Introduction to Spatial Regression

Glen Johnson, PhDLehman College / CUNY School of Public Health

glen.johnson@lehman.cuny.edu

Typical scenario:

• Have a health outcome and covariables aggregated at a common geographic level, such as counties, census tracts, ZIP codes …

• Want to measure association between the outcome and the covariables.

• Specific Question is: Are there variables that co-vary spatially with the outcome variable ?

Benzene in ambient air Smoking rate

Lung Cancer Rates (observed)

+ … + ? =

+

+ residual

2

For 1, ,

or, more generally, ( [ ])

and ~ (0, )

with cov( , ) 0 for all ,

ˆwhere

i i i

i i

i

i j

i i i

i n

y

g E y

iid

i j

y y

x β

x β

Consider the linear model:

This is the point of departure.

When applying regression modeling to spatial units that are connected in space (lattice data), the critical assumption that residuals are independently distributed with constant variance is typically violated.

Tobler’s First Law of Geography: Things closer in space tend to be more similar than things further apart

When we model the expected value, E[y], as a function of spatially-varying covariates, it is possible that we may explain all of the spatial variation of the observed response, y, with the covariates, leaving uncorrelated residuals.

When this is not the case, as is typical, the assumption of iid residuals is violated and we will obtained biased estimates of the variance – typically biased downward, leading to underestimating our standard errors and concluding that some covariates are significant when in fact they are not.

Tests for spatial autocorrelation should be applied to residuals if a “conventional” regression model is applied.

This may be done with various software packages or GIS add-ons.

A common statistic is Moran’s I, which equals

1 1

2

1 1 1

( )( )

1( )

n n

ij i ji j

n n n

i iji i j

w Y Y Y Y

IY Y w

n

When residual spatial autocorrelation is present, several approaches may be taken to adjust for it.

The simplest is to add a fixed effect dummy variable to allow the model intercept to change with spatial location. For example, an adjustment is made that depends on a county of membership.

i i c iy x β

This is essentially stratifying the analysis by locationAnd can be done with any statistical software

Since spatial location is a proxy for unobserved randomly varying covariables, it is more correctly treated as a random effect in a mixed effect model, such as

2

[ | ( )] ( )

where S(i) ~ N(0, )

i i

s

E y S i S i

x β

Which can be solved for through pseudo-likelihood methods, using software likePROC GLIMMIX or PROC MIXED in SAS, orR with appropriate library (?)

Illustration: Community Teen Pregnancy Rates vs. Socioeconomic

Status and Demographic Composition

For each ZIP code: Response (i.e. Teen Pregnancy cases)

Predictors:• % pop. > age 24 w/ 4-year or greater college

degree

• % single-parent households out of households w/ at least one child < 18 years old

• % of tot. pop. that is Black Alone

• % of tot. pop. that is Hispanic, regardless of race

• % of tot. pop. that is a foreign-born naturalized citizen

• % of tot. pop. with income below poverty

Population at Risk

County (crude indicator of neighborhood effect)

Teen Preg rate vs Education

0

50

100

150

200

250

300

350

0 20 40 60 80 100

% adults with 4-yr college degree

Pre

gn

anci

es p

er 1

000

fem

ales

ag

e 15

-19

Teen Preg rate vs Single-Parent Households

0

50

100

150

200

250

300

350

0 10 20 30 40 50

% households with one parent at home

Pre

gn

anci

es p

er 1

000

fem

ales

ag

e 15

-19

Teen Preg rate vs % Immigrants

0

50

100

150

200

250

300

350

0 5 10 15 20 25 30 35

% forein-born naturalized citizens

Pre

gn

an

cie

s p

er

10

00

fe

ma

les

a

ge

15

-19

. . .

Teen Preg rate vs Race

0

50

100

150

200

250

300

350

0 20 40 60 80 100

% black alone (regardless of hispanic ethnicity)

Pre

gn

anci

es p

er 1

000

fem

ales

ag

e 15

-19

Teen Preg rate vs % Hispanic

0

50

100

150

200

250

300

350

0 20 40 60 80 100

% hispanic (regardless of race)

Pre

gn

anci

es p

er 1

000

fem

ales

ag

e 15

-19

The Model …

For i = 1, …,n ZIP codes, let

yi = observed caseload

ni = population at risk

{x1, …, xp}i = community predictors

{β1, …, βp} = coefficients

Li = location effect, arising from a random process such that Li

~ N(0, σL2)

Then, the expected value of yi, given {x1, …, xp, L}i =

E[yi| {x1, …, xp, L}i ] = niexp(β1x1i + … + βp xpi + Li)

• Values for the unknown coefficients {β1, …, βp, σL2

} are estimated with SAS PROC GLIMMIX, assuming yi

arose from a Poisson random process, conditional on location.

• … thus allowing risk adjusted estimates of caseload for each ZIP code.

• Incorporating the “location effect”- adjusts for unidentified covariables that co-vary spatially with the response, thus reducing residual spatial autocorrelation and potential confounding- also provides a “smoothing” effect, in that the predicted caseload is adjusted towards a common local value

Teen Pregnancy Association with Select Covariables

No Spatial Effect with Spatial Effect

coefficient name estimate p-value estimate p-value

intercept -3.423 <0.0001 -3.262 <0.0001

% adults w/ Bachelors -0.016 <0.0001 -0.018 <0.0001

% Black Alone 0.008 <0.0001 0.01 <0.0001

% Hispanic 0.009 <0.0001 0.012 <0.0001

% Foreign Born 0.003 0.2884 0.002 0.5906

% single-parent households 0.04 <0.0001 0.027 <0.0001

model parameters

scale 0.166 0.009

chi-square / d.f. 1.13 0.91

-2 log likelihood 7934.5 2706.6

Residual Spatial Autocorrelation (Moran's I) 0.92 0.31

Deviation of Observed from Model-Predicted Teen Pregnancy Rates

(3-Year Average for the Year 2005)

No Spatial Correction

Moran's I = 0.92

county boundaries

Pearson ResidualsNo Spatial Correction

9.6 - 23

3.8 - 9.5

0.74 - 3.7

-1.2 - 0.73

-4.4 - -1.3

New York City

April, 2009

Deviation of Observed from Model-Predicted Teen Pregnancy Rates

(3-Year Average for the Year 2005)

with Spatial Random Effect

Moran's I = 0.31

county boundaries

Pearson Residualswith Spatial Random Effect

2.3 - 8.0

0.83 - 2.2

0.0093 - 0.82

-0.62 - 0.0092

-2.1 - -0.63

New York City

April, 2009

Other approaches include …

A spatial lag model, where

i ij j i ij

y w y x β

and a spatial error model, where

i ij j i ij

y w x β

for a spatial autoregressive coefficient ρ.

These two models differ by whether the adjustment is made by a weighted sum of the response variable or the residuals.

The spatial lag and spatial error models can be solved for in Geoda, a simple, well supported freeware found at

http://geodacenter.asu.edu/

… but only for gaussian responses.

For generalized linear models (i.e. Poisson and logistic regression), see R with appropriate libraries

http://www.r-project.org/

Another approach is hierarchical modelling, which treats the response as conditional on the weighted average of local neighborhood errors.

Frequentist solutions exist, but these hierarchical models lend themselves well to a fully Bayesian solution, as used by many geographic epidemiologists

Main advantages include

* flexibility offered by Generalized Linear Mixed Models

* obtain full distribution of possible outcomes - allows many ways to view the outcome (mean, median, percentiles)

- inference based on actual probability distributions, instead of confidence intervals

Main limitation is level of conceptual difficulty; however, implementation is accessible through free software …

WINBUGS (Bayesian Inference Using the Gibbs Sampler)

1. Define a likelihood for the observations , where 1,..., regions :

~ Poisson( ), where

is the calculated expected value ai i i i i

i

y i ni

y E

E

nd is the relative risk

ˆ (note: the max. likelihood estimate of is / , the SIR)

i

i i i iy E

i1

ij

2. Link the Poisson expectation to both fixed and random effects:

log( ) log( )

for a common mean , fixed effect covariates with

ku s

i j ij i ij

E x

x

coefficients ,

and random effects (components of variance) due to

unstructured and spatially structured sources of variation

j

u si i

4 4 2u

2

3. Assign prior probability distributions to parameters in the linear model

~ N(0, 10 ), ~ N(0, 10 ) for all , ~ N(0, )

and [ | ] ~ ( , ) for spatial neighborii

uj i

s ssi i

j

N

hood i

2 2

4. Assign hyperprior distributions to the hyperparameters

1 / ~ Gamma(a,b) and 1 / ~ Gamma(c,d)u u s s

A Hierarchical Model

is distributed conditionally on location, such that

2

22

[ | ] ~ N( , )

and

i

i

i

i i

si i

ij jj

iij ij

j j

w

w w

Focus on the random effect that captures local spatial autocorrelation

si

for(i IN 1 : n)

X(covariate)

beta

tau.s

tau.u

epsilon.u

epsilon.s[i]

alpha

E[i]

mu[i]

y[i]

A Directed Acyclic Graph of the Bayesian Model

Gibbs sampling basic procedure

- All stochastic parameters in the model are assigned an initial value (somewhat arbitrarily).

- The values for each parameter are updated by random simulation from a conditional probability distribution, given all other parameters in the model.

- After all terms have been updated, completing one cycle (of what is called a Markov Chain), the cycle is repeated.

- After many iterations, the simulated values for each term converge to a stationary posterior distribution (further iterations don’t change the distribution)

Estimation and inference can then be made from these posterior distributions

For example, a simulated sample of 1000 fitted SIR values (μi / Ei) can be used to yield a point estimate (typically the median)and an interval estimate, such as the 95 %-tile range (credible set)

SIR

0.00

0.02

0.04

0.06

0.08

0.10

0.12

50th %-tile5th %-tile 95th %-tile

An illustration for geospatial analysis of prostate cancer incidence in New York State, USA …

Prostate Cancer Incidence by ZIP codeadjusted for age and raceNew York State1994-1998

SIRhat[25] sample: 1000

0.4 0.6 0.8 1.0

0.0

2.0

4.0

6.0

SIRhat[26] sample: 1000

0.6 0.7 0.8 0.9 1.0

0.0 2.0 4.0 6.0 8.0

SIRhat[27] sample: 1000

0.8 1.0 1.2 1.4

0.0 2.0 4.0 6.0 8.0

SIRhat[28] sample: 1000

0.8 1.0 1.2

0.0

2.0

4.0

6.0

SIRhat[29] sample: 1000

0.6 0.8 1.0 1.2

0.0

2.0

4.0

6.0

SIRhat[30] sample: 1000

0.6 0.8 1.0 1.2

0.0

2.0

4.0

6.0

Example Output: Posterior Kernel Densities of Prostate Cancer Incidence

(`94-`98) for Some Manhattan ZIP Codes

some references

• Waller, L.A. and Gotway, C.A. 2004. Applied Spatial Statistics for Public Health Data. Wiley. 494 pp.

• Johnson, G.D. 2004. Smoothing Small Area Maps of Prostate Cancer Incidence in New York State (USA) using Fully Bayesian Hierarchical Modelling. International Journal of Health Geographics 2004, 3:29 ( http://www.ij-healthgeographics.com/content/3/1/29 )

• Elliot, P., Wakefield, J.C., Best, N.G. and Briggs, D.J. 2000. Spatial Epidemiology: Methods and Applications. Oxford. 475 pp.

• Statistics in Medicine. 2000. Vol. 19 (special issue on disease mapping)

• Lawson, A. et al. 1999. Disease Mapping and Risk Assessment for Public Health. Wiley. 482 pp.

GeoDa

http://geodacenter.asu.edu/

(with links to R and R-Geo)

WINBUGS for Bayesian Modeling

http://www.mrc-bsu.cam.ac.uk/bugs/welcome.shtml

Both of these freewares are supported by large international community with active listserves

Method and Software Sources