Peter Congdon, Centre for Statistics and Department of Geography, Queen Mary University of London. 1 Spatial Path Models with Multiple Indicators and Causes: Population Psychiatric Outcomes in US Counties

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 X 1,…X P to produce area scores for small set of underlying latent constructs F 1,…F Q Spatial structuring in latent construct typically not considered though Hogan/Tchernis (2004, JASA) provide Bayesian model for spatially structured Townsend deprivation score F. 2

Flow chart for Townsend Deprivation Score 3

Another Theme: Latent Spatial Constructs for Composite Morbidity Seek composite morbidity index: e.g. index of cardiovascular morbidity underlying J different observed outcomes Y j, either Normal, Poisson or Binomial (Wang & Wall, Biostatistics, 2003) Example: Y ji are counts,P i are Population offsets Then : Y ji ~ Poisson(P i ji )j=1,..,J log( ji )=α j +λ j F i F i ~ spatial(W, ,  2 F ) over areas i=1,..,I W =neighbourhood adjacencies,  = spatial correlation Loading λ j expresses influence of common factor F i on observed outcomes 4

Representing the impact of social structural constructs on morbidity: both X and Y indicators May seek area structural constructs F 1,…F Q measured by socioeconomic indicators X 1,…X P but oriented to explaining particular health outcomes Y 1,…Y J. 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) 5

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 F 1,..,F Q (and maybe also by known predictors W). May need iid random effects also for Y-model (e.g. overdispersed count responses) 6

Example: Psychiatric Morbidity for US Counties Y variables: suicide deaths (y₁) (Poisson), self-rated poor mental health (y₂) (Normal with varying precision). Source for y 2 is BRFSS (Behavioral Risk Factor Surveillance System) Q=4 latent constructs: social capital F 1, deprivation F 2, social fragmentation F 3, and rurality F 4, 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 W 1 =% White non-Hispanic and W 2 =% native American. 7

Expected effects of F variables and W variables on y-variables 8

Postulated Links (with Direction), Confirmatory Model Relating Constructs F1,F2,F3,F4 to X-indicators 9

Extending Model for Latent Factors 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 10

Causal Indicators of Constructs Assume latent constructs may be influenced by known (possibly partially observed) exogenous variables {Z 1i,..,Z Ki } Alternative terms: Z k 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 F 1 (e.g. income inequality –ve influence). Incorporating these into model improves measurement of latent construct. Here we use measure of income inequality Z 1, ethnic fractionalization index Z 2, and measure of religious adherence Z 3 11

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 F 1,..,F Q without any causal sequence. Plausible sequence among constructs in US county application: social capital F 1 depends on deprivation F 2 (expected -ve impact), fragmentation F 3 (expected -ve impact ), and rurality F 4 (expected +ve impact). See Rupasingha et al (2006) on substantive basis. So have separate models for F 1 and for {F 2,F 3,F 4 }. 12

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Antecedent and Dependent F variables Take {F 2,F 3,F 4 } to be trivariate CAR. These effects have zero means obtained by centering during MCMC sampling. Model for F 1 is separate univariate spatial prior with regression on other F variables and on Z variables Can include nonlinear effects of {F 2,F 3,F 4 } on F 1, and maybe Z-F interactions. 14

Mediating Effect of Dependent F variables Implications: effects on health (Y) variables of antecedent constructs {F 2,F 3,F 4 } may be partly or totally mediated by social capital. Total effect (e.g. direct effect of poverty F 2 on Y plus indirect effect through mediator F 1 ) may increase if mediation only partial From Baron-Kenny 1986: 15

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Actual Estimates, Multiple Causes (Formative Indicators) for Social Capital CauseParameterMean2.5%97.5% Deprivation  (s)  Fragmentation  (s)  Rurality  (s)  Rurality x High Deprivation  (s)  Income Inequality  (s) Ethnic Fractionalisation  (s) Religious Adherence  (s)

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Developments-Options Other possible model features: (a) predictor selection in regression model for F 1 and Y j (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. 23