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Transcript
Non-technical Overview of Geospatial
Statistical Methods
GIS/Mapping and Census Data Second Annual
Census Workshop Series
Workshop 3: Spatial Statistics, Spatial Research &
Confidential Census Data
New York Census Research Data Center (CRDC)
Baruch College, CUNY, May 8, 2008
A Survey of Topics
1.
2.
3.
4.
5.
6.
7.
8.
Points (Events) vs. Polygons (Areal Units)
Software Packages
Methods of Point Pattern Analysis
1.
Centrographic description
2.
Distance analysis
3.
Spatial clusters
Methods of Spatial Data Analysis
1.
Thematic mapping vs. exploratory spatial data analysis (ESDA)
2.
Spatial autocorrelation: how do we know if it is present and, if it is, why
do we care?
3.
Making neighbors: spatial weights
Spatial Regression Models
1.
Spatial error vs. spatial lag
2.
Spatial heterogeneity vs. spatial dependence
Spatial Interpolation
Space/Time Dependence
Spatial mixed and spatial generalized linear models
Atlanta metro region
with locations of selected homicides
Where to Look for Spatial Analytic Tools
ESRI home page, with links to resources for digital maps, data sets, utilities,
courses, etc. http://www.esri.com/
CrimeStat III: A Spatial Statistics Program for the Analysis of Crime Incident
Locations software, manual, and sample data
http://www.icpsr.umich.edu/NACJD/crimestat.html/
SaTScan v7.0.3 software, manual, and sample data available at
http://www.satscan.org
GeoDa home site, with links to GeoDa installation, manuals, tutorials, data sets
and other supporting materials https://www.geoda.uiuc.edu/
R Spatial Projects: packages (e.g., spdep) to carry out spatial data analysis using
the R language http://sal.uiuc.edu/csiss/Rgeo/
SpaceStat: A Program for the Statistical Analysis of Spatial Data, SpaceStat
tutorial and instructional manual, all available at
http://www.terraseer.com/products_spacestat.php
Stata: tools for spatial data analysis (spat* routines)
SAS: spatial error covariance structures in mixed and glimmix procedures
POINT PATTERN ANALYSIS
Using CrimeStat
1.
Spatial distribution
1.
2.
2.
3.
projected or spherical coordinates
polar coordinates
Distance analysis
Spatial clusters
Spatial distribution: spherical or projected
coordinate system
1.
2.
3.
4.
5.
6.
7.
Mean center
Median center
Center of minimum distance
Standard deviation of X and Y coordinates
Standard distance deviation
Standard deviational ellipse
Average density
Spatial distribution: polar coordinate
system
1. Directional mean and variance
2. Convex Hull
Centrographic statistics
These statistics originate in the 1920’s
e.g., Lefever, D. 1926. “Measuring geographic
concentration by means of the standard deviational
ellipse.” American Journal of Sociology 32(1): 88-94
They are called centrographic in that they are
two-dimensional analogs to the basic statistical
moments of the univariate distribution
Standard deviational ellipse
1.
2.
3.
Because we are working in 2 dimensions,
standard distance deviation distorts dispersion
by ignoring skew
Standard deviational ellipse gives dispersion in
2 dimensions
Derived from the bivariate distribution
Geometric and Harmonic Means
1.
These statistics “hug” the center of the
distribution and as such are useful measures
of central tendency when the distribution is
skewed
1.
2.
Geometric mean is the anti-log of the mean of
logarithms of X and Y
Harmonic mean is the inverse of the mean inverse
of X and Y
Average Density
1.
2.
3.
Number of incidents divided by area
Sometimes called “intensity”
Area can be defined in measurement
parameter page in CrimeStat
1.
otherwise minimum and maximum X and Y
Spatial Distribution Using Polar Coordinates
1.
2.
3.
4.
5.
Centrographic statistics discussed thus far have been
based on spherical or projected coordinates
Another metric can be used, called a polar coordinate
Statistics are calculated using trigonometric functions
Input is a set of vectors defined as angular deviations
from a reference vector and a distance vector
CrimeStat can convert X and Y coordinates into
angles with a bearing from an origin
Directional Mean
1.
2.
3.
Calculated as the intersection of the mean
angle and the mean distance
Directional mean is dependent on the choice of
origin
Triangulated mean is the intersection of the
directional mean from the lower-left and upperright origins
Measures of Spread
1. Convex Hull
1. Boundary drawn around the distribution of points
2. Represents the polygon that circumscribes all points
in the distribution such that no point lies outside the
polygon
2. Circular Variance
1. Standardized variance
2. 0 shows no variability; 1 indicates maximum
variability
Distance analysis
1. Nearest neighbor analysis
2. Ripley’s K statistic
Nearest Neighbor Index (NNI)
1. NNI = d(NN) / d(ran)
2. If observed average (nearest) distance is same
as mean random distance, ratio is 1.0
1. If ratio < 1, points are closer together than
expected on the basis of chance
2. If ratio > 1, evidence of dispersion more widely
than expected on basis of chance
K-Order Nearest Neighbors
1.
2.
3.
4.
NNI is only an indicator or first-order spatial
randomness.
What about second nearest neighbor?
What about Kth nearest neighbor?
Mean random distance to Kth nearest neighbor
1.
It is suggested that no more than 100 nearest
neighbors be calculated
Linear Nearest Neighbor Index
1.
2.
3.
4.
Uses indirect distances by applying a grid to
the region
Called Manhattan distances
Must supply a length of street network
Also a K-order LNNI
The Problem of Edge Effects
1.
2.
3.
An incident occurring near the border of the
study area may actually have its nearest
neighbor on the other side of the border.
Since there is no information on incidents
outside the study area, the program selects
another point as nearest neighbor
The observed nearest neighbor distance is
probably greater than it should be
Ripley’s K Statistic
1.
2.
3.
Index of non-randomness for different scale
values
It is a “super-order” nearest neighbor statistic
Provides a test of randomness for every
distance from the smallest up to some
specified limit area
Cluster Analysis
1. Mode
1. Fuzzy Mode
2. Nearest Neighbor Hierarchical Clustering
1. Risk-Adjusted NNHC
3. Spatial and Temporal Analysis of Crime
(STAC)
4. K-means Partitioning Clustering
Mode and Fuzzy Mode
1. Locations (points) with the highest number of
incidents are defined as “hot spots” (clusters)
2. Definition of a cluster is based on frequency
3. Usefulness depends on degree of resolution
4. Fuzzy mode allows a search radius
1. Caution: points are counted multiple times
Nearest neighbor hierarchical clustering
1. Identifies groups of incidents that are spatially close
2. It is a hierarchical clustering routine that clusters
points together on the basis of a criterion called a
threshold distance
1. determined by random distance algorithm or used defined
2. Minimum number of events may also be defined
3. Clustering is repeated until either all points are
grouped or the clustering routine fails
Spatial and Temporal Analysis
of Crime (STAC)
1.
2.
STAC is a combination of a scan statistic,
counting the number of events within a circle,
and the hierarchical clustering technique just
described.
The results are visualized as a standard
deviational ellipse (or convex hull) computed
for the points associated with each cluster
How Does STAC Work?
1.
2.
STAC lays out a 20 x 20 grid structure on the plane
defined by the area boundary
STAC places a circle on every node of the grid, with a
radius equal to 1.414 (square root of 2) times the
specified search radius. This ensures that the circles
overlap
1.
3.
The user can specify different search radii
STAC counts the number of points falling within each
circle and ranks the circles in descending order (the top
25 search areas are selected)
How Does STAC Work (cont.)?
4.
5.
6.
For the 25 circles (or all circles with at least 2 data
points) the X and Y coordinates of any node within
the search radius are recorded, along with the
number of data points found for each node
If a point belongs to 2 different circles, the points
within the circles are combined.
The process is repeated until there are no
overlapping circles
How Does STAC Work (cont.)?
7.
8.
9.
Using the data points in each cluster (hot spot),
STAC calculates the best-fitting standard
deviational ellipse (or convex hull)
Because the standard deviational ellipse is a
statistical summary, it may not contain every point
in the cluster or it may contain points that are not
in the cluster
The convex hull creates a polygon around all
points in the cluster
K-Means Partitioning Clustering
1.
2.
3.
4.
K-means clustering is a partitioning procedure
where the data are grouped into K groups
defined by the user
The routine tries to find the best positioning of
the K centers and then assigns each point to
the center that is nearest
All points are assigned to clusters
Useful when the user wants to control grouping
K-means routine
1.
Step 1: identification of an initial guess for the location
of the K clusters
1.
2.
3.
Grid is overlaid on the data set and number of points in
each grid cell is counted
Grid with highest count is (initial) first cluster
Second cluster is grid cell with next most points that is
separated by separation criterion
1.
4.
2.358 * 0.5 * SQRT(A / N); where 2.358 is student t for 0.01
significance level
Third (initial) cluster is selected and so on
K-means routine (cont.)
2.
3.
Step 2: local optimization assigns each point to
the nearest of the K seed locations
Then repeats by minimizing distance of points
to center of assigned cluster
Thematic Maps Using ArcGIS
1. Default setting is natural breaks
1. Also called Jenks’ optimization
2. Partitions data into user-defined number of
classes by calculating breaks based on
smallest possible total error (smallest sum of
squared deviations about the class mean)
Classification choices in ArcGIS
1. Other choices are:
1.
2.
3.
4.
5.
Quantiles
Standard deviation
Equal interval
Defined interval
manual
Functionality of GeoDa
Outlier Maps in GeoDa
1.
2.
Percentile maps
Box maps
Exploratory Spatial Data Analysis
1.
2.
ESDA is the spatial analog of Tukey’s EDA
ESDA is the visualization of spatial non-randomness
(i.e., positive or negative spatial autocorrelation).
Also used to discover:
1.
2.
3.
3.
Spatial trends
Spatial regimes (non-stationarity)
Spatial outliers
ESDA may also involve the visualization of spatial
covariates
Percentile maps provide an intuitively
appealing display of geographic patterns of
homicide rates. However, the simple visual
inspection of maps is potentially unreliable in
the detection of clusters and patterns in the
data. Human perception is not sufficiently
rigorous to assess "significant" clustering and
tends to be biased towards finding patterns,
even in spatially random data.
Moran Scatterplot
Why Do (Should) We Care?
1.
2.
3.
First Law of Geography (Tobler’s Law): everything
depends on everything else, but closer things more so
Clusters and correlated errors
Classical linear regression model assumes:
COV ui u j   0,
i j
COV  X i ui   0
An Example from Temporal Analysis
1.
2.
In time series analysis autocorrelation is a common
concern: first-order autoregressive scheme AR(1)
This can be shown by writing the two variable PRF as
Yt  1  2 X t  ut
where ut denotes error at time t and the AR(1)
scheme is given by
ut  ut 1   t ,
3.
1    1
 is known as the coefficient of autocovariance and t
satisfies the usual OLS assumptions
From temporal to spatial autocorrelation
1. In AR(1) scheme:
uˆ  uˆ 

ˆ 
 uˆ
t 1
t
2
t
2. Moran’s I Spatial Autocorrelation Statistic:
w z  z 

I
z
i
j
ij
i
j
2
i i
where wij is an element of a row-standardized spatial
weights matrix and zi is yi - m, the variable of
interest centered on m, the sample mean
How do we know if autocorrelation is present?
Durbin-Watson d statistic:
N
d
 uˆ  uˆ 
t 2
2
t
t 1
N
2
ˆ
u
t
t 1
0d 4
Bounds on d:
Relationship between d and ̂ :
d
d  21  ˆ , therefore ˆ  1 
2
So if
ˆ  0, d  2; ˆ  1, d  0; ˆ  1, d  4
From temporal to spatial autocorrelation
1.
Geary’s C:


C
i


w
z

z
ij
i
j
j
2i z
2
2
i
How do we represent the relationship
between areal units in the W matrix?
1.
Probably the most common choice is the
construction of an “adjacency” (contiguity)
matrix, in which spatially adjacent areal units
(where area i shares a common border with
area j) are assigned scores of 1, and 0
otherwise (including the main diagonal)
Spatial weights based on contiguity
Spatial weights matrix
1.
2.
3.
From X and Y coordinates, we can calculate
distance from centroids
Minimum distance ensures that all units will be
connected with at least one other unit (no
“islands”)
Another choice is k-nearest neighbors
Threshold based spatial weights and
visualization of connectivity
Spatial weights matrix
1.
2.
3.
Another popular choice is the inverse of the distance
between the geographic centers of the areal units
(again with the main diagonal set to zero)
This establishes a decay function that will weigh the
effect of events in geographically closer units more
heavily than those in more distant units
Inverse distance matrices are particularly useful
partitions of geographic space when the phenomenon
of interest involves the transfer or exchange of
information
Inverse Distance
county B
county C
county K
1/25
1/30
1/150
1/60
1/200
county A
county A
0
county B
1/25
0
county C
1/30
1/60
1/120
0
county K
1/150
1/200
1/120
0
Spatial weights matrices
The choice of matrix representations is by no
means limited to the examples given here. In
fact there is an infinite number of weights
matrices, but some representations will be
more substantively and historically compelling
than others.
How do we measure neighbors
among areal units?
1.
2.
3.
4.
5.
Sparse vs. full weights matrix?
Does size of matrix matter?
Row-standardization?
Unequal Size of areal units?
Edge effects?
Imagine the Moran Scatterplot as a
Thematic Map:
Local Moran’s I
The formal expression of the local Moran’s I is statistic
is:
Ii = (zi / i zi2) j wij zj,
where wij is an element of a row-standardized weights
matrix, observations zi and zj refer to the variable in
question in standardized form.
Is Ii “Significant”?
The significance of this statistic is based on a
permutation approach whereby 999 Ii statistics are
generated to form a reference distribution. Each
county’s Ii statistic is compared to this reference
distribution for inference purposes. Counties with
non-significant Ii values appear white in the Moran
Scatterplot Map.
Conditional randomization or
permutation approach
1.
2.
3.
Randomization is conditional in the sense that the value at a
given location is held fixed (that is, not used in the permutation)
and the remaining values are randomly permuted over the
locations in the data set
For each of these resampled data sets, the value of Li can be
computed
The resulting empirical distribution function provides the basis for
a statement about the “extremeness” of the observed statistic
relative to (and conditional on) the values computed under the
null hypothesis (the randomly permuted values)
LISA map of (log)
median housing
value in DC metro
area
LISA map of
percent black in
DC metro area
Stata’s spatgsa
Stata’s spatlsa
Other Uses of Spatial Autocorrelation
1.
2.
Logan used neighborhood clusters of high poverty to
identify target areas for community development
funding by NYC DYCD (Department of Youth and
Community Development)
Spatial Autocorrelation of Regression Residuals
Violates OLS Assumption
Deane, Glenn, Steven F. Messner, Thomas Stucky,
Charis Kubrin, and Kelly McGeever. “Not ‘Islands
Unto Themselves’: Exploring the Spatial Context of
City-Level Robbery Rates.” Journal of Quantitative
Criminology (forthcoming)
Spatial Dependence in City (log) Robbery Rates
OLS assumption violation: correlated error
Choosing the Appropriate Form
of Spatial Dependence
1.
2.
Lagrange multiplier tests as a diagnostic tool
Baller, Robert D., Luc Anselin, Steven F.
Messner, Glenn Deane, and Darnell F.
Hawkins. 2001. “Structural Covariates of U.S.
Country Homicide Rates: Incorporating Spatial
Effects.” Criminology 39: 561-590.
Spatial Heterogeneity vs.
Spatial Dependence
1.
2.
3.
Spatial data analysts too readily associate
spatial nonrandomness with spatial
dependence
Spatial heterogeneity should be the first choice
Spatial chow test for spatial regimes as an
alternative to spatial regression models
Interpreting the local effect of X
Deane, Glenn, E. M. Beck, and Stewart E.
Tolnay. 1998. “Incorporating Space into Social
Histories: How Spatial Processes Operate and
How We Observe Them.” International Review of
Social History, Supplement 6, 43:57-80. Also
reproduced in New Methods for Social History,
edited by Larry J. Griffin and Marcel van der
Linden (1999).
Estimation of Spatial Models
1.
The Mixed Spatial Lag Model:
y  Wy  X  
If the autoregressive parameter is known:
y  Wy  X  
This is a spatial filter model
If autoregressive parameter is not known
SpaceStat is still the most comprehensive package available:
1. Maximum likelihood estimation
2. Instrumental variables estimation
1.
2.
3.
3.
4.
via 2SLS
via generalized method of moments (GMM)
Spatial Seemingly Unrelated Regressions (SUR)
Conditional Autoregressive Model (CAR)
OLS estimation
1.
2.
3.
4.
Spatially lagged X can be estimated by OLS
Trend Surface Model
Spatial Expansion Model
Spatial Regime Model (Spatial Chow)
Problem of ML Estimation
1.
2.
The spatial weights matrix W is n x n and is
used ind matrix addition and multiplication
operations in the ML estimators
To obtain the variance-covariance matrix of
these estimators, which is necessary for
hypothesis testing, we must compute the
inverse of the n x n matrix A
Instrumental Variables Estimation
1.
2.
3.
Formally this is a two-stage-least squares process in
which the vector y* is estimated in the first-stage
(where y* is Wy): y* is regressed on the complete set
of fully exogenous variables
the actual observations on the y* variable are
replaced by their corresponding predicted values
In the second stage, y is regressed on the predicted
values of y* and the submatrix X
Estimation of Spatial Models Using SpaceStat
1. The models in the Regress module are
organized along 2 dimensions
2. There are 4 general classes of models, which
each correspond to a menu in the Regress
module
1.
2.
3.
4.
Classic regression model
Model with spatial error dependence
Model with heteroscedastic errors
Model with spatially lagged dependent variable
Second Dimension Pertains to the Form
of the Model Specification
1. 5 distinct forms in SpaceStat
1.
2.
3.
4.
5.
Generic regression
Trend surface
Spatial regimes
Spatial expansion
Spatial ANOVA
File Structure for Spatial Regression Models
1.
Trend surface: you must select the model specification as 2
(only two explanatory variables)
1.
2.
3.
4.
X coordinates and Y coordinates
Spatial regimes: you must select the model specification as 3
(you must also specify an indicator variable to define the
regimes)
Spatial expansion: you must select the model specification as
4 (you must also specify the expansion variables and the order
for the expansion polynomial)
Spatial ANOVA: you must select the model specification as 5
(the explanatory variables must be categorical)
File Structure by Estimation Method
1. Spatially lagged dependent variable
1. For spatial lag or spatial error models you may
specify the spatial lag Wy explicitly
2. However the default is that SpaceStat computes
the lag internally based on the weights matrix and
dependent variable
2. Spatially lagged explanatory variables
3. Instrumental variables
Other Software Packages and
Spatial Models
1.
2.
3.
4.
GWR: geographically weight regression
Stata has spat* routines: spatwmat, spatgsa,
spatcorr, spatlsa, spatdiag, spatreg
R has a variety of packages including spdep
Spatial mixed model (e.g., SAS)
1.
5.
Including spatial panel model
Spatial generalized linear model (and mixed
model)
Spatial Interpolation
1.
2.
3.
There are many interpolation techniques
These methods require point locations (X- and
Y- coordinants) and intensities
These methods include trend surface models,
spatial expansion, local regression models
(e.g., spatial spline models), kriging, and kernel
density estimation
How Do We Model the Data Generating Process?
1.
Trend Surface Method:
zi = a + b1ui + b2vi
2.
Spatial Expansion Method:
zi = a + b1X1i + … + bkXki
ai = a0 + a1ui + a2vi
b1i = b10 + b1ui + b2vi
bki = bk0 + b1ui + b2vi
Spatial Expansion
1.
Spatial expansion is a mixed model
zi = a0 + a1ui + a2vi + b10x1i + b11uiX1i +
b12vix1i + bk0xki + bk1uiXki + bk2vixki
Kernel Density Estimation
1.
2.
3.
Strictly speaking, kernel density estimation is not an
interpolation technique, it is the estimation of a
probability surface
A smooth (and symmetrical) kernel function is placed
over each point
The underlying density is estimated by summing the
functions at all locations (points) on the surface to
produce a smooth cumulative density function
How Does Kernel Density
Estimation Work?
1.
2.
3.
“Kernels” are the chosen density functions placed
over each point
There are many kernel density functions (CrimeStat
supports 5: normal, uniform, quartic (spherical),
triangular (conical), and negative exponential
(peaked))
Main difference is that the normal includes all points
in the pattern, whereas the others have a
circumscribed radius (a cut-off distance) that is
included in the summation
How Does Kernel Density
Estimation Work (cont.)?
4.
5.
6.
7.
The smoothness of the density function is a
consequence of the bandwidth size (as shown in the
figures)
Luckily, as long as the kernel function is symmetrical,
choice of kernel function generally doesn’t make too
much difference
Edge effects may also cause distortion
An intensity, or an exposure to risk, at each location,
results in 3-dimensional kernels
Geographically Weighted Regression
1.
2.
3.
GWR is very similar to kernel density
estimation in concept
But also like EB (empirical bayes) estimation of
multilevel models in its extraction of
information and parameter estimation
But its really just WLS with locational
coordinates doing the weighting
What is GWR?
1.
OLS regression:
y = B0 + B1x1 + B2x2 + u
2.
GWR regression:
y = B0(g)+ B1(g)x1 + B2(g)x2 + u
(g) indicates location of estimated parameters
GWR is weighted least squares
1. Matrix form of B:
1. In OLS regression: (X’X)-1 X’Y
2. In GWR regression: (X’ W(g) X)-1 X’ W(g) Y
2. W weights the connection of locational
coordinates ui, vi
GWR Regression Output
1.
2.
GWR will output a text file with regression
estimates (both global and local)
Includes standard regression diagnostics (e.g.,
ANOVA table) and random coefficient
diagnostics
GWR Casewise Regression Output
PARM_1 … PARM_n Values of the estimates of the parameters at each
regression point. n is one more than the number of independent variables
with PARM_1 containing the values of the intercept term.
SVAL_1 … SVAL_n Values of the estimates of the standard errors of the
parameters at each regression point. The numbering of these variables is as
for the parameter estimate variables.
TVAL_1 … TVAL_n Pseudo-t values
OBS Observed y variable value
PRED Predicted y variable value
RESID Unstandardised residual
HAT Leverage value
STDRES Standardised residual
COOKSD Cook’s Distance
LOCRSQ Pseudo-R2 values
GWR as a Diagnostic Tool
for Non-Stationarity
1.
2.
Compare normal sampling distribution of
regression parameters to spatial distribution
1. 2 x s.e. vs. IQR
Visualization of Non-Stationarity
1. Save uncompressed ESRI export file (.e00)
2. Join to attribute file and create choropleth
(thematic) maps
Example: GWR Regression
parameter
relative deprivation
residential instability
%nh black
% young male
(ln) pop size
% divorce
West
proactive police
Intercept
2 x s.e.
0.0489
0.006424
0.00337
0.017283
0.118453
0.018289
0.092001
0.060788
0.614187
IQR
0.14921
0.029074
0.011177
0.045791
0.266346
0.085312
0.549081
0.193305
1.828351
Detecting Space-Time Clusters
1. Knox Index
1. Discrete treatment of space-time
2. Mantel Index
1. Continuous treatment of space-time
3. Space/Time Scans
Knox Index
1.
2.
3.
4.
5.
Knox statistic is simply a chi-square statistic
Each pair of points is compared in terms of
distance and time interval
Distance is categorized as close in distance
and not close in distance
Time interval is categorized as close in time
and not close in time
N*(N-1)/2 pairs
Methods for Dividing Distance and Time
1. Mean distance and mean time interval
1. This is default in CrimeStat
2. Median distance and median time
3. User defined criteria for distance and time
separately
Observed Frequencies for Knox Index
Expected Frequencies for Knox Index
Chi-Square Based Test
1.
2.
Monte Carlo simulation for chi-square value
under spatial randomness
Random simulation is repeated K times (K
selected by user)
1.
where distance and time interval are selected from
range of min and max distance and time
Mantel Index
1.
2.
3.
Mantel Index is the continuous measure
counterpart to Knox statistic
It is a correlation between distance and time
interval for pairs of incidents
More formally it is a general test for the
correlation between two dissimilarity matrices
Mantel Index
1.
2.
Where Xij is an index of similarity between two
observations, i and j, for distance and Yij is an
index of similarity between the same
observations for time interval
T is a covariance measure
Mantel Index
1.
2.
The covariance is then normed by dividing by the
product of the standard deviations of X and Y
Thus the Mantel Index is a correlation coefficient
Limitations of Mantel Index
1.
2.
3.
Like usual correlation coefficient, Mantel Index is
sensitive to distributional form and outliers
Because test is a comparison of all pairs, N*(N-1)/2,
the correlations tend to be very small, which makes it
less intuitive for most analysts
Continuous treatment of time and space means that
sample size must be quite large to produce a stable
estimate
Kulldorff’s Scan: SaTScan
1.
2.
H0: the null spatial model is an inhomogeneous
Poisson point process with an intensity, mu,
proportional to the population-at-risk
H1: in some locations in the multidimensional
space, the number of cases exceeds that
predicted under the null model
How Does SaTScan Work?
1.
2.
3.
A cylindrical window is moved systematically through the
study’s geographic and temporal space
The window is centered on an individual region centroid
at a particular time and expanded to include neighboring
regions and time intervals until it reaches a maximum
size
The number of cases observed and expected within the
window is calculated at each window size
How Does SaTScan Work (cont.)?
4.
5.
6.
The maximum size will not exceed 50% of the
average population-at-risk for the study period
and 50% of the study period span
The window is then centered on the next region
centroid and the process is repeated
The hypotheses are evaluated with a maximum
likelihood ratio test that examines whether the null
or alternative model better fits the data