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Analysis of urban social structure often oriented to producing indices of unobserved constructs

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- 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.

- 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,Piare 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

- 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)

- 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)

- 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.

- Typical paradigm considers only responsive X-indicators, 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

- 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

- 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}.

- 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.

- 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:

- Other possible model features: (a) predictor selection in regression model for F1 and Yj (b) nonlinear effects of Fvariables 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.