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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. objectives. viewers should gain familiarity with: common terminology for multilevel models the need to account for clustered data

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Contextual analysis understanding and interpreting multilevel statistical models l.jpg

Contextual Analysis: Understanding and Interpreting Multilevel Statistical Models

Jay S. Kaufman, PhD

University of North Carolina at Chapel Hill

June 2007


Objectives l.jpg

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 l.jpg

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 l.jpg

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 l.jpg

    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 l.jpg

    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 l.jpg

    linear and logistic regression

    • 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


    Model assumptions l.jpg

    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 l.jpg

    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 l.jpg

    Graphical Depiction of the idea of “shrinkage”


    Efron morris 1977 1 l.jpg

    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 l.jpg

    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 l.jpg

    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 l.jpg

    efron & morris 1977 [4]


    The estimation problem 1 l.jpg

    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 l.jpg

    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 l.jpg

    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 l.jpg

    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.

    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.


    A logistic random intercept models of preterm delivery 1 l.jpg

    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 l.jpg

    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 0jis 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 l.jpg

    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


    Slide22 l.jpg

    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 l.jpg

    Empirical Bayes Graphs


    A logistic random intercept models of preterm delivery 224 l.jpg

    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 l.jpg

    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 l.jpg

    multilevel models: random and fixed

    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


    Random effects model interpretation l.jpg

    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 l.jpg

    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 l.jpg

    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 l.jpg

    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 l.jpg

    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


    Slide32 l.jpg

    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 l.jpg

    neighborhood deprivation and odds of preterm birth

    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.


    Tract high unemployment is associated with preterm birth for black women l.jpg

    tract high unemployment is associated with preterm birth for Black women

    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)


    Example causal interpretations 1 l.jpg

    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 l.jpg

    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 l.jpg

    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)


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