Spatial path models with multiple indicators and multiple causes: Mental health in US counties

Spatial and Spatio-temporal Epidemiology 2 (2011) 103–116

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Spatial path models with multiple indicators and multiple causes: Mental health in US counties Peter Congdon Centre for Statistics and Department of Geography, Queen Mary University of London, Mile End Rd., London E1 4NS, United Kingdom

a r t i c l e

i n f o

Article history: Received 26 January 2011 Revised 30 March 2011 Accepted 31 March 2011 Available online 30 April 2011 Keywords: Spatial Social capital Latent construct Path model Bayesian

a b s t r a c t This paper considers a structural model for the impact on area mental health outcomes (poor mental health, suicide) of spatially structured latent constructs: deprivation, social capital, social fragmentation and rurality. These constructs are measured by multiple observed effect indicators, with the constructs allowed to be correlated both between and within areas. However, in the scheme developed here, particular latent constructs may also be influenced by known variables, or, via path sequences, by other constructs, possibly nonlinearly. For example, area social capital may be measured by effect indicators (e.g. associational density, charitable activity), but influenced as causes by other constructs (e.g. area deprivation), and by observed features of the socio-ethnic structure of areas. A model incorporating these features is applied to suicide mortality and the prevalence of poor mental health in 3141 US counties, which are related to the latent spatial constructs and to observed variables (e.g. county ethnic mix). Ó 2011 Elsevier Ltd. All rights reserved.

1. Introduction Ecological variations in health outcomes, including mental health outcomes, are often related to overarching aspects of area social structure (e.g. area deprivation). Typically these aspects are not observed directly but are best viewed as latent constructs, which are measured by a range of observed proxy indicators. Measurement of neighborhood influences on health is important since ‘‘incorrect estimates of neighborhood health effects [may result] if neighborhood socioeconomic characteristics are poor proxies for the true neighborhood construct of interest’’ (Diez-Roux and Mair, 2010). However in modeling area data, it is usually no longer valid to assume units are independent, and so spatial correlation in latent constructs should also be considered. For example, Wang and Wall (2003) consider discrete area health outcomes (e.g. cancer deaths) and use a spatially structured common factor to account for inter-outcome correlations, while Hogan and Tchernis (2004) use a confirmatory model with a spatially E-mail address: [email protected] 1877-5845/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.sste.2011.03.003

correlated deprivation factor to summarize the correlation between several observed socio-economic indicators (e.g. unemployment and poverty rates). Existing area health applications which acknowledge spatial structure are oriented to estimating univariate latent constructs, either a common latent morbidity underlying several health outcomes (e.g. Zhu et al., 2005; Tzala and Best, 2008; Wang and Wall, 2003), or a common latent population risk underlying several observed socio-economic indicators (e.g. Hogan and Tchernis, 2004). By contrast, as in the application here to US mental health outcomes, there may be plural constructs of relevance, especially in defining latent population risks, and the full observation set relevant to measuring such constructs encompasses both health outcomes and socio-economic indicators. Thus area variations in both serious mental illness (SMI) and common mental disorders (CMDs) have been linked to deprivation, social capital and to rurality, all latent constructs (Allardyce et al., 2005; De Silva et al., 2005; Reijneveld and Schene, 1998), while suicide variations have been linked to deprivation, social fragmentation and to rurality (Corcoran et al., 2007; Evans et al., 2004).

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Different health outcomes may be differentially linked to latent constructs (e.g. the effect of area deprivation may be greater on SMI than on CMDs), and the health outcomes included contribute (as an additional form of indicator) to defining the posterior latent factor scores. So rather than deriving construct scores previously and separately (i.e. abstracted from the application context), it is preferable to combine both multiple socio-economic indicators and relevant health outcomes in a single model. It would be possible to pool information over health outcomes, and so form an endogenous health construct that can be related to the exogenous risk constructs in a structural model. However, here health outcomes are regressed separately and directly on the latent risk constructs, so that one can assess (say) whether social capital is a more important influence on poor mental health than on suicide (Arminger and Muthén, 1998). The simplest form of plural latent risks model that may be adapted to area data takes the latent constructs to follow a multivariate spatial form, such as the multivariate conditional autoregressive scheme (Mardia, 1988; MacNab, 2011), with the constructs simply measured by a broader set of multiple effect indicators. However, particular latent constructs may also be influenced by known exogenous variables, not only by other constructs. Such influences are sometimes referred to as formative indicators, namely ‘‘observed variables that are assumed to cause a latent variable’’, as opposed to effect indicators (Diamantopoulos and Winklhofer, 2001). In particular, spatial structural equation schemes may involve multiple causes of constructs as well as multiple indicators, which in turn implies possible path sequences between constructs. For example, area social capital may be measured by indicators such as associational density or charitable activity levels, but influenced as causes both by other constructs (e.g. social fragmentation), and by observed (exogenous) features of the demo-ethnic structure of areas, such as income inequality and ethnic fractionalization (Rupasingha et al., 2006). Hence one may generalize simple multivariate linear covariation between constructs to allow for effects of exogenous variables on constructs, and for recursive dependencies among the constructs themselves, including possibly nonlinear effects of one construct on another. A case study application considers two response variables, suicide deaths ðy1 Þ and self-rated poor mental health ðy2 Þ in 3141 US counties, in relation both to latent spatial constructs and to observed ecological risk variables (e.g. county ethnic mix). One among four latent constructs (social capital) is taken to depend in a path model formulation on three other constructs (deprivation, fragmentation and rurality), and on predetermined variables (e.g. ethnic fractionalization, income inequality). This model form reflects prior evidence for causal impacts on social capital (e.g. of fragmentation) (Fagg et al., 2008). A fully Bayesian approach is used in specifying and estimating the model, as this has benefits in estimation for a relatively complex model involving several random effects. Hence prior densities of parameters are specified, and these densities are updated via the likelihood of the observations. Iterative Monte Carlo Markov Chain techniques are used in estimation

(Gelfand and Smith, 1990), via the WINBUGS program (Lunn et al., 2009). 2. Spatial path models with multivariate constructs Let yji ði ¼ 1; . . . ; I; j ¼ 1; . . . ; JÞ be health outcomes (e.g. averages, counts) by area i and outcome j, observed from sources such as surveys or registered mortality. Assume sampling from the exponential family density,

" # yji hji  bj ðhji Þ pðyji jhji ; /j Þ ¼ exp þ cðyji ; /j Þ ; aji ð/j Þ

ð1Þ

with canonical parameters hji , and dispersion function aji ð/j Þ. The mean of yji is lji ¼ Eðyji jhji Þ, linked to predictors via a link function gðlji Þ. In the context of spatial health variations, some predictors X i ¼ ðX 1i ; . . . ; X Ki Þ0 of lji may be known, in the sense of being directly observed. However, often population risk factors (constructs) are not directly observed, but measured by other observed variables (indicators). Examples are rurality and area deprivation. For generality, let lji be predicted both by observed and latent risk variables, with the latter denoted F i ¼ ðF 1i ; . . . ; F Qi Þ0 . So the typical regression model for health outcome j (i.e. the jth equation in the structural model) would be

gðlji Þ ¼ aj þ bj F i þ uj X i þ uji ;

ð2Þ

where uji v Nð0; 1=/j Þ are Normal random residuals, including random errors for Poisson or binomial data with overdispersion. The measurement model to derive the F i is partly based on information provided by a collection of effect indicators Z i ¼ ðZ 1i ; . . . ; Z Pi Þ that are observed proxies for the latent F i . In the present application (described below) these are assumed to be derived from observed census indicators (e.g. migration in the precensal year) or intercensal indices of socioeconomic status or community organization (e.g. poverty rates, charitable activity). After possible preliminary transformation (e.g. log or logit transformation), the transformed variables zpi are taken to be Normal with factor loadings kp ¼ ðkp1 ; . . . ; kpQ Þ on the components of F i . One then has

zpi jF i v Nðxp þ kp1 F 1i þ    þ kpQ F Qi ; 1=#p Þ:

ð3Þ

Taking all kloadings as unknowns leads to an ‘‘exploratory’’ factor analysis. However, often there is substantial prior knowledge on links between indicators and constructs, and in fact indicators are chosen specifically to measure a particular construct. This leads to a confirmatory factor analysis, in which particular elements in the P  Q matrix K ¼ ½kpq  are set to zero. Let cp 2 ð1; . . . ; Q Þ denote that construct for which zp is an effect indicator. Then

zpi jF i v Nðxp þ kp;cp F cp i ; 1=#p Þ:

ð4Þ

Most simply one may assume indicators z1 to zP1 are chosen to measure the first construct, indicators zP1 þ1 to zP1 þP2 selected to measure the second construct, and so on, up to indicators zP1 þP2 þþPQ 1 þ1 to zP that measure the Q th construct. For example, in the case study below there are

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Q = 4 constructs, and P = 17 indicators in total, with P1 ¼ 4 indicators measuring area deprivation, P 2 ¼ 5 indicators measuring area social fragmentation, P3 ¼ 4 indicators of rurality and P4 ¼ 4 indicators of area social capital. 2.1. Prior structure of latent constructs One is then faced with modeling the F i . Many methods used to represent latent area variables use ad-hoc methods (e.g. sums of standardized scores on the observed indicators), or use generic multivariate methods that do not incorporate any spatial structure in the factor scores. For example, Rupasingha et al. (2006) derive a social capital score for US counties using principal component analysis, while Messer et al. (2006) use principal component analysis to derive an area deprivation score. However, with geographic clustering frequently apparent in the observed effect indicators, geographic correlation is likely in the latent construct or constructs, and this has implications for a suitable measurement model. For example, Hogan and Tchernis (2004) use P = 4 indicators to measure a spatially correlated material deprivation construct (Q = 1). Here we consider multiple spatially correlated constructs. To allow for correlation between the latent constructs F i ¼ ðF 1i ; . . . ; F Qi Þ0 within areas, while also allowing them to be spatially correlated between areas, one could assume the F qi follow a multivariate version of a spatial prior, such as the conditional autoregressive prior (CAR) prior (MacNab, 2011). Thus with F ¼ ðF 1 ; . . . ; F I Þ, one has (Gamerman et al., 2003)

PðFj F Þ / j F jI=2

As an example (and one relevant to the subsequent case study) assume H = 1, so that ðQ  1Þ of the Q constructs are precursor or antecedent constructs, following a ðQ  1Þ dimensional version of the scheme (5), while the remaining construct F Qi is dependent both on earlier constructs and on known predictors W i . This is one possible example scheme of dependence among the F variables, and others are possible with (say) distinct W variables applying to two or more intermediate and/or final construct variables, and the precursor scheme of lesser dimension. For F ðdÞ ¼ F Q , a linear model would involve regression terms gi with

gi ¼ dF ðaÞ i þ cW i ; ðaÞ

where F i ¼ ðF 1i ; . . . ; F ðQ 1Þi Þ0 denotes a ðQ  1Þ dimensional subset of F i , and d and c are regression parameters of dimension ðQ  1Þ and R, respectively. The regression term gi can be extended to include polynomial or interaction terms in antecedent constructs, and also interactions between W and F ðaÞ . For identifiability there is no intercept. To allow for spatial structure in F ðdÞ , one may adapt an appropriate spatial prior to accommodate regression features (e.g. Bell and Broemeling, 2000). Thus define residuals

ei ¼ F Qi  gi ;

 exp 0:5

X

# wik ðF i  F k Þ0  F ðF i  F k Þ ;

ð7Þ

and let F ½Qi ¼ ðF Q 1 ; . . . F Q ;i1 ; F Q ;iþ1 ; . . . F QI Þ denote the F Q scores excluding F Qi . Under a conditional autoregressive scheme, the conditional mean and precision of F Qi are

EðF Qi jF ½Qi Þ ¼ gi þ

X

wih eh =

h–i

PrecðF Qi jF ½Qi Þ ¼ "

ð6Þ

X

X

wih ;

ð8:1Þ

h–i

wih f;

ð8:2Þ

h–i

ð5Þ

i;k

with within area precision matrix  F of dimension Q, and adjacency matrix W ¼ ½wik  between areas i and k. An alternative scheme developed here, and arguably a more flexible one often relevant to problems encountered in spatial epidemiology (e.g. Berry, 2007; van Jaarsveld et al., 2007), involves construct path sequences, which allow (a) for extraneous (observed) causes of one or more constructs, and (b) for sequential relationships between constructs, that might include interactions or nonlinear effects. For example, such models may have relevance in assessing the relevance of environmental/contextual factors (e.g. superstore availability) to area obesity variations, and how far there is a direct effect of area deprivation on such variations (Ford and Dzewaltowski, 2008). ðaÞ Thus assume a ðQ  HÞ dimension subset F i ¼ ðF 1i ; . . . ; 0 F ðQ HÞi Þ of the F i to be antecedent in a path model, and H ðdÞ subsequent intermediate or final constructs F i ¼ ðF ðQ Hþ1Þi ; . . . ; F Qi Þ0 to depend on constructs earlier in the sequence (e.g. Bryman and Cramer, 1994). Later constructs in the path sequence may also depend on observed causal influences (or ‘‘formative indicators’’), denoted W i ¼ ðW 1i ; . . . ; W Ri Þ0 . Thus as well as multiple effect indicators Z of the latent F, there may be multiple formative indicators W of one or more F qi , and also hypothesized sequences among the F qi based on prior knowledge.

where f is a precision parameter. When the wih are binary contiguity indicators ðwih ¼ 1 for adjacent areas h and i, wih ¼ 0 otherwise), the expectations and precisions are

EðF Qi jF ½Qi Þ ¼ gi þ

X

eh =Li ;

ð9:1Þ

h2@ i

PrecðF Qi jF ½Qi Þ ¼ Li f;

ð9:2Þ

where @ i denotes the neighborhood (set of adjacent counties) of county i consisting of Li counties. Thus the complete model involves the structural model (1) and (2), the effect indicator model (4), the spatial prior (5) on F ðaÞ , and the formative indicator model(s) for F ðdÞ as exemplified by (6–9). Identification of the effect indicator model involves either fixing the variances of the F qi (a ‘‘standardized factors’’ analysis), or presetting appropriate loadings (Skrondal and Rabe-Hesketh, 2004, p. 66). Here the initial loadings in each indicator–construct sequence are set to 1. Thus k11 ¼ kP1 þ1;2 ¼    ¼ kP1 þP2 þþPQ 1 þ1;Q ¼ 1, which in the case study leads to k11 ¼ k5;2 ¼ k10;3 ¼ k14;4 ¼ 1. Variables chosen to have a fixed loading are sometimes known as marker variables, and in formal terms ‘‘choice of which marker variable to use is arbitrary’’ (Little et al., 2007, p. 358). In practice, it is not advisable to select as a marker an indicator that correlates weakly with other indicators (Kenny, 2011), and this criterion holds for the choices here. The model structure may be completed by Normal priors on unconstrained loadings kpq in (4), and on

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coefficients faj ; bjq ; ujk ; xp ; dq ; cr g, by positive variable priors (e.g. gamma or lognormal) on /j ; #p and f, and by a Wishart prior on  F . Table 1 contains a glossary of the more important coefficients for substantive inferences. 2.2. Missing data Particular health outcome variables y, or effect indicators Z may be subject to missing data. Suppose a particular outcome, say yi ¼ yki , is subject to partially missing values (this is the case for poor mental health in the subsequent case study), and let M i denote a binary indicator for missing data status ðM i ¼ 1 for yki missing; M i ¼ 0 for data present). Following Yang et al. (2008), an alternative to selection and pattern mixture models for missing data (Little, 2008), are shared parameter models, where the complete y data and missingness indicators M are conditionally independent given a group of shared parameters, such as common latent constructs. Additionally, let Di of dimension G be observed variables relevant to explaining missingness in the health responses. For example, if outcomes were more likely to be missing for small areas, one might include a measure of area population size (Monnat and Pickett, 2010). Defining wi ¼ PrðM i ¼ 1Þ, the shared factor model (sharing the vector F i Þ for missingness in y is then

logitðwi Þ ¼ t0 þ t1 F 1i þ t2 F 2i þ    þ tQ F Qi þ vDi :

ð10Þ

Missingness is informative or not according to whether or not ft1 ; . . . ; tQ g are nonzero (cf. Yang et al., 2008, p. 2835, for longitudinal data). For instance, if F 1 represented area deprivation, and the 95% interval for t1 were confined to positive values, then missing values would be more common in high deprivation areas. 3. Case study 3.1. Latent area constructs To set the framework for the case study, we first consider the substantive literature linking ecological variations in mental health outcomes such as depression, schizophrenia and suicide to area social structure. Of central importance here is a well-documented mental health gra-

Table 1 Glossary of parameters. Coefficients for observed outcomes (y, z) bjq

ujk wj mjq kpq

Coefficient of mean lj (of health outcome yj) on latent construct Fq Coefficient of mean lj (of health outcome yj) on known risk Xk Probability of missingness in yj Coefficient on Fq for logit regression for wj Loading of indicator zp on latent construct Fq

Coefficients in construct path model ghi Regression term for hth dependent construct (h = 1, . . . , H) dhq Coefficient in ghi for effect of antecedent factor Fq(q ¼ 1; . . . ; Q  H) chr Coefficient in ghi for effect of known formative cause Wr Cjq Total effect of antecedent factor q on outcome j qqr Correlation between antecedent constructs q and r

dient according to the level of area social deprivation, referring to geographic concentrations of material hardship and represented by observed indicators such as poverty, lower job skills, unemployment or welfare dependence. For example, Reijneveld and Schene (1998) mention that common mental disorders occur more frequently in deprived urban areas, partly due to the socioeconomic composition of the inhabitants of such areas; for example, low skill workers are more at risk of unemployment which is a confirmed individual level risk factor (Platt, 1984). There is also evidence of distinct contextual effects of area deprivation, after accounting for population composition, as demonstrated for depression by Yen and Kaplan (1999). A positive impact of area deprivation on suicide risk has also been reported, as in British and Irish studies (e.g. Gunnell et al., 1995; Corcoran et al., 2007). Also established as relevant to area variations in suicide and psychotic illness is ‘social fragmentation’, based in UK studies on four observed indices: one person households, residential turnover, non-married adults and concentrations of short stay private rented households (e.g. Allardyce et al., 2005; Evans et al., 2004). A similar measure (in terms of indicators used) but denoted as ‘residential instability’ is applied by Matheson et al. (2006). Fragmentation is an aggregate measure of confirmed individual level risk factors for psychiatric illness and suicide, such as living alone, social isolation, residential transience and being unmarried (e.g. Trout, 1980; Jackson and Cochran, 1991; Davey-Rothwell et al., 2008; Wu and De Maris, 1996), and may also encapsulate distinct contextual effects, such as adverse effects on mental illness of high neighborhood transience (McKenzie, 2008; Matheson et al., 2006). Social capital has also been proposed as an influence on mental health (Almedom, 2005; De Silva et al., 2005). While there is considerable debate about how social capital should be defined, it is generally considered as a feature of neighborhood social structure, namely ‘‘a web of cooperative relationships between citizens, high levels of interpersonal trust, and strong norms of reciprocity and mutual aid’’ (Veenstra et al., 2005). Of particular concern in the present study are ecological measures of social capital that do not require aggregation of individual responses (McKenzie et al., 2002; De Silva et al., 2005; Diez-Roux and Mair, 2010). Examples are the Petris measure of community-level social capital (Scheffler et al., 2008), and the index developed by Rupasingha et al. (2006), based on community level (effect) indicators of associational density, charitable activity and communal participation. Some of the potential links between community social capital and health are discussed by Scheffler et al. (2008), for example that ‘‘higher levels of community-level social capital, such as a higher density of voluntary organizations, may make social support, . . . more accessible’’. Of particular relevance to the current study is the secondary regression by Rupasingha et al. (2006) of their social capital index on potential causal influences (e.g. community stability, ethnic fractionalization, income inequality). These influences may include other constructs such as fragmentation: thus Fagg et al. (2008, p. 243) mention that ‘‘social fragmentation . . . implies that aspects of

P. Congdon / Spatial and Spatio-temporal Epidemiology 2 (2011) 103–116

social capital such as reinforcement of social norms, trust, and reciprocity may be more difficult to maintain’’. A number of studies report higher psychiatric morbidity in more urbanized residential settings (Weich et al., 2006; Peen et al., 2010). While general psychiatric morbidity may be higher in urban areas this is not necessarily true of particular psychiatric conditions or outcomes. Thus several studies show high suicide rates in more extremely rural areas (Singh and Siahpush, 2002; Levin and Leyland, 2005), possibly linked to poor access to psychiatric care and easier availability of lethal suicide methods (Miller et al., 2007). 3.2. Data on health outcomes and socioeconomic indicators in US counties The case study involves I = 3141 US counties, J = 2 health outcomes yji and P = 17 effect indicators which measure a Q = 4 dimensional construct vector F i ¼ ðF 1i ; . . . ; F Qi Þ0 . There are also known rather than latent influences on health and social capital: K = 2 observed risk predictors (for two race variables) are used in the structural regressions in (2), and R = 3 observed formative indicators W i (influences such as ethnic fractionalization) are used in a path model regression (6) involving Q  H ¼ 3 antecedent constructs. As outlined further below, of particular relevance for comparative model assessment are possible construct interactions in the path model regression. The first outcome is suicide deaths y1i over 2002–2006 and for ICD-10 Codes X60–X84, namely intentional selfharm. Suicide data are from Center for Disease Control Compressed Mortality database (http://wonder.cdc.gov/ mortsql.html) which contains mortality and population counts for the years 1979–2007. Death totals can be obtained by underlying cause of death, county, age, race, sex and year. The second outcome consists of county level survey averages y2i for days with poor mental health from the combined surveys for 2003–2005 under the behavioral risk factor surveillance system (BRFSS). The averages are based on the question ‘‘Now thinking about your mental health, which includes stress, depression, and problems with emotions, for how many days during the past 30 days was your mental health not good?’’ State level averages for this indicator can be found at http://apps.nccd.cdc.gov/HRQOL. Data from the BRFSS is used as it provides greater geographic detail and more extensive population coverage than other US population surveys, such as the National Institute of Mental Health’s National Comorbidity Survey. The effect indicator measurement model (4) for the latent risk constructs is based on 2000 census indicators, and intercensal data such as 2004 poverty and unemployment rates. The indicators of deprivation encompass education, occupation and economic well-being, as suggested by Ross and Mirowsky (2008) and Yost et al. (2001). Following Hofferth and Iceland (1998), indicators of rurality include rural occupation patterns, and population and housing density to represent spatial dispersion of residences. To measure social capital, the indicators used by Rupasingha et al. (2006) are updated, involving data from the 2004 County Business Patterns (providing density of employment in civic, voluntary and sports organizations),

107

data from the US Election Assistance Commission on the 2004 election turnout, 2000 Census response, and 2004 data on tax-exempt non-profit organizations from the National Center for Charitable Statistics. Indicators are log-transformed, so that for a percentage index Z between 0 and 100, z ¼ logð1 þ ZÞ. The confirmatory structure (4) is adopted, with observed indicators loading on only one of the four constructs. Thus the P = 17 effect indicators used are listed in Table 2, together with the expected sign (positive/negative) on the latent constructs to which they are linked: either deprivation ðF 1 Þ, fragmentation ðF 2 Þ, rurality ðF 3 Þ or social capital ðF 4 Þ. For example, the measures of fragmentation are one person households, one person renter households, family households, 1 year turnover and 5 year turnover (in years preceding the census); these are all positive measures of fragmentation, except for family households which is a negative measure. Similarly of the variables measuring rurality (indices 10–13 in Table 2), the first two are positive measures, while the population and housing density variables are negative measures. The structural regression models (2) also allow for the influence of county ethnicity on suicide and poor mental health. In particular X 1i ¼ logð%wnH þ 1Þ and X 2i ¼ log ð%natAm þ 1Þ denote transformed percents for white non-Hispanic (%wnH), and native Americans (%natAm). This choice of county race indices reflects evidence that, after controlling for socio-economic status, black Americans and Hispanics have lower rates of psychiatric disorder (Breslau et al., 2005). Similarly, recent data on ethnic suicide risks show a contrast between relatively high rates for white non-Hispanics and native Americans, and lower rates for black non-Hispanics, Hispanics and Asian Americans (CDC, 2010). 3.3. Model definition To define the structural model, a Poisson regression (including an overdispersion residual) is used for suicide deaths, so that

y1i v PoisðE1i l1i Þ;

ð11:1Þ

logðl1i Þ ¼ a1 þ b11 F 1i þ b12 F 2i þ b13 F 3i þ b14 F 4i þ u11 X 1i þ u12 X 2i þ u1i ;

ð11:2Þ

where u1i v Nð0; 1=/1 Þ, and expected suicides E1i are based on US-wide age specific rates applied to intercensal population estimates, specific for gender and 5 year age bands. A Normal model for average days of poor mental health y2i is adopted, with area specific precisions /2i depending on BRFSS subject totals N 2i in county i cumulated over 2003–2005. Thus /2i ¼ N 2i s20 þ s21 , where s20 and s21 are unknowns, so that counties with more subjects have higher precisions. This representation also allows for a high rate of missingness in this response (56% of counties have N 2i ¼ 0Þ, when /2i ¼ s21 . So the model for average days with poor mental health has the form

y2i v Nðl2i ; 1=/2i Þ;

ð12:1Þ

l2i ¼ a2 þ b21 F 1i þ b22 F 2i þ b23 F 3i þ b24 F 4i þ u21 X 1i þ u22 X 2i :

ð12:2Þ

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Table 2 Socioeconomic indicators (Zpi) used to measure latent area constructs, with expected sign of loading kpq. Index (p)

Indicator name

Construct (q)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

Poverty rate 2004 %Professional and managerial occupations, 2000 %Adults 25+, college graduates, 2000 %Workforce unemployed, 2004 %One person households, 2000 %Households that are one person household renters, 2000 %Family households, 2000 Residential turnover (change of address) in 1999–2000 Residential turnover (change of address) in 1995–2000 %Rural population in county, 2000 %Employed workers in agriculture, forestry, fishing (2000) Population density per square mile 2000 Housing density per square mile 2000 Response rate, 2000 Census Associational density per capita (2004, CBP) Tax-exempt non-profit organizations per capita (2004, NCCS) Turnout rates, 2004 Election

Deprivation

To allow comparison of the impacts of latent construct (over both constructs themselves q, and outcomes jÞ, standardized versions of bjq in (11.2) and (12.2) are used. The ðsÞ standardized form of bjq , denoted bjq , adjusts for the stan0:5 dard deviation rq ¼ ½v arðF qi Þ of construct q, and for variation in the health regression outcomes, namely in relative suicide risks, Sl1 ¼ ½v arðlog l1i Þ0:5 , and in average predicted days of poor mental health, Sl2 ¼ ½v arðl2i Þ0:5 , namely ðsÞ

bjq ¼ bjq rq =Slj :

ð13:1Þ

ð13:2Þ

The effect indicator measurement model is

zpi jF i v Nðxp þ kp;cp F cp i ; 1=#p Þ;

p ¼ 1; . . . ; P

ð14Þ

with cp 2 ð1; . . . ; 4Þ, and the P = 17 indicators as in Table 2. It remains to specify the models for missingness and for the latent constructs. For the poor mental health outcome, where missing data is over 50%, a formal missingness model is adopted to assess whether missingness is informative. Setting M i ¼ 1 for y2i missing, M i ¼ 0 for data present, and wi ¼ PrðM i ¼ 1Þ, the shared parameter model for missingness in y2 is then

logitðwi Þ ¼ t0 þ t1 F 1i þ t2 F 2i þ    þ t4 F 4i þ v1 Di ;

Rurality

Social captial

+ +  + + + + – – + + + +

PðF ðaÞ j F Þ / j F jI=2

"

 exp 0:5

X



ðaÞ wik F i



ðaÞ Fk

0

F



ðaÞ Fi



ðaÞ Fk

#  ;

i;k

ð16Þ where the wik are based on contiguity ðwik ¼ 1 if counties i and k are adjacent, wik ¼ 0 otherwise). Correlations between antecdent constructs are obtained as qqr ¼ RFqr =ðRFqq RFrr Þ0:5 where RF ¼  1 F . 3.4. Formative regression model

Similarly,the standardized form of ujk is 0:5 uðsÞ =Slj : jk ¼ ujk ½v arðX k Þ

Social fragmentation

+   +

ð15Þ

where Di consists of one predictor, the log of the total county population aged 18 and over in 2004. This reflects the fact that missing data on days with poor mental health is more likely for small counties. Defining Sw ¼ ½v ar ðlogitðwi Þ0:5 and SD ¼ ½v arðlogðDi Þ0:5 , standardized effects ðsÞ ðsÞ in (15) are obtained as tq ¼ tq rq =Sw and v1 ¼ vSD =Sw . For the antecedent latent constructs, F 1 ; F 2 ; F 3 (respectively, deprivation, fragmentation and rurality), a trivariate conditional autoregressive prior is adopted. With within ðaÞ area precision matrix  F of dimension 3; F i ¼ ðF 1i ; F 2i ; ðaÞ ðaÞ 0 ðaÞ F 3i Þ , and F ¼ ðF 1 ; . . . ; F I Þ, one has

The remaining construct F 4 (social capital) is dependent both on earlier constructs and on known predictors W i . The latter are potential causes (or formative indicators) of varying social capital. Following Rupasingha et al. (2006) and Rupasingha and Chilton (2009), the W variables are taken as a measure of income inequality ðW 1i Þ, an ethnic fractionalization index ðW 2i Þ, and a measure of religious adherence ðW 3i Þ, from the American Religious Data Archive. Income inequality (expected to reduce social capital) is measured by the ratio of mean county household incomes (2000 Census) to median income. The fractionalP ization index is ð1  m E2im Þ (where Eim is the population proportion of ethnic group m in county iÞ. This is a measure of ethnic divisions and is expected to reduce social capital. However, higher religious adherence is expected to be a positive source of social capital. So under binary adjacency, the model for the dependent latent construct, social capital, F 4 , assumes conditional means and precisions EðF 4i jF ½4i Þ and PrecðF 4i jF ½4i Þ as in (9.1) and (9.2) above, with @ i denoting the neighborhood of county i. A neighborhood is defined as all counties geographically contiguous to county i, except for Hawaii where it consists of the closest island. Assuming only linear ðaÞ effects of F i ¼ ðF 1i ; F 2i ; F 3i Þ, the regression terms gi in (9.1) are

gi ¼ d1 F 1i þ d2 F 2i þ d3 F 3i þ c1 W 1i þ c2 W 2i þ c3 W 3i :

ð17Þ

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The inclusion of the constructs of deprivation, fragmentation and rurality to explain varying social capital parallels the analysis of Rupasingha et al., 2006, Section 5. They use observed single variable proxies for area deprivation (county population with over 12 years schooling), community attachment (rate of residential mobility) and rurality (an urban–rural binary variable). However, using a single indicator (from a potentially broader set) to represent a latent construct may ‘‘reduce strength of the intended signal and thus under-estimate its association with the outcome of interest’’ (Shishehbor and Litaker, 2006), or as a possibly poor proxy for the construct, provide ‘‘incorrect estimates of neighborhood health effects’’ (Diez-Roux and Mair, 2010). When the regression (17) is linear in the antecedent constructs, one may straightforwardly combine (17) with (11.2) and (12.2), to obtain total effects of antecedent factors F ðaÞ (namely deprivation, fragmentation and rurality) on the mental health outcomes j in the structural regressions (11.2) and (12.2). The total effect includes both direct effects on the outcome j, and indirect effects via the dependent construct(s) F ðdÞ , here social capital. For the path scheme in the present case study, these total effects have (for H ¼ 1Þ the form

Cjq ¼ bjq þ bjQ dq ;

q ¼ 1; . . . ; Q  H:

ð18Þ

To allow comparison of impacts of the different causes in (17), and of total effects of F ðaÞ in (11.2) nad (12.2), standardized versions of dq ; cr and Cjq can be obtained. Setting Sg ¼ ½v arðgi Þ0:5 , and SW r ¼ ½v arðW ri Þ0:5 , one has

dqðsÞ ¼ dq rq =Sg ; ðsÞ r ¼ r SW r =Sg ; ðsÞ jq q =Slj : jq ¼

ð19Þ

c

c

ð20Þ

C

C r

ð21Þ

One drawback of the multivariate normal spatial prior (5) is that is implicitly confined to linear relationships between constructs. However, one might on substantive grounds expect nonlinear effects of the antecedent constructs (or of the W variables) on the dependent constructs, or interactive effects between them in the regression. For example, a single linear effect of rurality on social capital may not fully reflect adverse effects on social capital of rural poverty (Monnat and Pickett, 2010), and the concentration of poverty and other forms of disadvantage (especially among non-white race groups) in rural areas (Probst et al., 2004). To reflect this, the linear form in (17) can be modified to include an interaction between F 3i and F 1i . In the case study analysis below, this is specified via a step function in F 1 , namely

gi ¼ d1 F 1i þ d2 F 2i þ d3 F 3i þ d4 F 3i IðF 1i > 0Þ þ c1 W 1i þ c2 W 2i þ c3 W 3i ;

ð22Þ

so that d3 þ d4 represents the effect on social capital of living in a higher poverty rural area. Assume the impact of a particular antecedent construct F c on the dependent construct(s) is modeled nonlinearly, as are the impacts of F 1 and F 3 in (22). Then one may define that component Dci in gi involving the effect of F c . For example, in (22), one has

D1i ¼ ½d1 F 1i þ d4 F 3i IðF 1i > 0Þ=F 1i ¼ d1 þ d4 F 3i IðF 1i > 0Þ=F 1i

ð23Þ

and

D3i ¼ d3 þ d4 IðF 1i > 0Þ:

ð24Þ

If the effect of (say) F 1 was modeled by a quadratic with gi ¼ d1 F 1i þ d2 F 21i þ d3 F 2i þ d4 F 3i þ c1 W 1i þ c2 W 2i þ c3 W 3i , then D1i ¼ d1 þ d2 F 1i . Then one may obtain an estimated total effect Cjc of construct c on outcome j in the structural equations by averaging over individual areas, namely

Cjc ¼ bjc þ bjQ

1X Dci ; I i

ð25Þ ðsÞ

and Cjc , or its standardized version Cjc ¼ Cjc rc =Slj , may be monitored over MCMC iterations. 3.5. Model fitting and findings Two models are compared, the first (model (1)) involving a linear formative indicator regression for social capital as in (17), the other (model (2)) involving a nonlinear formative indicator regression with F 1  F 3 interaction, as in (22). The structural, effect indicator and missingness models are the same in both specifications. Identifiability in both models is achieved by setting k11 ¼ k5;2 ¼ k10;3 ¼ k14;4 ¼ 1 in (4). Prior assumptions about other unknowns are set out in Appendix 1. Inferences are based on the second halves of two chain runs of 20,000 iterations from dispersed initial values, with convergence achieved before iteration 10,000 using Brooks–Gelman criteria (Brooks and Gelman, 1998). The log of the pseudo marginal likelihood (PsML) is used to measure fit, based on Monte Carlo estimates of the conditional predictive ordinate (CPO), the predictive density based on all data except the ith observation – see Christensen et al. (2010, Section 4.9.2) for a justification of this fit measure in Bayesian applications. The CPO provides a form of cross-validation: each observation is excluded in turn, while the remaining data becomes the test data on which the model is fitted. In practice the CPO may be estimated without actually omitting cases as the harmonic mean of the likelihoods for each observation (Aslanidou et al., 1998). Here the PsML is evaluated for the J = 2 outcome regressions, and each of the P = 17 effect indicator measurement regressions; the PsML values for the effect indicators are then aggregated to the level of the Q = 4 constructs. Table 3 accordingly compares the PsML for the two models. It can be seen that introducing interaction in the formative indicator model leads to improved PsML values for the overall measurement model, though PsML values for the structural outcomes are similar between models (1) and (2). Table 4 shows parameter estimates under model (2). These are the standardized parameter estimates ðsÞ ðsÞ ðbjq ; ujk Þ for the structural model (11) and (12), loadings kpq for the measurement models, standardized coefficients for the missingness model (15) for poor mental health, and ðsÞ the standardized coefficients ðdðsÞ q ; cr Þ in the formative indicator model for social capital, as in Eq. (22). Also shown

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P. Congdon / Spatial and Spatio-temporal Epidemiology 2 (2011) 103–116

Table 3 Comparative model fit, log pseudo marginal likelihood.

Health outcomes total Suicide Poor mental health Missingness in health outcomes (poor mental health) Effect indicators All indicators Deprivation indicators Fragmentation indicators Rurality indicators Social capital indicators

ðsÞ

Model 1: Linear formative regression

Model 2: Formative regression including interaction

13129.5 10078.0 3051.6 1448.0

13122.3 10071.8 3050.5 1448.8

60542.7 14238.3 25177.6

60986.7 14246.1 25181.4

8527.4 12599.4

8976.8 12582.4

are total effects Cjc for c ¼ 1; . . . ; 3 and j ¼ 1; . . . ; 2 implied by (22) and (11) and (12). It is apparent that the first three constructs – deprivation, fragmentation and rurality – have positive impacts ðsÞ ðsÞ ðsÞ on suicide, as represented by b11 ; b12 and b13 . These impacts are significant, in the sense that the 95% interval is entirely positive. The positive impact of fragmentation on suicide is similar to that reported in UK and European studies, though previous US studies have only studied the impact of particular fragmentation variables such as the proportion of people living alone (e.g. Stack, 2000). The impact of rurality on suicide is consistent with differentials analyzed by Singh and Siahpush (2002) and Eberhardt and Pamuk (2004), and while assessing causation is problematic (Gessert, 2003), access to lethal methods of suicide (Miller et al., 2007; Hirsch, 2006) and elevated farmer suicide rates are relevant. Positive impacts on suicide are also found for the X variables in (2), namely county proportions white non-Hispanic and native-American. This accords with evidence of elevated suicide in counties with large native American minorities (e.g. Wissow et al., 2001). However, while the impact of social capital on suicide is negative, it is not significant. By contrast, for the poor mental health outcome, the ðsÞ impact of social capital, b24 , is significantly negative, and among the most important risk factors in terms of the absolute effect size. While a protective effect for community social capital has been reported for general health (Subramanian et al., 2002), similar findings specific to mental health in particular are limited. Also in contrast to its effect on suicide, rurality significantly reduces average days of poor mental health, consistent with review evidence relating to general mental health as opposed to particular conditions (Peen et al., 2010; Dohrenwend and Dohrenwend, 1974; Caracci, 2008; Berry, 2007; Rohrer et al., 2005). However, area deprivation is a positive risk factor for poor mental health. Positive impacts on levels of poor mental health are also found for the county ethnic variables, namely proportions white non-Hispanic and native-American. The estimates of the loadings kpq relating item indicators to constructs follow the hypothesized pattern in

Table 2. Whereas European studies report positive correlations between deprivation and fragmentation, possibly because both tend to be elevated in European urban settings (Congdon, 2004), the US-wide correlation between these two constructs is negative (at 0.61). Poverty and deprivation in the US are higher in rural areas (Holt, 2007; Weber, 2007), as confirmed by a correlation of 0.56 between deprivation and rurality constructs, whereas fragmentation is highest in urban areas (there is a correlation of 0.64 between fragmentation and rurality). The results involving the nonlinear formative indicator model (22) for social capital, are summarized in parameters dðsÞ and cðsÞ in Table 4. The impact of fragmentation – which essentially summarizes lower levels of family household orientation and lower residential stability in certain areas – is also a negative influence on social capital (cf. Rupasingha et al., 2006; Onyx and Bullen, 2000; Hofferth and Iceland, 1998). Table 4 also shows a significantly negative main effect d1 for deprivation. Ethnic fractionalization also significantly reduces social capital, while religious adherence increases it. However, the effect of income inequality is not significant. Table 4 shows a significant interaction coefficient d4 in (22), confirming a negative impact on social capital of rural poverty. By contrast, the rurality coefficient d3 under the linear formative model (17) in model 1 is found to be non-significant, with posterior mean 0.026, and 95% interval (0.11,0.06). However, as argued above, a single linear effect of rurality on social capital may not capture adverse effects on social capital of rural poverty, and other forms of ethnic economic disadvantage (Probst et al., 2004). The missingness model coefficients show significant impacts of three constructs (especially deprivation) on the chance of missing poor mental health averages, even after allowing for lower response in counties with smaller populations. At state level, there is a 0.58 correlation between average deprivation and the proportion of counties with missing data. Figs. 1–4 contain county maps of standardized latent construct scores. Highest average deprivation scores (Fig. 1) are in the South Eastern US, for example, Kentucky, Tennessee, and West Virginia, while highest rurality scores are in the mountain and Pacific states (reflecting high rurality averages in states such as Alaska, Montana and Wyoming). Modeling of geographic variation in social capital has so far only been attempted in a few studies, and so Fig. 4, containing social capital scores, adds to a limited evidence base. The map shows similarities with that presented by Rupasingha et al., 2006 (Fig. 2), with lower social capital averages in the south eastern US. The social capital state average scores correlate 0.81 with the ‘‘comprehensive social capital index’’ of Putnam (2000) (over 48 states), though the latter is based on aggregation of individual survey responses. Figs. 5 and 6 show posterior mean relative suicide risks l1i and average days of poor mental health l2i . The higher suicide risk in more deeply rural areas (e.g. in the Mountain and Pacific divisions) is apparent, as in official maps of actual age adjusted rates (CDC, 2009). The highest levels of poor mental health are in the south eastern US.

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P. Congdon / Spatial and Spatio-temporal Epidemiology 2 (2011) 103–116 Table 4 Parameter estimates, model 2, posterior means and 95% credible intervals.

Structural model Impacts on suicide

Risk factor

Parameter

Deprivation

b11

Fragmentation Rurality Social capital White non-hispanic Native American Impacts on poor mental health

Deprivation Fragmentation Rurality Social capital White non-hispanic Native American

Total effects of deprivation, fragmentation and rurality

Deprivation on suicide Fragmentation on suicide Rurality on suicide Deprivation on poor mental health Fragmentation on poor mental health Rurality on poor mental health

Impacts on probability of missing poor mental health (y2)

Deprivation Fragmentation Rurality Social capital

Measurement model (A) Multiple effect indicators for all constructs Loadings on F1 (deprivation)

Loadings on F2 (fragmentation)

Loadings on F3 (rurality)

Loadings on F4 (social capital)

Correlations between antecedent constructs

(B) Multiple causes (formative indicators) for social capital Causes of social captial

Mean

2.5%

97.5%

ðsÞ

0.076

0.059

0.092

ðsÞ b12 ðsÞ b13 ðsÞ b14 ðsÞ 11 ðsÞ 12

0.062

0.049

0.074

0.049

0.037

0.061

0.006

0.023

0.010

u u

0.054

0.044

0.064

0.083

0.073

0.093

ðsÞ b21 ðsÞ b22 ðsÞ b23 bs24 ðsÞ 21 ðsÞ 22

0.636

0.535

0.717

0.004

0.063

0.059

0.533

0.595

0.467

0.516 0.404

0.621 0.371

0.427 0.435

0.161

0.104

0.221

CðsÞ 11 CðsÞ 12 CðsÞ 13 CðsÞ 21 CðsÞ 22 CðsÞ 23

0.085

0.065

0.101

0.063

0.053

0.074

0.050

0.038

0.061

1.431

0.503

2.352

0.141

0.050

0.196

0.478

0.555

0.399

v ðsÞ 1 v ðsÞ 2 v ðsÞ 3 v ðsÞ 4

0.000

0.000

0.000

0.000

0.000

0.000

0.000

0.000

0.000

0.000

0.000

0.000

ðsÞ 1

0.000

0.000

0.000

u u

Population size

v

Indicator

Parameter

Mean

2.5%

97.5%

Poverty rate Professional–managerial College graduates Unemployment

k1,1 k2,1 k3,1 k4,1

1 0.81 1.75 0.61

-0.84 1.80 0.56

0.78 1.69 0.65

1 person households 1 person household renters Family households 1 year residential turnover 5 year residential turnover

k5,2 k6,2 k7,2 k8,2 k9,2

1 2.15 0.57 4.95 3.52

1.91  0.64 4.71 3.34

2.40 0.51 5.25 3.73

Rural population Agriculture etc. Population density Housing density

k10,3 k11,3 k12,3 k13,3

1 1.80 4.16 3.83

1.57 4.87 4.48

2.11 3.66 3.37

Census response Associational density Non-profit organizations Election turnout

k14,4 k15,4 k16,4 k17,4

1 1.71 2.41 0.40

1.60 2.26 0.37

1.84 2.58 0.42

Deprivation–fragmentation Deprivation–rurality Fragmentation–rurality

q12 q13 q23

0.61 0.56 0.64

0.64 0.54 0.66

0.58 0.59 0.61

ðsÞ

0.908

0.942

0.871

ðsÞ

0.364

0.416

0.282

ðsÞ

0.059

0.032

0.159

ðsÞ

0.130

Deprivation

d1

Fragmentation

d2

Rurality

d3

Rurality  high deprivation

d4

0.191

0.264

Income inequality

cðsÞ 1 cðsÞ 2 cðsÞ 3

0.021

-0.018

0.070

0.242

0.319

0.181

0.414

0.353

0.468

Ethnic fractionalisation Religious adherence

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P. Congdon / Spatial and Spatio-temporal Epidemiology 2 (2011) 103–116

Fig. 1. Deprivation, standardized scores.

Fig. 2. Fragmentation, standardized scores.

4. Discussion The strategy chosen for measurement of neighborhood influences on health is important, since latent area constructs relevant to ecological health variations may not be fully represented by univariate proxies (e.g. population density as a single variable proxy for rurality, or neighborhood poverty rate as a single measure of area socio-economic status) (Diez-Roux and Mair, 2010). Spatial structure in latent

constructs is also important, as evidenced by spatial clustering of poverty in the US (Holt, 2007). Standard multivariate techniques such as principal component analysis to derive latent constructs, such as community social capital or area deprivation, do take account of multiple effect indicators. However, such techniques do not take account of spatial correlation in the construct, or of the application context (e.g. are the constructs being used to explain health or labor market outcomes).

P. Congdon / Spatial and Spatio-temporal Epidemiology 2 (2011) 103–116

113

Fig. 3. Rurality, standardized scores.

Fig. 4. social capital, standardized scores.

As Rehkopf and Buka (2006) note, small area studies of neighborhood health effects are subject to other methodological problems, such as instability of rate estimates if conventional demographic techniques are used to measure the dependent variable for rare outcomes (Riggan et al., 1991; Langford, 1994). Hence multivariate linear regression of (say) county suicide death rates on predictors may be problematic (cf. Faria et al., 2006).

Distinct from aspatial analyses, the present study has adopted a fully specified statistical model, that pools strength spatially using Bayesian random effects methods, recognizes the application setting in the overall model, and combines both effect and formative indicators in deriving the latent constructs. The model is used to study the relationships between poor mental health and suicide mortality, on the one hand, and four latent constructs on

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P. Congdon / Spatial and Spatio-temporal Epidemiology 2 (2011) 103–116

Fig. 5. Suicide mortality relative risks.

Fig. 6. Poor mental health.

the other (deprivation, fragmentation, rurality and social capital). Existing spatial models for latent area constructs focus on univariate constructs. The application here involves plural latent risks, and a simple scheme would take these constructs to follow a multivariate linear spatial form, such as the multivariate CAR, with the constructs simply measured by a set of multiple effect indicators. However, particular latent constructs may also be influenced by known forma-

tive variables, as well as by other constructs, and nonlinear effects of particular constructs or formative variables on other constructs may exist. This paper has therefore allowed path sequences between constructs, and in particular area social capital is not only measured by effect indicators, but influenced as causes both by other constructs, and by observed county variables, such as ethnic fractionalization (Rupasingha et al., 2006). This scheme has potential application to other settings in spatial epidemiology.

P. Congdon / Spatial and Spatio-temporal Epidemiology 2 (2011) 103–116

5. Conclusion This paper has considered a spatial structural model relating suicide and poor mental health in US counties to unobserved ecological risk factors: deprivation, social capital, social fragmentation and rurality. Although aspects of the methodology are novel, such as the full recognition of spatial structuring in the latent constructs, and the use of a formative model for social capital, the findings are consistent with existing studies. Thus deprivation, fragmentation and rurality have positive impacts on suicide, and suicide is also elevated in areas with above average proportions of white non-Hispanic and native-American residents. Previous US studies of area suicide have only considered particular facets of fragmentation, so the present paper’s findings generalize UK and Irish studies showing a positive effect of fragmentation on suicide. For poor mental health, community social capital is found to be a protective factor, and rurality also significantly reduces poor mental health. By contrast, area deprivation is a positive risk factor for poor mental health. Social capital itself is found to be negatively related to deprivation, and to fragmentation, with a further interactive effect apparent between deprivation and rurality. Lower social capital scores are found in the south eastern US, and the geographic pattern of social capital is broadly similar to that described in Putnam (2000) and Rupasingha et al. (2006). Appendix A. Specification of prior densities The conditional form of the multivariate conditional autoregressive density for the antecedent constructs ðaÞ F i ¼ ðF 1i ; . . . ; F 3i Þ is ðaÞ

ðaÞ

1 F i jF ½i  N3 ðF i ; L1 i F Þ

where F i ¼ ðF 1i ; F 2i ; F 3i Þ is a vector of average construct scores in the neighborhood of county i, and Li is the number of counties in that neighborhood.  F is a precision matrix, assigned a Wishart prior with identity scale matrix and Q  H ¼ 3 degrees of freedom,

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