • If covariates are time-varying, can plot outcomes vs. covariates on same plot, connecting observations on same individuals • Plots can also be done using (within-person or within-time) outcome and covariate residuals. 0.8 0.6 q q q 0.4 0.2 log(FEV) 1.0 1.2 1.4 Log(FEV) vs. Height q qq q q q q q q q q 1.2 qq q qqqq q q qq qq q qqq q q q q qq qq q q q q q q q q qqq qq q qq q qq qq q qq q qq qq q q qq q qq q q q qq qqq q q qq qq q qqq qqq q q qq q q q qqqq qq qq q qq q qq q qq q qqq q qq q q qqq q q qq q qq qqq q qq q qq qq q q q q q q q qq q q q qq q qq q q q qq q qq qq q qq q q qq q q qq q q q q q q q 1.3 1.4 1.5 Height 1.6 1.7 ...
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