PHC6016 Social Epidemiology


Multilevel Approaches


October 20, 2016


Hui Hu, PhD



Linear Mixed-Effects Model


Generalized Linear Mixed-Effects Model

Hotspots Mapping


  • We usually assume the samples drawn from targeted population are independent and identically distributed (i.i.d.).


  • This assumption does not hold when we have data with multilevel structure:
    - clustered and nested data (i.e. individuals within areas)
    - longitudinal data (i.e. repeated measurements within individuals)
    - non-nested structures (i.e. individuals within areas and belonging to some subgroups such as occupations)

  • Samples within each group are dependent, while samples between groups stay independent
  • Two sources of variations:
    - variations within groups
    - variations between groups
  • A longitudinal study:
    - n = 3
    - t = 3
  • Complete pooling
    - poor performance
  • No pooling
    - infeasible for large n
  • Partial pooling
  • An alternative solution: include categorical individual indicators in the traditional linear regression model.
  • Why do we still need mixed-effects models?
  1. Account for both individual- and group-level variations when estimating group-level coefficients.
  2. Easily model variations among individual-level coefficients, especially when making predictions for new groups.
  3. Allow us to estimate coefficients for specific groups, even for groups with small n

Fixed and Random Effects

  • Random Effects: varying coefficients
  • Fixed Effects: varying coefficients that are not themselves modeled

How to decide whether to use fixed-effects or random-effects?

When do mixed-effects models make a difference?

Fixed and Random Effects

Two extreme cases:

  • when the group-level variation is very little
    - reduce to traditional regression models without group indicators (complete pooling)
  • when the group-level variation is very large
    - reduce to traditional regression models with group indicators (no-pooling)

Little risk to apply a mixed-effects model

What's the difference between no-pooling models and mixed-effects models only with varying intercepts?

  • In no-pooling models, the intercept is obtained by least squares estimates, which equals to the fitted intercepts in models that are run separately by group.
  • In mixed-effects models, we assign a probability distribution to the random intercept: 

Intraclass Correlation (ICC)

shows the variation between groups

ICC ranges from 0 to 1:

  • ICC -> 0: the groups give no information (complete-pooling)
  • ICC -> 1: all individuals of a group are identical (no-pooling)

Intraclass Correlation (ICC)

ICC ranges from 0 to 1:

  • ICC -> 0: "hard constraint" to 
  • ICC -> 1: "no constraint" to
  • Mixed-effects model: "soft constraint" to 

This constraint has different effects on different groups:

  • For group with small n, a strong pooling is usually seen, where the value of     is close to the mean      (towards complete-pooling)
  • For group with large n, the pooling will be weak, where the value of    is far away from the mean      (towards no-pooling)

Linear Mixed-Effects Model

Load the Packages and Data

1,000 participants

5 repeated measurements






race: 1=white, 2=black, 3=others

gender: 1=male, 2=female

edu: 1=<HS, 2=HS, 3=>HS


am: 1=measured in morning

ex: #days exercised in the past year

Varying-intercept Model with No Predictors

allows intercept to vary by individual

estimated intercept, averaging over the individuals

estimated variations

Varying-intercept Model with an individual-level predictor

Varying-intercept Model with both individual-level and group-level predictors

Varying Slopes Models

With only an individual-level predictor

Varying Slopes Models

Add a group-level predictor

Non-nested Models

Generalized Linear Mixed-Effects Model

Mixed-Effects Logistic Model

Empty model

Mixed-Effects Logistic Model

Add bmi and race

Mixed-Effects Poisson Model

Parameter Estimation Algorithms

  • ML: maximum likelihood
  • REML: restricted maximum likelihood
    - default in lmer()
  • PQL: pseudo- and penalized quasilikelihood
  • Laplace approximations
    - default in glmer()
  • GHQ: Gauss-Hermite quadrature
  • McMC: Markov chain Monte Carlo

Bolker BM, Brooks ME, Clark CJ, Geange SW, Poulsen JR, Stevens MHH, et al. 2009. Generalized linear mixed models: A practical guide for ecology and evolution. Trends in ecology & evolution 24:127-135.

Mixed-Effects Model vs. GEE

Mixed-Effects Model Marginal Model with GEE
Distributional assumptions Yes No
Population average estimates Yes Yes
Group-specific estimates Yes No
Estimate variance components Yes No
Perform good with small n Yes No

Hotspots Mapping

Introduction to Spatial Data

Data Models

A geographic data model is a structure for organizing geospatial data so that it can be easily stored and retrieved.

Geographic coordinates

Tabular attributes

Spatial Data Models

Vector Model

- points, lines, polygons

Raster Model
- exhaustive regular or irregular partitioning of space




.shp - the file that stores the geometry of the feature

.shx - the file that stores the index of the feature geometry

.dbf - the dBASE file that stores the attribute information

.prj  - the file that defines the shapefile's projection

.html, .htm, .xml - the files that usually contains metadata

.sbn and .sbx - store additional indices 

Coordinate Systems and Projections

3D sphere

Geographic Coordinate System

2D flat

Projected Coordiate System

Geographic Coordinate Systems

  • Longitude and latitude
  • Units: Degrees (DMS or DD)

Shape of the Earth

  • Surface: The Earth's real surface
  • Ellipsoid: Ideal, smooth surface
  • Geoid: Bumpy surface, where gravity is equal for all locations


  • Defines the position of the spheroid relative to the center of the earth.
  • Global datum:
    - uses the earth's center of mass as the origin
  • Local datum:
    - aligns its spheroid to closely fit the earth's surface in a particular area
    - a point on the surface of the spheroid is matched to a particular position on the surface of the earth
    - the coordinate system origin of a local datum is not at the center of the earth


Common Local Datum: North American Datum (NAD)

Common Global Datum: World Geodetic System (WGS)

Projected Coordinate Systems

  • A projected coordinate system is defined on a flat, two-dimensional surface
  • Unlike a geographic coordinate system, a projected coordinate system has constant lengths, angles, and areas across the two dimensions
  • A projected coordinate system is always based on a geographic coordinate system

The systematic rendering of a graticule on a flat map surface


Converting a sphere to a flat surface results in distortion

  • Shape (conformal) - If a map preserves shape, then feature outlines (like county boundaries) look the same on the map as they do on the earth.
    - Lambert Conformal Conic
    - UTM
  • Area (equal-area) - If a map preserves area, then the size of a feature on a map is the same relative to its size on the earth.
    - Alerts Equal Area Conic
  • Distance (equidistant) - An equidistant map is one that preserves true scale for all straight lines passing through a single, specified point.  If a line from a to b on a map is the same distance that it is on the earth, then the map line has true scale.  No map has true scale everywhere.  
  • Direction/Azimuth (azimuthal) – An azimuthal projection is one that preserves direction for all straight lines passing through a single, specified point.

Universal Transverse Mercator Coordinate System

  • World divided into 60 six-degree-wide zones
  • From 80S to 84N
  • Zones numbered 1-60 (N&S), W to E, starting at 180W


Differences between Projections

Spatial Patterns




Disease Cluster

  • The occurrence of a greater than expected number of cases of a particular disease within a group of people, a geographic area, or a period of time.
  • A collection of disease occurrence:
    - of sufficient size and concentation to be unlikely to have occurred by chance, or
    - related to each other through some social or biological mechanism, or having a common relationship with some other events or circumstance
  • Spatial aggregation of disease events may only be a function of the distribution of population
  • Disease cluster: residual spatial variation in risk after known influence have been accounted for


  • Confirmatory purpose
    - verify if a perceived cluster exists
  • Exploratory purpose
    - searching for spatial patterns
  • Identification of clusters can lead to interventions


  • Global clustering:
    - evaluate whether clustering exist as a global phenomena throughout the study region, without pinpointing the locaiton of specific cluster
    - aggregated data: Moran's I, Geary's C, etc.
    - points data: K-nearest neighbour method, etc.
  • Local clustering:
    - additionally specify the location and can be extended to specify spatial-temporal clusters

Local Clustering

  • Focused tests:
    - investigate whether there is an increased risk of disease around a predetermined point
    - e.g. Superfund site, power plant.
    - Lawson Waller score test
  • Non-focused tests
    - identify the location of all likely clusters in the study region
    - LISA, Getis-Ord's local statistics, spatial scan statistics 

LISA - Local Moran's I

  • Local indicators of spatial autocorrelation (LISA)
    - show similarity with neighbors and also test its significance
  • Divide the study region into 5 categories:
    - high-high locations: hot spots
    - low-low locations: cold spots
    - high-low locations: spatial outliers
    - low-high locations: spatial outliers
    - Locations with insignificant local autocorrelation
  • GeoDa

Spatial Scan Statstics

  • Search over a given set of spatial regions
  • Find those regions which are most likely to be clusters
  • Correctly adjust for multiple hypothesis testing
  • SatScan
  • A circular scanning window is placed at different coordinates with radius that vary from 0 to some set upper limit.
  • For each location and size of window
    H   = elevated risk within window as compared to outside of window


Multilevel Approaches - PHC6016

By Hui Hu

Multilevel Approaches - PHC6016

Slides for the Social Epidemiology guest lecture, Fall 2016

  • 1,404