1. EHS 655 Lecture 15: Exposure variability and modeling

2. What we’ll cover today Within- and between-subjects variability
Variance components
Fixed, random, and mixed effects

3. WITHIN- AND BETWEEN-PERSON VARIABILITY
We’ve already talked about within- and between group variability
What about variability at individual level?
Between-person variability
Reflects differences between workers beyond those explained by shared characteristics (e.g., workplace)
Within-person variability
Reflects variability due to day to day differences in exposure

4. Example: within- and between-person variability
A Burdorff 2005

5. Example: within-person variability between groups
Rappaport, 2008

6. How do we evaluate within- and between-person variability?
Step 1: measure it
Need repeated measurements on individuals
Step 2: analyze it
Need to evaluate variance components
Within-person
Between-person
Why do we care?
Relative importance of day-to-day differences in exposure vs between-worker differences
Useful when considering approaches to exposure control

7. Exercise Why do we care about variance components?
Think about how within-person variability applies to exposure controls
For which jobs do you think training and personal protective equipment might be most and least effective at controlling exposures?
Welder:  within-person,  between-person
Mechanic:  within-person,  between-person
Assembler:  within-person,  between-person

8. Within- and between-person variability and controls
We always adhere to the hierarchy of controls
But…theoretically…
Engineering controls
Could be more effective where within-subject variance >> between-subject
Administrative and personal protective equipment (PPE)
Could be more effective where between-subject variance >> within-subject

9. VARIANCE COMPONENTS σT2 = σW2 +σB2 Where Total variance, σT2
Within-person variance, σW2
Between-person variance, σB2
σW2 requires repeated measurements on individuals

10. Repeated measurements
Correlation between repeated measurements for person must be taken into account
Where ρ is correlation between any two repeated measurements on individual
We typically assume this is constant over time

11. Can we analyze variance via standard multivariable linear regression?
Not accurately
Problem
Ignores within-person variability and correlation between measurements on same person (i.e. errors not independent)
Appropriate only if people have exactly one measurement each

12. Repeated measurements – attenuation
Where
b = expected value of regression coefficient of dependent Y on independent X
Β = true regression coefficient
σW2 = within-person variance
σB2 = between-person variance
Heederik, Bolei, Kromhout, Smid, 1991

13. Variance components example – job title
Heederik, Bolei, Kromhout, Smid, 1991

14. Variance components example – different exposure predictors
Kromhout et al, 1993

15. So, how do we get estimates of within- and between-person variance?
Things to include in analyses
We’ve already been considering fixed effects
Shared characteristic (e.g., workplace, department)
Average response of group from common regression model
Now we can add random effects
Allows estimation of responses for each individual (i.e., accounts for individual variability)
Mixed models include both effects

16. Fixed vs. random effects
Rappaport et al, Ann Occup Hyg, 1999

17. Variance components approach 1: Oneway random effects ANOVA
Strength
Addresses between- and within-person variability
Weakness
Ignores work characteristics
Stata: loneway depvar idvar
Take within- and between-person SDs that are output and square them to get variance components
Compute ƛ, (σW2/σB2)

18. Variance components approach 1: Oneway random effects ANOVA
Can also look at within- and between-group variance
Stata: loneway depvar groupvar
Take within- and between-group SDs that are output and square them to get variance components

19. Reference: One-way random effects ANOVA model for between- and within-person variance
Where
Xij is exposure level for ith worker on jth day
μ is mean exposure
Β1 is random deviation of ith person’s true exposure μyi from μy
ε is random deviation of ith person’s exposure for the jth day from true expsure μyi

20. Variance components approach 2: Mixed effects regression
Strengths
Can address work characteristics
Can evaluate variance components
Can consider several sources of variability
One continuous dependent variable (linear regression)
One binary dependent variable (logistic regression)
One to many categorical or continuous predictors

21. Mixed effects regression
Includes both fixed and random effects
Fixed effect: same coefficient applied to all in group
Random effect: coefficients vary by individual, site, or some other factor
Estimates within- and between-person variance while adjusting for fixed effects
Linear regression
Stata: xtmixed depvar indepvar || idvar:

22. Mixed effects vs multivariable linear regression
Peretz et al, 2002

23. Comparison: random vs mixed effects models
Peretz et al, 2002

24. Comparison: random vs fixed effects models
Peretz et al, 2002

25. Comparing models Log likelihood
Absolute number meaningless
Lower number is better
To compare nested models only (logistic regression)
Wald test – basically chi-square test to evaluate whether coefficient = 0
Likelihood ratio test – measure of unexplained variance in the dependent variable
Smaller value = better fit, but absolute number is meaningless

26. Resources Linear models – repeated measurements analysis
Interpreting Stata linear mixed model output

27. Reference: Mixed effects regression model
Where
Xij is exposure level
i=worker 1 to k
j = 1 to n repetitions for the ith person
Β0 is intercept for background exposure group
Β1 to Βp are fixed effects
Xij1 to xijp are values of variables on ith person on jth day
bi to bk is ith worker random effects (discrepancy between their intercept and group intercept)
Z1 to zk are person’s indicators (0/1 – i.e., dummy variables)