Transcript Slide 1
Spatial Path Models with Multiple
Indicators and Causes: Population
Psychiatric Outcomes in US
Counties
Peter Congdon, Centre for Statistics and Department of
Geography, Queen Mary University of London.
[email protected]
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Latent constructs of urban structure (“urban
structure constructs”)
Analysis of urban social structure often oriented to
producing indices of unobserved constructs
Examples: area deprivation, social fragmentation, social
capital, familism, rurality, etc
Various multivariate (or other) methods use observed
indicators X1,…XP to produce area scores for small set of
underlying latent constructs F1,…FQ
Spatial structuring in latent construct typically not
considered though Hogan/Tchernis (2004, JASA) provide
Bayesian model for spatially structured Townsend
deprivation score F.
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Flow chart for Townsend Deprivation Score
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Another Theme: Latent Spatial Constructs for
Composite Morbidity
Seek composite morbidity index: e.g. index of cardiovascular
morbidity underlying J different observed outcomes Yj, either
Normal, Poisson or Binomial (Wang & Wall, Biostatistics, 2003)
Example: Yji are counts,Pi are Population offsets
Then :
Yji ~ Poisson(Piji)
j=1,..,J
log(ji)=αj+λjFi
Fi ~ spatial(W,,2F) over areas i=1,..,I
W =neighbourhood adjacencies, = spatial correlation
Loading λj expresses influence of common factor Fi on observed
outcomes
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Representing the impact of social structural
constructs on morbidity: both X and Y indicators
May seek area structural constructs F1,…FQ measured
by socioeconomic indicators X1,…XP but oriented to
explaining particular health outcomes Y1,…YJ.
Latent factors represent aspects of urban social
structure, environmental exposure, etc. These are
“mainly” measured by X indicators, but partly also
measured by the Y outcomes.
Example: Want not “general” deprivation score but a
context-specific score tuned to explaining variations in
psychiatric morbidity (Y)
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Social structure and morbidity model: defining
aspects
Usually assume confirmatory model relating X
variables to F variables (mutually exclusive subsets
of X indicators explained by only one F variable).
Usually extensive prior evidence to support such an
approach
By contrast, typically each Y variable potentially
explained by all constructs F1,..,FQ (and maybe also
by known predictors W). May need iid random
effects also for Y-model (e.g. overdispersed count
responses)
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Example: Psychiatric Morbidity for US Counties
Y variables: suicide deaths (y₁) (Poisson), self-rated poor mental
health (y₂) (Normal with varying precision). Source for y2 is
BRFSS (Behavioral Risk Factor Surveillance System)
Q=4 latent constructs: social capital F1, deprivation F2, social
fragmentation F3, and rurality F4, measured by P=17 X-indicators
of urban structure
Choice of X-indicators for social capital follows Rupasingha et al
(2006) The production of social capital in U.S. Counties, Journal
of Socio-Economics, 35.
Also relevant to explaining Y-outcomes are known predictors
W1=% White non-Hispanic and W2=% native American.
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Expected effects of F variables and W
variables on y-variables
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Postulated Links (with Direction), Confirmatory Model Relating
Constructs F1,F2,F3,F4 to X-indicators
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Extending Model for Latent Factors
• Typical paradigm considers only responsive Xindicators, i.e. caused by latent constructs
However, there may be indicators relevant to
measuring latent constructs that are better viewed
as causes of the construct.
Also some F-variables may be better viewed as
depending on other F variables: so one may want a
more flexible regression scheme for multiple latent
factors than that implied by multivariate normality
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Causal Indicators of Constructs
Assume latent constructs may be influenced by known (possibly
partially observed) exogenous variables {Z1i,..,ZKi}
Alternative terms: Zk sometimes called formative indicators, i.e.
"observed variables that are assumed to cause a latent variable", as
opposed to effect indicators X (Diamantopoulos & Winklhofer,
2001).
In US county application, literature suggests several possible
causes of social capital F1 (e.g. income inequality –ve influence).
Incorporating these into model improves measurement of latent
construct.
Here we use measure of income inequality Z1, ethnic
fractionalization index Z2, and measure of religious adherence Z3
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Sequences among F variables
Bayesian analyses generally consider only univariate F, and if
they consider multivariate F, assume multivariate normal
conditionally autoregressive (MCAR) prior.
MCAR has implicit linear regressions between F1,..,FQ
without any causal sequence.
Plausible sequence among constructs in US county
application: social capital F1 depends on deprivation
F2(expected -ve impact), fragmentation F3 (expected -ve
impact ), and rurality F4 (expected +ve impact). See
Rupasingha et al (2006) on substantive basis.
So have separate models for F1 and for {F2,F3,F4}.
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Antecedent and Dependent F variables
Take {F2,F3,F4} to be trivariate CAR. These effects have zero
means obtained by centering during MCMC sampling.
Model for F1 is separate univariate spatial prior with
regression on other F variables and on Z variables
Can include nonlinear effects of {F2,F3,F4} on F1, and maybe
Z-F interactions.
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Mediating Effect of Dependent F variables
Implications: effects on health (Y) variables of antecedent
constructs {F2,F3,F4} may be partly or totally mediated by
social capital.
Total effect (e.g. direct effect of poverty F2 on Y plus indirect
effect through mediator F1) may increase if mediation only
partial
From Baron-Kenny 1986:
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Actual Estimates, Multiple Causes (Formative
Indicators) for Social Capital
Cause
Parameter
Mean
2.5%
97.5%
Deprivation
d(s)1
-0.908
-0.942
-0.871
Fragmentation
d(s)2
-0.364
-0.416
-0.282
Rurality
d(s)3
0.059
-0.032
0.159
d(s)4
-0.191
-0.264
-0.130
Income Inequality
g(s)1
0.021
-0.018
0.070
Ethnic Fractionalisation
g(s)2
-0.242
-0.319
-0.181
Religious Adherence
g(s)3
0.414
0.353
0.468
Rurality x High Deprivation
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Developments-Options
Other possible model features: (a) predictor selection in
regression model for F1 and Yj (b) nonlinear effects of F
variables on Y variables (c) Informative missingness in Y
variables with spatial factors predicting probability of
missing data
Social capital likely to be important for explaining
variation in other health outcomes, such as mortality,
e.g. Social capital and neighborhood mortality rates in
Chicago, Lochner et al, 2003
May often be a case for general latent constructs that are
not context-specific.
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