# Presentation: 587442

##  Contextual Analysis: Understanding and Interpreting Multilevel Statistical Models

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

## 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

## 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

## motivation for multilevel models

- standard regression models are mis-specified for clustered data:
- 		yi = 0 + 1xi + εi;   ε ~ N(0,σ2) *i.i.d.*
- [next 3 slides]
- hierarchical models outperform unbiased models (i.e., lower mean squared error)
- [“shrinkage”]

## 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)

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

## linear and logistic regression 

- linear model review:
- logistic model review:

- yi = β0 + β1X1i + β2X2i...+ εi

- β0 = intercept	β1 = 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 = intercept	β1 = slope for exposure X1
- β2 = slope for exposure X2

## 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

## 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

## Graphical Depiction of the idea of “shrinkage”

## 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

## 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.

## 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.

## efron & morris 1977 [4]

## 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).

## 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.

## 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:
- Θb* = w(A/N) + (1-w)r

## the multilevel estimation problem

- Just need weights wj.   How much do you trust
- the cluster-specific proportions, versus how much
- you trust the grand proportion?  Depends on Nj.

- 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+ )

## 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, τ00)

## 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* = λj(β0j) + (1-λj) Ŷ00

## 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 > 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

## 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.

## Empirical Bayes Graphs

## 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, τ00)

## 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 + 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.

## 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

## random effects model interpretation 

- 1n (Pij/1-Pij) = Y00 + Y01Zj + β1Xij + μ0j
- note: conditioning on μj, the cluster-specific parameter
- -  β1Xij gives the effect parameters a conditional interpretation

## population average models

- 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

## 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)

## 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

## 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

## Highest quartile of neighborhood deprivation clusters in downtown Raleigh and in Northeast Wake county near Rolesville and Zebulon

## neighborhood deprivation and odds of preterm birth

*	*White women	Black women*	*
- 	_OR__	__95% CI__	__OR__	__95% CI__	_
**4****th**** quartile****	****1.28 (1.01, 1.61) ****	****1.48 (1.00, 2.18) **
**3****rd**** quartile****	****1.10 (0.94, 1.29)****	****1.37 (0.93, 2.04)**
**2****nd**** quartile****	****1.05 (0.90, 1.22)****	****1.39 (0.93, 2.08)**
**1****st**** 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 <20	1.09 (0.75, 1.59)	**0.69 (0.52, 0.92)**
- < 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)**
- > High school	1.00 (referent)	1.00 (referent)
- Not married	1.19 (0.95, 1.49)	**1.46 (1.21, 1.76)**
- Married	1.00 (referent)	1.00 (referent)

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

## 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_
**>5% 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)
- < 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)

## example causal interpretations [1]

- 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

## example causal interpretations [2]

- 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

## 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)