Generalized spatial structural equation models

Biostatistics (2005), 6, 4, pp. 539–557 doi:10.1093/biostatistics/kxi026 Advance Access publication on April 20, 2005

Generalized spatial structural equation models XUAN LIU∗ , MELANIE M. WALL, JAMES S. HODGES Division of Biostatistics, School of Public Health, University of Minnesota, MMC 303, Minneapolis, MN 55455, USA [email protected]

S UMMARY It is common in public health research to have high-dimensional, multivariate, spatially referenced data representing summaries of geographic regions. Often, it is desirable to examine relationships among these variables both within and across regions. An existing modeling technique called spatial factor analysis has been used and assumes that a common spatial factor underlies all the variables and causes them to be related to one another. An extension of this technique considers that there may be more than one underlying factor, and that relationships among the underlying latent variables are of primary interest. However, due to the complicated nature of the covariance structure of this type of data, existing methods are not satisfactory. We thus propose a generalized spatial structural equation model. In the first level of the model, we assume that the observed variables are related to particular underlying factors. In the second level of the model, we use the structural equation method to model the relationship among the underlying factors and use parametric spatial distributions on the covariance structure of the underlying factors. We apply the model to county-level cancer mortality and census summary data for Minnesota, including socioeconomic status and access to public utilities. Keywords: Bayesian; Factor analysis; Latent; Linear model of coregionalization (LMC); Markov chain Monte Carlo (MCMC); Spatial; Structural equation models (SEM).

1. I NTRODUCTION Structural equation modeling is a widely used statistical modeling method in social and behavioral sciences that escalated in use in the early 1980s with the improvement in computing and the introduction of the LISREL software (J¨oreskog, 1973; J¨oreskog and S¨orbom, 1996). Structural equation modeling is also becoming more widely used in public health, biological, and medical research (e.g. Bentler and Stein, 1992; Pugesek et al., 2003). Most generally, structural equation modeling combines the ideas of factor analysis with regression. That is, a researcher may be interested in possibly several regression-type relationships between variables, but some or all of the variables of interest cannot be measured directly (i.e. they are latent). So instead of measuring the latent variable directly, a set of observable variables is measured where each such observable is assumed to be an imperfect measure of the underlying latent variable of interest. Then, a structural equation model (SEM) assumes a factor analysis type of model to ‘measure’ the latent variables ∗ To whom correspondence should be addressed.

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via the multiple imperfect measures, while simultaneously assuming a regression type of model for the relationship among the latent variables. Traditionally, estimation methods were developed for SEMs restricted to have only normally distributed observed variables, linear relations among the latent variables, and independently observed individuals. Developments to broaden the SEM have been made in several directions and include developments made for factor analysis (as it is a submodel of SEM). For example, methods have been developed for allowing observed variables of mixed types from an exponential family (Muth´en, 1984; Sammel et al., 1997; Moustaki and Knott, 2000), for including nonlinear relationships among latent variables (Wall and Amemiya, 2000, 2001; Lee and Zhu, 2002; Lee and Song, 2003), and for considering clustered individuals, i.e. ‘multilevel’ sampling designs (McDonald and Goldstein, 1989; Muth´en, 1989; Dunson, 2000; Lee and Shi, 2001). Recently, Goldstein and Browne (2002) have proposed a general multilevel factor model for continuous or binary responses with fixed effects in the measurement model using a Bayesian approach implemented in MLwiN (Rasbash et al., 2000). This model also allows for dependency between the factors and allows the factors to be functions of covariates. This model is commonly referred to as the ‘multiple indicators multiple causes’ model (J¨oreskog and Goldeberger, 1975). Another possible extension to the SEM, and the focus of the current paper, should be used on geographically referenced or spatially sampled data. One group of methods for analyzing multivariate spatial data mainly uses principal component methods on the variance–covariance matrix of the data to generate components of different spatial scales (e.g. Switzer and Green, 1984; Wackernagel, 1988; Grunsky and Agterberg, 1992). These methods are descriptive in the sense that they perform direct operations on the data instead of making explicit statistical modeling assumptions allowing uncertainty. In the rest of this paper, we concentrate on statistical modeling methods and thus introduce more recent statistical literatures for modeling multivariate spatially sampled data. Christensen and Amemiya (2002, 2003) developed exploratory factor analysis frameworks for multivariate spatial data aiming to explore the relationship between the underlying factors and the observed variables, and applied their methods on agricultural data. Population-based data sources, such as the U.S. Census, vital records (birth and death certificates), and the Behavioral Risk Factor Surveillance Survey are commonly summarized at some geographic level (e.g. state, county, and census tract). This results in a large number of observed variables available at each geographic unit and provides the possibility of performing ecological-type regressions for investigating influences of risk factors on outcomes. Recently, Wang and Wall (2003) considered data of this type and developed a common spatial factor model for measuring a latent spatial factor underlying multiple spatially referenced observed variables (lung, pancreas, and esophageal cancer mortality by county). Using Wang and Wall’s (2003) spatial factor model, Hogan and Tchernis (2004) modeled the latent spatial factor ‘material deprivation’ via a set of four variables reported at the census tract level. Both these papers analyzed multivariate spatial data and described the correlations within location and across location using a single spatially distributed latent factor. A natural extension is then to consider regression-type relationships between multiple underlying spatial factors. For example, it could be of interest to model the relationship between the spatial factor ‘underlying cancer mortality’ and the spatial factor ‘material deprivation’, while also adjusting for other variables. Thus, this paper proposes a generalized spatial structural equation model (GSSEM) that incorporates the common spatial factor model of Wang and Wall (2003) as a means to ‘measure’ the latent variables of interest, and simultaneously models the regression relation between the latent variables while taking account of spatial correlation. Section 2 reviews the generalized common spatial factor model (Wang and Wall, 2003) and its applications to cancer mortality data and ‘material deprivation’ (Hogan and Tchernis, 2004). Section 3 introduces the GSSEM and proposes the use of a coregionalized spatial structure (see e.g. Banerjee et al., 2004) for modeling the multivariate underlying factors. A fully Bayesian framework is used for model fitting and is described in Section 4. Section 5 demonstrates the application of the GSSEM to data linking

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socioeconomic and public utility access spatial factors to a common spatial factor underlying cancer mortality in Minnesota counties. The discussion is left for Section 6. 2. T HE GENERALIZED SINGLE COMMON SPATIAL FACTOR MODEL AND ITS APPLICATION Wang and Wall (2003) proposed a general spatial factor analysis framework that can handle data from any exponential family distributions. A common factor is used to model the correlation among the variables, and a spatial distribution is used for the prior distribution of the common factor to account for the acrosslocation correlations. Alternatively, this spatial distribution can be understood as a model for the common factor, analogous to the familiar i.i.d. normal model for the common factor. The formulation of the generalized common spatial factor model can be summarized as follows. Let Z i j be the jth ( j = 1, . . . , p) random variable observed at location si (i = 1, . . . , n). Then, Z i j are assumed to be from an exponential family with distribution F, and are independent conditional on their mean θi j and variance σ j2 , i.e. ind

Z i j |θi j , σ j ∼ F(θi j , σ j2 ) (i = 1, . . . , n, j = 1, . . . , p),

(2.1)

with link function g(·), so that g(θi j ) = µ j + λ j f (si )

(i = 1, . . . , n, j = 1, . . . , p),

(2.2)

where µ j is the intercept, an unknown parameter to be estimated, and λ j ( j = 1, . . . , p) is also an unknown parameter called the factor loading. The f (si ) is introduced to represent the common spatial factor at location si . In particular, f (si ) represents a univariate spatial process underlying all p of the observed variables. Let f = ( f (s1 ) , . . . , f (sn ) ) ; then f is assumed to have a multivariate normal distribution, α )), f ∼ MVN(µf 1n , C(α α ) is the covariance matrix that captures the spatial structure of the where µf is the mean of f and C(α common factor f, with α representing the vector of parameters in the covariance structure. Modeling Minnesota cancer mortality data summarized by county, Wang and Wall (2003) used a Poisson distribution for F in (2.1) and a log link function for g(·) in (2.2), i.e. ind

Z i j |θi j ∼ Poi(θi j ) (i = 1, . . . , n, j = 1, . . . , p), and

log(θi j ) = log(E i j ) + λ j f (si )

(i = 1, . . . , n, j = 1, . . . , p),

where n = 87 represents the 87 counties of Minnesota; p = 4 is the number of observed variables, including county-level death counts due to cancer of the lung, pancreas, esophagus, and stomach; log(E i j ) is a known constant representing the log of the population standardized expected number of deaths in county si for cancer j. A conditional autoregressive (CAR) structure was used for the spatial covariance structure for f. They fitted the generalized common factor model using a full Bayesian approach, obtaining posterior summaries for all the unknown constants and the underlying factor. Estimates for factor loadings λ1 , λ2 , and λ3 were highly significant and λ4 was not significant, which implied that only the first three cancers were associated with the common underlying factor, and the fourth cancer, stomach cancer, was not strongly associated with the factor. This result was consistent with their exploratory results. They produced a map of E( f (si ) |{Z i j }), the factor scores, for Minnesota counties, and found a spatial pattern of the underlying factor, in which north-central Minnesota had higher scores than other areas. The map of the factor scores is of interest in that it describes how the underlying cancer mortality risk factor is distributed across Minnesota counties. Moreover, by examining the posterior distribution of the factor scores, we can

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evaluate the factor score estimates taking account of the posterior uncertainty and thus compare or test differences among counties to identify individual counties that have extreme scores. Hogan and Tchernis (2004) used the generalized common spatial factor model (2.1) and (2.2) to model census data from Rhode Island. This model assumed a normal distribution for F and identity link function, ind

Z i j |θi j , σ j ∼ N (θi j , σ j2 ) (i = 1, . . . , n, j = 1, . . . , p), and

θi j = αi j + λ j f (si )

(i = 1, . . . , n, j = 1, . . . , p).

α ) for the underlying factor, inHogan and Tchernis proposed several parametric covariance structures C(α cluding a spatially independent structure that ignores the spatial dependence among the factors, a marginal (geostatistical) model which allows the covariance between two f (si ) to depend on the Euclidean distance between the two si , a CAR model which defines the spatial covariance structure of the underlying factor based on neighborhood adjacencies, and finally an interesting combination of geostatistical and areal spatial analysis methods where a CAR model is used with a distance-dependent neighbor structure. They used a posterior predictive criterion to select the basic CAR model as the final model. Like Wang and Wall, they visualized the underlying factor using mapping tools, which aided interpretation of their results. 3. G ENERALIZED SPATIAL STRUCTURAL EQUATION MODELS 3.1

Motivations

The generalized spatial factor model introduced in the previous section is a flexible tool for modeling and interpreting a single underlying construct in multivariate spatial data, but one might be interested in the relationship among several underlying constructs measured by different groups of variables. For example, suppose we find that for Minnesota counties, several cancer mortality rates share a common spatial factor, and we hypothesize and find that several census variables related to income and education share a common spatial factor. Then, we may be interested in the research question of whether socioeconomic status (SES) is associated with the cancer mortality risk for Minnesota counties. SEMs offer a unified method for handling this, combining factor analysis and regression analysis. Without accounting for any spatial structure, Wall and Li (2003) used the SEM method to estimate how SES and public utilities accessibility are related to death rate from respiratory disease in Minnesota counties, where the response variable is the log age-adjusted respiratory disease death rate for each of the 87 Minnesota counties, and SES and public utilities accessibility are two latent variables measured by three and two observed variables, respectively. Their study compared a multiple linear regression using all five indicator variables as predictors to an SEM, and found that the SEM fit better and offered more interpretable results. One weakness of Wall and Li’s model was that it completely ignored the spatial structure in the data. They used the traditional SEM, which assumes that the underlying factors are uncorrelated across counties, which might be unrealistic in this kind of geo-referenced data. It may be reasonable to assume that counties are more similar when they are closer or adjacent to each other. Thus, we are motivated to combine the idea of structural equation modeling with the common spatial factor model and propose an SEM that accounts for spatial correlations. 3.2 The model A traditional SEM (J¨oreskog, 1979; Bentler and Weeks, 1980; Bollen, 1989) is defined in two stages. The first stage specifies the relationship between the manifest variables and the underlying factors: Zi =  fi +  i

(i = 1, . . . , n),

where Zi is a p × 1 vector of manifest (observed) random variables measured on an individual i,  is a matrix of unknown parameters (factor loadings), fi is a Q × 1 vector of underlying factors for individual i,

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and  i is a p vector of normal random errors which are independent of fi and have a diagonal covariance T , fT ), where f and f partition f and have lengths Q and Q , matrix. In the second stage, let fiT = (f1i 1i 2i i 1 2 2i respectively, Q 1 + Q 2 = Q. Then, the structural equation can be defined as f1i = Bf1i +  f2i + δ i ,

(3.1)

where B (with zeroes along the diagonal) and  are unknown parameter matrices, and δ i is a Q 1 ×1 vector of normal random errors with diagonal covariance matrix. The vector f1i is often called the endogenous (dependent) underlying factors, and f2i is often called the exogenous (independent) underlying factors. We propose the GSSEM by combining the ideas of the traditional SEM and the generalized common spatial factor model reviewed in Section 2. At the first stage of the traditional SEM, the manifest variables Zi are assumed to be from normal distributions; we relax this and allow them to be from any exponential family distribution. We do not require all the manifest variables to be from the same distribution, i.e. different manifest variables may be from different exponential family distributions. The model assumes that there are Q latent factors underlying the p observed variables. We then assume that each underlying (1) (1) factor has its own pq unique observed variables measuring it, q = 1, . . . , Q. That is, Z i1 , . . . , Z i p1 (2)

(2)

are the p1 manifest variables for the first underlying factor, Z i1 , . . . , Z i p2 are the p2 manifest variables (Q)

(Q)

for the second underlying factor, etc., and Z i1 , . . . , Z i p Q are the p Q manifest variables for the Qth underlying factor such that p1 + p2 + · · · + p Q = p. Since we assume Q underlying factors, we define Q separate single generalized common factor models at the first stage of the GSSEM, i.e. for the qth model (corresponding to the qth underlying factor): (q)

(q)

(q) ind

Z i j |θi j , σ j (q)

(q)

(q)

∼ F(θi j , σ j2 ) (i = 1, . . . , n, j = 1, . . . , pq ),

(q) (q)

(q) (si )

g(θi j ) = β j xi j + λ j f q (q)

where λ j

(i = 1, . . . , n, j = 1, . . . , pq ),

is the factor loading for the qth factor on the jth manifest variable, and we have allowed

(q) (q) for observed covariates xi j in the model, similar to Sammel and Ryan (1996), with β j the vector (q)

of regression coefficients for xi j . Note that in the first stage of the model, we define Q separate single common factor models, i.e. we assume that the Q underlying factors are related only to their own manifest variables. This structure is usually called simple structure and it is often assumed in practical problems because it allows for clear interpretation of the underlying factors. In the second stage of the model, as in traditional SEM notation, we partition the vector of underlying T , fT ). Besides considering factors fi into two vectors of endogenous and exogenous factors, fiT = (f1i 2i spatial structure, we also consider two changes to the form of the traditional structural model (3.1). We fix B = 0, so endogenous variables cannot depend on one another, and we introduce the possibility of observed covariates as predictors, i.e. f1i = Hi +  f2i + δ i , where Hi is a Q 1 × k matrix of observed covariates and and  are matrices of unknown constants. We assume δ i and f2i to be two independent multivariate spatial processes with dimensions Q 1 and Q 2 , respectively. While most generally δ i can be spatially distributed, in many problems it may be reasonable to consider it independently distributed, as it represents error. This can be a model choice issue. Since Q 1
1 and Q 2
1, in general, we must specify flexible multivariate covariance structures for f2i and δ i , i.e. we should model not only between-variable covariance but also across-location covariance.

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A useful method for modeling multivariate spatial data is the linear model of coregionalization (LMC; see Grzebyk and Wackernagel, 1994; Wackernagel, 2003; Banerjee et al., 2004). LMC was originally developed to transform a multivariate geostatistical process into linear combinations of components of different spatial scales by using principal component analysis on the variance–covariance matrix. Gelfand et al. (2004) adopted the LMC in a full Bayesian modeling approach to model multivariate geostatistical data. In their approach, they assumed the spatial ranges of the components unknown. In this paper we use the LMC method on the underlying factors of multivariate areal data analogously to the approach of Gelfand et al., and the aim is to generate a rich and flexible class of variance–covariance structures to model the multivariate spatial process of the underlying factors. We consider a relatively simple example to illustrate this constructive modeling strategy. Suppose Y(si ) is a p × 1 vector, a realization of a p variate spatial process at location si (i = 1, . . . , n). Define Y(si ) = Av(si ), where A is a p × p full-rank matrix and the components of v(si ), v j (si ) ( j = 1, . . . , p), are spatial processes with unit variances and independent across j. Let C be the n × n covariance matrix of each spatial process v j ( j = 1, . . . , p). Let YT = (YT (s1 ), . . . , YT (sn )) and  be the covariance matrix of Y, then  = C ⊗ T, where T = AAT . Note that this specification is equivalent to a separable covariance specification as in Mardia and Goodall (1993), where T is interpreted as the within-site covariance matrix between variables and C is interpreted as the across-site covariance matrix. In practice, A is commonly assumed to be lower triangular or upper triangular without losing any generality, since for any positive definite matrix T, there is a unique lower or upper triangular matrix A such that T = AAT . The model just introduced can be extended to a more general LMC if we allow the p independent processes v1 , . . . , v p to have different distributions with covariance matrices C1 , . . . , C p , so =

p


Cj ⊗ Tj,

j=1

where T j = A j ATj are called coregionalization matrices, and A j is the jth column of A. Returning to the GSSEM, we may specify a multivariate spatial process for δ i and f2i using the LMC method: δ i = A1 wi f2i = A2 vi

(i = 1, . . . , n), (i = 1, . . . , n),

where A1 and A2 are upper triangular matrices of sizes Q 1 and Q 2 ; wi and vi are independent zero-mean α ) as the covariance matrix and unit-variance spatial processes of dimensions Q 1 and Q 2 . If we denote C(α of some random variable from process wi or vi , then, in general, a parametric spatial model could be α ). One choice is the geostatistical model where wi or vi is assumed to be isotropic considered for C(α α ) = (ci j ), and exponentially correlated, i.e. C(α ci j = ce e−|si −s j |/φ , where |si − s j | is the distance between locations si and s j , ce is the ‘partial sill’, which is the variance of the spatial process, and φ is the range parameter. Here α = (ce , φ) . Another possible model is the CAR

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model, which is commonly used for areal data. In the CAR model, α ) = τ 2 (In − ρW)−1 , C(α where ρ is referred to as the ‘spatial association’ parameter, τ 2 is a parameter that is proportional to the conditional variance of v (si ) |v (−si ) , where v (−si ) denotes the neighbors of v (si ) , and W = (wi j ) is a matrix containing the neighboring information, where wi j = 1 implies that region i is adjacent to region j and otherwise wi j = 0. In the measurement model part of the GSSEM, we assume that the manifest variables are independent conditioned on the underlying factors, and consequently assume that the spatial structure in the data is completely explained by the underlying factors. This enables us to model only the spatial distributions of the underlying factors instead of the much higher dimensional manifest variables. This makes the method much more computationally efficient, since the analysis of high-dimensional multivariate spatial data is computationally demanding. Including the structural equation in the model explains the correlations among the underlying factors in a meaningful way, which is sometimes the most interesting question in a study and can be hard to answer by ordinary joint modeling of the observed variables. Moreover, by incorporating fixed covariates, GSSEM is able to explain the influence of the covariates on the underlying factors of interest, which can also be useful. Using the LMC method to model the joint distribution of the exogenous underlying factors, GSSEM also allows flexibility in modeling both the within- and across-site correlations among the underlying factors. 4. F ITTING THE GSSEM Some statistical packages fit the classic SEM, e.g. LISREL, AMOS, and PROC CALIS in SAS. The estimation methods used in these packages include maximum likelihood and generalized least squares methods. These packages mostly assume that the observed variables have normal distributions, and allow no correlations among individuals. The MPLUS package is more general in that it allows for multinomial observed variables and nested designs which induce correlations (including spatial) between individuals within the same cluster. However, it is not flexible enough to allow general correlations between individuals or observations from any exponential family. When non-Gaussian exponential distributions are assumed for manifest variables and spatial correlations are introduced between locations, the likelihood function no longer has a simple form and is difficult to maximize. To overcome this problem, we use a fully Bayesian approach. 4.1

Prior specifications

In the measurement model stage of GSSEM, the unknown constants in the mean structure of the man(q) (q) ifest variables, i.e. the elements of β j ( j = 1, . . . , p) and λ j ( j = 1, . . . , pq , q = 1, . . . , Q), are each assigned a normal prior, N (µ0 , σ02 ), where µ0 is usually taken to be 0, and σ02 is chosen to be a large number to make the prior vague (e.g. we use σ02 = 104 in the later example). Note that we assign the same vague normal prior to these parameters for convenience, but this is not necessary. The scale (q) parameters σ j2 of the observed variables, if any, are assigned an inverse gamma prior with parameter 0.01 and 0.01, as it is the most commonly used conjugate prior in Bayesian spatial methods for unknown variance parameters. In the structural equation stage of the model, elements of and  are also assigned the same vague normal priors N (µ0 , σ02 ). For upper diagonal matrix A, to ensure that it is a Cholesky decomposition of a positive definite matrix during updates, we assign inverse gamma priors to the diagonal elements, and normal priors to the off-diagonal elements. Let α δ be the vector of parameters involved in the spatial covariance matrices for the Q 1 independent spatial processes w and α v be the vector of

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parameters involved in the spatial covariance matrices for the Q 2 independent processes v. Since the data in our example are areal data, we consider the CAR covariance model for w and v, and thus α δ and α v consist of precision and spatial association parameters. We assign an inverse gamma prior IG(0.01, 0.01) with mean 1 to the variance components of α δ and α v , and a uniform distribution in (0, 1) for the spatial association parameters. Let  be the vector that contains all the unknown parameters in the GSSEM, then ), which is the product of the priors of all the parameters specithe joint priors can be expressed as P( fied as above. Note that we chose the inverse gamma distribution as the prior for the variance parameters because it is conditionally conjugate and is currently used widely in Bayesian modeling for spatial data. As pointed out recently by some authors (e.g. Kelsall and Wakefield, 1999; Gelman, 2005), the inverse gamma might be informative and makes the posterior distribution sensitive to the prior distributions when very small values of the variance parameters are possible. In our example, we do a sensitivity analysis, where consider two alternative priors, i.e. IG(0.001, 0.001) and a proper flat prior Uniform(0, 5000) for the variance parameters. The posterior distributions hardly differ in all these cases.

4.2 Posterior distribution Let Z be the np × 1 vector of all the manifest variables, f1 be a Q 1 n × 1 vector of endogenous factors, and f2 be a Q 2 n × 1 vector of exogenous factors over n locations, and let x and H be the covariates in the measurement and structural models, respectively. Then, the joint posterior of all the unknown parameters and the factors is (1) (Q) , f1 , f2 |Z, x, H) ∝ P(Z|x, f1 , f2 , β (1) , . . . , β (Q) , σ 2 , . . . , σ 2 , λ(1) , . . . , λ(Q) ) P( ). × P(f1 |H, f2 , ,  , α δ )P(f2 |v, α v )P( Posterior computations used Markov chain Monte Carlo (MCMC; Gelman et al., 2004). MCMC gives a random sample from the marginal posterior distribution for each parameter and underlying factor, and thus posterior summaries such as the posterior means and quantiles of the parameters can be obtained. The credible intervals (e.g. the interval formed by the posterior 0.025 and 0.975 quantiles) of the parameters can be used to assess the significance of the parameters. Computationally, we only need to calculate the full conditionals of each parameter given all other parameters, which is usually not hard, and we use the Metropolis–Hasting algorithm when closed-form full conditionals are not available. This model can be implemented using the Winbugs software (Gilks et al., 1994; http://www.mrc-bsu.cam.ac.uk/bugs/winbugs/contents.shtml).

5. A PPLICATION OF GSSEM

TO

M INNESOTA CANCER DATA

In this study, we are mainly interested in how SES, accessibility of public utilities, and smoking prevalence of Minnesota counties are associated with a common spatial factor underlying lung, pancreas, and esophageal cancer mortality in these counties. This general interest implies a regression relationship among those variables, where the factor underlying the three cancers’ mortality rates is the dependent variable, and SES, utility accessibility, and smoking prevalence are independent variables. The variables SES, utility accessibility, and the common factor underlying the three cancers’ mortality risks are complex latent constructs rather than measurable quantities. Thus, the GSSEM is a natural choice for treating these constructs as underlying latent factors and studying their relationships through manifest variables used to measure them. Figure 1 is a path diagram representing the GSSEM we consider, where ovals represent underlying factors or random errors, rectangles represent observed variables, and an arrow means that two items are related to each other. Note that the smoking variable is treated as a predictor in the SEM, and the other observed variables are manifest variables in the measurement model. Also, note that we use the

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Fig. 1. Diagram of the GSSEM for Minnesota cancer mortality data.

Table 1. The ranges for observed and expected death counts for the three cancers in Minnesota counties

Observed Expected

Lung cancer Minimum Maximum 15 3797 21 3528

Pancreas cancer Minimum Maximum 2 830 5 791

Esophageal cancer Minimum Maximum 0 319 298 2

name ‘cancer mortality risk factor’ for simplicity, and do not intend to claim direct causality by this factor of the mortality from the three cancers. The three cancers may have different etiologies, but there may exist some other factors that influence the mortality of patients with different cancers in a similar way, such as SES and accessibility of public utilities. Wang and Wall (2003) found that the mortality risk of the three cancers share a common spatial factor. Therefore, we interpret the name ‘cancer mortality risk factor’ as a common spatial factor shared by the three cancers’ mortality risks. 5.1

The data

In this example, as in Figure 1 we have a data set consisting of nine variables, eight of which are considered manifest variables in the measurement model and one of which is considered a covariate in the SEM. The variable serving as the covariate is the county-level smoking prevalence from the Minnesota Behavioral Risk Factor Survey System, based on 1998 data. The eight variables serving as manifest variables are death counts of three cancers, esophageal, pancreas, and lung cancer, for the 87 Minnesota counties, three variables measuring SES of Minnesota counties, percent with high-school education (eduhs), median household income (medhhin), and per capita income (percapit), and two variables measuring utility accessibility of Minnesota counties, percentage of households with access to public water (pubwater) and percentage of households using wood to heat the home (wood). Wang and Wall (2003) and Wall and Li (2003) used these same variables. In the Minnesota cancer example of Wang and Wall (2003), they found that the mortality rate of esophageal, pancreas, and lung cancer of the 87 counties share a common underlying factor which they interpreted as cancer mortality risk. Wall and Li (2003) used classic SEM to study the influence of SES and utility accessibility on respiratory disease death rate in Minnesota counties, using eduhs, medhhin, and percapit to measure SES and pubwater and wood to measure utility accessibility for each county. [Wall and Li (2003) referred to utility accessibility as ‘ruralness’.] We also consider ‘smoking prevalence’ as an additional observed predictor H . In our study, six observed variables, eduhs, medhhin, percapit, pubwater, wood, and smoking prevalence, are standardized so that they have average 0 and sample standard deviation 1. Based on histograms, we assumed normal distributions for these observed variables. Table 1 shows the minimum and maximum observed and expected counts for

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Fig. 2. Maps of the observed variables for underlying SES (left) and utility accessibility (right).

each cancer for Minnesota counties, where the expected counts are calculated by assuming that the death rates of all these counties under the same age group are all the same. We see that counts in some counties of Minnesota are very sparse, and thus a Poisson distribution is assumed for the cancer death counts. Figure 2 presents maps of the five manifest variables used to measure SES and utility accessibility. The maps show that eduhs, medhhin, and percapit have similar spatial patterns, supporting the assumption that they may share a common spatial factor; likewise, pubwater and wood have strongly opposite spatial patterns, suggesting that they share a common spatial factor to which one of them is negatively correlated. Figure 3 shows maps of raw standardized mortality ratios for lung, pancreas, and esophageal cancer, and a map of smoking prevalence. The death rate maps show similar spatial trends for the three cancers. Smoking prevalence has a spatial distribution like the three cancers, which supports treating it as a predictor of the underlying cancer mortality risk factor. 5.2

Model fitting

The combined data set has nine county-level variables, one of which is a covariate in the SEM and eight of which manifest three underlying factors; the three observed variables for the cancer mortality risk

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Fig. 3. Maps of the raw SMR of three cancers and smoking prevalence.

factor are death counts and the remaining five variables are continuous variables or proportions that can (1) (1) (1) be treated as continuous. We assume that for county i, Z 1 , Z 2 , and Z 3 are the esophageal, pancreas, (2) (2) (2) and lung cancer death counts, which manifest the first factor (cancer mortality risk); Z 1 , Z 2 , and Z 3 (3) (3) are eduhs, medhhin, and percapit, which manifest the second factor (SES); and Z 1 and Z 2 are wood and pubwater, which manifest the third factor (utility accessibility). For simplicity of notation, we drop the county index i here. The measurement model for this example is (1)

(1)

(1)

(1)

(1)

(1)

Z 1 |θ1 ∼ Poi(θ1 ), Z 2 |θ2 ∼ Poi(θ2 ), (1) (1) (1) Z 3 |θ3 ∼ Poi(θ3 ), (2)

Z 1(2) |θ1(2) , σ12

(2)

Z 2(2) |θ2(2) , σ22

(2)

(2) (2) Z 3 |θ3 , σ32 (3)

(3)

(3)

(3)

(3)

(3)

Z 1 |θ1 , σ12 Z 2 |θ2 , σ22

(2)

∼ N (θ1(2) , σ12 ), (2)

(2)

∼ N (θ2 , σ22 ), (2)

∼ N (θ3(2) , σ32 ), (3)

(3)

(3)

(3)

∼ N (θ1 , σ12 ), ∼ N (θ2 , σ22 ).

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X. L IU ET AL . (1)

(1)

(1)

(2)

(2)

(2)

(3)

(3)

Let θ T = (θ1 , θ2 , θ3 , θ1 , θ2 , θ3 , θ1 , θ2 ), then define the link and joint mean structure as g(θθ ) = µ +  f, ⎞ ⎛ (1) (1) ⎞ ⎛ log(θ1 ) λ log(E 1 ) ⎜ ⎟ ⎜ 1(1) ⎜log(θ (1) )⎟ ⎜ ⎟ ⎜ ⎜ 2 ⎟ ⎜λ2 ⎜ ⎟ ⎜log(E 2 )⎟ ⎟ ⎜ ⎜log(θ (1) )⎟ ⎜ ⎜log(E 3 )⎟ ⎜ 1 ⎜ 3 ⎟ ⎟ ⎜ ⎜ ⎟ ⎜ ⎟ ⎜ (2) ⎜ θ ⎟ ⎜ ⎜ ⎜ ⎜ 1 ⎟ ⎜ 0 ⎟ ⎟+⎜ 0 = ⎜ ⎟ ⎟ ⎜ ⎜ θ (2) ⎟ ⎜ 0 ⎟ ⎜ ⎜ ⎜ 2 ⎟ ⎜ ⎟ ⎜ 0 ⎜ ⎟ ⎜ ⎟ ⎜ (2) 0 ⎜ θ ⎟ ⎜ ⎟ ⎜ 0 ⎜ 3 ⎟ ⎜ ⎟ ⎜ ⎜ ⎟ ⎜ θ (3) ⎟ ⎝ 0 ⎠ ⎜ ⎝ 0 ⎝ 1 ⎠ 0 (3) 0 θ ⎛

2

0 0 0 (2)

λ1

(2)

λ2

1 0 0

0

⎞

⎟ 0 ⎟ ⎟ ⎟ 0 ⎟ ⎟⎛ ⎞ ⎟ f1 0 ⎟ ⎟ ⎝ f2⎠ , ⎟ 0 ⎟ ⎟ f3 ⎟ 0 ⎟ ⎟ (3) λ1 ⎟ ⎠ 1

where E 1 , E 2 , and E 3 are age-adjusted (but not sex-adjusted) expected number of deaths for the three cancers for the whole population of Minnesota counties (gender is not adjusted), which are treated as (2) (2) (2) (3) (3) known constants. We standardized the manifest variables Z 1 , Z 2 , Z 3 , Z 1 , and Z 2 , so they are on the same scale and their means (in µ) can be assumed to be 0. Some elements in  are fixed to 0 such that each observed variable is related to only one of the factors, and three factor loadings are fixed to 1 to fix the scale uncertainty of the corresponding factor. These constrains in  make the measurement model have a simple structure as in confirmatory factor analysis models, which removes the scale uncertainty and uncertainties arising in the exploratory factor analysis, which assumes that each observed variable is related to several underlying factors. Therefore, the model defined above is identified, which means that all possible observations do not have identical likelihoods for two different sets of parameters. The f 1 , f 2 , and f 3 are the underlying factors representing cancer mortality risk, SES, and utility accessibility, respectively. The relationships among the factors are then modeled in the SEM: f 1 = β H + γ1 f 2 + γ2 f 3 + δ,

(5.1)

where H is the fixed covariate, smoking prevalence and β, γ1 , and γ2 are unknown constants. (The subscript i is again dropped for simplicity.) Let δ be the vector of δ over the 87 Minnesota counties. Since we are only considering one structural model, i.e. Q 1 = 1, it is natural to assume that δ has a univariate CAR covariance structure with covariance parameters α δ = (τδ , ρδ ), where τδ is the precision parameter and ρδ the spatial correlation parameter. We use the LMC to model the joint distribution of ( f 2 , f 3 )T :    a1 a2 v1 f2 = Av = , f3 0 a3 v2 where v1 and v2 are vectors of v 1 and v 2 over 87 Minnesota counties (the subscript i is again suppressed). We assume that v1 and v2 are independent spatial processes with CAR covariance structures having overall scale parameter set to 1 and spatial correlation parameters ρv 1 and ρv 2 , respectively. Priors are specified as in Section 4.1, except we adopt an equivalent conditional reparameterization on the LMC (Section 7.2 of Banerjee et al., 2004) to improve computing efficiency. We use the Winbugs software to obtain posterior summaries for the parameters. (The Winbugs code can be found in the Appendix.) We ran three Markov chains simultaneously started from different initial points. The monitor plots show that the three chains mixed well within 5000 iterations. We discarded the first 10 000 iterations and took another 5000 iterations for the posterior calculation. The lag 1 posterior sample autocorrelations of most

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Table 2. Posterior summaries for Minnesota cancer data analyzed using GSSEM Posterior mean (posterior 0.025 and 0.975 quantiles) Measurement model parameters

(1) λ1 (1) λ2 (1) λ3 (2) λ1 (2) λ2 (2) λ3 (3) λ1 (3) λ2

0.82

(0.36, 1.28)

—

—

—

0.46

(0.19, 0.75)

—

—

—

—

—

—

—

0.23

(0.16, 0.33)

0.10

(0.06, 0.16)

0.047

(0.003, 0.094)

0.016

(0.0005, 0.079)

0.23

(0.16, 0.32)

1 0.90

(0.79, 1.02)

0.97

(0.88, 1.07)

1

—

−1.15

(−1.29, −0.99)

1

—

(2) σ12 (2) σ22 (2) σ32 (3) σ12 (3) σ22

Structural

β

0.04

(0.003, 0.073)

τδ

0.036

(0.004, 0.08)

equation

γ1

0.053

(0.008, 0.098)

ρδ

0.97

(0.88, 0.998)

parameters

γ2

−0.067

(−0.13, −0.008)

ρv 1

0.96

(0.88, 0.997)

a1

1.49

(1.25, 1.76)

ρv 2

0.97

(0.91, 0.998)

a2

0.72

(0.45, 1.02)

a3

1.06

(0.86, 1.3)

parameters (e.g. λs, β, γ s, and ρs) are lower than 0.5, and the Gelman–Rubin diagnostic plot of the chains are mostly within the 0.8–1.2 band, which suggests satisfactory convergence. We also calculate the posterior summaries at different points of the Markov chain, and the result summaries are almost identical after burn-in period. No thinning on the draws seems necessary. The posterior samples are summarized in Table 2. In the measurement model stage, all the factor loadings, the λs, are highly significant (their 95% credible intervals do not cover 0), with values similar to the results in Wang and Wall (2003) and Wall and Li (2003). The interpretation of the significant factor loadings is that all the manifest variables are significantly related to their respective underlying factors. For example, the measurement model for the (1) (1) (1) (1) (1) first observed variable (esophageal cancer) is Z 1 |θ1 ∼ Poi(θ1 ), where log(θ1 ) = log(E 1 ) + λ1 f 1 . (1)

Equivalently, we can write it as log(

θ1 E1

(1)

) = λ1 f 1 , and log(

(1)

θ1 E1

) is usually interpreted as the log-relative (1)

risk of esophageal cancer in the disease mapping literature. From Table 2, the posterior mean of λ1 is 0.82, which implies that a unit increase in f 1 (cancer mortality risk) incurs a 0.82 unit increase in logrelative risk of esophageal cancer. In the structural equation, β is significant, which implies that smoking prevalence is associated with the cancer mortality risk factor in Minnesota counties. The estimates for γ1 and γ2 are also both significant, which implies that both SES and utility accessibility are important predictors of the cancer mortality risk factor in Minnesota counties after adjusting for smoking. Positive γ1 implies that counties with higher SES tend to have higher cancer mortality risk; negative γ2 implies that counties with more access to public utilities tend to have lower cancer mortality risk. The posterior estimates for the spatial correlation parameters ρδ , ρv 1 , and ρv 2 are very close to their upper limit 1. Figure 4 gives maps of the posterior means of f 1i , f 2i , and f 3i , which show spatial associations between neighboring counties, and hence that the spatial covariance structures for the underlying factors are necessary.

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In addition to fitting the model defined in (5.1), two additional models were fitted to assess the assumed spatial model. The three models are compared using the deviance information criteria (DIC) (Spiegelhalter et al., 2002). We call the model defined in (5.1) Model A, and the two alternative models Model B and Model C, where Model B only differs from Model A by assuming that δ in (5.1) is independent across locations, and Model C is the classic SEM which assumes that each underlying factor is independent across locations. DIC are 1980.6, 2035.7, and 2047.6 for Models A, B, and C, respectively; DIC prefer the model with the smallest value, so Model A is preferred. That is, the model fit is improved by accounting for the spatial distribution of the underlying factors. This result matches the strong spatial patterns in Figure 4, the maps of the fitted underlying factors. 5.3

Predicting the underlying spatial factors

Figure 4 gives maps of the posterior means of the underlying factors. These maps are one of the most interesting results of the GSSEM analysis, and can be useful summaries for public health researchers. The map of f 1 (cancer mortality risk) shows that counties in the northeastern triangle of Minnesota, including part of the Minneapolis–St. Paul metro area, have higher risk, and the southwestern counties have lower cancer mortality risk. The map of f 2 shows that SES is highest in the Minneapolis–St. Paul metro area and generally decreases with distance from the metro area. The map of f 3 shows that counties that have high utility accessibility include the metro counties and the farming counties south and west of the Twin Cities, while the north-central and far northeastern counties, which mostly consist of forests and lakes, have low utility accessibility. Therefore, the maps of f 2 and f 3 seem to be reasonable summaries of geographic and demographic features of Minnesota counties. Besides the point estimates of the parameters in the SEM part of the model, we are also interested in how well the SEM predicts the underlying factor of interest, cancer mortality risk. As an analog to an ordinary linear regression model, define the SEM expected value for county i: f 1SEM,i = β Hi + γ1 f 2,i + γ2 f 3,i .

(5.2)

In each iteration of the MCMC, we compute a draw of f 1SEM,i using that iteration’s draw of β, γ1 , γ2 , f 2,i , and f 3,i . Then, we define the SEM predicted value fˆ1SEM,i as the posterior mean of the draws of f 1SEM,i and also define (5.3) δˆi = fˆ1,i − fˆ1SEM,i , where fˆ1,i is the posterior mean of f 1 for county i. Conceptually, we treat fˆ1,i as the data, fˆ1SEM,i as the predicted value from the SEM, and δˆi as the residual. In fact, computationally, δˆi is the posterior predicted value of the δ in (5.1). ˆ Generally speaking, the Figure 5 gives maps of the SEM predicted value fˆ1SEM,i and SEM residual δ. map of fˆ1SEM,i has a similar spatial trend to the map of fitted f 1 in Figure 4, i.e. the northeastern part of Minnesota has higher values than the southwestern part of the state. This is consistent with the significant point estimates of β, γ1 , and γ2 in the SEM part of the model, which implies that we capture some of the important predictors for the cancer mortality risk. Nevertheless, for many counties fˆ1,i and fˆ1SEM,i differ, suggesting that there are some unobserved spatial-clustered covariates we are missing in the SEM for predicting the cancer mortality risk. We can see this in the map of the SEM residuals (delta) in Figure 5. As an analog to ordinary regression, if the prediction is good, then the residual plot should show no pattern and thus no spatial pattern, but the map of the residual δˆi shows a strong spatial pattern. Figure 5 also implies that the map of δˆi and fˆ1SEM,i have similar scales. In other words, the residuals have sizes similar to those of the predicted values, which also suggests that there are important predictors missing in the SEM.

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Fig. 4. Maps of the posterior means for the three underlying factors.

6. S UMMARY AND DISCUSSION This paper proposed a GSSEM which enables us to model the relationships among the underlying factors. GSSEM is a very general method that can be applied to a large class of multivariate spatial data. First, GSSEM can handle multivariate data from different exponential families. The Minnesota cancer example had both cancer death counts and continuous variables. Second, GSSEM reduces the dimensionality of multivariate data from different exponential family distributions to a smaller dimensional Gaussian process. The most important feature of GSSEM is that it enables us to model a spatially distributed underlying

Fig. 5. Maps of the SEM predicted value and SEM residual of the cancer risk factor.

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quantity of interest, cancer mortality risk in Minnesota counties, accounting for the direct effect of some fixed covariates and of other variables that are not directly observable. This produces an answer to the research question of most direct interest. In our example, we could answer the research question by simply performing inferences on the regression coefficients in the structural equation. Thus, GSSEM offers a very flexible tool for modeling multivariate spatial data and answering research questions about latent factors underlying these data. As mentioned earlier, the GSSEM can be extended to allow for additional spatial correlation to be included in the measurement part of the model, but as we have presented it, the observed measures are considered independent conditional on the latent factors. Christensen and Amemiya (2003) pointed out that the validity of the estimation procedure of their two-step approach was based on asymptotic theory (Amemiya and Anderson, 1990) which requires the error terms in the measurement model to be spatially independent. In our Bayesian approach, there is no theoretical obstacle to putting a spatial or other correlation structure on the measurement errors. This is in contrast to Christensen and Amemiya (2003): they had to assume independent measurement errors or they could not derive the large-sample approximation they needed to do inference. It is possible, though, that in our Bayesian approach, there may be practical obstacles to fitting a model with a spatial or other correlation structure on the measurement errors, e.g. poor identification of the unknown parameters in the correlation structure and associated difficulties with the MCMC algorithm. We have not explored this elaboration of our model so we cannot comment further. APPENDIX

### Winbugs code for the analysis of cancer mortality data model { for ( i in 1:N) { ##### First specify the measurement model part O1[i] ˜ dpois( w1[i] ) O2[i] ˜ dpois( w2[i] ) O3[i] ˜ dpois( w3[i] ) z1[i] ˜ dnorm( w4[i], taue[1]) z2[i] ˜ dnorm( w5[i], taue[2]) z3[i] ˜ dnorm( w6[i], taue[3]) z5[i] ˜ dnorm( w7[i], taue[4]) z4[i] ˜ dnorm( w8[i], taue[5]) log(w3[i]) <- log(E3[i]) + lambda[1] * f1[i] log(w2[i]) <- log(E2[i]) + lambda[2] * f1[i] log(w1[i]) <- log(E1[i]) + f1[i] w4[i] <- lambda[3] * f2[i] w5[i] <- lambda[4] * f2[i] w6[i] <- f2[i] w7[i] <- lambda[5] * f3[i] w8[i] <- f3[i] m[i]<- 1/num[i] # scaling factor for variance in each cell ##### structural equation ###predicted f1 by SEM muf1[i] <- beta[1] * f2[i] + beta[2] * f3[i] + beta[3] * smk[i] delta[i] <- f1[i] - muf1[i] ## spatial error ##### coregionalization# # The coregionalization formulas are: # f3[i] <- a[1] * v1[i] + a[2] * v2[i] # f2[i] <- a[3] * v2[i]

Generalized spatial structural equation models

555

#Where v1 and v2 are two ind unit variance "CAR" process with #correlation parameter alpha[2] and alpha[3]. To improve computing #efficiency and convergence of the MCMC algorithm, we use an #alternative equivalent formulation by integrating v1 and v2 out. muf3[i] <- f2[i] * a2/a[1] ## The conditional mean of f3[i] given f2[i] muf2[i] <- 0 ## Mean of f2[i] is 0 (same as v2[i]) } a[1] <- 1/sqrt(a1.prec) a[2] <- a2 a[3] <- 1/sqrt(a3.prec) f1[1:N] ˜ car.proper(muf1[], C[], adj[], num[], m[], precf, alpha[1]) f3[1:N] ˜ car.proper(muf3[], C[], adj[], num[], m[], a3.prec, alpha[3]) f2[1:N] ˜ car.proper(muf2[], C[], adj[], num[], m[], a1.prec, alpha[2]) ### Define the variances as the reciprocal of precisions sigmaf <- 1/precf for ( i in 1:5) { sigmae[i] <- 1/taue[i] } ### set up the weight matrix C for the CAR distributions cumsum[1] <- 0 for(i in 2:(N+1)) { cumsum[i] <- sum(num[1:(i-1)]) } for(k in 1 : sumNumNeigh) { for(i in 1:N) { pick[k,i] <- step(k - cumsum[i] - epsilon) * step(cumsum[i+1] - k) } C[k] <- 1 / inprod(num[], pick[k,]) # weight for each pair of neighbours } epsilon <- 0.0001 ###Priors for the parameters for ( i in 1:3) { beta[i] ˜dnorm( 0,0.001) } precf ˜ dgamma(0.01,0.01) a2˜dnorm(0,0.001) a1.prec ˜dgamma(0.01,0.01) a3.prec ˜ dgamma(0.01,0.01) for (i in 1:3){ alpha[i] ˜ dunif(0,1) } for (i in 1:5) { taue[i ] ˜ dgamma(0.01,0.01) lambda[i]˜ dnorm(0, 0.001) } }

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[Received September 23, 2004; revised March 9, 2005; accepted for publication April 11, 2005]