Spatial statistics presentation Texas A&M Census RDC

An Introduction to Spatial Analysis in Socio-Economic

and Health ResearchCorey S. Sparks, PhD

Department of DemographyThe University of Texas at San Antonio

Outline

1) What’s special about spatial? Good news Bad news

2) Modes of thinking about spatial analysis Macro and Micro

3) Concepts of space

4) Spatial analysis is NOT statistics, but spatial statistics needs spatial analysis

5) Using this for something meaningful

What’s special about spatial?

Spatial data have more information than ordinary data Think of them as a triplet

Y, X, and Z, where Y is the variable of interest, X is some other information that influences Y and Z is the geographic location where Y occurred

If our data aren’t spatial, we don’t have Z Spatial information is a key attribute of behavioral

data This adds a potentially interesting attribute to any

data we collect

What’s special about spatial?

Spatial data monkey with models Most analytical models have assumptions, spatial

structure can violate these models We typically want to jump into modeling, but

without acknowledging or handling directly, spatial data can make our models meaningless

Some of the problems are..

The ecological fallacy

The tendency for aggregate data on a concept to show correlations when individual data on a concept do not.

In general the effect of aggregation bias, whereby those studying macro-level data try to make conclusions or statements about individual-level behavior

This also is felt when you analyze data at a specific level, say counties, your results are only generalizeable at that level, not at the level of congressional districts, MSA’s or states.

The often-arbitrary nature of aggregate units also needs to be considered in such analysis.

The modifiable area unit problem (MAUP)

This is akin to the ecological fallacy and the notion of aggregation bias.

The MAUP occurs when inferences about data change when the spatial scale of observation is modified. i.e. at a county level there may be a significant association between

income and health, but at the state or national level this may become insignificant, likewise at the individual level we may see the relationship disappear.

This problem also exists when we suspect that a characteristic of an aggregate unit is influencing an individual behavior, but because the level at which aggregate data are available, we are unable to properly measure the variable at the aggregate level. E.g. we suspect that neighborhood crime rates will the recidivism hazard

for a parolee, but we can only get crime rates at the census tract or county level, so we cannot really measure the effect we want.

Spatial structure

Structure is the idea that your data have an organization to them that has a specific spatial dimension

Think of a square grid Each cell in the grid can be though of as being

neighbors of other cells base on their proximity, distance, direction, etc.

This structure generally influences data by making them non-independent of one another

At best, you can have a correlation with your neighbor

At worst, your characteristics are a linear or nonlinear function of your neighbors

Spatial heterogeneity

Spatial heterogeneity is the idea that characteristics of a population or a sample vary by location

This can manifest itself by generating clusters of like observations

Statistically, this is bad because many models assume constant variance, but if like observations are spatially co-incident, then variance is not constant

Modes of thinking about spatial analysis

Macro Observations are areas, or aggregates of individuals Processes generally involve interaction among

these areas We can’t really get at variation within these areas

Can’t see the people within the county All we have are counts of some variable of interest Think of you stereotypical Census tract, or county

In social science, we often refer to such analyses as “ecological”

Analyzing places Looking for trends an associations at the population

level

Macro models

An example I think the infant mortality rate in US counties is a

function of the socioeconomic status of the county residents Furthermore, I expect that characteristics of the built

environment in counties will further influence the infant mortality rate

I might hypothesize that areas that are more “built up” may have higher rates of infant mortality than less “built up” places

My outcome is a rate My predictors are other rates Everything is measured at the county level

Modes of thinking in spatial analysis

Micro

Observations are individuals

Processes may still involve interactions between observations, but we often go beyond that

Look at behaviors of people Why do we do what we do?

Multi-level models

The concept that individuals are nested within a hierarchy and the behavior of interest is determined by both the characteristics of the individual and the level of nesting.

This can be though of broadly as individuals within any hierarchy, such as a city block, an organization, a villages, or a community.

The spatial idea of multi-level analysis is that the individual is nested within a distinct geographic unit. Substantively, something about that area is thought to influence our

behavior of interest. Sometimes, we have a specific idea how that happens, other times we

have a weak conceptual model

Statistically these methods correct for the common occurrence of dependence within aggregate units (i.e. people from the same area may be more alike than expected by random chance alone) which violates the assumptions of most linear regression models (iid residuals)

Concepts of space

To start, let’s think about how to define space as a concept i.e. what levels of space are important for human

behavior? Neighborhoods Villages/towns Social networks>social “space” Households Climatic zones Political/administrative boundaries

What we really want to get at

What concept of space is important in my work? That’s for you to decide

More often than not, as social scientists, we will undertake a macro analysis or some form of nested micro analysis

At the macro level, we have to conceptualize the space of our units Where are counties relative to one another Is the spatial heterogeneity in my outcome at the county level

At the micro level, we have to justify why we believe the level of nesting we can measure is in fact the right level in terms of our outcome Just because I know what MSA a child lives in, doesn’t tell me much

about the neighborhood in which they grew up

Spheres of social integration

Individuals exist within formal or more often informal spheres of interaction. You can think of these as areas where common

ideas unite people together, or the range (meaning distance) of social norms.

Common ideology often unites individuals into groups and groups into larger aggregates, look at the concept of nationalism for example.

They can likewise have a strictly a-spatial representation, such as a belief network.

Some parting thoughts about space

Realization that human behavior is a dynamic process, not a static phenomena

Use of the cross-section as opposed to the cohort (longitudinal data) or with respect to changes in the neighborhood.

Defining “meaningful” neighborhoods, not just available sampling areas, how do individuals interact within these areas?

How do individuals interact with their neighborhood?

Ideas of agency and how an individual is an agent of change

Do people choose neighborhoods? Or are some chosen by the neighborhoods they are born into, or only the ones they can afford?

Context or Composition?

Is it the places where people live that is important? (Context)

Or

Is it the nature of the people that live in the place? (Composition)

Again, when justifying a spatially oriented analysis, these things need to be addressed

Spatial analysis is NOT statistics

Enter the GIS

GIS is the manager of spatial information Goes beyond the realm of relational databases by

incorporating the spatial context of all data Uses this as the overarching infrastructure for organizing

information

The use of space as a tool The GIS allows us to edit, merge, split, add, delete all

types of information based purely on the spatial location of the data

But, it is not statistics!

GIS operations as spatial analysis

We use the GIS to help us manage and visualize spatial data

There are thousands of specific tools offered in a standard GIS

We can organize the tools of spatial analysis by what type of data they use: Point patterns Areas (polygons) Images (raster) Interactions Networks Sure I’m missing something

Spatial statistics needs spatial analysis

1) Without spatial analysis, we would miss out on the interactions between our spatial information Unable to construct important variables for

statistical analysis

2) Without spatial analysis, we would be limited in our visualization of our data, and our results From maps to 3-D visualizations

Wrap Up

Spatial data is special For better or for worse

Ignoring spatial structure can invalidate models

Exploratory Spatial Data Analysis

Corey S. Sparks, PhDDepartment of Demography

The University of Texas at San Antonio

Exploratory Spatial Data AnalysisWhat is ESDA? In exploratory data analysis, we are looking for trends

and patterns in data

We are working under the assumption that the more one knows about the data, the more effectively it may be used to develop, test and refine theory

This generally requires we follow two principles: skepticism and openness

One should be skeptical of measures that summarize data, since they can conceal or misrepresent the most informative aspects of data,

and..

We must be open to patterns in the data that we did not expect to find, because these can often be the most revealing outcomes of the analysis.

We must avoid the temptation to automatically jump to the confirmatory model of data analysis.

The confirmatory model says: Do my data confirm the hypothesis that X causes Y.

We would normally fit a linear model (or something close to it) and use summary measures (means and variances) to test if the pattern we observe in the data is real.

The exploratory model says,

To the contrary, “What do the data I have tell me about the relationship between X and Y. This lends to a much more open range of alternative explanations

The principle of EDA is explained best in the simple model:

Data = smooth + rough

The smooth bit is the underlying simplified structure of a set of observations. This is the straight line that we expect a relationship to follow in linear regression for example, or the smooth curve describing the distribution of data: the pattern or regularity in the data.

You can also think of the smooth as a central parameter of a distribution, the mean or median for example

The rough is the difference between our actual data, and the smooth. This can be measured as the deviation of each point from the mean.

Outliers, for example are very rough data, they have a large difference from the central tendency in a distribution, where all the other data points tend to clump

Doing some ESDA We will use new tools (namely GeoDa) to do some Spatial EDA.

The way GeoDa differs from traditional ways of viewing data (on paper, or with summary statistics), is that it goes beyond presenting the data, to visualizing the data.

Data visualization is a dynamic process that incorporates multiple views of the same information, i.e. we can examine a histogram that is linked to a scatter plot that is

linked to a map to visualize the distribution of a single variable, how it is related to another and how it is patterned over space.

We can also do what is called brushing the data, so we can select subsets of the information and see if, there are distinct subsets of the data that have different relational or spatial properties. This allows us to really visualize how the processes we study unfolds over space and allows us to see locations of potentially influential observations.

Some examples of ESDA items

Histograms

Boxplots

Scatter plots

Parallel coordinate plots

Area statistics/Local statistics

Histograms Histograms are

useful tools for representing the distribution of data

They can be used to judge central tendency (mean, median, mode), variation (standard deviation, variance), modality (unimodal or multi-modal), and graphical assessment of distributional assumptions (Normality)

Box Plots

Box plots (or box and whisker plots) are another useful tool for examining the distribution of a variable

You can visualize the 5 number summary of a variable Miniumum, Maximum,

lower quartile, Median, and upper quartile

Upper Quartile

Median

Lower Quartile

Maximum

Minimum

IQR

Scatter Plots Scatterplots show

bivariate relationships

This can give you a visual indication of an association between two variables

Positive association (positive correlation)

Negative association (negative correlation)

Also allows you to see potential outliers (abnormal observations) in the data

Slight positive association

Potential Outlier

Parallel Coordinate Plots

Parallel coordinate plots allow for visualization of the association between multiple variables (multivariate)

Each variable is plotted according to its “coordinates” or values

Local Statistics Local statistics

allow you to see how the mean, or variation, of a variable varies over space

You can generate a local statistic by weighting an ordinary statistic by some kind of spatial weight

Example ->

Property crime in San Antonio, TX

Global vs. Local Statistics

By global we imply that one statistic is used to adequately summarize the data

i.e. the mean or median

Or, a regression model that is suitable for all areas in the data

Local statistics are useful when the process you are studying varies over space, i.e. different areas have different local values that might

cluster together to form a local deviation from the overall mean

Or a regression model that accounts for the same level of variation in the outcome in all locations

Autocorrelation This can occur in either space or time

Really boils down to the non-independence between neighboring values

The values of our independent variable (or our dependent variables) may be similar because

Our values occur closely in time (temporal autocorrelation) closely in space (spatial autocorrelation)

Preliminaries to assessing autocorrelation

Basic Assessment of Spatial Dependency

Before we can model the dependency in spatial data, we must first cover the ideas of creating and modeling neighborhoods in our data.

By neighborhoods, I mean the clustering or connectedness of observations

The exploratory methods we will cover depend on us knowing how our data are arranged in space, who is next to who.

This is important (as we will see later) because most correlation in spatial data tends to die out as we get further away from a specific location (Tobler’s law)

Tobler's first law of geography Waldo Tobler (1970) suggested the “jokingly” first law

of geography “Everything is related to everything else, but near things

are more related than distant things”

We can see this better in graphical form: We expect the correlation between the attributes of two points to diminish as the distance between them grows

One thing about autocorrelation

Autocorrelation is typically a local process

Meaning it typically dies out as distance between observations increase

plot(exp(-.05*seq(1:100)), type="l", lwd=2, ylab="Correlation", xlab="Distance")

W

To identify which observations are close to one another, we use W

W is the spatial weight matrix Square n by n matrix with 1/0 entries identifying if

any two observations are spatial neighbors

How is this constructed?

There are two typical ways in which we measure spatial relationships

Distance and contiguity

In a distance based connectivity method, features (generally points) are considered to be contiguous if they are within a given radius of another point. The radius is really left up to the researcher to decide.

Likewise, we can calculate the distance matrix between a set of points

This is usually measured using the standard Euclidean distance

Where x and y are coordinates (lat/long) of the point or polygon in question, this is the as the crow flies distance

Distance based neighbors

More on distances There are lots of distance measures

Manhattan distances

d = |x1 – x2| + |y1- y2|

These are city-block distances

We can use these distances to create a neighborhood of points using certain criteria

Who is who's neighbor There are many different criteria for deciding if two

observations are neighbors

Generally two observations must be within a critical distance, d, to be considered neighbors.

This is the Minimum distance criteria, and is very popular.

This will generate a matrix of binary variables describing the neighborhood.

We can also describe the neighborhoods in a continuous weighting scheme based on the distance between them

Inverse Distance Weight wij=1d ij

or

Inverse Squared Distance Weight wij=1

d ij2

Contiguity/Polygon Adjacency

Polygons are contiguous if they share common topology, like an edge (line segment) or a vertex (point)

Neighborhoods are created based on which observations are judged “contiguous”

This is generally the best way to treat polygon features

Distances aren’t typically used for polygons for several reasons Spacing of observations What centroid? Is it a good measure?

• Rook adjacency

• Neighbors must share a line segment

• Queen adjacency

• Neighbors must share a vertex or a line segment

• If polygons share these boundaries (based on the specific definition: rook or queen), they are given a weight of 1 (adjacent), else they are given a value 0, (nonadjacent)

Order of adjacency

Observation of interest

First order neighbors (Rook)

Second Order neighbors (Rook)

Observation of interest

First order neighbors (Queen)

Second Order neighbors (Queen)

First and second orderRook adjacency

First and second orderQueen adjacency

What does a spatial weight matrix look like?

Set of polygons

Spatial Weight Matrix, W

Standardized W

Most times, in analytical work, W is standardized

Typically this is done by dividing each 1/0 element in the row by the sum of the row, which would yield this matrix:Polygon 1 2 3 4

1 0 0.5 0.5 02 0.5 0 0 0.53 0.5 0 0 0.54 0 0.5 0.5 0

Forms of autocorrelation

Positive autocorrelation

This means that a feature is positively associated with the values of the surrounding area (as defined by the spatial weight matrix), high values occur with high values, low with low

Negative autocorrelation

This means that a feature is negatively associated with the values of the surrounding area (as defined by the spatial weight matrix), high with low, low with high

The (probably) most popular global autocorrelation statistic is Moran’s I

Measuring spatial autocorrelation:Common measures

Moran’s I

Geary’s C

Getis-Ord G

All typically give similar results

Moran's I

Measure of standardized spatial autocovariance

with xi being the value of the attribute at location i, xj being the

value of the attribute at location j,

S0 is the sum of all spatial weights

wij is the weight for location ij (0 if they are not neighbors, 1 otherwise)

Approximately bound on 0,1 with a similar interpretation as a Pearson or Spearman correlation

Geary’s C

Measure of spatial covariance

with xi being the value of the attribute at location i, xj being the value of the attribute at location j,

S0 is the sum of all spatial weights

wij is the weight for location ij (0 if they are not neighbors, 1 otherwise)

Bound on 0,2 0 to 1 = positive autocorrelation 1= no autocorrelation 1 to 2=negative autocorrelation

Getis – Ord G

Unlike I and C, the interpretation of G focuses on whether high values of x tend to cluster with other high values of x

The measure is directional

Likewise, low with low

Local Moran’s I

We can also describe the local trends in autocorrelation in our data by constructing local versions of our autocorrelation statistics

Each has a local version

This is useful to see where the autocorrelation is high or low within our data Goes beyond the global statistics to help us

visualize where in our data associations are strong

How do you determine if the dataare in a cluster?

This is done via the Moran scatterplot.

If an observation is in the top right quadrant of the plot = high-high

Bottom left quadrant = low – low

Top left quadrant = high – low

Bottom right quadrant = low-high

Moran scatterplot (univariate)

It is sometimes useful to visualize the relationship between the actual values of the dependent variable and their lagged values. This is the so called

Moran scatterplot

Lagged values are the average value of the surrounding neighborhood around location I

lag(x) = wij*x = Wx in matrix terms

The Moran scatterplot shows the association between the observation of interest and its neighborhood's average value

The variables are generally plotted as z-scores, to avoid scaling issues

Moran scatterplot for San Antonio neighborhood deprivation

I = .688

High Positive autocorrelation

Local Moran cluster map

Significant high neighborhood deprivation on the west, east and south sides

Significant low neighborhood deprivation on the north side

Spatially lagged values If we have a value xi at location i and a spatial weight

matrix wij describing the spatial neighborhood around location i, we can find the lagged value of the variable by:

Wxi = xi * wij

This calculates what is effectively, the neighborhood average value in locations around location i, often stated x-i

Moran scatterplot (Bivariate) We can also compare the lagged value of one variable

verses the value of another variable

Wy versus x

This is the so-called “multivariate Moran scatterplot”

It is often useful in so-called space-time analysis, when we compare the value of one variable measured at two different time points. This will show the correlation between space over time.

Positive autocorrelation between a tract’s deprivation and the average minority concentration in the neighboring tracts

I = .537

What these methods tell you

Moran's I is a descriptive statistic It simply indicates if there is spatial

association/autocorrelation in a variable

Local Moran's I tells you if there is significant localized clustering of the variable Where spatial clusters and what type of cluster is

present

Two examples of spatial analysis and statistics

Goals

To show how you can use R, software that is free and available in the RDC to: 1) Perform interesting spatially-oriented analysis 2) Merge spatial data from multiple sources

Spatial join 3) Visualize results from a spatial analysis much as

you would in a GIS environment

Without a GIS!

Residential Segregation and Crime in San Antonio, TX

Four data sources: American Community Survey 5 year summary file

Census tract level Census 2000 Summary File 1

Block level San Antonio Police Department call data

Point data based on geocoded addresses All calls received by the SAPD in a year

Census TIGER file for San Antonio tracts

Methods

1)Merge ACS calculated fields to shapefile based on tract FIPS code

2) Perform spatial join of crimes to tracts Count # of crimes in each tract

3) Create a thematic map of the crime rate using quantile breaks

4) Create tract-level indices of residential segregation by aggregating up from the block level Merge these to the shapefile

5) Perform ESDA using Local Moran statistics on the segregation index and crime rate

Thematic Map/Histogram

Local Moran Cluster Maps

Spatial Disease Mapping

Application to crime data, but principles remain the same Trying to locate areas of excess disease risk

Finding spatial clusters of disease Kulldorff and Nagarwalla’s Spatial Scan Statistic DCluster library

Modeling spatial relative risk Bayesian Hierarchical Regression Posterior model summaries INLA library Approximate Bayesian inference by Integrated Nested

Laplace Approximation Great for Gaussian random field models

Methods

Spatial Scan Statistic Constructs a grid over the area, centered on each

observation Using a set radius (defined based on a proportion of

the population size at risk), a circle is created on each point

The likelihood function

is used, and the area with the maximum value of the function is the most likely cluster of cases Allows for ranked clusters (primary, secondary, etc)

Scan StatisticYou know where this is?The Riverwalk

Take home message:Watch your wallet!

Basics of Bayesian Modeling

Some statistical ideas: Likelihood

Usually we have some data that we assume follow some distribution

For counts, maybe Y ~ Poisson(λ) L (y, λ) is the joint probability distribution of the data and

the parameters The model likelihood is the product of this distribution for

all observations We get the estimate of λ that is the most likely to have

generated our data (y) Maximum likelihood estimate of λ

This is traditional statistics

Bayesian statistics

Now, lets consider the case where we have some knowledge about λ, say we believe it comes from a certain distribution, so : λ ~ Gamma(α,β) And we can either assume some value for these

parameters (α=β=1) or allow them to also come from some distribution

α~Exp(u) ,β~Exp(p) Why a Gamma distn? Because Gamma is >0, and

we know θ is >0 because it is a relative risk

Combining information

When we combine the likelihood and the prior, we form what is called the posterior distribution

Thus we have Bayes Theorem which in the continuous case is:

Which states, the posterior distribution of θ, conditional on y is the product of the likelihood and the prior distribution of θ

The denominator in Bayes theorem is a constant, and this is generally written as:

Which says the posterior is proportional to the likelihood times the prior

A simple Poisson – Gamma model

A common relative risk model is the Poisson - Gamma model

yi ~ Pois(eiθ)θ ~Gamma(α,β)The posterior distribution of θ (the relative risk) is[θ |yi ,α,β ] =L(y|θ,α,β)p(θ ) which is fully written:

[θ |yi ,α,β ] =β *α*

Γ(α*)θα*−1 exp(−θβ*)

where α* = yi +∑ α, β* = ei +∑ β

Which, barring the constant gamma function is:

[θ |yi ,α,β ] ∝Gamma( yi +∑ α, ei +∑ β)

Simple Models

Let’s now consider a simple realistic model

yi~Poisson(ei θi) Assume the usual log link function

log (θi) = C1+C2

C1=a0+x’β Linear Poisson regression with intercept

C2=other terms Could add random effect - > this would be like an individual frailty

model, or a group frailty model or a random intercept model Generally this is called a Generalized Linear Mixed Model

Now we just need to code it up!

Mapping and spatial modeling

We just achieved a better, more accurate SMR map with a simple Poisson model with a random effect This model “drew strength” from counties with good

data to help smooth counties with poor data, but it didn’t account for correlation between neighboring counties

We can also build a model that includes a spatially correlated random effect between counties Spatially smoothed rates

Building a Spatial Model

Take a Poisson-Lognormal model y~Pois(ei θi)

θi=α+ x’β + ui

What if we introduce more interesting structure on u? A common form of spatial correlation structure is the

CAR model, or also called the Besag model This is:ui ~ N (u j ,τ /nj )

Where uj is the mean of the random effects of

the neighboring j observations, and nj is the

number of neighboring observations

More models

Another model that is commonly used in practice is the convolution model, or the Besag, York and Mollie model y~Pois(ei θi)

θi=α+ x’β + ui + vi

Where now ui is a correlated heterogeneity (CH) term and vi is an Uncorrelated Heterogeneity (UH) term

Posterior Parameter Estimates

Why Use R?

R is free (So are GeoDA and QGIS)

R is available in the RDC (So are GeoDA and QGIS)

R is extremely capable and flexible (So is QGIS)

R is a scripting and statistical language (Neither GeoDA or QGIS are)

R is one stop shopping for many geospatial techniques Spatial joins, projections, data merging, raster

analysis, vector operations