here - The BIAS project

Report
Models for small area data
with applications in health care
Nicky Best
Department of Epidemiology and Biostatistics
School of Public Health, Imperial College London
http://www.bias-project.org.uk
http://www1.imperial.ac.uk/medicine/people/n.best/
Outline
1. Introduction
2. Mapping and spatial smoothing of health data
3. Classifying areas and summarising geographical
variations in health outcomes
4. Modelling and mapping multiple outcomes
5. Modelling and mapping temporal trends
6. Hierarchical related regression models for
combining individual and small area data
Outline
1. Introduction
2. Mapping and spatial smoothing of health data
3. Classifying areas and summarising geographical
variations in health outcomes
4. Modelling and mapping multiple outcomes
5. Modelling and mapping temporal trends
6. Hierarchical related regression models for
combining individual and small area data
A brief history of disease mapping
Health indicator maps have a long history in epidemiology and
public health
•
Spot maps:
– Yellow fever pandemic New York (Seaman, 1798)
– Cholera and the Broad Street Pump (Snow, 1854)
Spot map of cholera
cases (Snow, 1854)
A brief history of disease mapping
Health indicator maps have a long history in epidemiology and
public health
•
Spot maps:
– Yellow fever pandemic New York (Seaman, 1798)
– Cholera and the Broad Street Pump (Snow, 1854)
• Chloropeth maps:
– Geographical distribution of mortality from heart disease, cancer and TB in
England & Wales (Haviland, 1878)
– Cancer mortality by county in England & Wales, adjusted for age and sex
(Stocks, 1936, 1937, 1939)
Female cancer 1851-60
(Haviland 1878)
Female lung cancer
SMR 1921-30
(Stocks, 1939)
A brief history of disease mapping
Health indicator maps have a long history in epidemiology and
public health
•
Spot maps:
– Yellow fever pandemic New York (Seaman, 1798)
– Cholera and the Broad Street Pump (Snow, 1854)
• Chloropeth maps:
– Geographical distribution of heart disease, cancer and TB in England & Wales
(Haviland, 1878)
– Cancer rates by county in England & Wales, adjusted for age and sex (Stocks,
1936, 1937, 1939)
• National and international disease atlases, e.g
– Atlas of Cancer Incidence in England & Wales 1968-85 (Swerdlow & dos
Santos Silva, 1993)
– Atlas of Mortality in Europe 1980/81 & 1990/91 (WHO, 1997)
Female lung cancer
incidence 1968-85
(Swerdlow and dos
Santos Silva, 1993)
Age-standardised mortality from IHD, 1980-81 (WHO)
Recent developments in disease mapping
• Development of Geographical Information Systems (GIS)
– Geographically indexed relational database
– Computer program to map and analyze spatial data
• Increasing availability of geo-referenced data
– Ability to geocode, use GPS
– Disease outcomes, demographics, environmental quality, health
services
• Development of statistical methods
– Sophisticated techniques for separating signal from noise
– Ability to account for spatial (and temporal) dependence
– Methods for cluster detection and classification of areas
Interest in mapping health events at small-area scale
Small area health data in the UK
• Administrative geography in UK includes
– Postcodes (10-15 households)
– Census Output Areas (COA; ~300 people)
– Electoral wards (~500 to 2000 people)
– Local authority districts, Health authority districts (10’s of thousands)
• Postcoded data on mortality, births/still births, congenital anomalies,
cancer incidence, hospital admissions
• Population and socio-economic indicators from Census (COA)
• Increasing availability of modelled environmental data at fine geographical
resolution (grids)
• Limited access to geographical identifiers for certain individual-level
cohorts (e.g. Millenium Cohort, British Household Panel Survey) and
health surveys (e.g. Health Survey for England)
Small area health data in Spain
• Administrative geography in Spain is divided into:
– 17 regions
– 52 provinces
– ~8000 municipalities, ranging from small villages to large cities
– Census tracts (finer sub-division in large cities)
• Geocoded (place of residence) data on births, mortality
(national), cancer incidence (regional; ~26% population),
hospital discharge administrative data (national; public
hospitals)
• Small area (municipality) data on population and socioeconomic indicators from Census
Examples of recent disease atlases
and health-related maps for Spain
Atlas of cancer mortality and other causes of death in Spain
1978-1992 (López-Abente et al., 1996)
Maps Ageadjusted Rates
and
Standardised
Mortality Ratios
(SMR) at
province level
Atlas of cancer mortality at municipality level in Spain 1989-1998
(López-Abente et al., 2007)
Maps (Bayesian)
smoothed relative risks
of mortality and
probability of excess
risk, at municipality
level
Also produced maps of mortality from selected
causes other than cancer, e.g. Influenza
…… + contextual maps of
socioeconomic variables and
environmental hazards
Outline
1. Introduction
2. Mapping and spatial smoothing of health data
3. Classifying areas and summarising geographical
variations in health outcomes
4. Modelling and mapping multiple outcomes
5. Modelling and mapping temporal trends
6. Hierarchical related regression models for
combining individual and small area data
Why map small area disease rates?
• Interest in mapping geographical variations in health
outcomes a the small area scale
– Highlight sources of heterogeneity and spatial patterns
– Suggest public health determinants or aetiological
clues
• Small scale
– UK: electoral ward or census output area
– SPAIN: municipality
– less susceptible to ecological (aggregation) bias
– better able to detect highly localised effects
Why smooth small area disease rates?
• Typically dealing with rare events in small areas Ai
Yi is the observed count of disease in area Ai
Ei is the expected count based on population size,
adjusted for age, sex, other strata ….,
• Relative risk usually estimated by SMRi = Yi / Ei
• Standard practice is to map SMRs
BUT sparse data need more sophisticated
statistical analysis techniques
Why smooth small area disease rates?
• SMR represents estimate of ‘true’ (underlying) risk
in an area, Ri, i.e. Ri = SMRi
• Statistical uncertainty about estimate based on
assuming Poisson sampling variation for data
Yi ~ Poisson(Ri Ei)
• SE(Ri) = SE(SMRi)  1 / Ei
– SMRi very imprecise for rare diseases and small
populations
– precision can vary widely between areas
Why smooth small area disease rates?
• SMRi in each area is estimated independently
– ignores possible spatial correlation between
disease risk in nearby areas due to possible
dependence on spatially varying risk factors
– leads to problems of multiple significance testing
Map of SMR of adult leukaemia in West Midlands Region,
England 1974-86
(Olsen, Martuzzi and Elliott, BMJ 1996;313:863-866).
Is the variability real
or simply reflecting
unequal expected
counts ?
Have the red
highlighted areas
truly got a raised
relative risk?
Methods for smoothing disease maps
• These problems may be addressed by spatial
smoothing of the raw data
• Idea is to “borrow information” for neighbouring
areas to produce better (more stable, less noisy)
estimates of the risk in each area
• Similar principle to scatter plot smoothers, moving
average smoothers….
• Many methods available
Methods for smoothing disease maps
• Ad hoc, local smoothing algorithms
e.g. spatial moving averages, headbanging algorithm
+ quick and simple to implement
– can be very sensitive to ad hoc choice of weights etc.
– no uncertainty estimates (standard errors)
• Trend surface analysis
e.g. kriging, polynomial/spline smoothing
+ estimation of ‘smoothing parameters’ based on tradeoff between fit and smoothness
– can be sensitive to choice of penalty for trade-off
+ standard errors usually available
Methods for smoothing disease maps
• Random effects models
e.g. empirical Bayes, hierarchical Bayes
+ data-based estimation of model parameters that
control smoothing
+ full power of statistical modelling available: standard
errors, prediction, probability calculations, inclusion of
covariates
– more complex to understand and implement
Bayesian Approach
• Use probability model to obtain smoothed risk
estimate Ri in area i that is a compromise (weighted
average) of
– observed area-level risk ratio (Yi/ Ei)
– local or regional mean relative risk (m)
• Weights depend on the precision of the SMR
( 1 / Ei) in area i and the variability (heterogeneity)
of the true risks across areas (v) local or regional
mean relative risk (m)
• Aim is to estimate posterior probability distribution
of the unknown model parameters (Ri, m, v)
conditional on the data (Yi/ Ei)
Bayesian disease mapping model
• Typical Bayesian disease mapping model:
Yi ~ Poisson(Ri Ei),
log (Ri) ~ Normal (m, v)
• Hierarchical Bayesian model also requires
specification of a (prior) probability distribution for
m and v
– These are often taken to be ‘non-informative’
• Empirical Bayes involves 2-step process:
– Estimate m and v empirically from observed data
– Ignore uncertainty in estimates of m and v and plug
these values into the Bayesian model above
Software
• Estimation of Bayesian hierarchical models
requires computationally intensive simulation
methods (MCMC)
– Implemented in free WinBUGS and GeoBUGS
software: www.mrc-bsu.cam.ac.uk/bugs
• Free software INLA (Rue et al, 2008) implements
fast approximation: www.r-inla.org
• Empirical Bayes smoothing implemented in Rapid
Inquiry Facility (RIF):
www.sahsu.org/sahsu_studies.php#RIF
Map of occurrences of adult leukaemia in West Midlands Region,
England 1974-86
(A) unsmoothed SMR
(B) smoothed by
Bayesian methods
(Olsen, Martuzzi and Elliott, BMJ 1996;313:863-866)
0
5
Expected
count
Expected
0count
2 4
Comparison of estimation methods
relative risk
0 2 4 6 8
SMR
Hierarchical Bayes
Empirical Bayes
5
10
area
15
20
Including spatial dependence in
disease risk
• Ri are typically spatially correlated because
they reflect, in part, spatially varying risk factors
Incorporation of spatial dependence in the
distribution of the Ri’s
Conditional Autoregressive (CAR) model
log (Ri ) ~ Normal (mi , vi)
mi = k Rk / ni = average risk in neighbouring areas
vi = v / ni → variance inversely proportional to number of
neighbours
Besag, York, Mollie (1991) Annals of the Institute of Statistics and Mathematics, 43: 1-59
Childhood leukaemia incidence in London, 1986-1998
(summary)means for RR
Non-spatial
smoothing
(posterior mean Ri)
N
SMR
Raw data
(SMR)
(418) <
N
(7)
0.5
0.5 -
RR
(30)
0.7 0.5
<0.5
(55)
0.9 -
0.7
0.9
(0) <
(39)
0.5 0.5-0.7
(71)
1.1 0.7 0.7-0.9
(109)
1.4 -
(192)
1.1
0.7
1.4
RR
0.9
2.0
N
0.9-1.1
(264)
(183) 0.9
>= - 2.01.1
1.1 1.1-1.4
(251)
1.4 1.4-2.0
(96)
>2.0
(31) >=
20.0km
1.4
2.0
2.0
20.0km
Spatial smoothing
(posterior mean Ri)
20.0km
20.0km
Mapping uncertainty
• Mapping the mean posterior value of Ri does
not make full use of the posterior distribution
• Map posterior SD
• Map
Probability (Ri > 1)
Note – this is not the same
as a classical p-value
0.5
0.5
1.0
1.0
1.5
1.5
Relative Risk
2.0
2.0
Relative Risk, Ri
2.5
2.5
Posterior SD of relative risk estimates
Posterior mean relative risk
Posterior sd of relative risk
RR
values for sd(0) <
N
(39)
0.5
0.5 -
0.7
(0) <
0
(324)
0
0
N
(192)
0.7 -
0.9
(418)
(264)
0.9 -
1.1
(91)
0.
(251)
1.1 -
1.4
(20)
0.
(5)
1.0
(96)
(31) >=
1.4 -
2.0
(15) >=
2.0
RR
SD
(0) <
0.5
values
for sd
<0.5
(39)
N
0.5 0.5-0.7
0.2
<0.2
0.7
(324)
0.2 0.4
0.2-0.4
0.7 0.7-0.9
0.9
0.9 0.9-1.1
1.1
(91)
0.6 - 0.8
0.6-0.8
1.1 1.1-1.4
1.4
(20)
0.8 - 1.0
0.8-1.0
(5)
1.0 1.2
1.0-1.2
(192)
20.0km
(0) <
(264)
(251)
1.4 1.4-2.0
(96)
>2.0
(31) >=
2.0
2.0
20.0km
(418)
0.4 0.6
0.4-0.6
>1.21.2
(15) >=
Posterior probability that relative risk > 1
Posterior mean relative risk
Posterior probability that relative risk > 1
values for pp(0) <
RR
N
(39)
N
(163)
0.5
0.5 -
(321)
0.7
(192)
0.7 -
0.9
(250)
(264)
0.9 -
1.1
(139)
(251)
1.1 -
1.4
(96)
(31) >=
1.4 -
2.0
2.0
RR
Prob
(0) <
<0.50.5
values for pp
(39)
0.5 0.5-0.7
20.0kmN
(163) < 0.25
<0.25
0.7
(321)
0.25 -
0.9
0.9 0.9-1.1
1.1
(250)
1.1 1.1-1.4
1.4
(139) >= 0.75
(264)
(251)
1.4 1.4-2.0
(96)
>2.0
(31) >=
2.0
20.0km
2.0
0.5
0.25-0.50
0.7 0.7-0.9
(192)
0.5 - 0.75
0.50-0.75
>0.75
Atlas of cancer mortality at municipality level in Spain 1989-1998
(López-Abente et al., 2007)
Outline
1. Introduction
2. Mapping and spatial smoothing of health data
3. Classifying areas and summarising geographical
variations in health outcomes
4. Modelling and mapping multiple outcomes
5. Modelling and mapping temporal trends
6. Hierarchical related regression models for
combining individual and small area data
Classifying areas with excess risk
• Richardson et al (2004): Simulation study
investigating use of posterior probabilities in
disease mapping studies
• Classify an area as having an elevated risk if
[Prob (Ri > 1)] > 0.8
• High specificity
(false detection < 10%)
• Sensitivity 60%-95% for Ei of 5-20 and true Ri of
1.5-3.0
Childhood leukaemia in London
Posterior prob(RR>1)
Posterior mean RR
probability of(0)
RR
< greater
0.5 than 1.0
RR
N
N
(39)
0.5 -
0.7
(192)
0.7 -
0.9
(264)
0.9 -
1.1
(251)
1.1 -
1.4
(96)
(31) >=
1.4 -
2.0
2.0
(0) <
<0.50.5
(39)
0.5 0.5-0.7
0.7 0.7-0.9
(192)
0.7
0.9
probability of RR greater than 1.0
0.9 -
0.9-1.1
1.1
1.1 1.1-1.4
1.4
(264)
20.0km
N
(251)
1.4 1.4-2.0
(96)
>2.0
(31) >=
2.0
2.0
20.0km
(772)
<0.8< 0.8
(101) >=
>0.8
0.8
Comparison of SaTScan and Bayesian classification
rule
SaTScan (Kuldorff,
www.satscan.org):
Location of most likely cluster
Bayesian:
probability
of RR greater than 1.0 of excess risk
Probability
N
probability of RR greater than 1.0
Most likely cluster;
p<0.001
2nd most likelyN cluster; p = 0.2
20.0km
(772) < 0.8
< 0.8
(101) >= 0.8
≥ 0.8
Summarising geographic variation
• Often interested in providing overall summary
measure of variability between areas, e.g.
– to compare variability of different outcomes
– to quantify how much variation can be explained by
covariates
• Percentile Ratio: Ratio of outcomes (relative risks)
in areas ranked at the qth and (100-q) th percentiles
– e.g. 90th Percentile Ratio, PR90 = R95%/R5%
– Posterior distribution of PR90 easily calculated from
MCMC output
Relative survival from colon cancer, England
Data
• Survival/censoring times for all 7007 cases of colon cancer diagnosed in
England in 1995 and followed for 5 years (provided by B Rachet, LSHTM)
• Covariates: sex, age at diagnosis, clinical stage, deprivation score, Health
Authority (95 area, 1-300 cases per HA)
• Population mortality rates by age and sex for England and Wales, 19851995.
Questions of interest
• Is there evidence of differences between Health Authorities in relative
survival that may indicate differences in effectiveness of care received?
– Relative survival measures difference between age/sex-adjusted
mortality rate in general population and in patients with disease of
interest
• How do these geographical differences change when we adjust for
socioeconomic deprivation and clinical stage of cancer?
Relative survival from colon cancer, England
ykit ~ Poisson(mkit)
(subject k, area i, time interval t)
log(mkit – Ekit) = log nkit + at + bxki + Hi
Standard model for relative survival
Area spatial effect
Relative survival from colon cancer, England
ykit ~ Poisson(mkit)
(subject k, area i, time interval t)
log(mkit – Ekit) = log nkit + at + bxki + Hi
Area spatial effect
Standard model for relative survival
adjustment
relative Without
exces s hazard
(model 6)
for
deprivation and clinical stage
(7)relative
< 0.85
(31)
After
adjustment
exces
s hazard
(model 3)for
deprivation
and clinical stage
0.95
0.85 -
(7)
(16
N
N
Relative
(28) 0.95 excess
(9) 1.05 mortality
(36
1.05
(25
1.15
(11
(20) >= 1.15
relative exces s hazard (model 6)
(7)
< 0.85
<0.85
(31)
0.85 0.85-0.95
0.95
(28)
0.95 0.95-1.05
1.05
N
(9)
1.05 - 1.15
1.05-1.15
PR90 = 1.95
(95 % CI 1.62-2.38)
(20)
>= 1.15
≥1.15
PR90 = 1.83
(95 % CI 1.54-2.24)
Ranking and classifying extreme areas
• Interest in ranking areas for e.g. policy evaluation,
‘performance’ monitoring
• Rank of a point estimate is highly unreliable
– Would like to measure uncertainty about rank
• Straightforward to calculate posterior distribution of
ranks (or any function of parameters) using MCMC
– Obtain interval estimates for ranks
• Can also calculate posterior probability that each area is
ranked above a particular percentile
Rank (posterior mean and 95% CI) of the 95
Health Authorities
Without adjustment for deprivation
and clinical stage
caterpillar plot: rank.HA
After adjustment for deprivation
and clinical stage
caterpillar plot: rank.HA
Upper quartile
0.0
25.0
50.0
Rank
75.0
Upper quartile
0.0
25.0
50.0
Rank
75.0
Posterior probability that HA is ranked in top 5%
After adjustment for deprivation
and clinical stage
Without adjustment for deprivation
and clinical stage
(samples)means for p.bottom5
(samples)means
(20) < 0.0 for p.bottom5
N
N
(64)
(6)
(samples)means for p.bottom5
0.0 0.1 -
0.0 (4) 0.2 0.0-0.1
(1) >= 0.5
0.1-0.2
0.2-0.5
>0.5
(20) <
0.2
0.0
(64)
N
0.1
0.0 -
(6)
0.1 -
0.2
(4)
0.2 -
0.5
(1) >=
0.5
0.1
0.5
200.0km
200.0km
200.0km
Outline
1. Introduction
2. Mapping and spatial smoothing of health data
3. Classifying areas and summarising geographical
variations in health outcomes
4. Modelling and mapping multiple outcomes
5. Modelling and mapping temporal trends
6. Hierarchical related regression models for
combining individual and small area data
Joint spatial variation in risk of
multiple diseases
Disease 1
Disease 2
RR
SMR
0.7
0.7
SMR
0.7
Specific component 1
RR
0.7
1.0
1.5
1.0
1.5
SMR
0.7
Shared component
1.0
1.0
1.0
1.5
Specific component 2
RR
Knorr-Held
and1.5Best (2001) RR
0.7
1.0
0.7
1.0
1.5
1.5
1.5
Statistical model
Y1i ~ Poisson(R1i E1i);
log R1i = Si + U1i
Y2i ~ Poisson(R2i E2i);
log R2i = Si + U2i
Si ~ spatial model (shared component of risk)
U1i ~ spatial model (component of risk specific to disease 1)
U2i ~ spatial model (component of risk specific to disease 2)
• Extends to >2 diseases (Tzala and Best, 2006)
• Extends to shared variations in space and time
(Richardson et al, 2006)
© Imperial College London
Joint variation in COPD and lung cancer in GB
values for SMR1
COPD SMR
N
values for SMR2
(42 ) <
0.66
(10 0)
0.66 -
0 .8
(13 9)
0.8 -
1 .0
(12 1)
1.0 -
(40 )
1 .25 -
(17 ) >=
Lung cancer SMR
N
(11 ) <
0.6
(10 1)
0.6
(20 1)
0.8
1.25
(10 3)
1.0
1.5
(33 )
(10 ) >=
1.5
200.0km
200.0km
Best and Hansell (2009)
1 .25
1
Modelled risk estimates
values for shared
Shared risk
N
Shared risk interpreted as
mainly reflecting
geographical variations in
community-level smoking
behaviour
(16 8) <
values for log.COPD.spec.MALE.model2
0.9
(40 )
0 .9 -
(32 )
0 .95 -
(34 )
1 .0 -
(25 )
1 .05 -
(28 )
1 .1 -
(13 2) >=
COPD specific risk
0 .9 5
1.0
N
1 .1 5
1.15
-0.1 -
(111)
COPD specific risk
interpreted as reflecting
smoking-adjusted
variations in COPD
mortality
200.0km
200.0km
(39)
(71) -0.05 -
1 .0 5
1.1
(80) < -0.1
0.0 -
(89)
0.05 -
(36)
0.1 -
(33) >=
0.15
Joint variation in relative survival of colon and
breast cancer by English Health Authority
• Shared spatial patterns of relative survival may reflect
variations in effectiveness of health care system
Observed 5-year relative
survival: Breast
MLE estimate of relative survival: breast
Observed 5-year relative
estimate of relative survival: colon
survival: Colon
65.0 - 70.0
(3) < MLE
65.0
(13)
N
N
(36)
70.0 -
75.0
(3)
65.0
< <65%
(38)
75.0 -
80.0
(13)
70.0
65%65.0
to -70%
(5) >= 80.0
MLE estimate of relative survival: colon
(2)
< 20.0
< 20%
(10)
- 30.0
20%20.0
to 30%
N
(36)
75.0
70%70.0
to -75%
(31)
- 40.0
30%30.0
to 40%
(38)
80.0
75%75.0
to -80%
(36)
- 50.0
40%40.0
to 50%
(5)
>= 80.0
>80%
(16)
>= 50.0
>50%
200.0km
200.0km
Difference in relative survival in each HA
compared to England as a whole
Posterior Prob that
shared difference > 0
Shared difference
shared differential in relative s urvival
(3) < -3.0 of shared > 0.0
probability
N
N
(13)
-3.0 -
-1.5
(17)
< -1.5
0.2 <(63)
0.2
1.5
(13) -3.0
-1.5
-15%
to --30%
(15)
(64)
0.2
3.0
0.8
(63) -1.5
1.5
-15%
to -15%
(14)
>= 0.8
> 0.8
< -3.0
-30%
prob<(3)
that
relative excess hazard
N
1.5-–0.2
0.8
(1) >=
(11) <
(73)
(11) >
3.0
(15) to
1.530%
3.0
15%
(1) >= 3.0
>30%
200.0km
200.0km
Difference in relative survival in each HA
compared to England as a whole
Difference specific to
colon cancer
Difference specific to
breast cancer
breast specific differential in rel. surv
(0) < -3.0
colon s pecific differential in rel surv.
(1)
N
-3.0 -
-1.5
N
< -3.0 differential in relative s urvival
<(0)shared
-30%
(94)
(3)-30%
<- -3.0
<
-1.5
1.5
(1) -3.0
-1.5
-15%
to- -30%
(0)
(13)
1.5
-
(94) -1.5
1.5
-15%
to 15%
(0) >= (63)
3.0
N
-3.0
- -1.5
-15%3.0
to -30%
1.5
-15%-1.5
to 15%
(0)
1.5
3.0
15%
to -30%
(15) to
1.530%
3.0
15%
(0) >=
>30%
(1) >=
>30%
3.0
200.0km
3.0
200.0km
Diet-related cancers in Greece
Cancer-specific
spatial residuals
Spatial common factor
1
2
3
4
5
values for p
oesophagus
(13) <-0.1184
values
for p
N
N(10) -0.1184 - -0. 02204
(6) -0.11
(8) -0.02204 - 0.03013
(1) -0.01
(8) 0.03013 - 0.1332
(7) 0.030
(12) >= 0.1332
(19) >=
(10) -0.005441 - -0.002873
N
stomach
values for p
(11) <-0.005441
(10) -0.002873 - -1.145E-4
(18) <-0
(10) -1.145E-4 - 0.001841
(10) >=0.001841
200.0km
values for p
N
colorectal
200.0km
values
for p
(14) <-0.1234
N(10) -0.1234 - -0. 01909
pancreas
(6) <-0.1
(4) -0.12
(7) -0.01909 - 0. 0284
(2) -0.01
(17) 0.0284 - 0. 1359
(15) 0.0
(3) >= 0.1359
(24) >=
200.0km
200.0km
200.0km
values for p
N
(0) <-0.1234
values
for p
prostate
(18) -0.1234 - -0. 01909
N
Cut-points based on quintiles of
distribution of factor values and
of residuals across all cancers
(10) <-0.11
bladder
(9) -0.1184
(30) -0.01909 - 0.0284
(14) -0.022
(3) 0.0284 - 0.1359
(15) 0.0301
(0) >= 0.1359
(3) >= 0.1
Tzala and Best (2006)
200.0km
200.0km
Outline
1. Introduction
2. Mapping and spatial smoothing of health data
3. Classifying areas and summarising geographical
variations in health outcomes
4. Modelling and mapping multiple outcomes
5. Modelling and mapping temporal trends
6. Hierarchical related regression models for
combining individual and small area data
Extensions of disease mapping to
space time modelling
Noisy data
in each area
Noise model: Poisson/Binomial
Latent structure: Space + Time + (Residuals)
+
joint Bayesian estimation
Inference
Basic space-time model set-up
Yit ~ Poisson(Rit Eit);
log Rit = Si + Tt + Uit
Si ~ spatial CAR model (common spatial pattern)
Tt ~ random walk (RW) model (common temporal trend)
Uit ~ Normal(0, v) (space-time residual reflecting
idiosyncratic variation)
• Extends to shared variations of 2 outcomes in
space and time
© Imperial College London
Space-time variations in Male and Female lung
cancer incidence (Richardson et al, 2006)
• Lung cancer, with its low survival rates, is the
biggest cancer killer in the UK
– Over one fifth of all cancer deaths in UK are from lung
cancer (25% for male and 18% for female)
• Major risk factor is smoking.
– Smoking time trends different for men/women: uptake of
smoking started to decrease in cohorts of men after 1970,
while for women the levelling off was later, after 1980
• Other risk factors include exposure to workplace
agents, radon, air pollution …
Interested in similarity and specificity of patterns
between men and women
Space-time analysis of Male and Female lung
cancer incidence
Male/Female lung cancer incidence in Yorkshire:81-85, 86-90, 91-95, 96-99
(Richardson, Abellan, Best, 2006)
Shared and specific patterns and time trends
Shared component
Female/Male differential
Time trend for male RRs
in 10 wards
Time trend for female RRs
in 10 wards
Detection of space-time interaction patterns
Detection of space-time interaction patterns
Noisy data
in each area
Noise model: Poisson/Binomial
Latent structure: Space + Time + (Residuals)
+
joint Bayesian estimation
Inference
Detection of space-time interaction patterns
Noisy data
in each area
Noise model: Poisson/Binomial
Latent structure: Space + Time + Interactions
Any patterns?
+
joint Bayesian estimation
+
Inference
Detection of space-time interaction patterns
• Study the persistence of patterns over time
– Interpreted as associated with stable risk factors,
environmental effects, socio-economic determinants
• Highlight unusual patterns, via the inclusion of space
time interaction terms, which are modelled by a mixture
model
• Unusual patterns in some areas may be linked to
recording changes, emerging environmental hazards,
impact of new policy or intervention program, …
a general tool for surveillance ?
Detection of space-time interaction patterns
Yit ~ Poisson(Rit Eit);
log Rit = Si + Tt + Uit
Si ~ spatial CAR model (common spatial pattern)
Tt ~ random walk (RW) model (common temporal trend)
Uit ~ Normal(0, v) (space-time interaction;
idiosyncratic variation)
© Imperial College London
Detection of space-time interaction patterns
Yit ~ Poisson(Rit Eit);
log Rit = Si + Tt + Uit
Si ~ spatial CAR model (common spatial pattern)
Tt ~ random walk (RW) model (common temporal trend)
Uit ~ q Normal(0, v1) +(1-q) Normal(0, v2); v2 > v1
(mixture model to characterise ‘stable’ and ‘unstable’ patterns
over time)
• Compute posterior probability, pit, that interaction parameter
Uit comes from the Normal(0, v2) component
• Classify area as ‘unstable’ if pit > 0.5 for at least one time, t
(simulation study → 10% false positive rate; 20% false negative rate)
© Imperial College London
Detecting unusual trends in congenital
anomalies rates in England (Abellan et al 2008)
• Annual postcoded data on congenital anomalies (non
chromosomal) recorded in England for the period 1983 –
1998
• Annual postcoded data on total number of live births, still
births and terminations
• 136,000 congenital anomalies  84.5 per 105 birth-years
• Congenital anomalies are sparse:
 Grid of 970 grid squares with variable size, to equalize the
number of births and expected cases per area
• Variations could be linked to socio-economic or environmental
risk factors or heterogeneity in recording practices
 Interest in characterising space time patterns
© Imperial College London
Congenital anomalies in England, 1983-1998
Spatial main effect:
evidence of spatial
heterogeneity, linked
to deprivation and
maternal age
Temporal main effect:
downward trend
around 1990 reflects
implementation of
“minor anomalies”
exclusion policy
Congenital anomalies: Space-time interactions
Most areas are stable
(cluster 1)
Some have a change
around 90-91 where
modifications in the
classification of
anomalies occurred
(clusters 2 and 3)
Identified one very unusual time profile
due to a change of local recording practice
Outline
1. Introduction
2. Mapping and spatial smoothing of health data
3. Classifying areas and summarising geographical
variations in health outcomes
4. Modelling and mapping multiple outcomes
5. Modelling and mapping temporal trends
Summary
6. Hierarchical related regression models for
combining individual and small area data
Summary
• Smoothing of small area risks is important to help
separate ‘signal’ (spatial pattern) from ‘noise’
–Allows meaningful inference even when data are sparse
• Achieved by ‘borrowing’ information from
neighbouring regions
• Bayesian hierarchical modelling provides formal
method for carrying out this ‘borrowing of information’
–Provides rich output for statistical inference (estimation,
quantification of uncertainty, hypothesis testing)
–But, depends on “structural” assumptions built into the
model (e.g. spatial dependence)
–Computationally intensive
© Imperial College London
Summary
Bayesian approach extends naturally to allow:
• Adjustment for covariates (see later)
• Joint mapping of 2 or more health outcomes
• Joint modelling of spatial and temporal variation
Benefits of Joint Analysis of related
health outcomes
Joint analysis of two related health outcomes is of
interest in several contexts:
• Epidemiology: quantify ‘expected’ variability linked
to shared risk factors and tease out specific
patterns
• Health planning: assess the performance of the
health system, e.g. for health outcomes linked to
screening policies
• Data quality issues: uncover anomalous patterns
linked to a data source shared by several
outcomes
Benefits of Space Time Analysis for
(non-infectious) health outcomes
• Study the persistence of patterns over time
– Interpreted as associated with stable risk factors,
environmental effects, distribution of health care access …
• Highlight unusual patterns in time profiles via the
inclusion of space-time interaction terms
– Time localised excesses linked to e.g. emerging
environmental hazards with short latency
– Variability in recording practices
 Increased epidemiological interpretability
 Potential tool for surveillance
Outline
1. Introduction
2. Mapping and spatial smoothing of health data
3. Classifying areas and summarising geographical
variations in health outcomes
4. Modelling and mapping multiple outcomes
5. Modelling and mapping temporal trends
6. Hierarchical related regression models for
combining individual and small area data
Introduction
• Models and applications discussed so far have
focused on:
– describing geographical and temporal patterns in health
outcomes
– partitioning sources of variation into e.g. systematic and
idiosyncratic, spatial and temporal, shared and specific,
…
• Growing interest in trying to explain geographical
variations at level of areas and individuals
→ Build regression models linking health outcomes
and explanatory variables
Regression models for small area data
Standard regression for individual-level outcomes
Individual exposure
xij
Aggregate exposure
Zi, Xi
yij
Individual
outcome
Ecological regression
Aggregate exposure
Zi, Xi
Yi
Aggregate
outcome
Hierarchical Related Regression (HRR; Jackson et al, 2006, 2008a,b)
Individual exposure
Aggregate exposure
xij
Zi, Xi
yij
Yi
Individual
outcome
Aggregate
outcome
Case study: Socioeconomic inequalities in health
Jackson, Best and Richardson (2008b)
• Geographical inequalities in health are well documented
• One explanation is that people with similar characteristics
cluster together, so area effects are just the result of differences
in characteristics of people living in them (compositional effect)
• But, evidence suggests that attributes of places may influence
health over and above effects of individual risk factors
(contextual effect)
– economic, environmental, infrastructure, social
capital/cohension
Question
• Is there evidence of contextual effects of area of residence on
risk of limiting long term illness (LLTI) and heart disease, after
adjusting for individual-level socio-demographic characteristics
Data and Methodological Issues
Methodological issues
• Surveys typically contain sparse individual data per area so
difficult to estimate contextual effects
• Can’t separate individual and contextual effects using only
aggregate data (ecological bias)
• Improve power and reduce bias by combining data using new
class of multilevel models developed by BIAS
Our goal: data synthesis using
• Individual-level data
– Health Survey for England, 1997-2001
• Area-level (electoral ward) data
– 1991 census small-area statistics
– Hospital Episode Statistics
Data sources
INDIVIDUAL DATA
Health Survey for England
AREA (WARD) DATA
Census small area statistics
• Self-reported limiting long term illness
• Self reported hospitalisation for heart
disease
• Carstairs deprivation index
(area-level material deprivation)
• age and sex
• ethnicity
• social class
• car access
• income
• etc.
Ward codes made
available under
special license
Individual-level
Health outcomes
Contextual effect
Individual predictors
Multilevel model for individual data
yij = disease (1) / no disease (0)
xij = non-white (1) / white (0)
Zi = deprivation score
b
c
m,v2
ai
xij
yij
person j
Zi
area i
yij ~ Bernoulli(pij), person j, area i
logit pij = ai + b xij + c Zi
ai ~ Normal(m, v2)
b = relative risk of disease
for non-white versus white
individual
c = contextual effects
ai = “unexplained” area effects
Results from analysis of individual survey
data: Heart Disease (n=5226)
Univariate regression
Area deprivation
Multiple regression
Area deprivation
No car
Social class IV/V
Non white
0.2
0.5
1.0
2.0
odds ratio
5.0
10
Results from analysis of individual survey
data: Limiting Long Term Illness (n=1155)
Univariate regression
Area deprivation
Multiple regression
Area deprivation
Female
Non white
Doubled income
0.5
1.0
odds ratio
2
3
Comments
• CI wide and not significant for most effects
• Some evidence of contextual effect of area deprivation
for both heart disease and LLTI
– Adjusting for individual risk factors (compositional
effects) appears to explain contextual effect for
heart disease
– Unclear whether contextual effect remains for LLTI
after adjustment for individual factors
– Survey data lack power to provide reliable answers
about contextual effects
• What can we learn from aggregate data?
Area-level data
AREA (WARD) DATA
Contextual effect
Census small area statistics
• Carstairs deprivation index
• population count by age and sex
• proportion reporting LLTI
Aggregate health
outcomes
Aggregate
versions &of
denominators
individual
predictors
• proportion non-white
• proportion in social class IV/V
• proportion with no car access
PayCheck (CACI)
• mean & variance of household income
Hospital Episode Statistics
• number of admissions for heart disease
Standard ecological regression model
Yi = number with disease
Ni = population
Xi = proportion non-white
Zi = area deprivation score
Yi ~ Binomial(qi, Ni), area i
logit qi = Ai + BXi + CZi
Ai ~ Normal(M, V2)
B = association between
disease prevalence and
proportion non-white
C = contextual effects
Ai = “unexplained” area effects
B
M,V2
C
Ai
Zi
area i
Yi
Xi
Ni
Comparison of individual and ecological
regressions: Heart Disease
Individual
Area deprivation
Ecological
No car
Social class IV/V
Non white
0.2
0.5
2.0
1.0
odds ratio
5.0
10
Comparison of individual and ecological
regressions: Limiting Long Term Illness
Individual
Area deprivation
Ecological
Female
Non white
Doubled income
0.5
1.0
2
odds ratio
3
6
Ecological bias
Ecological bias (difference between individual and aggregate
level effects) can be caused by:
• Confounding
– confounders can be area-level (between-area) or individuallevel (within-area).
→ include control variables and/or random effects in model
• Non-linear covariate-outcome relationship, combined with
within-area variability of covariate
– No bias if covariate is constant in area (contextual effect)
– Bias increases as within-area variability increases
– …unless models are refined to account for this hidden
variability
Standard ecological regression model
Yi = number with disease
Ni = population
Xi = proportion non-white
Zi = area deprivation score
Yi ~ Binomial(qi, Ni), area i
logit qi = Ai + BXi + CZi
Ai ~ Normal(M, V2)
B = association between
disease prevalence and
proportion non-white
C = contextual effects
Ai = “unexplained” area effects
B
M,V2
C
Ai
Zi
area i
Yi
Xi
Ni
Integrated ecological regression model
Yi = number with disease
Ni = population Average of the
Xi = proportionindividual
non-whiteprobabilities
of disease,
Zi = area deprivation
scorepij, in area i 2
m,v
Yi ~ Binomial(qi, Ni),
b
c
area i
qi =  pij(xij,Zi,ai, b, c)fi(x)dx
ai ~ Normal(m, v2)
b = relative risk of disease
for non-white versus white
individual
c = contextual effects
ai = “unexplained” area effects
ai
Zi
area i
Yi
Xi
Ni
Combining individual and aggregate data
Multilevel model
for individual data
b
xij
c
yij
person j
Integrated
ecological model
m,v2
m,v2
ai
ai
Zi
Zi
area i
area i
b
Yi
c
Xi
Ni
Combining individual and aggregate data
Hierarchical Related
Regression
(HRR) model
b
c
m,v2
Joint likelihood for yij
and Yi depending on
shared parameters
ai, b, c, m, v2
(Jackson, Best, Richardson,
2006, 2008a,b)
ai
xij
yij
Zi
person j
area i
Yi
Xi
Ni
Combining individual and aggregate data
Hierarchical Related
Regression
(HRR) model
b
c
m,v2
Joint likelihood for yij
and Yi depending on
(Jackson, Best, Richardson,
Estimation carried out using
shared parameters
2006, 2008a,b)
ai, b, c, m, v2
R software (maximum likelihood)
or WinBUGS (Bayesian)
ai
xij
yij
Zi
person j
area i
Yi
Xi
Ni
Comparison of results from different regression
models: Heart Disease
Individual
Area deprivation
Standard ecological
Integrated ecological
No car
HRR
Social class IV/V
PR95 = 10.1; 95% CI(5.3, 18.1)
Non white
PR95 = 4.2; 95% CI(3.6, 5.1)
0.2
0.5
1.0
2.0
odds ratio
5.0
10
Comparison of results from different regression
models: Limiting Long Term Illness
Individual
Area deprivation
Standard ecological
Integrated ecological
Female
HRR
PR75 = 2.7;
95% CI(1.7, 4.1)
Non white
PR75 = 2.9;
95% CI(2.4, 3.7)
Doubled income
0.5
1.0
odds ratio
2
3
6
Comments
• Integrated ecological model yields odds ratios that are
consistent with individual level estimates from survey
• Large gains in precision achieved by using aggregate data
• Significant contextual effect of area deprivation for LLTI but
not heart disease
• More unexplained between-area variation (PR95) for heart
disease than LLTI
• Little difference between estimates based on aggregate data
alone and combined individual + aggregate data
– Individual sample size very small (~0.1% of population
represented by aggregate data)
• In other applications with larger individual sample sizes
and/or less informative aggregate data, combined HRR
model yields greater improvements (simulation study)
Strengths of HRR approach……
 Aims to provide individual-level inference using aggregate
data by:
 Fitting integrated individual-level model to alleviate one
source of ecological bias
 Including samples of individual data to help identify effects
 Uses data from all geographic areas (wards, constituencies),
not just those in the survey
 Improves precision of parameter estimates
 Improves ability to investigate contextual effects
…..and limitations of HRR approach
 Integrated individual-level model relies on large contrasts in
the predictor proportions across areas
 e.g. limited variation in % non-white across constituencies:
(median 2.7%, 95th percentile 33)
 Our estimates may not be completely free from ecological
bias (Jackson et al, 2006)
 If individual level data too sparse, may be overwhelmed by
aggregate data
Data requirements for HRR models
• Individual data
– requires geographical (group) identifiers for individual
data
• Aggregate data
– requires large exposure contrasts between areas
– requires information on within-area distribution of
covariates
• Important to check compatibility of different data sources
when combining data
Thank you for
your attention
Acknowledgements:
Sylvia Richardson, Juanjo Abellan, Virgilio Gomez-Rubio,
Chris Jackson
Training courses in Bayesian Analysis of Small Area Data
using WinBUGS and INLA, London, July 13-16 2010
See www.bias-project.org.uk for details
References
Abellan JJ, Richardson S and Best N. Use of space-time models to investigate the stability of patterns of
disease. Environ Health Perspect 116(8), (2008), 1111-1119.
Best N and Hansel A. Geographic variations in risk: adjusting for unmeasured confounders through joint
modelling of multiple diseases. Epidemiology, 20(3), (2009), 400-410.
Best, N.G., Richardson, S. and Thomson, A. A comparison of Bayesian spatial models for disease
mapping. Statistical Methods in Medical Research 14, (2005), 35-59.
Jackson C, Best N and Richardson S. Hierarchical related regression for combining aggregate and
individual data in studies of socio-economic disease risk factors. J Royal Statistical Society Series A:
Statistics in Society 171 (2008a) ,159-178
Jackson C, Best N and Richardson S. Studying place effects on health by synthesising individual and arealevel outcomes. Social Science and Medicine, 67, (2008b), 1995-2006
Jackson C, Best N and Richardson S. Improving ecological inference using individual-level data. Statistics
in Medicine, 25, (2006), 2136-2159
Knorr-Held, L. and Best, N.G., A shared component model for detecting joint and selective clustering of two
diseases, Journal of the Royal Statistical Society, Series A 164, (2001), 73-86.
Richardson, S., Thomson, A., Best, N.G. and Elliott, P. Interpreting posterior relative risk estimates in
disease mapping studies. Environmental Health Perspectives 112, (2004), 1016-1025.
Richardson, S., Abelan, J.J.,and Best, N. Bayseian spatio-temporal analysis of joint patterns of male and
female lung cancer in Yorkshire (UK). Statistical Methods in Medical Research 15, (2006), 385-407.
Tzala, E. Best, N. Bayesian latent variable modelling of multivariate spatio-temporal variation in cancer
mortality. Stat Methods Med Res 17, (2008), 97-118.
Spanish disease atlases and other related resources
López-Abente G, Pollan M, Escolar A, Errezola M, Abraira V. 1996. Atlas of cancer mortality and other
causes of death in Spain 1978-1992. Madrid: Fundación Científica de la Asociación Española contra el
Cáncer. http://www2.uca.es/hospital/atlas92/www/Atlas92.html. Maps of SMRs and Directly
Standardised Rates at province level
Benach J, Yasui Y, Borrell C, Rosa E, Pasarin MI, Benach N, Español M, Martinez JM, Daponte A. 2001.
Atlas of mortality at small area level in Spain 1987-1995. Barcelona: Universitat Pompeu Fabra.
Empirical Bayes smoothing of relative risks at municipality level (aggregated some municipalities to
reduced sparseness).
All of the following produce maps of Bayesian smoothed relative risks at municipality level:
López-Abente G, Ramis R, Pollán M, Aragonés N, Pérez-Gómez B, Gómez-Barroso D, Carrasco JM, Lope
V, García-Pérez J, Boldo E, García-Mendizabal MJ. 2007. Atlas of cancer mortality at municipality
level in Spain 1989-1998. Área de Epidemiología Ambiental y Cáncer del Centro Nacional de
Epidemiología, ISCIII.
Botella P, Zurriaga O, Posada de la Paz M, Martinez-Beneito MA, Bel E, Robustillo A, Ramalle E, Duran E,
Sanchez-Porro P. 2006. National-Provincial Atlas of Rare Diseases 1999-2003.
Martinez-Beneito MA, Lopez-Quilez A, Amador A, Melchor I, Botella P, Abellan C, Abellan JJ, Verdejo F,
Zurriaga O, Vanaclocha H, Escolano M. 2005. Atlas of Mortality in the Valencian Region 1991-2000.
Benach J, Matinez JM, Yasui Y, Borrell C, Pasarin MI, Español E, Benach N. 2004. Atlas of mortality at
small area level in Catalonia 1984-1998. Barcelona: Universitat Pompeu Fabra / Fundació Jaume
Bofill / Editorial Mediterrània
Spanish disease atlases and other related resources
DEMAP group, Andalusian School of Public Health, Granada.
Produced interactive mortality atlas for Andalucia, and socioeconomic indices at municipality level for
Spain. See http://www.demap.es/Demap/index.html
MEDEA: Research network on Epidemiology and Public Health, working on socioeconomic and
environmental inequalities in health at small area level. See
http://www.proyectomedea.org/medea.html
VPM Atlas Project (Atlas de Variaciones en la Práctica Médica). Studying and mapping variations in
provision and usage of health care at small area level in 16 of the 17 regions of Spain, using data on
hospital discharges. See http://www.atlasvpm.org/avpm/inicio.inicio.do
Smoothing of the RRs of hot spots (4 contiguous areas with
average expected counts ≈ 5) for different spatial models
Richardson et al (EHP, 2004)
True RR = 3
True RR = 2

similar documents