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Contextual Analysis: Understanding and Interpreting Multilevel Statistical Models

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Contextual Analysis: Understanding and Interpreting Multilevel Statistical Models

Jay S. Kaufman, PhD

University of North Carolina at Chapel Hill

June 2007

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

- 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

- standard regression models are mis-specified for clustered data:
yi = 0 + 1xi + εi; ε ~ N(0,σ2) i.i.d.

- [next 3 slides]

- [“shrinkage”]

- 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)

- 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’ aren’t empirically clustered, article and grant reviewers may demand it

- linear model review:
- logistic model review:

yi = β0 + β1X1i + β2X2i…+ εi

β0 = interceptβ1 = slope for exposure X1

β2 = slope for covariate X2

ε = error term (assumed normal and i.i.d.)

ln[ P(y) / (1-P(y))] = α + β1X1 + β2X2…

α = interceptβ1 = slope for exposure X1

β2 = slope for exposure X2

- 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

- 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

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

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.

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.

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).

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.

- 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:
- Θb* = w(A/N) + (1-w)r

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.

ΘEB* = (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.

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, τ00)

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 0jis formed by

taking the weighted average of these two quantities, with

intra-class correlations for weights:

0j* = λj(β0j) + (1-λj) Ŷ00

- 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 > 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

So better to shrink toward some prior knowledge, or

empirical prior based on the aggregate proportion.

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, τ00)

Replacing the second-level equation into the first

level equation yields the combined equation:

1n (Pij/1-Pij) = Y00 + Y01Zj + β1Xij + 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.

random intercept models: context specific mean realized from a random distribution

- random effects models
- random intercept
- random slope
- random slope and random intercept

random slope models: exposure effect realized from a random distribution

1n (Pij/1-Pij) = Y00 + Y01Zj + β1Xij + μ0j

note: conditioning on μj, the cluster-specific parameter

- β1Xij gives the effect parameters a conditional interpretation

- Pr (Y ij=1 | 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

- 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)

- 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

- 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

White womenBlack women

OR95% CIOR95% CI

4th quartile1.28 (1.01, 1.61) 1.48 (1.00, 2.18)

3rd quartile1.10 (0.94, 1.29)1.37 (0.93, 2.04)

2nd quartile1.05 (0.90, 1.22)1.39 (0.93, 2.08)

1st quartile1.00 (referent)1.00 (referent)

Age 35+1.13 (0.89, 1.44)2.07 (1.57, 2.72)

Age 30-341.00 (0.80, 1.44)1.66 (1.30, 2.11)

Age 25-291.19 (0.95, 1.48)1.30 (1.04, 1.61)

Age 20-241.00 (referent)1.00 (referent)

Age <201.09 (0.75, 1.59)0.69 (0.52, 0.92)

< High school1.31 (0.96, 1.78)1.87 (1.46, 2.39)

High school1.31 (1.10, 1.56)1.36 (1.12, 1.64)

> High school1.00 (referent)1.00 (referent)

Not married1.19 (0.95, 1.49)1.46 (1.21, 1.76)

Married1.00 (referent)1.00 (referent)

Messer LC, Buescher PA, Laraia BA. Kaufman JS. SCHS study No. 148. Nov 2005.

LogisticLogistic (PA)Logistic (RE)

OR 95% CIOR 95% CIOR 95% CI

>5% unemployment1.29(1.08, 1.55) 1.29(1.04, 1.61) 1.31(1.04, 1.64)

Age 25-291.31 (1.05, 1.64)1.31 (1.04, 1.61)1.31(1.05, 1.64)

Age 30-341.69 (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 school1.37 (1.13, 1.66)1.37 (1.10, 1.70)1.38(1.14, 1.67)

< High school1.74 (1.36, 2.26)1.74 (1.33, 2.27)1.76(1.34, 2.29)

Not married1.49 (1.23, 1.80)1.49 (1.25, 1.77)1.49(1.23, 1.80)

- population average logistic model (>5% 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

- random effects logistic model (>5% 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

- 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)