Contextual Analysis: Understanding and Interpreting Multilevel Statistical Models

1
Contextual Analysis Understanding and
Interpreting Multilevel Statistical Models

• Jay S. Kaufman, PhD
• University of North Carolina at Chapel Hill
• June 2007

2
objectives

• viewers should gain familiarity with
• common terminology for multilevel models
• the need to account for clustered data
• potential advantage of a biased estimator
• the idea of a shrinkage estimator
• specification of random effects
• interpreting the different types of multilevel
  models

3
definition and synonyms

• multi-level regression models allow for
  investigation of the effect of group or place
  characteristics on individual outcomes while
  accounting for non-independence of observations
• synonyms different models
• multilevel models - fixed effects models
• contextual models - random effects models
• hierarchical models - marginal models (e.g., GEE)
• longitudinal (panel) data, repeated measures
  designs use the same methods

4
motivation for multilevel models

• standard regression models are mis-specified for
  clustered data
• yi ?0 ?1xi ei e N(0,s2) i.i.d.
• next 3 slides
• hierarchical models outperform unbiased models
  (i.e., lower mean squared error)
• shrinkage

5
when observations are not independent

• dependence arises when data are collected by
  cluster / aggregating unit
• children within schools
• patients within hospitals
• pregnant mothers within neighborhoods
• cholesterol levels within a patient
• why care about clustered data?
• two children / observations within one school are
  probably more alike than two children /
  observations drawn from different schools
• knowing one outcome informs your understanding of
  another outcome (i.e., statistical dependence)

6
when you need multilevel models

• reality 1 anytime you have data collected from
  some aggregate unit / clusters, you will have to
  use ml models
• reality 2 calculating an intraclass correlation
  coefficient will quantify your clustering (in
  absence of running a ml model)
• reality 3 even if your clustered data arent
  empirically clustered, article and grant
  reviewers may demand it

7
linear and logistic regression

• linear model review
• logistic model review

yi ß0 ß1X1i ß2X2i ei
ß0 intercept ß1 slope for exposure X1 ß2
slope for covariate X2 e error term (assumed
normal and i.i.d.)
ln P(y) / (1-P(y)) a ß1X1 ß2X2
a intercept ß1 slope for exposure X1 ß2
slope for exposure X2
8
model assumptions

• baseline outcome means (mean values when exposure
  and covariates 0) differ only due to
  variability between subjects
• individual differences from the mean (i.e.,
  errors) are independent and identically
  distributed
• all non-specified variables (e.g., area-level
  variables those confounders you did not measure)
  assumed 0

9
the idea of shrinkage

• trade-off between bias and precision in the
  estimation of parameter ? using estimator ?
• MSE(?) E?? ?2
• VAR(?) E?? E?2
• BIAS(?) (E? ? ?)
• MSE(?) VAR(?) BIAS(?)2

10
Graphical Depiction of the idea of shrinkage
11
efron morris 1977 1
Problem predicting future batting performance of
baseball players based on past performance, when
there are 3 or more players. Data on 18
major-league players after their first 45 times
at bat in the 1970 season. What is known To
be predicted Player 1 hits /
Proportion of hits first 45
times at bat at end of season Player 18
hits / Proportion of hits
first 45 times at bat at end of
season
12
efron morris 1977 2
For each player i, the unbiased estimate of the
proportion of hits at end of season is simply
the observed proportion of hits out of the first
45 unbiased estimate of ?i ( hits /
first 45 times at bat for player i) Intuition
about regression to the mean 1) player
performances fluctuate at random around their
own individual means, and 2) players who have
done well in the first 45 times at bat are more
likely to have done better than their own
player-specific means during this period.
13
efron morris 1977 3
If you had to bet, you'd wager that the worst
performing players would do a bit better in the
long run and the best players would do a little
bit worse in the long run. WHY? Because the
player-specific means are more narrowly
distributed than the means of the first 45 times
at bat, since these estimates include the random
sampling variability of each player around his
own mean in addition to the natural variation of
the player-specific means.
14
efron morris 1977 4
15
the estimation problem 1
Consider estimation of the average risk of
preterm delivery among women enrolled in a cohort
study. Denote this average risk by ? (target
parameter). Data observation of A preterm
deliveries in a cohort of N enrolled women.
Observed proportion (A/N) is the usual
estimator of the risk parameter ? under standard
validity assumptions (maximum-likelihood
estimator, MLE).
16
greenland 2000
Greenland 2000 Figure 1 Rifle 1 shots
X Rifle 2 shots Rifle 3 shots
Greenland 2000 Figure 2 How cluster from
Rifle 1 could be made better by pulling toward
a point r.
17
the estimation problem 2
In Figure 2, the usual estimator A/N is shrunk
toward the point r. A Bayesian estimator is an
example of a "shrinkage estimator" because it
combines prior information with the data. For
example, for prior guess r, weight the observed
proportion A/N and prior guess r by their sample
sizes N and n. Define weight w N/(N n),
then this estimator is the weighted average
Tb w(A/N) (1-w)r
18
the multilevel estimation problem
When you have Aj/Nj for j different clusters,
you can avoid relying on prior information by
using the grand mean A/N as the prior to
shrink toward. TEB
(wjAj/Nj) (1-wj)(A/N )
Just need weights wj. How much do you trust
the cluster-specific proportions, versus how
much you trust the grand proportion? Depends on
Nj.
19
a logistic random intercept models of preterm
delivery 1
The simplest hierarchical logistic model
expresses the tract-level intercepts ?0j as a
function of an overall intercept ?00 and
tract-specific random deviation terms ?0j. For
probability of preterm delivery pij Pr(yij 1)
for individuals i in tracts j 1n
(Pij/1-Pij) ?0j
?0j Y00 µ0j, µ0j N(0, t00)
20
a logistic random intercept models of preterm
delivery 2
?00 is the mean of the distribution of random
coefficients, estimated as the weighted average
of tract intercepts. So both the log-odds of
outcome in each tract and ?00 (the weighted
average of tract-specific log-odds) are estimates
for the true tract-specific log-odds. An
optimal (minimum MSE) estimator for ?0j is formed
by taking the weighted average of these two
quantities, with intra-class correlations for
weights
?0j ?j(ß0j) (1-?j) Y00
21
intraclass correlation coefficient

• estimates the degree of clustering by unit of
  aggregation
• icc between cluster variance / total variance
• icc 0 no clustering -- people within a
  cluster are just the same as people in the other
  clusters
• icc gt 0 people in the same cluster are more
  similar to each other than to people in other
  clusters
• total variance within cluster between cluster
  variance

22
The observed proportions in small clusters are
not realistic values for the true risk too
highly variable
So better to shrink toward some prior knowledge,
or empirical prior based on the aggregate
proportion.
23
Empirical Bayes Graphs
24
a logistic random intercept models of preterm
delivery 2
Add individual-level or neighborhood-level
covariates to explain some of the between tracts
variance. For probability of preterm delivery
pij Pr(yij 1) for individuals i in tracts
j 1n (Pij/1-Pij) ?0j ?1Xij
?0j Y00 Y01Zj µ0j,
µ0j N(0, t00)
25
a logistic random intercept models of preterm
delivery 3
Replacing the second-level equation into the
first level equation yields the combined
equation 1n (Pij/1-Pij) Y00 Y01Zj
ß1Xij µ0j These models have
random effects only for the intercept, but one
could also specify models with random effects
for one or more of the slope terms.
26
multilevel models random and fixed

• random effects models
• random intercept
• random slope
• random slope and random intercept

random intercept models context specific mean
realized from a random distribution
random slope models exposure effect realized
from a random distribution
27
random effects model interpretation
1n (Pij/1-Pij) Y00 Y01Zj ß1Xij µ0j note
conditioning on µj, the cluster-specific
parameter - ß1Xij gives the effect parameters a
conditional interpretation
28
population average models

• Pr (Y ij1 Xij) f (Xij ?)
• note no conditioning on cluster
• Yij preterm birth (1) versus term birth (0) for
  woman i in tract j
• Xij low (1) or high (0) ses for woman i in
  tract j
• no locations specified, just averaged over all
  tracts
• allows you to compare average low versus
  average high ses women

29
fixed effects models

• context-specific variables not allowed to vary
  held fixed
• controls for observed and unobserved contextual
  variables
• usually accomplished by creating an indicator
  (i.e., dummy variable) for each unit of
  analysis (e.g., block group)

30
partitioning variance

• random-effects models allow you to decompose the
  total variance in individual-level outcomes into
  within-group and between-group components
• In the ANOVA context, has an explanatory
  interpretation as identifying the mechanism as
  being contextual or compositional

31
deciding which model to use

• depends on what you want to say
• if you want to look at risk / odds for the
  average individual with some exposure compared
  with average individual with some other exposure,
  use a population averaged model (e.g., GEE)
• if you want to talk about how changes in context-
  specific exposures will change the risk / odds in
  that context, use the random-effects
• if you want to want to consider the effect of
  some variable holding all observed and unobserved
  variables contextual factors constant, use a
  context fixed effect model

32
Highest quartile of neighborhood deprivation
clusters in downtown Raleigh and in Northeast
Wake county near Rolesville and Zebulon
33
neighborhood deprivation and odds of preterm birth

• White women Black women
• OR 95 CI OR 95 CI
• 4th quartile 1.28 (1.01, 1.61) 1.48 (1.00, 2.18)
  
• 3rd quartile 1.10 (0.94, 1.29) 1.37 (0.93, 2.04)
• 2nd quartile 1.05 (0.90, 1.22) 1.39 (0.93, 2.08)
• 1st quartile 1.00 (referent) 1.00 (referent)
• Age 35 1.13 (0.89, 1.44) 2.07 (1.57, 2.72)
• Age 30-34 1.00 (0.80, 1.44) 1.66 (1.30, 2.11)
• Age 25-29 1.19 (0.95, 1.48) 1.30 (1.04, 1.61)
• Age 20-24 1.00 (referent) 1.00 (referent)
• Age lt20 1.09 (0.75, 1.59) 0.69 (0.52, 0.92)
• lt High school 1.31 (0.96, 1.78) 1.87 (1.46, 2.39)
  
• High school 1.31 (1.10, 1.56) 1.36 (1.12, 1.64)
• gt High school 1.00 (referent) 1.00 (referent)
• Not married 1.19 (0.95, 1.49) 1.46 (1.21, 1.76)

Messer LC, Buescher PA, Laraia BA. Kaufman JS.
SCHS study No. 148. Nov 2005.
34
tract high unemployment is associated with
preterm birth for Black women

• Logistic Logistic (PA) Logistic (RE)
• OR 95 CI OR 95 CI OR 95 CI
• gt5 unemployment 1.29 (1.08, 1.55) 1.29 (1.04,
  1.61) 1.31 (1.04, 1.64)
• Age 25-29 1.31 (1.05, 1.64) 1.31 (1.04,
  1.61) 1.31 (1.05, 1.64)
• Age 30-34 1.69 (1.33, 2.15) 1.70 (1.35,
  2.10) 1.68 (1.32, 2.14)
• Age 35 2.10 (1.60, 2.76) 2.10 (1.60,
  2.77) 2.10 (1.60, 2.75)
• High school 1.37 (1.13, 1.66) 1.37 (1.10,
  1.70) 1.38 (1.14, 1.67)
• lt High school 1.74 (1.36, 2.26) 1.74 (1.33,
  2.27) 1.76 (1.34, 2.29)
• Not married 1.49 (1.23, 1.80) 1.49 (1.25,
  1.77) 1.49 (1.23, 1.80)

35
example causal interpretations 1

• population average logistic model (gt5
  unemployment versus ?5 unemployment)
• OR 1.29 (95 CI 1.04, 1.61)
• the odds of preterm delivery will increase by
  29 for a randomly selected woman in a low
  unemployment if she were to be relocated to a
  tract with high unemployment

36
example causal interpretations 2

• random effects logistic model (gt5 unemployment
  versus ?5 unemployment)
• OR 1.31 (95 CI 1.04, 1.64)
• the odds of preterm delivery will increase by
  31 for a randomly selected woman in a specific
  census tract with low unemployment if that tract
  is somehow manipulated to have high unemployment

37
summary

• standard regression models assume that data is
  not clustered by a higher level grouping
• one can model clustered data by either using
  methods robust to this violation of assumptions,
  or else by modeling this clustering directly
• random effects models estimate conditional
  parameters (i.e., the effect of exposure given a
  particular cluster)