Advanced Spatial Analysis
Spatial Regression Modeling
Paul R. Voss
and
Katherine J. Curtis
Day 5
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Review of yesterday
• Dealing with heterogeneity in relationships
across space
• Discrete & continuous spatial heterogeneity
in relationships
• GWR
- concept & motivation
- how to do it
- what it all means
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Questions?
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Plan for today
• Shift away from spatial econometrics
– away from the classical, frequentist perspective
– away from the MLE world
• Spatial data analysis from a Bayesian
perspective
– example using count data & Poisson likelihood
– meaning of Bayes’ rule
– MCMC
• Afternoon lab
– brief demonstration of spatial modeling in WinBUGS & R
– your analyses; your data; your issues
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Bayesian orientation…
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Situate this in a statistical framework (1)
Back in my day…
OLS linear regression models
continuous dependent variables
independent & normally distributed errors
(Legendre 1805; Gauss 1809; Galton 1875)
Adiren-Marie Legendre
(1752-1853)
Carl Friedrich Gauss
(1777-1855)
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Francis Galton
(1822-1911)
Situate this in a statistical framework (2)
Back in my day…
OLS linear regression models
continuous dependent variables
independent & normally distributed errors
(Legendre 1805; Gauss 1809; Galton 1875)
ascendency of the “frequentists”; NHST; Maximum Likelihood Estimation (MLE)
decision theory (Neyman & E. Pearson 1933)
evidential interpretation of statistical data; p-values (Fisher 1922 – 1960s)
Statistical
significance tests:
What should I do?
Jerzy Neyman
(1894-1981)
Egon Sharp Pearson
(1895-1980)
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Statistical
significance tests:
What’s the
Ronald Aylmer Fisher
evidence?
(1890-1962)
Situate this in a statistical framework (3)
Back in my day…
OLS linear regression models
continuous dependent variables
independent & normally distributed errors
ascendency of the “frequentists”;
Generalized Linear Models…
fixed effects regression
accommodate many kinds of outcomes:
Gaussian, Poisson, binomial
maximum likelihood estimation
independent observations
(Nelder & Wedderburn 1972; McCullagh & Nelder 1989)
John Ashworth Nelder
(1924 – 8/7/10)
R. W. M. Wedderburn
(1947 – 1975)
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Peter McCulloch
University of Chicago
Situate this in a statistical framework (4)
Back in my day…
OLS linear regression models
continuous dependent variables
independent & normally distributed errors
ascendency of the “frequentists”
Generalized Linear Models…
fixed effects regression
accommodate many kinds of outcomes:
Gaussian, Poisson, binomial
maximum likelihood estimation
(Nelder & Wedderburn 1972; McCullagh & Nelder 1989)
Generalized Linear Mixed Models…
linear predictor contains random effects
MLE (difficult integrals over the random effects)
Iterative approximations of the likelihood:
Laplace, PQL, MQL, numerical quadrature
(Breslow & Clayton 1993)
Norman E. Breslow
Univ. of Washington
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David G. Clayton
Cambridge Univ.
Situate this in a statistical framework (5)
Back in my day…
OLS linear regression models
continuous dependent variables
independent & normally distributed errors
(Legendre 1805; Gauss 1809; Galton 1875)
Generalized Linear Models…
fixed effects regression
accommodate many kinds of outcomes:
Gaussian, Poisson, binomial
maximum likelihood estimation
(Nelder & Wedderburn 1972; McCullagh & Nelder 1989)
Generalized Linear Mixed Models…
linear predictor contains random effects
MLE (difficult integrals over the random effects)
Iterative approximations of the likelihood:
Laplace, PQL, MQL, numerical quadrature
(Breslow & Clayton 1993)
MCMC…
1950s onward
simulations on Bayesian posterior distributions
samplers:
Gibbs, M-H, others
(enormous literature)
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While this was going on, indeed,
before all this, there was a
somewhat parallel development of
what originally was referred to as
“inverse probability”
19th Century:
Inverse Probability &
Linear regression
20th Century:
Frequentist paradigm
21st Century:
Bayesian paradigm?
And, at times, a rather fierce debate between what
came to be known as “classical (likelihood)
statisticians” and “Bayesians”
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The Reverend Thomas Bayes
• First discovered the theorem that
now bears his name
• Written up in paper, “An Essay
Towards a Problem on the Doctrine
of Chances”
• Published posthumously in the
Philosophical Transactions of the
Royal Society in 1763
• Independently(?) discovered and
formalized by Laplace (1774;
1781); “theory of inverse
probability” became the dominant
paradigm
• Interest (theoretical) renewed in
mid-20th century; interest (applied)
exploded around 1985-90
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Thomas Bayes
(1702 – 1761)
Pierre Simon,
Marquis de Laplace
(1749 – 1827)
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Bayesian methods are often turned to when
likelihood estimation becomes difficult or
unrealistic (especially in the world of spatial
data analysis)
• Advantages of a Bayesian perspective
-
more natural interpretation of parameter intervals & probability
intervals
clear & overt assumptions regarding prior assumptions
ease with which the true parameter density is obtained
ability to update as new data become available
likelihood maximization not required (for probit & logit models,
maximization of the simulated likelihood function can be difficult)
desirable estimation properties (consistency & efficiency)
• Disadvantages of a Bayesian perspective
-
steep learning curve; for those of us trained in a classical
perspective, things don’t come easy
determining a reasonable prior
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From early understanding of
the laws of probability…
P(AB) = P(A|B)P(B) = P(B|A)P(A)
from which we obtain “Bayes’ formula”:
P(B|A) = P(A|B)P(B)
P(A)
This statement is true whether you take a
Bayesian perspective or not. It’s only in a
certain understanding of this formula that
you become a Bayesian
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So… Bayes’ formula
P(B|A) = P(A|B)P(B)
P(A)
Now, from the “Law of Total Probability”,
we have for mass probabilities:
c
P( A)  P( A | B) P( B)  P( A | B ) P( B )
c
or, for density probabilities:

P( A)  P( A | B) P( B)dB
B
P(B|A)  P(A|B)P(B)
“Posterior” probability
“Likelihood”
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“Prior” probability
Usually the focus is on using the Bayes
formulation to estimate model parameters
(rather than expressing relationships
among event probabilities
P(B|A)  P(A|B)P(B)
P(parameters|data)  P(data|parameters)P(parameters)
“Posterior” probability
“Data Likelihood”
“Prior” probability
The “Bayesian mantra”
The goal generally is to achieve “optimal”
parameter estimates where attention is focused
on precision (std. errors of the parameters)
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From this we have the
“Bayesian mantra”
POSTERIOR  LIKELIHOOD x PRIOR
P[θ|D]  P[D|θ] P[θ]
In full Bayesian modeling the prior
distribution is formulated before
consulting the data
In so-called empirical Bayesian
modeling the prior distribution is
derived from the data
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Approaches to Model Building
i.e., approaches to making good “guesses” about
the values of the parameters in the model and
inferences about them
• Classical statistics
• Bayesian statistics
• Regardless of approach, the process of
statistics involves:
– formulating a research question
– collecting data
– developing a probability model for the data
– estimating the model
– assessing the quality of the “fit” & modifying the model as necessary
– summarizing the results in order to answer the research question
(“statistical inference”)
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Classical approach (1)
• Three basic steps:
– specify the model
– estimate & evaluate the model (MLE)
– inference
• Fundamental idea:
– a good choice for the estimate of a parameter (viewed as fixed)
is that value of the parameter that makes the observed data
most likely to have occurred
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Classical approach (2)
• The process of MLE involves:
1. construction of a likelihood function of the parameter(s) of interest
L( | D)  f ( D |  )  P( D |  )
2. for example, normal likelihood:
L( | D)  L(  ,  | X )  f ( X |  ,  )
n


i 1
 ( xi   ) 2 
exp 
2 
2
2
2


1
– Note: observations {xi, x2,…,xn) are assumed (conditionally)
independent
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Classical approach (3)
• The process of MLE involves:
(3) Simplify the likelihood function:
– for example, for the normal likelihood:
L(  ,  | X )  (2 2)
n

2
 1
exp  2
 2

( xi   ) 2

i 1
n

(4) Take logarithm of likelihood function:
(  ,  | X )  n ln( ) 
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1
2 2
n

i 1
( xi   )2
Classical approach (4)
• The process of MLE involves:
(5) Take partial derivative of the log-likelihood function with respect to
each parameter (for example, for the normal log likelihood):
 n( x   )


2
and

n 1
  3

 
n

( xi   )2
i 1
(6) Set the partial derivatives to zero and solve for the parameters:
n
ˆ  x and ˆ 2 
These should look familiar
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
( xi   )2
i 1
n
Classical approach (5)
• The process of MLE involves:
(7) Take partial second derivatives to obtain estimates of the variability
in the estimates for the mean & standard deviation (Hessian matrix):
  2
  2
 2

T 
  2

  2 

 2 
  2 

2
  
 4 
n



2


 n( x   )

4


n
2 4
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n( x   )




4

n
( xi   ) 2 
i 1


6
Bayesian approach
• Three basic steps (same as classical approach),
with a rather different notion about the parameters
• Fundamental idea:
– a good choice for the estimate of the parameters (not fixed) are those
values of the parameter estimated from a probability distribution
involving both the observed data and our subjective notion of the
value of the parameter
– Bayesian approach tells you how to alter your beliefs
• The modern Bayesian approach involves:
– specification of a “likelihood function” (“sampling density”) for the data,
conditional on the parameters
– specification of a “prior distribution” for the model parameters
– derivation of the “posterior distribution” for the model parameters,
given the likelihood function and the prior distribution
– obtain a sample (Monte Carlo simulation) from the posterior
distribution of the parameters
– Summarize the parameter samples and draw inferences
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I’m going to use as a running
example an application in spatial
epidemiology
South Carolina congenital
abnormality deaths 1990
Data set developed & much used by
Andrew Lawson,
Spatial epidemiologist,
Biostatistics, Bioinformatics, and
Epidemiology program
Medical University of South Carolina
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South Carolina congenital
abnormality deaths 1990
4
11
13
6
5
0
0
2
0
2
1
3
5
7
3
0
4
0
14
7
0
1
1
3
3
0
7
8
6
11
0
8
1
1
2
8
16
Some considerations:
1
1
3
• Data are a complete realization, not a sample
17population
1
• Need to know something about
the underlying
1
0
to understand elevated counts;
understand the
relative risk
0 just
5 a list of numbers
• Spatial structure is relevant; it’s not
• Areal units are arbitrary; have nothing to do with health
outcomes
• Areal units are irregular in shape and
size
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Mapping issues
• Relative risk estimation
– key concept
– we want to capture & understand the data structure to get good
estimates of relative risk
– generally measured by comparing observed counts to expected
counts; standardization.
– helps to guide health resource allocations
• Disease clustering
– main issue is to understand apparent clustering of risk
– clustering of both data and risk is common (Tobler’s 1st Law)
– extent of clustering of risk and location of clusters of high risk
• Ecological analysis
– more research oriented
– identifying covariates to find causes of elevated risk or clusters
– especially useful when hypothesis involves a putative source of
pollution
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Relative risk estimation
• SMRs (Standardized Mortality/Morbidity
Ratios); or SIRs (Standardized Incidence
Ratios)
– yi :
Observed counts
– ei :
Expected counts (derived through standardization)
– yi / ei : Estimated SMR; estimate of i (a “saturated est.”)
• Some issues:
– the SMR is notoriously unstable; it can blow up when ei is
small; SMR is unbiased estimate of i but is inefficient;
need to compute and study the std. errors
– zero counts aren’t differentiated; estimated SMR is zero
when yi is zero, even when ei may vary considerably
– confusing notation: ei has different meanings
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If we assume the underlying process
is a Poisson process…
• Individuals within the study population behave
independently wrt disease propensity, after allowance is
made for observed or unobserved confounding variables;
“conditional independence”
• The underlying “at risk” background intensity (RR) has a
continuous & constant (i.e., i =  ) spatial distribution within
the study area
• Events are unique (occur as spatially separate events)
• When these (Poisson process) assumptions are met, the
data can be modeled via a likelihood approach with an
intensity function representing the “at risk” background
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MLE of Relative Risk parameters…
• Likelihood: We usually assume that the counts of disease
result from a Poisson process such that yi has a Poisson
distribution with expected value eii
• We write: yi ~ Pois(eii)
• The counts have a Poisson likelihood:
L(eii | yi ) 
(eii | yi ) 
m

i 1
m
(eii ) yi exp( eii )
yi!
m
m
 y (ln( e  ))   e    ln( y !)
i
i 1
i i
i i
i 1
i
i 1
• Differentiate wrt eii and set to zero yields:
MLE
ˆ
i  yi ei  SMR
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So, with MLE…
• The counts have a joint probability of arising
based on the likelihood, L(eii |yi)
• L(eii |yi) is the product of Poisson probabilities for
each of the areas (again, why can we say this?)
• It tells us how likely the parameter estimates of  i
are given the data, yi , under a set of Poisson
process assumptions
• This is a common approach for obtaining
estimates of parameters
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Cautions:
• Visual message can change with change
in quantile
declaration
Mean
= 0.9703
• Large literature on cognitive aspects of
Std. dev. = 0.8024
disease mapping
• Our eye is drawn to the extreme values;
so remember: unstable variances
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So… what have we got here?
• Choropleth map of SMRs of
infant mortality from congenital
abnormalities, SC counties, 1990
• SMRi = yi/ei, where yi is the
observed count & ei is the
expected count (note: we permit
ei to be fractional)
• In this example, the ei are simple expectations generated
using “indirect standardization”; i.e., by applying the crude
rate (per 1,000 births) for SC, aggregated over the years
1990-98, to county births in 1990:
46
1990
ei
1990
 Births i

*
1990  98
(congenita l abnormalit y deaths) i
i 1
46

1990  98
(births) i
i 1
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The SMR are the ML estimates of
the true unknown relative risks
• yi is the count of disease in the ith region (these are fixed
data assumed to follow a Poisson(ei i ))
ind
yi | i ~ Poisson(eii )
• ei is the expected count in the ith region (also fixed)
• We will say that i is the “true,” fixed, relative risk in the ith
region (a parameter); this is what we wish to estimate
• E(yi) = ei E(i) = ei i under assumptions of Poisson process
• yi / ei : SMR; an unbiased estimate of i
• So… we have some estimates of the relative risks in each
county; what about the variance of these estimates?
• It turns out that there are different opinions about this
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We now move beyond MLE…
In particular, we drop the classical notion that
i is a vector of fixed parameters; we assume
that the i are random variables (having a
distribution) and we will make some parametric
assumptions concerning these parameters
Which as Bayesians we will call “priors”
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We will establish a simple hierarchy…
• yi ~ Pois(eii)
• i ~ Gamma(, )… (don’t panic; next slide)
• This is a very simple example which allows the
risk to vary according to a distribution
•  and  are unknown here and we can either try
to estimate them from the data, or…
• Perhaps also give them a distribution:
e.g.,  ~ Exp(),  ~ Exp()
• This is how Bayesian hierarchies are established:
– yi is the lowest level in hierarchy
– i is the next level
–  &  affect i ; i affects yi ;  &  affect  & 
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Why assume a gamma prior?
• That is, why i ~ Gamma(, ) ?
• It makes conceptual sense; as estimates of
relative risk, we want a distribution for i that
requires the i are non-zero, and have a mean
probably not too far from unity while allowing for
some observations well above unity
– Gamma is one such distribution, although in its
simplicity has some shortcomings
– Log-normal distribution shares these distributional
attributes (and opens up opportunities not permitted by
the Gamma distribution
• Important: Gamma prior is the “conjugate” prior
for the Poisson likelihood
• And what’s  &  ?
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We look at the notion of “conjugacy” in this
context by first examining the distributions
• We started out with a Poisson likelihood:
L ( | Y ) 
m

i 1
y
 ie  
yi!


 im1 yi  m
e

m
y!
i 1 i
• and then defined Gamma “prior”:
    1  
p ( ) 
 e
( )
• Excluding the normalizing constants, the product
of Poisson likelihood & Gamma prior is:
p ( | Y )  
 im1 yi    1  ( m   ) 
e
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which is
Gamma(yi+, m+)
Now…define “conjugacy”:
• When the prior & likelihood are of such a form that
the posterior distribution follows the same form as
the prior, the prior and likelihood are said to be
“conjugate”
• By way of illustration, the previous slide reveals
that a Poisson likelihood and a Gamma prior yield
a Gamma posterior
• Very important matter prior to modern MCMC
estimation
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A basic hierarchy
Hyperparameter
Data
Parameter
Hyperparameter
Data
1st level
Likelihood
Distribution
2nd level
Prior
Distribution
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Our model hierarchy
Defined here by a DAG (Directed Acyclic Graph)


 = k is a shape parameter;
 =1/ is an inverse scale
parameter



y
g ( |  ,  )  
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 1
  
e
(  1)!
for   0
What do we do when
conjugacy analysis fails us?
• In general, a posterior distribution will be so
complex that we must use simulation to
obtain samples
• So… (cue the music!) Modern Posterior
Inference (MCMC) to the rescue!
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Modern Posterior Inference
• Unlike the ML parameters (estimates of risk), Bayesian
model parameters are described by a distribution;
consequently a range of values of risk will arise (some more
likely than others); not just the single most likely value
• Posterior distributions are sampled to give a range of these
values (posterior sample). This sample will contain a large
amount of information about the parameter of interest
• Again, a Bayesian model consists of a likelihood and prior
distribution(s)
• The product of the likelihood and the prior distribution(s)
yields the posterior distribution
• In Bayesian modeling, all the inference about parameters is
made from the posterior distribution
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Posterior Sampling
• There are several methods used for this.
The two most common are:
– Gibbs Sampling
– Metropolis-Hastings Sampling
• Both are examples of Markov Chain Monte
Carlo (MCMC) methods
Nicholas Constantine
Metropolis
(1915-1999)
W. Keith Hastings
(1930 - )
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Josiah Willard Gibbs
(1839-1903)
Gibbs Sampler
• Fast algorithm; it yields a new sample value at
each iteration
• Requires knowledge of the conditional
distributions of the parameters given the other
parameters
• For parameter set [{i }, ,  ]:
– Fix the i ‘s and  to estimate 
– Then, fix the new  and , say, to estimate 1
– etc. etc. etc.
• BUGS (later WinBUGS) was developed for this
method (Bayesian inference Using Gibbs
Sampling)
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Gibbs sampler…
Posterior distribution is a joint pdf of the unknown parameters:
{ ,  ,  1,  2, ..., n}
ˆ  { |  ,  1,  2, ..., n}
we can do this because we have assigned
probability distributions to the parameters
and have initialized them
ˆ  { | ˆ ,  1,  2, ..., n}
ˆ1  { 1 | ˆ , ˆ ,  2, ..., n}

ˆn  {n | ˆ , ˆ , ˆ2, ..., ˆn  1}
one update cycle is complete
return to the top and begin next update cycle; do
this many times until convergence is achieved
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Metropolis-Hastings (MH) sampler
• This is a simple algorithm (more simple than
Gibbs sampling) for updating parameters
and sampling posterior distributions
• It does not require knowledge of the
conditional distributions, but it does not
guarantee a useful new sample at each
iteration
• Simple to implement
• WinBUGS now includes MH updating
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Metropolis-Hastings (MH) sampler
Works off a very simple rule: accept an updated
parameter value if it moves the simulation toward the
Posterior limiting distribution:
Let (, ’) be the acceptance probability of moving from
current parameter estimate  to proposed update  ’:
 p{ ' k |  1,  2 , ..., k  1, k  1, ..., n} 
 (k , 'k )  min 1,

 p{k |  1,  2 , ..., k  1, k  1, ..., n} 
Update is accepted if the proposal function is > 1
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Brief history
• Thomas Bayes (1702 – 1761); Oh… him
• Gibbs sampler: Geman & Geman (1984)
• MCMC: Gelfand & Smith (JASA, 1990)
• BUGS: Biostatistics Unit, Medical Research
Council, Cambridge University; 1989
• WinBUGS: Imperial College School of Medicine
at St Mary's, London; 2000
• Continued development:
–
–
–
–
–
OpenBUGS
DoodleBUGS
GeoBUGS
BRUGS
R2WinBUGS
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Using WinBUGS
• WinBUGS is a windows version of the
BUGS package. BUGS stands for
Bayesian inference using Gibbs
Sampling
• The software must be programmed to
sample form Bayesian models
• For simple models there is an
interactive Doodle editor; more complex
models must be written out fully.
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WinBUGS Introduction
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Doodle Editor
• The doodle editor allows you to visually
set up the ingredients of a model
• It then automatically writes the BUGS
code for the model
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BUGS code and Doodle stages
for simple Poisson-Gamma model
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BUGS code and Doodle stages
for simple Poisson-Gamma model
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Doodle model code:
Original model code:
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Sometimes we may choose to just
set the hyperparameters
Or let the data determine the values
(so-called “Empirical Bayes”)
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For example, we might set the Gamma
hyperparameters to, say, 0.1 & 0.1
These are somewhat “uninformative” or “diffuse” priors
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Even a very non-informative prior has the
effect of slightly pulling the likelihood estimates
(the SMRs) toward the mean of the prior
“Bayesian smoothing”
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Returning to the South Carolina
congenital abnormality deaths 1990
4
11
13
6
5
0
2
0
2
0
1
3
3
0
4
5
7
0
14
7
0
1
1
3
3
8
6
11
and recalling how
the expected deaths
were calculated…
(indirect
standardization)
1
1
0
0
8
2
8
7
16
1
1
3
1
1
17
0
0
46
1990
ei
1990
 Births i

*
5
1990  98
(congenita l abnormalit y deaths) i
i 1
46

1990  98
(births) i
i 1
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The data (including 2 covariates)
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Berkeley County, SC
1990 Deaths: 16; Expected: 8.54
SMR: 1.87 (1.07 – 3.04)
2000 pop: 143,000
68% white
Poverty rate: 11.8%
Density: 129 pers/mi2
Part of the Charleston-North Charleston-SommervilleMetropolitan Statistical Area
Principal employer: U.S. Naval Weapons Station (18,450)
Saluda County, SC
1990 Deaths: 3; Expected: 0.77
SMR: 3.92 (0.81 – 11.46)
2000 pop: 18,000
66% white
Poverty rate: 15.6%
Density: 41 pers/mi2
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Confidence intervals for
the SMRs
It turns out that there
are many options
We’ll examine some of
this in lab this afternoon
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Saluda Co.
Deaths = 3
McCormick Co.
Deaths = 0
Allendale Co.
Deaths = 1
Calhoun Co.
Deaths = 0
Lee & Bamberg Cos.
Deaths = 1 each
Barnwell Co.
Deaths = 1
Berkeley Co.
Deaths = 16
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This afternoon we will see what happens
when various Bayesian models are fit to
these data to estimate relative risk
•
•
•
•
•
•
•
Simple Poisson-Gamma
Poisson-Log-Normal with pov. covariate
Poisson-Log-Normal with inc. covariate
Poisson-Log-Normal with both pov. & inc.
Poisson-Log-Normal with pov. & UH
Poisson-Log-Normal with pov., UH & CH
Preview: Bayesian smoothing tells us there’s
probably nothing interesting in these data
regarding the incidence of infant congenital deaths
in 1990
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Mean
= 0.9703
Std. dev. = 0.8024
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Mean
= 1.0373
Std. dev. = 0.0814
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Original SMR MLE
saturated estimates of
relative risk
Mean
= 0.9703
Std. dev. = 0.8024
Full Bayes mean
posterior estimates of
relative risk (model:
PLNPovUHCH)
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Mean
= 1.0373
Std. dev. = 0.0814
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Things I’ve learned
• This stuff is not easy
– the price of entry is high, despite the free software
• Learning to efficiently code in R takes practice
– my R script is not particularly elegant
• Look at carefully at the WinBUGS log file
– the model doesn’t always converge (see next slide)
– may have to try it 3-4 times
Good
• Because full Bayes modeling
is based on a
simulation, the results vary from one run to
another
– This takes some time getting used to
Not good
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Readings for today
• Besag, Julian, Jeremy York, & Annie Mollié.
1991. “Bayesian Image Restoration with Two
Applications in Spatial Statistics.” Annals of the
Institute of Statistical Mathematics 43(1):1-20. [In
the beginning…]
• Package ‘R2WinBUGS: Running WinBUGS and
OpenBUGS from R / S-PLUS. Version 2.1-18.
March 22, 2011. [Useful acces to R functionality
in a Bayesian framework]
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Afternoon Lab
Bayesian Modeling in
WinBUGS & R
Presentations & discussion
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Areas of needed research:
• Spatial panel models; space-time
interactions
• Latent continuous variables; binary
dependent variable; counts; etc.
• Flow models
• Endogenous weighting matrices
• More users of the Bayesian perspective in
the social sciences
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Thanks for your participation!!
Special thanks Stephen Matthews, Don
Janelle, & the terrific PSU support!
See you this afternoon
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