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.
p.congdon@qmul.ac.uk
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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(Piji)
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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