Bayesian space-time models for
surveillance and policy evaluation
using small area data
Nicky Best
Department of Epidemiology and Biostatistics
Imperial College, London
Joint work with Guangquan (Philip) Li, Sylvia
Richardson, Bob Haining, Anna Hansell,
Mireille Toledano, Lea Fortunato
Outline

Introduction

Policy Evaluation: Evaluating Cambridgeshire
Constabulary’s ‘no cold calling’ initiative

Surveillance: Detecting unusual trends in chronic
disease rates
Introduction



Bayesian space-time modelling of small-area data is
now common in many application areas
 disease mapping
 small area estimation (official statistics)
 mapping crime rates
 modelling population change
 .....
Key feature is that data are sparse
Bayesian hierarchical model allows smoothing over
space and time → improved inference
Introduction


Many different inferential goals
 description
 prediction
 surveillance
 estimation of change / policy impact
 .....
Many different ways of formulating the space-time model
 space + time (separable effects)
 space + time + interaction
 space-time mixture models
 .....
Our set-up

Inferential goals: detection of areas with ‘unusual’ time trends
 Goal 1: Policy evaluation


Goal 2: Surveillance


a policy or intervention has been implemented in a known subset of
areas, and we wish to evaluate whether this has had a
measureable impact on the event rate in those areas
no a priori subset of areas of interest; we just wish to identify any
areas whose event rate differs markedly from the general time trend
General modelling framework


Assume most areas exhibit a common temporal trend (separable
space and time effects) – the ‘common trend’ model
For a small subset of areas, assume time trend is unusual (space-time
interaction) – the ‘local trend’ model
Goal 1: Policy Evaluation
Evaluating Cambridgeshire Constabulary’s
‘No Cold Calling’ initiative
In collaboration with
Guangquan Li*, Robert Haining+, Sylvia Richardson
+University
*Imperial
of Cambridge
College, London
Definition of a “cold call”




A visit or a telephone call to a consumer by a trader, whether
or not the trader supplies goods or services, which takes
place without the consumer expressly requesting the
contact.
Not illegal but often associated with forms of burglary and
“rogue trading”.
To discourage cold calling police have targeted specific
neighbourhoods as “no cold calling” (NCC) areas: street and
house signage; information packs for residents; informal
follow-up meetings.
Cambridgeshire Constabulary initiated NCC scheme in parts
of Peterborough in 2005 and extended it in 2006.
Locations of the NCC areas in Peterborough
Summary of NCC-targeted areas
Data for evaluation

All reported “burglary in a dwelling” events (Home Office
classification code 18, sub-codes 0-10, and code 29) used
as outcome

Surrogate for rouge trading and distraction burglary (very small
number or recorded events)

Data aggregated to annual counts by Census Output Area
(COA) in Peterborough

Time period: 2001-2008

Total of 9388 recorded burglaries

Median burglaries per area per year = 2

5th and 95th percentiles: 0 – 8
Raw data: individual and aggregated time trends
Poisson test
RR01-04 = 1.06, p=0.56
RR05-08 = 0.85, p=0.19
Positive impact
of policy?
Strategy for evaluation

Compare burglary rates before and after implementation of NCC
scheme


Comparison is done after adjustment for systematic changes in
burglary rate in other non-NCC areas


difference between 2 time periods is indicative of impact of policy
use of ‘control’ areas helps to differentiate how much of the change is
due to the policy and how much to other external factors
Deal with sparsity of the data (i.e. small number of burglary
events) by


Data aggregation → assessing overall impact
Hierarchical modelling of local impacts → assessing both overall and
local impacts
→ Separate signal from noise
Constructing the control group


Control areas are selected to have similar local characteristics (e.g.
burglary rates; deprivation scores) to those in the NCC-targeted group
Control areas are chosen to be Lower Super Output Areas (LSOA) to
obtain reliable control data (results are similar with COA-level controls)
Control
Criterion
Description
No. of
LSOAs
1
All LSOAs in Peterborough
88
2
±10% burglary rate of the NCC group in 2005
9
3
±20% burglary rate of the NCC group in 2005
20
4
±30% burglary rate of the NCC group in both 2004 and 2005
7
5
LSOAs containing the NCC-targeted COAs (but excluding the
NCC-targeted COAs)
10
6
LSOAs that had “similar” multiple deprivation scores (MDS) as
those for the NCC LSOAs in 2004
46
Evaluation procedure
Evaluation procedure
Evaluation procedure
The impact function

We consider various functional forms for the impact function
(Box and Tiao, 1975)

The impact of the policy is quantified through the estimation
of the function parameter(s)

Model selection via DIC
Name
No change
Step change
A linear function of time
A generalization
function
Functional form
Full model specification
Control areas
+ NCC areas pre-scheme
yit ~ Poisson(ni it )
log(it )    ui   t   it
 ~ N(0,1000) (overall intercept)
 1:T ~ RW1 ( W,  2 ) (time effect)
NCC areas post-scheme
(t ≥t0)
ykt ~ Poisson(nk   kt )
log( kt )   *  uk*   t*   kt*
 I (t  t0 )  f (t , bk )
*  
 t*   t
ui ~ N(0,  u2 ) (area effect)
uk*  ui  k
 it ~ N(0,  2 ) (overdispersion)
 kt*  N(0,  2 )
f (t , bk )  bk  (t  t0  1)
bk ~ N( b ,  b2 )
Implementation
yit
2
Common trend
model
u2
2
it

t
ui


Model fitted in
WinBUGS
Common trend model
fitted to control areas
(all years) plus NCC
areas (years before
scheme only)
Implementation
yit
2
2*
Common trend
model
u2
2
it

t
*
t *
uk*
b
b 2
bk
ykt

ui
for k=i
kt*

Local trend
model , t ≥ t0

Model fitted in
WinBUGS
Common trend model
fitted to control areas
(all years) plus NCC
areas (years before
scheme only)
Local trend model
(impact function) fitted
to NCC areas (years
after scheme)
Implementation
‘cut’ link
* distributional
constant
(no learning)
yit
2
2*
Common trend
model
u2
2
it

t
*
t *

uk*
b
b 2
bk

ui
for k=i
kt*


ykt
Local trend
model, t ≥ t0
Model fitted in
WinBUGS
Common trend model
fitted to control areas
(all years) plus NCC
areas (years before
scheme only)
Local trend model
(impact function) fitted
to NCC areas (years
after scheme)
‘Cut’ function used to
prevent NCC area
(post-scheme) data
influencing estimation
of common trend model
parameters
Results: choice of impact function
No Change
Step
Linear
Generalization
function
Dbar
15.27
14.32
9.77
11.75
pD
1.21
2.29
2.25
2.57
DIC
16.49
16.61
12.02
14.33

Linear impact function has smallest DIC
No change
Posterior
probability of
“success”
i.e. Pr(bk < 0)
Heterogeneity of local impacts
b = -1.1
95% CI(-2.6, 0.2)
f (t, bk) = bk∙(t  t0+1); bk =  + b xk + dk ;


dk ~ N(0,  2)
Some of the variability in local NCC impacts may be due to coverage
The larger the proportion of properties that were visited in a COA, the
greater the impact of the NCC scheme
Heterogeneity of local impacts
Two possible explanations for coverage effect
 A “threshold” effect


A “dilution” effect


NCC scheme does not have a measurable impact (in terms of
reducing burglary rates) unless a sufficient number of households in
the local area are visited
Because the COA is the unit of analysis, the NCC scheme impact
could be diluted when the households that are visited are only a small
proportion of the total households in the COA
Neither of these explanations for the coverage effect
undermines our overall assessment of the policy’s success
Conclusions: NCC scheme

NCC scheme led to overall “success”

Overall, NCC-targeted areas experienced a 16% (95% CI: -2% to
34%) reduction in burglary rate per year

This suggests a positive impact of the NCC policy which had
the effect of stabilizing burglary rate in the targeted areas
while overall burglary rates were going up

Linear impact function is better at describing the data than
the other 3, suggesting a gradual and persistent change

There exist different impacts between targeted COAs,
perhaps due to local differences in implementing the
schemes
Assessing NCC impact for whole of Cambridgeshire

The NCC scheme was extended to the whole of
Cambridgeshire for the period 2005-08

We applied our evaluation model to assess impact of NCC
scheme separately for urban and rural areas

Overall, schemes in urban areas were more successful than
those in rural areas.
% change in burglary rates after 1st year of NCC scheme
No change
Urban
Rural
No change
12UBFW0011 (0.74)
12UEHH0013 (0.76)
00JAPA0012 (0.81)
12UCGA0013 (0.83)
00JAND0001 (0.85)
12UDGS0014 (0.86)
12UCGD0002 (0.88)
12UDGQ0006 (0.87)
00JANG0024 (0.88)
12UCGA0003 (0.88)
12UDGQ0024 (0.89)
00JANG0009 (0.92)
NCC2007
00JAPB0010 (0.76)
00JANG0025 (0.96)
NCC2006
00JANY0010 (0.55)
00JANC0016 (0.54)
00JANG0013 (0.85)
00JANE0006 (0.83)
00JANE0010 (0.88)
MDI
MatchRate
00JANT0027 (0.92)
00JANQ0023 (0.91)
NCC2005
Overall
Overall(0.96)
(0.96)
−100
−50
0
50
Percentage change in burglary rate
compared to controls
Overall (0.38)
100
29
Conclusions: Model

Hierarchical model allows borrowing of strength across NCC
areas


Joint estimation of common trend and local trend models
enables full propagation of uncertainty



enables evaluation of local impacts even when data are
sparse
Parameters of common trend model treated as
‘distributional constants’ in local trend model
Facilitated using ‘cut’ function in WinBUGS
More complex impact functions could be implemented, but
need sufficient time points post-policy for reliable estimation
Goal 2: Surveillance
Detecting unusual trends in chronic disease
rates
In collaboration with
Guangquan Li, Sylvia Richardson, Anna Hansell, Mireille
Toledano, Lea Fortunato
Imperial College, London
Surveillance of small area data

For many areas of application, such as small area estimates
of income, unemployment, crime rates and rates of chronic
diseases, smooth time changes are expected in most areas

However, policy makers and researchers are often
interested in identifying areas that ‘buck’ the national trend
and exhibit unusual temporal patterns

These abrupt changes may be due to emergence of
localised predictors/risk factors(s) or the impact of a new
policy or intervention

Detection of areas with “unusual” temporal patterns is
therefore important as a screening tool for further
investigations
Motivating example 1: COPD mortality

Chronic Obstructive Pulmonary Disease (COPD) is a
common chronic condition characterized by slowly
progressive and irreversible decline in lung function


responsible for approximately 5% of deaths in the UK
Main risk factors include

Smoking

Occupational exposure to high levels of dusts and fumes

Outdoor air pollution

“Umbrella” term for broad range of disease phenotypes

Time trends may reflect variation in risk factors and also
variation in diagnostic practice/definitions
Motivating example 1: COPD mortality

Objective 1: Retrospective surveillance


to highlight areas with a potential need for further investigation and/or
intervention (e.g. additional resource allocation)
Objective 2: Policy assessment

Industrial Injuries Disablement Benefit was made available for miners
developing COPD from 1992 onwards in the UK

As miners with other respiratory problems with similar symptoms
(e.g., asthma) could potentially have benefited from this scheme,
there was debate on whether this policy may have differentially
increased the likelihood of a COPD diagnosis in mining areas
Data



Observed and agestandardized expected annual
counts of COPD deaths in
males aged 45+ years
 374 local authority districts
in England & Wales
 8 years (1990 – 1997)
Difficult to assess departures of the local temporal patterns by eye
Need methods to
 quantify the difference between the common trend pattern and the
local trend patterns
 express uncertainty about the detection outcomes
Bayesian Space-Time Detection: BaySTDetect

BaySTDetect (Li et al 2011) is a novel detection method for
short time series of small area data using Bayesian model
choice between two competing space-time models



Model 1 assumes space-time separablility for all areas → one
common temporal pattern across the whole study region
Model 2 provides local time trend estimates for each spatial unit
individually
For each area, a model indicator is introduced to decide
whether Model 1 or Model 2 is supported by the data
→ Quantifying the difference

A Bayesian procedure of controlling the false discovery rate
is employed
→ Expressing uncertainty about detected areas
BaySTDetect: modelling framework
yit ~ Poisson( it  Eit )
log( it )    i   t model 1 for all i, t
i ~ spatial BYM model (common spatial pattern)
The temporal trend
pattern is the same
for all areas
 t ~ random walk (RW[ 2 ]) model (common temporal trend)
log( it )  ui  it model 2 for all i, t
ui ~ N(0,1000) (area-specific intercept)
Temporal trends are
independently
estimated for each area.
it ~ random walk (RW[ i2 ]) (area-specific temporal trend)
Model selection
 A model indicator zi indicates for each area whether
Model 1 (zi =1) or Model 2 (zi =0) is supported by the data
Implementation
Model 2: Local trend
Model 1: Common trend
t
i
it
ui
it[C]
it[L]
Eit
Eit
yit
yit
zi
it
Selection model
it  zi  it[C ]  (1  zi )  it[ L ]
Eit
yit
Prior on model indicator

For the model indicator zi, we have
zi ~ Bernoulli( ) where   0.95

This prior on zi

reflects the surveillance nature of the analysis
where we expect to find only a small number of
unusual areas a priori

ensures that a common trend can be
meaningfully defined and estimated
Classifiying areas as “unusual”






Classification of areas as “unusual” is based on the posterior
model probabilities pi = Pr(zi | data)
Small values of pi indicate low probability that area i fits the
common trend → high probability of being “unusual”
Need a rule for calibrating the pi that acknowledges the multiple
testing setting
 How low does pi need to be in order to declare area i as
unusual?
False Discovery Rate (FDR) is the proportion of detected areas
that are false (i.e. not truly unusual) (Benjamini & Hochberg,
1995)
Various methods to estimate or control FDR
Here we control the posterior expected FDR (Newton et al 2004)
Detection rule based on FDR control


First rank the areas according to increasing values of pi
At a nominal FDR level of , the first k ranked areas are
declared as unusual where k is the maximum integer
k
satisfying
p
j 1
( j)
 k 
where p(j) is the jth ranked posterior common-trend model
probability

This procedure ensures that (posterior) expected number
of false positives is no more than (k ×) of the k declared
unusual areas
Simulation study to evaluate operating
characteristics of BaySTDetect





Simulated data were based on the observed COPD
mortality data
Three departure patterns were considered
When simulating the data, either the original set of
expected counts from the COPD data or a reduced set
(multiplying the original by 1/5) were used
15 areas (approx. 4%) were chosen to have the unusual
trend patterns
 areas were chosen to cover a wide range expected
count values and overall spatial risks
Results were compared to those from the popular
SaTScan space-time scan statistic
Simulation Study: Departure patterns
Common trend, exp(t)
Departure pattern, exp(t ∙)
2 different departure magnitudes:  =1.5 and  =2.0
Simulation Study: expected counts
Table: Summary of the original set of age-adjusted expected counts used
in the simulation
Simulation Study: FDR control
Empirical FDR vs corresponding pre-defined level: Pattern 2
0.15
0.20
Pre-set FDR level
0.8
0.6
0.4
0.2
0.0
0.0
0.10
Empirical FDR
1.0
1.0
0.8
0.6
95% sampling
interval
0.4
Empirical FDR
mean
0.2
1.0
0.8
0.6
0.4
0.2
0.0
Empirical FDR
0.05

Reduced expected;
=2.0
Original expected;
=2.0
Original expected;
=1.5
0.05
0.10
0.15
0.20
Pre-set FDR level
0.05
0.10
0.15
0.20
Pre-set FDR level
SaTScan: Empirical FDR = 0.19 (0.00 to 0.78) for scenario with original
expected counts and  =2.0
Sensitivity of detecting the 15 truly unusual areas
SaTScan (p=0.05)
True
departure
magnitude:
=1.5
E=24 E=33 E=42 E=52 E=80
Expected count quantiles
Expected count quantiles
Sensitivity
0.0 0.2 0.4 0.6 0.8 1.0
E=24 E=33 E=42 E=52 E=80
0.0 0.2 0.4 0.6 0.8 1.0
Sensitivity
Pattern 2
0.0 0.2 0.4 0.6 0.8 1.0
Sensitivity
0.0 0.2 0.4 0.6 0.8 1.0
Sensitivity
BaySTDetect (FDR=0.1)
True
departure
magnitude:
=2.0
E=24 E=33 E=42 E=52 E=80
E=24 E=33 E=42 E=52 E=80
Expected count quantiles
Expected count quantiles
Sensitivity of detecting the 15 truly unusual areas:
reduced expected counts
Pattern 2; True departure magnitude: =2.0
SaTScan (p=0.05)
0.8
0.6
0.2
0.4
Sensitivity
0.6
0.4
0.2
0.0
0.0
Sensitivity
0.8
1.0
1.0
BaySTDetect (FDR=0.1)
E=5
E=6
E=8
E=11
E=16
Expected count quantiles
E=5
E=6
E=8
E=11
E=16
Expected count quantiles
COPD application: Detected areas (FDR=0.05)
COPD application: Interpretation

Results provide little support for hypothesis regarding the
industrial injuries policy



only 3 out of 40 ‘mining’ districts detected (Barnsley,
Carmarthenshire and Rotherham)
unusual trend patterns in these areas are not consistent
Two unusual districts (Lewisham and Tower Hamlets) with
an increasing trend (against a national decreasing trend)
were identified in inner London



These areas are very deprived, with high in-migration and ethnic
minorities → might expect different trends to rest of country
In fact, Tower Hamlets has been commissioning various local
enhanced services to tackle high rates of COPD mortality since 2008.
This rising trend could potentially have been recognised earlier in the
1990s through using BaySTDetect as a surveillance tool.
COPD application: SaTScan


Primary cluster: North (46 districts) – excess risk of 1.05 during 1990-92
Secondary cluster: Wales (19 districts) – excess risk of 1.12 during 1995-96
Example 2: Data mining of cancer registries

The Thames Cancer Registry (TCR) collects data on newly
diagnosed cases of cancer in the population of London and
South East England

It is one of the largest cancer registries in Europe, covering
a population of over 12 million, and holds nearly 3 million
cancer registration records.

We perform a retrospective surveillance of time trends for
several cancer types using BaySTDetect

aim to provide screening tool to detect of areas with
“unusual” temporal patterns

automatically flag-up areas warranting further
investigations
Cancer data

Cancer incidence for population aged 30+ years





Breast (female only)
Colon (males and females combined)
Lung (males and females, separately)
South East England, ward level (1899 areas)
Period 1981-2008


Data were aggregated by 4-year intervals
7 time periods for the detection analysis
Cancer data summary
OBS
EXP
OBS
colon
EXP
Female OBS
lung
EXP
Male lung OBS
EXP
breast

Min
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
Q1
10.0
11.3
5.0
5.7
3.0
4.0
6.0
7.6
Median Mean
16.0
17.6
16.5
17.6
8.0
9.1
8.5
9.1
5.0
6.4
5.9
6.4
10.0
11.8
11.2
11.8
Q3
24.0
23.0
12.0
11.8
9.0
8.3
16.0
15.2
Max
69.0
56.5
42.0
34.6
34.0
24.5
66.0
39.5
Comparable to reduced expected count scenario in simulation study
Results: Number of detected areas (out of 1899)
Cancer type
FDR=0.05
FDR=0.1
FDR=0.15
FDR=0.2
Breast
9
19
35
54
Colon
0
3
5
8
Lung (female)
0
1
2
4
Lung (male)
6
14
24
39
54
Detected areas:
breast cancer
Summarising the unusual trends

With a relatively large number of detected areas (e.g.,
breast and male lung cancer), examination of the
individual trends becomes difficult

For the detected areas, the estimated RR trends from
the local trend model are fed into a standard hierarchical
clustering method (hclust in R)
  i   t
log( it )  
ui  it

model 1
model 2
The cluster-specific trends are then compared to the
overall RR trend
56
1
2
3
Period
4
5
6
7
0.6
0.6
0.6
0.8
0.8
0.8
1.2
1.4
1.6
1.8
1.6
1.8
1
2
3 clusters
2
3
3
4
Period
4
5
5
6
Overall trend
20 areas
12 areas
10 areas
12 areas
6
7
1.8
1
1.6
d
hclust (*, "complete")
1.2
1.4
1.4
0.4
0.4
0.6
0.6
0.8
0.8
29UMGT
22UHHP
26UFGH
22UQGT
00AWGC
26UJFX
00ASHB
26UCHD
26UJGQ
00AGGE
43UDGA
43ULGR
00LCPB
00ADGW
43UMFU
00AUFY
00AJGY
00ALHF00APGK
00ANGA
45UDGQ
00BBGX
29UBHR
29UCGF 26UJGC
00AYGL26UEGJ
00ASGJ
00BKGQ
00AKGP
00BHGR
29UHHE
00BCGU
00BHGK
00BCFZ
00BKGR
00BFGE
43UFGN
00BAGM
00MLNP
00BFGN
45UBFT
00BJGG
00BFGS
21UDFU
00AUGM
00ANGC
29UNHA
21UGGJ
00BJGM
00ATGB
00ALGP
00AFGM
00AFGG
0.0
0.5
1.2
1.0
1.2
Relative Risk
1.0
Relative Risk
1.0
Height
1.5
1.6
1.6
1.4
1.4
2.0
2.5
1.8
1.8
Overall trend
54 areas
1.0
Relative Risk
1.0
Relative Risk
1.2
Relative Risk
1.0
Overall trend
30 areas
12 areas
12 areas
0.4
0.4
0.4
Cluster Dendrogram
1 cluster
2 clusters
7
Overall trend
42 areas
12 areas
Breast
cancer
FDR=0.2
Period
1
1
2
4 clusters
2
3
Period
3
4
Period
4
5
5
6
6
7
5 clusters
Overall trend
20 areas
12 areas
10 areas
5 areas
7 areas
7
Black line =
common trend
Coloured lines =
average local trend
in each cluster
BaySTDetect: Conclusions and Extensions



We have proposed a Bayesian space-time model for
retrospective detection of unusual time trends
Simulation study has shown good performance of the model
in detecting various realistic departures with relatively
modest sample sizes
Possible extensions include:



Spatial prior on zi to allow for clusters of areas with unusual
trends
Time-specific model choice indicator zit, to allow longer time
series to be analysed
Alternative approaches to calibrating posterior model
probabilities, e.g. decision theoretic approach (Wakefield,
2007; Muller et al., 2007)
References

G. Li, R. Haining, S. Richardson and N. Best. Evaluating Neighbourhood Policing using
Bayesian Hierarchical Models: No Cold Calling in Peterborough, England. Submitted

G. Li, N. Best, A. Hansell, I. Ahmed, and S. Richardson. BaySTDetect: detecting unusual
temporal patterns in small area data via Bayesian model choice. Submitted

G. Li, S. Richardson , L. Fortunato, I. Ahmed, A. Hansell and N. Best. Data mining cancer
registries: retrospective surveillance of small area time trends in cancer incidence using
BaySTDetect. Proceedings of the International Workshop on Spatial and Spatiotemporal
Data Mining, 2011.
www.bias-project.org.uk
Funded by ESRC National Centre for Research Methods