Methods for Small Area Analyses of Spatial and Space-time Data Evan - - PowerPoint PPT Presentation

Methods for Small Area Analyses

• f Spatial and Space-time Data

Evan Carey Robert Penfold Elisabeth Dowling Root AcademyHealth Conference, Seattle, WA June 25, 2018

Outline

• Introduction
• Challenges of spatial data
• Representing space and defining spatial

relationships

• Spatial autocorrelation
• Focus on analysis techniques for area data

– Disease mapping & BYM CAR Models

• Focus on analysis techniques for continuous

data

Part 1: Foundational Concepts

• Why do I care about space: is space a parameter of interest,
• r a nuisance parameter?
• What are different ways spatial data can be represented in

my data?

• How do I define ‘near’ and ‘far’?
• What does autocorrelation mean?
• How does spatial autocorrelation differ from spatial trends?
• Why is data irregularly distributed across space challenging

to model?

• How is this connected to small area analysis?
• What does ‘shrinkage’ mean, and why does it improve

models?

Why do you care about space?

I am interested in the relationship between location and my outcome. I am not interested in the effect of location, but my data has spatial nature…

• I want to identify areas with

high or low disease rates.

• Potentially create maps

showing above/below average outcomes.

• I want to estimate the effect
• f space!
• Ignoring space in your

models may give you biased results/incorrect p-values

• Correctly modeling space

fixes the issue.

• Space is a ‘nuisance’

parameter here.

Geospatial Data & Public Health

Geographic data, Geographic Information Systems (GIS), and spatial analysis provide public health officials with the capability to perform two unique types of analysis:

• 1. Find statistically significant areas of high or low

incidence

• 2. Examine the spatial relationship between health
• utcomes and population/contextual factors

Geographic Variation in Health

• People (demographics) and the risk factors

contributing to health are dispersed unevenly across communities and regions

• Often we are interested in identifying patterns of

disease (or some other health outcome) across space

• We are also interested in understanding the reasons for

these patterns:

– Composition: differences in kinds of people who live in places – Context: differences in neighborhood or area-level physical

• r social environments

But…“spatial is special”

• Data that are referenced to location bring important

additional information to your data analysis

• But, spatially referenced data also bring special

problems to your analysis

– heterogeneity of observational units → heteroskedasticity – spatial autocorrelation → residual dependence

• A consequence of these “special problems” is that

traditional assumptions of standard regression techniques are violated

– statistical inference from such a model is not valid

Spatial data is complex

• The methods we chose to cope with the

complexities of spatial data depend on how we define space

– Discrete geographic phenomena have spatial bounds. Locations may be within or outside a geographic feature.

• Areal data: census tracts, counties, states

– Continuous geographic phenomena have properties continuously distributed across the landscape. Locations are specific and have value.

• Point data
• These definitions of space are represented by

different geographic data types

What are Spatial Data

ID Tract ChildDth Race DistPCP 1 1237 Yes White 5000 2 1237 No AA 3560 3 1238 No White 10789 4 1238 No Asian 7689 Object: Home

Spatial data: longitude, latitude (x, y) 76.9147, 107.6098 Attribute data: Survey data Spatial Relationships:

• Proximity to physician
• “Contained in” census tract

Tract PctPov PctAA Foreclose PCP 1237 .056 .241 .011 1 1238 .079 .443 .043 3 1239 .151 .078 .225 10 1240 .224 .011 .105

Attribute data: Census tract/PCSA characteristics

Object: Health Center

• Location
• Attributes
• Spatial Relationships

Spatial Data Types

Event Data (Points) Lattice Data (Areas) Geostatistical Data (Grid)

It’s important to understand that these designations are not mutually exclusive

Points can be geolocated in some relevant areal units

These aggregations can be used to produce rates

0.02 0.16 0.7 0.14 0.09 0.18 0.11 0.05 0.00

GIS Spatial Data Analysis Spatial Analysis

“Spatial Statistics”

Event (Point) Data Geostatistical Data Lattice (Area) Data

Regional Count data Spatial Econometrics Spatial Regression Analysis

|

Point Pattern Analysis Spatial Epidemiology Crime Analysis

|

Spatial Prediction

|

“Spatial Data Production”

Thinking in one dimension: Does time effect the outcome?

Thinking in one dimension: Does time effect the outcome?

Thinking in one dimension: Does time effect the outcome?

Thinking in one dimension: Is there a time trend?

Spatial Autocorrelation and Trends (2D)

• Correlation in space

– Is a variable in a location correlated with the values in nearby places?

• Spatial trends in the outcome

– The outcome differs systematically as a function of spatial location. These are distinct concepts! * Humans are pretty bad at identifying spatial trends by eye. We tend to over interpret noise when it is on a map ☺

“Everything is related to everything else, but near things are more related than distant things.”

Defining spatial relationships

• What is a neighbor? What’s next to what?
• These spatial relationships can be defined in a

number of ways

– Contiguity (common boundary, K-nearest neighbors)

• What is a “shared” boundary?
• How many “neighbors” to include?

– Distance (distance band)

• What distance do we use?

Contiguity based neighbors

• For areas:

– All polygons that share a common border

• For points

– Distance

k=1 k=2 k=3 1 km 1.5 km K-nearest neighbors (KNN) Euclidean distance

Thinking in one dimension: Does time effect the outcome?

The problem with sparse data…

The problem with sparse data…

General Shrinkage Idea

If we have observed last year’s hospital mortality rate, what is your best prediction of next year’s hospital mortality rate?

High Low

If we have observed last year’s hospital mortality rate, what is your best prediction of next year’s hospital mortality rate?

Only use information from each hospital to predict mortality. No pooling of information (no shrinkage!)

High Low

If we have observed last year’s hospital mortality rate, what is your best prediction of next year’s hospital mortality rate?

Share (pool) information across hospitals. Prediction is ‘shrunk’ towards the mean.

High Low

Sharing Spatial Data (Shrinkage)

2/8 = 0.25 Census Tract A 4/20 = 0.2 Census Tract B 1/10 = 0.1 Census Tract E 1/45 = 0.02 Census Tract C 3/30 = 0.1 Census Tract F 2/25 = 0.08 Census Tract D

Focus on methods for continuously indexed data

Spatial models implemented with R- INLA

Motivating example: Outcomes of Veterans in Colorado

Goal: Identify areas of high and low event probability. What does the ideal method need to have?

Ideal method

• Identify spatial trend and make predictions at

all points.

• Resilient to irregularly spaced data (small area

analysis!)

• Exhibit shrinkage / stabilization
• Incorporate other patient level traits in the

model (‘adjust’)

• Converge in reasonable time in medium to

large datasets

Point pattern analysis versus point referenced models.

Binary Outcome = Patient Location Patient Demographics

+

http://open.lib.umn.edu/mapping/chapter/6-analysis/

Community care utilization in Colorado (data simulation – no PHI here!)

Simulating Success of Community care Referrals in the VHA

• Simulation 1:

– no spatial trend (pure spatial noise)

• Simulations 2-4:

– Spatial trend of varying strengths. How successful are different methods at recovering the underlying spatial trends of the binomial process??

Method 1: Simple Interpolation (2D Smoother)

• Use a 2D smoother:

– Gaussian kernel weighting – Allows smoothing of binary process at irregularly space locations. – Can compute mean and variance across space. – Nadaraya-Watson smoother (Nadaraya, 1964, 1989; Watson, 1964)

• What results do you expect to get using this

method?

Results for data with no spatial trend.

Results for data with a spatial trend (simulation 2)

Results for data with a spatial trend (simulations 3 and 4)

Spatial Models with R-INLA

• Integrated Nested Laplace Approximation (INLA). An

alternative to MCMC for fitting Bayesian models.

• Latent Gaussian models

– Fixed effects, structured and unstructured Gaussian random effects combined linearly with likelihoods specified. – ‘focus on the continuous representation of the GRF through an (stochastic partial differential equation) SPDE’

• Coding is straightforward via R-INLA package.
• Convergence is fast in medium to larger datasets.
• The problem with large spatial data…most traditional

methods of spatial inference require inversion of the covariance matrix, which is an n3 calculation!

Bakka, Haakon, Håvard Rue, Geir-Arne Fuglstad, Andrea Riebler, David Bolin, Elias Krainski, Daniel Simpson, and Finn Lindgren. “Spatial Modelling with R-INLA: A Review.” ArXiv:1802.06350 [Stat], February 18, 2018. http://arxiv.org/abs/1802.06350.

R-INLA process

• Import data, use R-INLA package for ease of model

specification and fitting.

• Construct mesh for notion of spatial location:

– Helper functions in R-INLA. – Expand mesh beyond boundaries of data – Experiment with density of nodes.

• Connect mesh to observations (output is matrix)
• Create the model

– Spatial effect is connected to the mesh/observations object – Other patient level effects not connected to location matrix.

• Fit the model.
• Results: Summarize hyperparameter distributions.
• Results: Make predictions on a dense grid of the region.

Construct mesh.

Results for data with no spatial trend.

Results for data with a spatial trend (simulation 2)

Results for data with a spatial trend (simulations 3 and 4)

Comparison to other methods

• R-INLA works is easy to implement and works

well in larger datasets.

• Bayesian framework allows hierarchical model

specification, and flexible summary of the posterior.

• Review article evaluated 7 possible approaches

to this problem:

– R-INLA and Fixed Rank Kriging performed optimally in larger datasets (memory usage and PU time) – Methods generally provided similar estimates.

Bradley, Jonathan R., Noel Cressie, and Tao Shi. “A Comparison of Spatial Predictors When Datasets Could Be Very Large.” Statistics Surveys 10, no. 0 (2016): 100–131. https://doi.org/10.1214/16-SS115.

Focus Area Name

Focus on methods for areal data

Spatial smoothing: Headbanging, Locally weighted averaging, and Bayesian CARs Elisabeth Dowling Root, MA, PhD

Department of Geography & Division of Epidemiology The Ohio State University

Focus Area Name

Mapping Rates

• For small areas, rates and mortality ratios are

very instable and maps of rates can be misleading

– AND rates are spatially correlated

• Trade-off between geographic resolution and the

variability of mapped estimates

• Spatial smoothing can reduce the random noise

in maps of observable health data

– Highlight meaningful geographic patterns in the underlying risk

Focus Area Name

Shrinkage Estimation and Spatial Smoothing

• Shrinkage methods are often used to stabilize rates

across small areas

– Smoothed estimates for each area “borrow strength” (precision) from data in other areas by an amount depending on the precision of the raw estimate of each area

• Estimated rate in area A is adjusted by combining

knowledge about:

– Observed rate in that area – Average rate in surrounding areas

• The two rates are combined using some form of

weighted average, weights depend on the population size in area A

Focus Area Name

There are many techniques for spatial smoothing

• Locally Weighted Average (Anselin, 2006)

– Smooths toward the mean – Area value replaced by population weighted average of surrounding areas

• Headbanging (Mungiole, Pickle, Simonson, 1999)

– Smooths toward the median – Area values replaced if large deviation from the median and population is not large

• Bayesian Hierarchical (CAR) Models (Lawson, 2013)

– Smooths toward the mean – Area values calculated using a CAR model with a spatial random effect term

Focus Area Name

Headbanging

Rate=0.3 N=8

Census Tract A

Rate=0.2 N=20 Rate=0.1 N=10 Rate=0.02 N=45 Rate=0.1 N=30 Rate=0.08 N=25 Rate=0.12 N=35 Rate=0.15 N=22 Rate=0.02 N=45 Rate=0.04 N=55 Rate=0.03 N=60

Headbanging uses the median, but this technique can also be applied to the neighborhood mean

Focus Area Name

Headbanging

Rate=0.3 N=8

Census Tract A

Rate=0.2 N=20 Rate=0.1 N=10 Rate=0.02 N=45 Rate=0.1 N=30 Rate=0.08 N=25 Rate=0.12 N=35 Rate=0.15 N=22 Rate=0.02 N=45 Rate=0.04 N=55 Rate=0.03 N=60

Is center value between high and low medians? -- NO Is the population much greater than neighbors? -- NO RATE = 0.09

REPLACE!!

Rate N Weighted Rate Median 0.02 45 0.027 0.02 45 0.027 0.10 10 0.030 low 50% 0.20 8 0.048 0.03 60 0.054 0.08 25 0.060 0.04 55 0.066 0.10 30 0.090 high 50% 0.15 22 0.099 0.12 35 0.125

Focus Area Name

Headbanging

Rate=0.3

N=200 Census Tract A

Rate=0.2 N=20 Rate=0.1 N=10 Rate=0.02 N=45 Rate=0.1 N=30 Rate=0.08 N=25 Rate=0.12 N=35 Rate=0.15 N=22 Rate=0.02 N=45 Rate=0.04 N=55 Rate=0.03 N=60 Rate N Weighted Rate Median 0.02 45 0.027 0.02 45 0.027 0.10 10 0.030 low 50% 0.20 8 0.048 0.03 60 0.054 0.08 25 0.060 0.04 55 0.066 0.10 30 0.090 high 50% 0.15 22 0.099 0.12 35 0.125

Is center value between high and low medians? -- NO Is the population much greater than neighbors? -- YES

DON’T REPLACE!!

Focus Area Name

Example: Data Privacy and Spatial Smoothing

Focus Area Name

Focus Area Name

Focus Area Name

Standardized Mortality Ratio

• Standardized Mortality Ratios show locations on a map with higher than

expected rates given the age-, sex-, etc- distribution of the population in that area 𝑇𝑁𝑆𝑗 = 𝑍

𝑗

𝐹𝑗 ∗ 1000 𝑍

𝑗 is the observed number of events

𝐹𝑗 is the expected number of events 𝐹𝑗 = ෍

𝑘

𝑞𝑘𝑜𝑗𝑘 j is the population stratum (e.g., age*sex*race) 𝑞𝑘 is the frequency of the reference population 𝑜𝑗𝑘 is the number of people in area i in stratum j

• Spatial SMRs also smooth rates using surrounding area observed/

expected rates

Focus Area Name

Model for spatially smoothed SMRs

𝑍

𝑗|𝜈𝑗

~ 𝑄𝑝𝑗𝑡𝑡𝑝𝑜(𝜈𝑗) log 𝜈𝑗 = log 𝐹𝑗 + 𝑐𝑗 𝑐𝑗|𝑐

𝑘≠𝑗 ~ 𝑂

σ𝑘≠𝑗 𝑥𝑗𝑘𝑐

𝑘

σ𝑘≠𝑗 𝑥𝑗𝑘 , 𝜏2 1 σ𝑘≠𝑗 𝑥𝑗𝑘

– 𝑐𝑗 are area-specific random effects with a correlated random effect distribution – 𝑥𝑗𝑘 are weights defining which regions j and i are neighbors – 𝜏2 is the variance controlling how similar 𝑐

𝑘 is to its neighbors

• In a Bayesian framework, weights depend on the precision of the

SMR (1/𝐹𝑗) in area i and the variability (heterogeneity) of the true risks across areas local or regional mean

Focus Area Name

Spatially smoothed SMRs

• The raw and smoothed standardized mortality ratio (𝑇𝑁𝑆𝑗

and ෣ 𝑇𝑁𝑆𝑗) are: 𝑇𝑁𝑆𝑗 = 𝑍

𝑗

𝐹𝑗 ෣ 𝑇𝑁𝑆𝑗 = Ƹ 𝜈𝑗 𝐹𝑗

• For areas with lots of data:

𝑇𝑁𝑆𝑗 ≈ ෣ 𝑇𝑁𝑆𝑗

• For areas with sparse data:

෣ 𝑇𝑁𝑆𝑗 ≈weighted average of SMR in the neighboring areas

Focus Area Name

𝑇𝑁𝑆𝑗 vs. ෣ 𝑇𝑁𝑆𝑗 (Age/Sex/Race Adjusted Suicides)

Focus Area Name

𝑇𝑁𝑆𝑗 vs. ෣ 𝑇𝑁𝑆𝑗 in Dayton

Focus Area Name

Classifying areas with excess (or lower) risk

• Classify an area as having an elevated/lower

risk if:

– Posterior probabilities [Prob (SMRi > 1)] > 0.8 – Outside 95% credible interval

• High specificity

– (false detection < 10%)

• Sensitivity 60%-95% for Ei of 5-20 and true

SMRi of 1.5-3.0

Focus Area Name

Areas of excess/less risk

Focus Area Name

Thoughts on when to smooth

Smoothing should be considered when:

1. The addition of one event or one more person at risk results in a large difference in the rate (e.g., a change

• f 25% or more)

2. The number of events that form the numerator is ≤ 3 3. The number of persons at risk per region is small and the numbers change by an order of magnitude across a region (e.g., 10 people in tract A vs. 100 people in tract B)

*Smoothing reduces noise and makes trends and patterns more clear

Focus Area Name

Methodology can be extended to multivariate models

Bayesian CAR model with maternal demographics Bayesian CAR model with demographics + tract-level SDHs

Focus Area Name

Software

• Estimation of Bayesian models requires

computationally intensive simulation methods (MCMC)

– Implemented in free WinBUGS and GeoBUGS software: www.mrc-bsu.cam.ac.uk/bugs – Also R package CARBayes

• R package INLA implements fast approximation:

www.r-inla.org

– R package diseasemapping calls INLA specifically for disease mapping