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)
  - 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

Fixed and Random Effects

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: 

Linear Mixed-Effects Model

Pull the codes and dataset: https://github.com/benhhu/R-Mixed-Effects-Model

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

Varying-intercept Model with No Predictors

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

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 

ftp://ftp2.census.gov/geo/tiger/TIGER2016/BG/tl_2016_12_bg.zip

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, 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

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 (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

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: 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