lecture4 - CoxRegressionII 1 1.Stratification Violations of...

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1 Cox Regression II
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2 1. Stratification  Violations of PH assumption can be resolved by: Adding time*covariate interaction Adding other time-dependent version of the covariate Stratification
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3 Stratification  Different stratum are allowed to have different baseline  hazard functions. Hazard functions do not need to be parallel between  different stratum. Essentially results in a “weighted” hazard ratio being  estimated: weighted over the different strata. Useful for “nuisance” confounders (where you do not care  to estimate the effect). Does not allow you to evaluate interaction or confounding  of stratification variable (will miss possible interactions).
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4 Males: 1, 3, 4, 10+, 12, 18 (subjects 1-6) Females: 1, 4, 5, 9+   (subjects 7-10) Example: stratify on gender ) .... ) 5 ( ) 5 ( ) 5 ( ( ) ) 1 ( ) 4 ( ) 4 ( ) 4 ( ( ) ) 4 ( ) 4 ( ) 4 ( ) 4 ( ) 4 ( ( ) ) 3 ( ) 3 ( ) 3 ( ) 3 ( ) 3 ( ) 3 ( ( ) ) 1 ( ) 1 ( ) 1 ( ) 1 ( ) 1 ( ( ) ) 1 ( ) 1 ( ) 1 ( ) 1 ( ) 1 ( ) 1 ( ) 1 ( ( ) ( 10 9 10 9 8 8 6 5 4 3 3 6 5 4 3 2 2 10 9 8 7 7 6 5 4 3 2 1 1 1 h h h x h h h h x h h h h h x h h h h h h h h h h h x h h h h h h h L L m i i p + + + + + + + + + + + + + + + + + + + + + = = = β
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5 The PL .... ) ) 1 ( ) 1 ( ) 1 ( ) 1 ( ) 1 ( ( ) ) 1 ( ) 1 ( ) 1 ( ) 1 ( ) 1 ( ) 1 ( ) 1 ( ( ) ( 10 9 8 7 7 6 5 4 3 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 1 βx βx βx βx βx βx βx βx βx βx βx βx β e e e e e x e e e e e e t L L f f f f f m m m m m m m m i i p λ λ λ λ λ λ λ λ λ λ λ λ + + + + + + + + = = = = )... ( ) ( ) ( 10 9 8 7 7 6 5 4 3 2 1 1 1 βx βx βx βx βx βx βx βx βx βx βx βx β e e e e e x e e e e e e L L m i i p + + + + + + + + = = =
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6 Age is a common confounder in Cox  Regression, since age is strongly related to  death and disease. You may control for age by adding baseline  age as a covariate to the Cox model. A better strategy for large-scale longitudinal  surveys, such as NHANES, is to use age as  your time-scale (rather than time-in-study). You may additionally stratify on birth cohort to  control for cohort effects.    2. Using age as the time-scale  in Cox Regression 
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7 Age as time-scale The risk set becomes everyone who was at  risk at a certain age rather than at a certain  event time. The risk set contains everyone who was still  event-free  at the age  of the person who had  the event. Requires enough people at risk at all ages  (such as in a large-scale, longitudinal survey).
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8 The likelihood with age as time Event times: 3, 5, 7+, 12, 13+ (years-in-study) Baseline ages: 28, 25, 40, 29, 30   (years) Age at event or censoring: 31, 30, 47+, 41, 43+ ) ) 41 ( ) 41 ( ) 41 ( ) 41 ( ( ) ) 31 ( ) 31 ( ) 31 ( ) 31 ( ( ) ) 30 ( ) 30 ( ) 30 ( ) 30 ( ) 30 ( ( ) ( 5 4 3 4 5 4 1 1 5 4 2 1 2 1 h h h h x h h h h x h h h h h L L m i i p + + + + + + + + = = = β
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9 3. Residuals Residuals are used to investigate the  lack of fit of a model to a given subject.  For Cox regression, there’s no easy  analog to the usual “observed minus  predicted” residual of linear regression
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10 Martingale residual c i
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