Introduction
Linear MixedEffects Model
Generalized Linear MixedEffects Model
Hotspots Mapping
Introduction
 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)
 nonnested 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 mixedeffects models?
 Account for both individual and grouplevel variations when estimating grouplevel coefficients.
 Easily model variations among individuallevel coefficients, especially when making predictions for new groups.
 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 fixedeffects or randomeffects?
When do mixedeffects models make a difference?
Fixed and Random Effects
Two extreme cases:
 when the grouplevel variation is very little
 reduce to traditional regression models without group indicators (complete pooling)  when the grouplevel variation is very large
 reduce to traditional regression models with group indicators (nopooling)
Little risk to apply a mixedeffects model
What's the difference between nopooling models and mixedeffects models only with varying intercepts?
 In nopooling models, the intercept is obtained by least squares estimates, which equals to the fitted intercepts in models that are run separately by group.
 In mixedeffects 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 (completepooling)
 ICC > 1: all individuals of a group are identical (nopooling)
Intraclass Correlation (ICC)
ICC ranges from 0 to 1:
 ICC > 0: "hard constraint" to
 ICC > 1: "no constraint" to
 Mixedeffects 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 completepooling)
 For group with large n, the pooling will be weak, where the value of is far away from the mean (towards nopooling)
Linear MixedEffects Model
Pull the codes and dataset: https://github.com/benhhu/RMixedEffectsModel
Load the Packages and Data
1,000 participants
5 repeated measurements
bmi
time
id
age
race: 1=white, 2=black, 3=others
gender: 1=male, 2=female
edu: 1=<HS, 2=HS, 3=>HS
sbp
am: 1=measured in morning
ex: #days exercised in the past year
Varyingintercept Model with No Predictors
allows intercept to vary by individual
estimated intercept, averaging over the individuals
estimated variations
Varyingintercept Model with an individuallevel predictor
Varyingintercept Model with both individuallevel and grouplevel predictors
Varying Slopes Models
With only an individuallevel predictor
Varying Slopes Models
Add a grouplevel predictor
Nonnested Models
Generalized Linear MixedEffects Model
MixedEffects Logistic Model
Empty model
MixedEffects Logistic Model
Add bmi and race
MixedEffects 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: GaussHermite 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 24:127135.
MixedEffects Model vs. GEE
MixedEffects Model  Marginal Model with GEE  

Distributional assumptions  Yes  No 
Population average estimates  Yes  Yes 
Groupspecific 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
Points
Lines
Shapefiles
.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
Datum
 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
Datum
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, twodimensional 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
Distortion
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 (equalarea)  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 sixdegreewide zones
 From 80S to 84N
 Zones numbered 160 (N&S), W to E, starting at 180W
Differences between Projections
Spatial Patterns
Random
Cluster
Regular
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
Why
 Confirmatory purpose
 verify if a perceived cluster exists
 Exploratory purpose
 searching for spatial patterns
 Identification of clusters can lead to interventions
Methods
 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: Knearest neighbour method, etc.
 Local clustering:
 additionally specify the location and can be extended to specify spatialtemporal 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
 Nonfocused tests
 identify the location of all likely clusters in the study region
 LISA, GetisOrd'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:
 highhigh locations: hot spots
 lowlow locations: cold spots
 highlow locations: spatial outliers
 lowhigh 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
A
Multilevel Approaches  PHC6016
By Hui Hu
Multilevel Approaches  PHC6016
Slides for the Social Epidemiology guest lecture, Fall 2016
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