L16handout - PUBH 7430 Lecture 16 J Wolfson Division of...

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Unformatted text preview: PUBH 7430 Lecture 16 J. Wolfson Division of Biostatistics University of Minnesota School of Public Health October 27, 2011 Random/mixed effects models, continued Example: Wallaby growth Data Data on growth (weight, length of ears/head/feet/arms/tail) of 77 wallabies over four years Question How fast do a wallaby’s ears grow during the second year of life? Example: Wallaby growth A simple linear model E (EarLengthij | Ageij ) = β0 + β1 Ageij Fitting this model (age starts counting from zero at one year of age), we get Variable (Intercept) Age (days) Coef 627.9 0.15 95% CI (618.0, 637.7) (0.10, 0.21) (very similar results obtained if we fit GEE w/ independence working correlation) Example: Wallaby growth Wallabies: Ear length vs. Age q q q 700 q q q q qq q q qq q q 650 qq q q q q q q 600 q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q qq q q q q q q q 0 q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q qq q q q q q q q q q q q q q q q Ear length q q qq q q q q qq q qq q q qq q q q q q q q q q q 50 100 150 200 250 Age in days since one year old 300 350 Example: Wallaby growth But... ... wallabies start with different ear lengths! Wallabies: Ear length vs. Age q q q 700 q qq q q qq q 650 qq q q q q q q 600 q q q q q 0 q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q qq q q q q q q q q q q q q q q q q q qq q q q q q Ear length q q q qq q q q q q q qq q q qq q q q q q q q q q q 50 100 150 200 250 Age in days since one year old 300 350 Example: Wallaby growth May want to include a “wallaby effect” to account for this heterogeneity: Random effects model E (EarLengthij | Ageij , wi ) = β0 + β1Ageij + wi wi = wallaby effect, distributed as N (0, D ) Example: Wallaby growth The regression line for ear length as a function of age has • An intercept which is unique to each individual: ˆ ˆ E (EarLengthij | Ageij = 0) = β0 + wi ˆ • A slope which is common to all individuals: ˆ ˆ ˆ E (EarLengthij | Ageij = a+1)−E (EarLengthij | Ageij = a) = β1 Random effects Ear length vs. Age, Animal 81 q q 650 0.000 600 0.004 Ear length 700 0.008 q −100 −50 0 50 100 q q q q q q q q q qq qq q q q qq q q q qq qqq q q qq q q qq q q q qq q q qq qq q q q q q q q qq q q q q qq q q q q q q q q qq q q qq q q qq q q q q q q q q qq q q q q qq q q qq q q q q qq q q q q q qqq q qqq q q q q q q qq q qq q q qq q q qq qq q q qq qq q q q qq q q qq q q qq qq q qq q qq q q q qqq qq q q q qqq q qq q q q q q q q q q 0 50 100 150 200 250 300 350 Age in days since one year old Random effects Ear length vs. Age, Animal 87 q q 0.000 650 600 0.004 Ear length 700 0.008 q −100 −50 0 50 100 q q q q q q qq q q q qq qq q q q q qq q q qqq q qq q q q qq q q q q q q q q qq q q q q q q q q qq q q q q qq q q q q q q q q qq q q qq q q qq q q q q q q q q qq q q q qq qq q q q q qq q qq q q q q q q qqq q qq q q q q q q q qq q qq q q qq q q q q qq q q qq q q q q q qq q q qq q q qq q q qq q qq qq q q q qqq q qq q q qqq q qq q q q q q q q q 0 50 100 150 200 250 300 Age in days since one year old 350 Random effects Ear length vs. Age, Animal 45 q q 650 0.000 600 0.004 Ear length 700 0.008 q −100 −50 0 50 100 q q q q q q qq q q q qq qq q q q q qq q q qqq q qq q q q qq q q q q q q q q qq q q q q q q q q qq q q q q qq q q q q q q q q qq q q qq q q qq q q q q q q q q qq q q q q qq q q qq q q q q q qq q q q q qqq q qqq q q q q q q q qq q qq q q q qq q qq qq q qq q q qq q qq q q qq q q qq qq q qq q qq q q q qqq qq q q q qqq q qq q q q q q q q q 0 50 100 150 200 250 300 350 Age in days since one year old Random effects Ear length vs. Age, Animal 47 q q 0.000 650 600 0.004 Ear length 700 0.008 q −100 −50 0 50 100 q q q q q q qq q q q qq qq q q q q qq q q qqq q qq q q q qq q q q q q q q q qq q q q q q q q q qq q q q q q q q qqq q q q q qq q q qq q q qq qq q q q q q q qq q q q q q qq q q qq qq q qq q q q q q qqq q qqq q q q q q q qq q qq q q qq q q qq qq q q qq qq q q qq q q q qq q q qq qq q qq q qq q q q qqq qq q q q qqq q qq q q q q q q q q 0 50 100 150 200 250 300 Age in days since one year old 350 Random effects fit Fitting the model (details later), we obtain Variable (Intercept) Age (days) Coef 626.5 0.15 95% CI (614.6, 638.4) (0.13, 0.16) Coefficient interpretation ˆ E (EarLengthij | Ageij ) = 626 + 0.145Ageij + wi Question How do we interpret the Age effect? “Conditional on wi , a one-day difference in age is associated with a 0.145-millimeter (?) increase in ear length” Interpreting coefficients Two possible interpretations: Interpretation A: Within-cluster effect “On average, a wallaby’s ears grow by 0.145 millimeters a day between one and two years of age.” Interpretation B: Between-cluster effect for two “average” clusters “Two ’average’ wallabies who differ by one day in age are expected to have ear lengths which differ by 0.145 millimeters.” Linear vs. random effects model Comparisons • Estimates of age effect: • Simple linear model: 0.152 • Random effects model: 0.145 • 95% confidence intervals for age effect: • Simple linear model: (0.10, 0.21) • Random effects model: (0.13, 0.16) Random effects as variance partitioning ˆ Standard error for age effect (β1 ) using simple linear model: 0.0269 Standard error for age effect using random effects model: 0.0096 What happened? Random effect has “soaked up” some of the variability and given us a more precise estimate of the age effect Random effects as variance partitioning Another way of viewing random effects models is as a way of partioning sources of variability (a la ANOVA). EarLengthij = β0 + β1 Ageij + δij Error term δij reflects two sources of variability: • Variability in ear length between wallabies • Variability in ear length within the same wallaby Random effects as variance partitioning Another way of viewing random effects models is as a way of partioning sources of variability (a la ANOVA). Rewrite the simple linear regression model as: EarLengthij = β0 + β1 Ageij + wi + ij where • wi ∼ N (0, D ) reflects the variablity in ear length measurements due to individual “wallaby” effects • ij ∼ N (0, σ 2 ) reflects the residual variability for each individual wallaby’s ear length measurements • wi and ij are all independent • Both D and σ 2 are unknown Random effects as variance partitioning EarLengthij = β0 + β1 Ageij + wi + ij • In addition to estimating fixed effects β0 and β1 , we also want to estimate Var (wi ) = D and Var ( ij ) = σ 2 • Relative size of D and σ 2 reflects relative contribution of cluster-level (i.e. wallaby) and observation-level (i.e. residual) variation to overall variability in the outcome. σ 2 ⇒ Cluster effects account for most of variability in outcome • σ2 D ⇒ Variability in within-cluster observations accounts for most variability in outcome •D Random effects Ear length vs. Age, Animal 81 q q 650 0.000 600 0.004 Ear length 700 0.008 q −100 −50 0 50 100 q q q q q q q q q qq qq q q q qq q q q qq qqq q q qq q q qq q q q qq q q qq qq q q q q q q q qq q q q q qq q q q q q q q q qq q q qq q q qq q q q q q q q q qq q q q q qq q q qq q q q q qq q q q q q qqq q qqq q q q q q q qq q qq q q qq q q qq qq q q qq qq q q q qq q q qq q q qq qq q qq q qq q q q qqq qq q q q qqq q qq q q q q q q q q q 0 50 100 150 200 250 300 350 Age in days since one year old Random effects Ear length vs. Age, Animal 87 q q 0.000 650 600 0.004 Ear length 700 0.008 q −100 −50 0 50 100 q q q q q q qq q q q qq qq q q q q qq q q qqq q qq q q q qq q q q q q q q q qq q q q q q q q q qq q q q q qq q q q q q q q q qq q q qq q q qq q q q q q q q q qq q q q qq qq q q q q qq q qq q q q q q q qqq q qq q q q q q q q qq q qq q q qq q q q q qq q q qq q q q q q qq q q qq q q qq q q qq q qq qq q q q qqq q qq q q qqq q qq q q q q q q q q 0 50 100 150 200 250 300 Age in days since one year old 350 Random effects as variance partitioning How does this explain the narrower CI for age effect in random effects model? ˆ • Estimation error in β1 is a function of residual variability • In linear model, all variability (error) is residual • In random effects model: • We condition on wi • Variation due to cluster (wallaby) is accounted for • Residual variability is much smaller ˆ • Estimate of β1 is more precise ...
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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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