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Environmental and Ecological Statistics 7, 93±111, 2000
Using structural equation modeling to investigate relationships among ecological variables ZIAD A. MALAEB U.S. Department of the Interior, U.S. Geological Survey/National Wetlands Research Center, Gulf Breeze Project Of®ce, 1 Sabine Island Drive, Gulf Breeze, Florida 32561-5239. Currently at the US Department of Defense/US Army Safety Center, bldg 4905, 5th Ave. Fort Rucker, AL 36362-5363 U.S.A.
J. KEVIN SUMMERS U.S. Environmental Protection Agency, NHEERL-Gulf Ecology Division, 1 Sabine Island Drive, Gulf Breeze, Florida 32561-5239
BRUCE H. PUGESEK U.S. Department of the Interior, U.S. Geological Survey/National Wetlands Research Center, 700 Cajundome Boulevard, Lafayette, Louisiana 70506. Currently at USGS/Northern Rocky Mountain Research Center, 1648- S. 7th Street, MSU, Bozeman, MT 59717 Received July 1998; Revised April 1999 Structural equation modeling is an advanced multivariate statistical process with which a researcher can construct theoretical concepts, test their measurement reliability, hypothesize and test a theory about their relationships, take into account measurement errors, and consider both direct and indirect effects of variables on one another. Latent variables are theoretical concepts that unite phenomena under a single term, e.g., ecosystem health, environmental condition, and pollution (Bollen, 1989). Latent variables are not measured directly but can be expressed in terms of one or more directly measurable variables called indicators. For some researchers, de®ning, constructing, and examining the validity of latent variables may be the end task of itself. For others, testing hypothesized relationships of latent variables may be of interest. We analyzed the correlation matrix of eleven environmental variables from the U.S. Environmental Protection Agency's (USEPA) Environmental Monitoring and Assessment Program for Estuaries (EMAP-E) using methods of structural equation modeling. We hypothesized and tested a conceptual model to characterize the interdependencies between four latent variablessediment contamination, natural variability, biodiversity, and growth potential. In particular, we were interested in measuring the direct, indirect, and total effects of sediment contamination and natural variability on biodiversity and growth potential. The model ®t the data well and accounted for 81% of the variability in biodiversity and 69% of the variability in growth potential. It revealed a positive total effect of natural variability on growth potential that otherwise would have been judged negative had we not considered indirect effects. That is, natural variability had a negative direct effect on growth potential of magnitude ÿ 0.3251 and a positive indirect effect mediated through biodiversity 1352-8505 # 2000
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of magnitude 0.4509, yielding a net positive total effect of 0.1258. Natural variability had a positive direct effect on biodiversity of magnitude 0.5347 and a negative indirect effect mediated through growth potential of magnitude ÿ 0.1105 yielding a positive total effects of magnitude 0.4242. Sediment contamination had a negative direct effect on biodiversity of magnitude ÿ 0.1956 and a negative indirect effect on growth potential via biodiversity of magnitude ÿ 0.067. Biodiversity had a positive effect on growth potential of magnitude 0.8432, and growth potential had a positive effect on biodiversity of magnitude 0.3398. The correlation between biodiversity and growth potential was estimated at 0.7658 and that between sediment contamination and natural variability at ÿ 0.3769. Keywords: exploratory and con®rmatory factor analysis, indicator variables, latent variables, measurement errors, measurement models, path analysis, regression analysis, structural equation modeling, structural models 1352-8505 # 2000
Kluwer Academic Publishers
1. Introduction Most parametric and non-parametric statistical techniques are deviced to analyze data from controlled experiments where there is typically a dependent (response) variable, where measurement errors are negligible in the independent variable(s), and where uncontrolled variation is at a minimum. A substantial portion of statistical techniques comes under the heading of linear models where regression analysis and the analysis of variance are the two major topics (Hocking (1996)). In linear models, there is at least one response (dependent) variable conjectured to depend on one or more explanatory (independent) variables. One is typically interested in studying or predicting the behavior of the response from other variables in the model. For most applications, the functional relationship between the dependent and independent variables can be well described with a linear model, which can be tested using statistical techniques based on normal theory. The essential feature of these techniques is that only the dependent variable is assumed to be subject to measurement error or other uncontrolled variation. Levels and ranges of independent variables in regression analysis and the analysis of variance where one is interested in predicting the response, for example, are either speci®cally chosen or manipulated by the investigator. Since the prediction is conditional on predetermined values, independent variables may be considered ®xed quantities and thus measured without error. Ordinary regression methods based on normal theory can serve best if the purpose of the inquiry is to predict the behavior of the response from ®xed, predetermined values of independent variables assumed to be measured without error. They no longer suf®ce and indeed, may give misleading results in purely observational studies (e.g., environmental monitoring). In observational studies where experimentation is impossible or impractical, all variables are subject to measurement error and uncontrolled variation, and the purpose of the inquiry is to estimate relationships that account for variation among the variables in question and not to predict a particular response from uncontrolled values of ``independent'' variables (Joreskog and Sorbom (1996a)). In the assessment of ecological conditions, common univariate and multivariate statistical techniques based on normal theory have been used to investigate potential associations between variables. Environmental data seldom satisfy statistical assumptions of the normal theory and often interact multivariately in complex ways. Understanding the
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mechanism underlying complex covariance structures between ecological variables has been limited not only by the statistical techniques created to analyze data from controlled experiments but also by the way we have been accustomed to formulating hypotheses. Thinking only in terms of directly observable variables con®nes our horizons and limits our assessment of complex systems. Lacking from traditional statistical techniques is a process by which an investigator is able to formulate and test hypotheses about theoretical ecological issues and concepts in the presence of measurement errors and uncontrolled variation, where both direct and indirect effects are considered and where regression relationships can be simultaneously evaluated. Structural equation modeling is such a process.
2. Structural equation modeling The concept of structural equation modeling was ®rst introduced nearly eighty years ago by the population biologist Sewell Wright (Wright (1934)) and later developed most notably by Karl Joreskog and Dag Sorbom (Joreskog (1973); Joreskog (1977); Joreskog (1981); Joreskog and Sorbom (1982)). The development of software packages (e.g., LISREL, EQS, AMOS, and CALIS) speci®cally made to ®t and test SEM models has made structural equation modeling a prominent form of data analysis for the past twenty years, particularly in the ®elds of psychology, sociology, and economics. Structural equation modeling assumes that there is an underlying mechanism which leads to a theoretical covariance structure between a vector of random variables. The goal is to propose and test a model that will mimic this underlying mechanism. The statistical objective of structural equation modeling then is to recreate the sample covariance matrix R with a more parsimonious model-implied covariance matrix R
h that estimates fewer parameters than there are variances and covariances in the sample (Bollen (1989)). The research objective is to have theoretical and measurement precision, provide a consistent way of integrating complex systems, and test hypotheses about interdependencies of variables and concepts in complex systems. The fundamental null hypothesis
H0 in structural equation modeling is that the covariance matrix of the observed variables, R, is a function of a set of parameters h (Bollen (1989); Joreskog (1973)). That is, H0 :
R R
h
Unlike testing hypotheses in linear models where we typically would like to reject the null in favor of the alternative hypothesis, here we seek the acceptance of the null hypothesis. Rejecting the null hypothesis above amounts to rejecting the proposed model. To test whether a hypothesized covariance structure is consistent or inconsistent with the data, analysis is conducted on either the sample correlation or, the more recommended, sample covariance matrix of the observed variables which may be continuous, categorical, or a combination thereof. Structural equation modeling methods minimize the differences between the observed covariances and the covariances predicted by the model. The observed covariances minus the predicted covariances make up the residuals. Structural equation modeling is a combination of path analysis, factor analysis and regression analysis. Structural equation methods may be applied in ®ve steps: (1) model speci®cation, (2) model identi®cation, (3) parameter estimation, (4) testing model ®t, and
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(5) respeci®cation of the model (Bollen and Long (1993)). Model speci®cation consists of formulating latent variables, hypothesizing their interdependencies, and choosing their indicators; model identi®cation is concerned with whether unique estimates of the model parameters can be obtained; parameter estimation is the process of estimating the model parameters using one of several available methods of estimation; testing model ®t is the statistical process of assessing whether or not the model ®ts the data; and respeci®cation of the model refers to modifying the initial model to ®t the data better. We only give a brief introduction to structural equation models in matrix form in the Appendix. Structural equation modeling is well documented in several excellent books, e.g., Bollen (1989), Bollen and Long, (1993), Hayduk (1996), Hayduk (1987), Hoyle (1995), Joreskog and Sorbom (1996a and b), and Marcoulides and Schumacker (1996). We would also like to point out that there is an active discussion group on the Web with knowledgeable and helpful participants. The relevant information can be obtained at the following Web site http://www/gsu.edu/ * mkteer/semnet.html.
3. EMAP data example Data used in this paper were collected by the U.S. Environmental Protection Agency's (USEPA) Environmental Monitoring and Assessment Program for Estuaries (EMAP-E) in the Louisianian Province over 60-day periods from July to September, 1991 to 1994. The EMAP-E program is part of a larger program (EMAP) implemented by the USEPA in 1990 to assess the current status, extent, changes, and trends of the nation's environmental conditions (Summers et al. (1995)). The following 11 continuous EMAP variables were initially considered for analysis: (1) total PAHs in sediments, (2) total PCBs in sediments, (3) total pesticides in sediments, (4) contaminants 4 ER-L in sediments (i.e., number of sediment contaminants greater than ER-L in sample, see explanation of ER-L below), (5) sediments silt/clay percent, (6) dissolved oxygen (mg/l) at the bottom, (7) salinity ( ppt) at the bottom, (8) depth (m) at time of sampling, (9) transmissivity (%) at 1 meter (i.e., water clarity), (10) benthic species richness (i.e., the total number of taxa in n grabs), and (11) benthic abundance (i.e., the mean number of organisms in n grabs). All contaminants were compared to guidelines established by Long et al. (1995) that were developed from a biological effects database (BEDS) containing the concentrations of contaminants at which adverse biological effects occurred (i.e., altered benthic communities, sediment toxicity, and histopathological disorders in demersal ®sh). The guidelines are referred to as effects range-low (ER-L) and effects range-median (ER-M) which delineate concentrations at which adverse biological effects occur rarely (5 ER-L), occasionally (ER-L±ERM), or frequently (4 ER-M).
3.1 Data collection The EMAP has an elaborate sampling strategy under which these data were collected. It uses a global grid to identify sampling sites (Stevens (1994)). This strategy is also employed to study speci®c ecosystems or resource types (e.g., estuaries, forests, lakes and streams, wetlands) by dividing the grid into smaller sub-grids. The EMAP-E divides its estuaries into three categories, large estuaries (surface area 4 250 km2), small estuaries
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(surface area 5 250 km2), and large tidal rivers, and it uses slightly different sampling strategies for each. In the Louisianian Province, large estuary sampling sites were selected by randomly placing the sampling stations within the hexagonal space associated with each grid point (Summers et al. (1995)).
3.2 Statistical assumptions and data transformations For maximum likelihood, the w2 test is correct when the observed variables have a joint multivariate normal distribution. Though univariate normality would not imply multivariate normality, we examined the univariate normality for each of the observed variables and applied appropriate Box-Cox power transformations as needed using Malaeb's BoxCox SAS code (Malaeb (1997)). Transformations were mainly applied to achieve linearity and homoscedasticity even though they helped achieve exact and approximate univariate normality for eight of the variables. We shall refer to the transformed variables by their original untransformed names. Their transformations are indicated in Table 1. Furthermore, each observed variable was examined for ``unusual'' observations before and after transformations were applied since outliers can have great effects on linear correlation coef®cients. No unusually large outliers were encountered, and all 481 observations for each variable were used in the analysis. The overall mean of each observed variable (i.e., mean of each column vector in the data matrix) was also substituted for the few missing values encountered in the data. Since only about six observations (out of 481) were missing in each column, substituting the mean for missing values did not have any effects on shrinking the variance.
3.3 The conceptual model In a traditional model of estuarine dynamics, biodiversity and growth potential may be represented by species richness and abundance, respectively, of benthic macroinvertebrates. Because benthic macroinvertebrates reside in the sediments, they are often used as indicators of environmental perturbations, both natural and anthropogenic (Boesch and Rosenberg (1981); Diaz and Rosenberg (1996); Engle, Summers, and Gaston (1994)). Benthic diversity and community structure may be altered by the presence of sediment contamination (e.g., PAHs, PCBs, pesticides, metals), some of which are acutely toxic to benthic organisms and others of which may reduce growth or reproduction or lead through competitive interactions to dominance by a few pollution-tolerant species. Natural habitat variability (e.g., salinity, sediment silt-clay content, water clarity, and dissolved oxygen) is also an important factor in determining both the geographical distribution of benthic species and gradients in diversity and abundance. We formulated a structural model based on our knowledge of estuarine dynamics relationships among four latent variablesÐ sediment contamination, natural variability, biodiversity, and growth potential (Fig. 1). The sign and magnitude of direct and indirect effects of sediment contamination, natural variability, and biodiversity on growth potential were of particular interest. The structural model in Fig. 1 re¯ected our understanding of these relationships and was a good start even though it was overparameterized (i.e., unidenti®ed). We hypothesized that sediment contamination had both direct and indirect effects on biodiversity of magnitudes g11 , and
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Table 1. The analyzed correlation matrix with model-predicted values and residuals. Log Log Log Log Log Log (Total (Contaminants > p p (Richness) (Abundance) (Total PAHs) (Total PCBs) Pesticides) ER-L) Silt/Clay Percent Salinity Water Clarity Observed Predicted Residual
1.00 1.00 0.00
Log (abundance)
Observed Predicted Residual
0.77 0.77 0.00
1.00 1.00 0.00
Log (total PAHs)
Observed ÿ 0.47 Predicted ÿ 0.46 Residual ÿ 0.01
ÿ 0.27 ÿ 0.28 0.01
1.00 1.00 0.00
Log (total PCBs)
Observed ÿ 0.36 Predicted ÿ 0.37 Residual 0.01
ÿ 0.20 ÿ 0.22 0.02
0.67 0.68 ÿ 0.01
1.00 1.00 0.00
Log (total pesticides)
Observed ÿ 0.42 Predicted ÿ 0.43 Residual 0.01
ÿ 0.26 ÿ 0.25 ÿ 0.01
0.59 0.59 0.00
0.66 0.63 0.03
1.00 1.00 0.00
Observed ÿ 0.26 Log (contaminants > ER-L) Predicted ÿ 0.26 Residual 0.00
ÿ 0.17 ÿ 0.16 ÿ 0.01
0.52 0.49 0.03
0.38 0.39 ÿ 0.01
0.42 0.45 ÿ 0.03
1.00 1.00 0.00
Silt/Clay Percent
Observed ÿ 0.47 Predicted ÿ 0.39 Residual ÿ 0.08
ÿ 0.31 ÿ 0.18 ÿ 0.13
0.42 0.39 0.03
0.32 0.31 0.01
0.32 0.36 ÿ 0.04
0.21 0.22 ÿ 0.01
1.00 1.00 0.00
p Salinity
Observed 0.39 Predicted 0.42 Residual ÿ 0.03
0.11 0.16 ÿ 0.05
ÿ 0.25 ÿ 0.21 ÿ 0.04
ÿ 0.20 ÿ 0.17 ÿ 0.03
ÿ 0.13 ÿ 0.19 ÿ 0.06
ÿ 0.02 ÿ 0.12 0.10
ÿ 0.22 ÿ 0.27 0.05
1.00 1.00 0.00
p Water Clarity
Observed 0.29 Predicted 0.31 Residual ÿ 0.02
0.09 0.12 ÿ 0.03
ÿ 0.20 ÿ 0.16 ÿ 0.04
ÿ 0.11 0.13 ÿ 0.02
ÿ 0.010 ÿ 0.15 0.05
ÿ 0.05 ÿ 0.09 0.04
ÿ 0.22 ÿ 0.21 ÿ 0.01
0.31 0.27 0.04
1.00 1.00 0.00
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Log (richness)
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Figure 1. The hypothesized structural model.
g21 6b12 , respectively, and direct and indirect effects on growth potential of magnitudes g21 , and g11 6b21 , respectively. Similarly, we hypothesized that natural variability had both direct and indirect effects on biodiversity of magnitudes g12 , and g22 6b12 , respectively, and direct and indirect effects on growth potential of magnitudes g22 , and g12 6b21 , respectively. Finally, we hypothesized that biodiversity had a direct effect on growth potential of magnitude b21 , and that growth potential had a reverse direct effect on biodiversity of magnitude b12 . We let the model estimate F21 , the correlation between sediment contamination and natural variability, as well as z1 and z2 , the random disturbances in biodiversity and growth potential, respectively. The next step in the analysis was to ®nd sets of indicators which ``best'' described each latent variable in the model. The eleven observed variables were proposed to represent the four latent variables as follows: total PAHs, total PCBs, total pesticides, contaminants 4 ER-L, and silt/clay percent for sediment contamination; dissolved oxygen, salinity, depth, and water clarity for natural variability; benthic species richness for biodiversity; and benthic abundance for growth potential. Measurement models were used to test the measurement reliability of indicators representing sediment contamination and natural variability, both individually and collectively. Biodiversity and growth potential were each represented by only one indicator, namely, benthic species richness and benthic abundance, respectively. We ®xed their loadings and their variances to 1, in essence making the (somewhat unrealistic) assumption that the two concepts biodiversity and growth potential were identical to their corresponding indicators, and hence, no measurement models were needed to test their measurement reliability. We conducted the analysis using the correlation matrix for ease of interpretation but hasten to point out that correct standard errors of parameter estimates are obtained only when the covariance matrix is used (Cudeck (1989)). Having speci®ed the initial structural model (Fig. 1) based on our knowledge of estuarine dynamics, the strategy of our analysis,
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Figure 2. Measurement model for sediment contamination.
as recommended by JoÈreskog (1993, p. 313), was to estimate the measurement models for sediment contamination and natural variability separately (Figs. 2 and 3) and jointly (Fig. 4), and ®nally, estimate the structural equation model for biodiversity and growth potential jointly with the measurement models (Fig. 5).
Figure 3. Measurement model for natural variability.
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Figure 4. Measurement model for sediment contamination and natural variability combined.
The overall ®t of the measurement model for sediment contamination (Fig. 2 ) was excellent
w2 1.77, df 3, p-value 0.62; GFI 1.0; AGFI 0.99; RMSEA 0.0; PGFI 0.2; NFI 1.0; NNFI1.0; PNFI 0.3; CFI 1.0; IFI 1.0; RFI 0.99; CN 3074, (Recall that a small p-value rejects the null hypothesis of a good model,
Figure 5. Final structural equation model.
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and the CN is the minimum sample size at which the w2 test rejects the model) with two signi®cantly correlated pairs of measurement errors, namely, those of total PAHs and total PCBs with that of total pesticides. This indicated that the ®ve observed variables formed an acceptable set of indicators of sediment contamination, with some indicators being stronger than others as indicated by their loading coef®cients (Fig. 2). Even though this set of indicators provided an acceptable representation of sediment contamination, it by no means implies a uniform endorsement of the indicators for there may be other sets that provide an equivalent ®t. Similarly, the measurement model for natural variability (Fig. 3) exhibited a good ®t
w2 0.0024, df 1, p-value 0.96; GFI 1.0; AGFI 1.0; RMSEA 0.0; PGFI 0.10; NFI 1.0; NNFI 1.0; PNFI 0.17; CFI 1.0; IFI 1.0; RFI 1.0; CN 1325544 with only one pair of signi®cantly correlated measurement errors between those of dissolved oxygen and water clarity. Here too, natural variability was well represented by the four indicators. Having established satisfactory measurement models for sediment contamination and natural variability individually, we next combined Figs. 2 and 3 in one measurement model containing all nine indicator variables to study their joint behavior before assembling them in the ®nal structural model. This model (not shown) had several signi®cantly correlated measurement errors most notably between those of dissolved oxygen and depth with those of the other seven variables. Unless theoretically substantiated, we would like for measurement errors not to correlate in crosssectional studies. Too many correlated measurement error terms between two or more indicators complicate and even obscure the interpretation of latent variables. There are various possibilities for the several signi®cant correlations between measurement errors of dissolved oxygen and depth with other measurement errors. One may be due to the exclusion of an underlying concept common to both of dissolved oxygen and depth that may be responsible for their joint covariances with other variables in the model. Another is that when we combine all nine indicators in one measurement model, the covariability of dissolved oxygen and depth with other variables is adequately accounted for by salinity and water clarity alone. To eliminate or minimize the number of signi®cantly correlated measurement errors, one could (1) introduce another latent variable, or (2) remove dissolved oxygen and depth. Since we were conducting exploratory and not con®rmatory analyzes where initial models may be modi®ed, we removed dissolved oxygen and depth from the model. The resulting model (Fig. 4) exhibited a good ®t
w2 20.17, df 11, p-value 0.043; GFI 0.99; AGFI 0.97; RMSEA 0.042; PGFI 0.39; NFI 0.98; NNFI 0.98; PNFI 0.51; CFI 0.99; IFI 0.99; RFI 0.96; CN 590, and had only two cases of correlated measurement errorsÐone between total PCBs and total pesticides and another between contaminants 4 ER-L and silt/clay percent.
3.4 Final model Having been satis®ed with the overall ®t of all measurement models, we ultimately combined the structural and the measurement parts (i.e., Figs. 1 and 4, respectively) in a one structural equation model and tested its ®t. LISREL18 gave a warning message that the error variance for growth potential was not identi®ed indicating that the model was overidenti®ed, as we originally suspected. Identi®cation would be corrected by either removing the path from sediment contamination to growth potential or ®xing the variance of either growth potential or biodiversity to a constant. We corrected the identi®cation
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problem by removing the path from sediment contamination to growth potential, particularly since it had a very small estimated coef®cient of ÿ 0.018 (or equivalently, a very low R2 of 0.0003). The resulting model suggested, through a signi®cant ®t index, that a path from sediment contamination to silt/clay percent needed to be established. We modi®ed the model accordingly and the resulting ®nal model is shown in Fig. 5. The ®nal structural equation model (Fig. 5) had a good agreement with the data
w2 53.5, df 21, p-value 0.00012; GFI 0.98; AGFI 0.95; RMSEA 0.057; PGFI 0.46; NFI 0.97; NNFI 0.97; PNFI 0.57; CFI 0.98; IFI 0.98; RFI 0.95; CN 351. Even though the w2 test statistic of 53.5 with 21 degrees of freedom and an associated p-value of 0.00012 called for the rejection of the model at the 5% level of signi®cance, this w2 rejection was in part due to the large sample size
N 481 used in the analysis. Note that the w2 test may reject a plausible model when the sample size is moderately large. If we set the sample size to 200, as suggested by Hoelter (1983), for example, the resulting w2 becomes 22.18 with 21 degrees of freedom and an associated p-value of 0.39, thereby not rejecting the model. The minimum size of the sample at which the w2 test would reject the model is CN 351. The model had only one signi®cant pair of correlated measurement errors, namely, between total PAHs and total pesticides. All parameter estimates in the ®nal model (Fig. 5) were examined to ensure their admissibility and that they made substantive sense. We also examined the model identi®ability by satisfying the two necessary but not suf®cient conditions, namely that (1) the scale of every latent variable must be set, and (2) the number t of the free parameters estimated by the model is less than the number of the sample variances/covariances for the observed variables
p q
p q 1=2, where p is the number of indicators of the dependent latent variables Z, and q is the number of indicators of the independent latent variables x (Hayduk (1996)). The scale of each latent variable in our model was equal to 1 by default since the correlation and not the covariance matrix was used; and the number of free parameters estimated by the model (i.e., 24) was less than
2 7
2 7 1= 2 45. We did not establish suf®ciency. Instead, we followed Hayduk's (1987, p. 149) suggestion to reestimate the model coef®cients with a few (here 3) different sets of starting values and obtained the same parameter estimates each time. Further evidence that parameters were uniquely estimated was that LISREL18 did not issue a warning message about the identi®cation of any of the estimated parameters in this model. Therefore, we accept the statistical ®t and conclude that the model adequately describes the covariance structure between the nine observed variables.
4. Results and interpretation The main purpose of our inquiry was to examine the hypothesized pattern of effects of sediment contamination, natural variability, and biodiversity on growth potential by recreating the matrix of observed correlations. Model-predicted correlations were very similar to the observed correlations, as re¯ected by the residuals (i.e., observed minus predicted), 41 out of 45 total predictions were within 0.05 of the observed values, and only two residuals were relatively large, i.e., ÿ 0.13 between silt/clay percent and abundance, and 0.10 between salinity and contaminants 4 ER-L (Table 1). Our initial hypothesized structural relationships in Fig. 1 were con®rmed except that the direct path from sediment contamination to growth potential was not signi®cant and, thus, was removed. The model
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Malaeb, Summers, Pugesek Table 2. Simultaneous regression relationships of the measurement models. Note the relative low R2 values of the indicators of natural variability despite the acceptable overall ®t of the model. Log (total PAHs) 0:936sediment contamination d1 , Log (total PCBs) 0:746sediment contamination d3 , Log (total pesticides) 0:866sediment contamination d2 , Log (contaminants4ER-L) 0:536sediment contamination d4 , Silt/clay percent 0:286(sediment contamination)
ÿ 0:35 6Natural variability d6 , p Salinity 0:606natural variability d8 , p Water clarity 0:456natural variability d10 , Log (richness) 16biodiversity e1 , Log (abundance) 16growth potential e2 ,
R2 R2 R2 R2 R2
0:86 0:55 0:73 0:28 0:28
R2 R2 R2 R2
0:36 0:20 1:00 1:00
accounted for 69% of the variability in growth potential and 81% of the variability in biodiversity. All regression and path coef®cients were signi®cantly different from zero. Table 2 gives the estimated regression coef®cients and their corresponding R2 values. Direct effect of natural variability on growth potential was estimated at ÿ 0.3251, while that of the indirect effect via biodiversity at 0:4509 0:534760:8432, giving the total effects a positive estimate of 0.1258. The direct, indirect (via growth potential) and total effects of natural variability on biodiversity were estimated as 0:5347; ÿ 0:1105 ÿ 0:325160:3398, and 0:4242 0:5347 ÿ 0:1105, respectively. Sediment contamination, on the other hand, had only an indirect effect on growth potential via biodiversity and was estimated at ÿ 0:1649 ÿ 0:195660:8432, and its direct effects on biodiversity was estimated at ÿ 0.1956. The model also estimated the effects of biodiversity on growth potential at 0.8432 and a reverse effect of growth potential on biodiversity at 0.3398. The estimated correlation between biodiversity and growth potential was 0.7658, and that between sediment contamination and natural variability at ÿ 0.3769.
5. Conclusion and discussion Structural equation modeling is an advanced statistical process that, when applied properly, helps provide an insight into complex theoretical issues. Structural equation modeling techniques are typically used to con®rm or disprove an a priori hypothesized model as in con®rmatory factor analysis. They can, however, be used to generate models in an exploratory sense, as was done in this paper. Both approaches can be rewarding. When using structural equation modeling to con®rm or disprove a hypothesized theory about structural relationships between latent variables, one must completely specify a model, including correlated measurement errors, before data are collected. If structural equation modeling methods are used for generating and exploring models, one should have a starting model in which a theory is suspected to hold. The ®nal accepted model, particularly one which has been modi®ed in the course of analysis, should be tested and con®rmed with new data. Here we presented an example of an exploratory model since our models were modi®ed in the process. Though our ®nal model in Fig. 5 has not been tested
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with new data, it nonetheless has con®rmed our suspicion of at least part of an underlying complex phenomenon that gave rise to the observed correlations between the nine environmental variables in the model. Structural equation modeling offers an unparalleled treatment of measurement errors and their possible correlations. By taking measurement errors into account, structural equation modeling gives more reliable and accurate estimates of parameters than multiple regression for example (Joreskog and Sorbom (1996a)). Correlated measurement errors might shed some light on the appropriateness of the number of latent variables used to represent a set of observed variables. The presence of unsubstantiated correlated measurement errors often indicates that the observed covariance structure between variables may be in part due to other latent variables not included in the model, or perhaps some indicators need to be excluded. When we combined the two measurement models of Figs. 2 and 3, for example, several unexplainable correlated measurement errors resulted. Many such correlations were associated with depth and/or dissolved oxygen. Excluding depth and dissolved oxygen resulted in a much better ®t to the data. In structural equation modeling, regression relationships are assessed simultaneously, allowing the researcher to compare the relative importance of indicator variables. Comparisons of the relative importance of indicators could be made only on those indicators not shared with more than one latent variable. For example, the relative importance of four of the ®ve indicators representing sediment contamination could be evaluated through examining their corresponding regression coef®cients. Total PAHs had the highest regression coef®cient of 0.93, followed by 0.86, 0.74, and 0.53 for total pesticides, total PCBs, and contaminants 4 ER-L, respectively Table 2). Also, regression coef®cients for the two unshared indicators of natural variability were 0.60 and 0.45 corresponding to salinity and water clarity, respectively, and indicating that salinity was a more important indicator of natural variability than water clarity. Note, however, that both salinity and water clarity were weak indicators of natural variability, as re¯ected by their low R2s (Table 2), despite the acceptable overall ®t of the model. Indirect effects are not treated in traditional statistical methods but are taken into account in structural equation modeling. Consideration of both direct and indirect effects in the ®nal model led to an interesting revelation in examining the relationship between natural variability and growth potential. That is, the direct effect of natural variability on growth potential was estimated to be negative ( ÿ 0.3251) which was offset by an indirect positive effect of 0.4509 through biodiversity. This yielded a total positive effect of 0.1258. Had we not considered indirect effects, we would have been led to believe that natural variability had a moderate negative effect on growth potential when in fact the overall effect was positive. Perhaps the most attractive feature of structural equation modeling is that it offers a new way of thinking about data analysis that greatly complements existing statistical methods. This new way of thinking constitutes formulating latent concepts, examining their validity, testing their measurement reliability, and hypothesizing and testing structural relationships among them. Formulating latent constructs and testing their measurement reliability with measurement models is an achievement of its own. Hypothesizing and testing their interdependencies and ``causal'' relationships enables a researcher to examine fundamental issues and theoretical concepts at a greater depth, a task that is dif®cult to achieve with traditional statistical methodologies. The process of formulating latent variables, choosing their appropriate indicators, testing their measurement reliability (both individually and
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collectively), and hypothesizing and testing their structural (or ``causal'') relationships is rather rigorous, particularly since the statistical properties of the observed variables (e.g., univariate and multivariate distributions, linearity, unusual observations, covariances, etc.) are thoroughly examined prior to their use in a structural equation model. In this study, for example, we formulated four latent constructs, hypothesized their directional interdependencies, chose their indicators, tested the measurement reliability of two of them individually and collectively, assembled them in a structural equation model, and performed the statistical test. The univariate and multivariate statistical properties of all chosen indicator variables were examined. This rigorous process of model examination and testing can be rewarding regardless of whether or not a researcher arrives at a model which con®rms what is already known or reveals something new. When the end result of a structural equation model is nothing new to the investigator, then con®rming what is already known is reassuring, particularly when approaching the problem from a different perspective. When the model does reveal something new, its results can be trusted with more con®dence since direct and indirect effects as well as measurement errors have been considered. An example of developing ``new'' approaches to existing paradigms can be seen even in the simplistic model described in this manuscript. The creation of conceptual ``latent'' variables to characterize sediment contamination and natural variability permits us to examine basic structural and functional relationships among biodiversity, growth and the environment. Such simple tools allow us to begin to examine to broad-scale relationships between ecological phenomena and the environment. The strong effect of natural variability on changes on biodiversity with ameliorating effects of anthropogenic source of contamination shows that while the basic drivers for diversity in Gulf of Mexico benthic communities appears to be natural variability, contaminants can clearly reduce that diversity as a secondary effector. Growth patterns of variability, at least in terms of population size, seems largely affected by biodiversity levels in the community and clearly augmented by reduced levels of variation in natural variability. Such ®ndings can be used to promote our understanding of the underlying principles that govern changes in biodiversity and growth, particularly if the proposed model for the Gulf of Mexico is similar in structure and result to models for other geographic regions. Structural equation modeling will undoubtedly meet some skepticism, but our hope is to raise the curiosity of environmental researchers enough to enquire about this important tool of data analysis for dealing with complex ecological and environmental issues.
Disclaimer The mention of trade names or commercial products in this manuscript does not constitute endorsement or recommendation for use by either the U.S. Department of the Interior/U.S. Geological Survey or the U.S. Environmental Protection Agency (USEPA).
Appendix Matrix representation of structural equation models The following is a brief matrix introduction to structural equation modeling using the so called LISREL Model notation.
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For deviations of observations about their mean, a structural equation model can be succinctly described by the following set of three equations: X Kx n d; Y Ky g e; g Bg Cn f
1
where X is a q61 vector of observable indicators of the independent latent variables n; Y is a p61 vector of observable indicators of the dependent latent variables g; g is an m61 random vector of dependent latent variables; n is an n61 random vector of independent latent variables; e is a p61 vector of measurement errors in Y; d is a q61 vector of measurement errors in X; Ky is a p6m matrix of coef®cients of the regression of Y on g; Kx is a q6n matrix of coef®cients of the regression of X on n; C is an m6n maatrix of coef®cients of the n variables in the structural relationship; B is an m6m matrix of coef®cients of the g variables in the structural relationship. Typically, B has zeros in the diagonal, and
I ÿ B is required to be nonsingular; is an m61 vector of equation errors (random disturbances) in the structural relationship between b and n. The statistical assumptions are as follows: (1) E
g E
n E
e E
d E
f 0, where E(?) denotes the expected value, (2) e is uncorrelated with g, (3) d is uncorrelated with n, (4) f is uncorrelated with n, (5) f, e, and d are mutually uncorrelated, and (6)
I ÿ Bÿ1 is nonsingular. The associated covariance matrices are as follows: Cov
n U
n6n: Cov
e He
p6p: Cov
d Hd
q6q: Cov
f W
m6m:
covariances covariances covariances covariances
between the between the between the between the
independent variables n; measurement errors in Y; measurement errors in X; structural errors f.
With these assumptions, the model-implied covariance matrix R
h of the observed variables is given by: 2 R
h 4
Ky
I ÿ B
ÿ1
CUC0 W
I ÿ B
Kx UC0
I ÿ B
ÿ1
K0y
ÿ1
K0y He
Ky
I ÿ B
ÿ1
CUK0x
Kx UK0x Hd
3 5:
2
The
p q model-implied symmetric covariance matrix R
h can be partitioned into four matrices Ryy
h; Rxx
h; Rxy
h Ryx
h0 denoting the model-implied covariances between the Y observed variables, the X observed variables, and the X and Y observed variables, respectively. That is,
108
" R
h
Ryy
h
Ryx
h
Rxy
h Rxx
h
Malaeb, Summers, Pugesek
# :
Assuming zero means and using Equation (2), these partitions are derived as follows: Ryy
h E
YY 0 E
Ky g e
Ky g e EKy gg0 Ky Ky ge0 eg0 K0y ee0 Ky E
gg0 K0y Ky E
ge0 E
eg0 K0y E
ee0 0
Ky E
gg K0y E
ee0 ;
since E
ge0 E
eg0 0:
where E
gg0 Ef
I ÿ Bÿ 1
Cn f
I ÿ Bÿ 1
Cn f0 g Ef
I ÿ B
ÿ1
0
Cn f
Cn f
I ÿ B 0
ÿ 10
g
I ÿ B
ÿ1
E
Cn f
Cn f
I ÿ B
I ÿ B
ÿ1
E
Cnn0 C0 E
Cnf0 E
fn0 C0 E
ff0
I ÿ B
ÿ1
ÿ1
I ÿ B E
Cnn0 C0 E
ff0
I ÿ B
ÿ10
since E
Cnf0 E
fn0 C0 0 ÿ1
I ÿ B CE
ff0 C0 E
ff0
I ÿ B
ÿ10
:
Letting F E
xx0 ; C E
zz0 , and Ye E
ee0 , it follows that Ryy
h Ky
I ÿ Bÿ1 CUC W
I ÿ Bÿ1 K0y He: Similarly, Ryx
h EYX 0 E
Ky g e
Kx n d0 E
Ky g e
n0 K0x d0 0
Ky E
gn0 K0x Ky E
gd E
en0 K0x E
ed0 Ky E
I ÿ Bÿ 1
Cn fn0 K0x Ky
I ÿ B
ÿ1
CE
nx0 K0x ;
since E
fn0 0
Ky
I ÿ Bÿ 1 CK0x ; where U E
nn0 :
ÿ10
Using structural equation modeling
109
Finally, Rxx
h E
XX0 E
Kx n d
Kx n d0 EKx n d
n0 K0x d0 Kx E
nn0 K0x E
dd0 ; since E
nd0 0 Kx UK0x Hd ; since Hd E
dd0 : Assembling the above in one matrix yields R
h in Equation (2).
Acknowledgments The authors would like to thank the editor and the four anonymous reviewers for their invaluable comments and suggestions. We are particularly indebted to the third reviewer whose constructive detailed comments and suggestions greatly improved this manuscript. We also would like to thank Dr Alexander von Eye, Professor of Family and Child Ecology and Professor of Psychology, Michigan State University; Dr Adrian Tomer, Professor of Psychology, Shippensburg University; Dr Jim Grace,1 Ecologist; and Dr Sijan Sapkota,2 Statistician, for their technical input. Many thanks go to Ms. Beth Vairin,1 Writer/Editor, and Mr Daryl McGrath2 for their editorial comments, Pete Bourgeois1, Geographic Information System Specialist, Virginia Engle, Ecologist; USEPA, Gulf Breeze, Florida, Thomas Heitmuller,1 Quality Assurance Specialist for their input, Renee Conner,2 Technical Illustrator, and Lois Haseltine,2 Desktop Publisher for preparing the ®gures and tables.
Note Contribution Number 999 of the Gulf Ecology Division of the National Health and Environmental Effects Research Laboratory and Contribution Number 97-019 of the National Wetlands Research Center.
References Boesch, D.F. and Rosenberg, R. (1981) Response to stress in marine benthic communities. In Stress effects on natural ecosystems, G.W. Barret and R. Rosenberg (eds), John Wiley & Sons, New York, pp. 179±200. Bollen, K.A. (1989) Structural Equations with Latent Variables, Wiley, New York. Bollen, K.A. and Long, J.S. (1993) (eds) Testing Structural Equation Models, Sage, Newsburg Park, California. 1
U.S. Department of the Interior, U.S. Geological Survey/National Wetlands Research Center, Gulf Breeze Project Of®ce, 1 Sabine Island Drive, Gulf Breeze, Florida 32561-5239. 2 Johnson Controls World Services, Inc., National Wetlands Research Center, Gulf Breeze Project Of®ce, 1 Sabine Island Drive, Gulf Breeze, Florida 32561-5239.
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Cudeck, R. (1989) Analysis of correlation matrices using covariance structure models. Psychological Bulletin, 105, 317±27. Diaz, R.J. and Rosenberg, R. (1996) The in¯uence of sediment quality on functional aspects of marine benthic communities. In Development and progress in sediment quality assessment rationale, challenges, techniques and strategies Ecovision World of Monograph Series, M. Munawar and G. Dave (eds), Academic Publishing, Amsterdam, pp. 57±68. Engle, V.D., Summers, J.K., and Gaston, G.R. (1994) A benthic index of environmental condition in Gulf of Mexico estuaries. Estuaries, 17, 372±84. Hayduk, L.A. (1987) Structural Equation Modeling with LISREL, Essentials and Advances, The Johns Hopkins University Press Ltd., London. Hayduk, L.A. (1996) LISREL Issues, Debates, and Strategies, The Johns Hopkins University Press, Baltimore and London. Hocking, R.R. (1996) Methods and Applications of Linear Models, Regression and the Analysis of Variance, Wiley, New York. Hoelter, J.W. (1983) The analysis of covariance structures: Goodness-of-®t indices. Sociological Methods and Research, 11, 324±44. Hoyle, R.H. (1995) (eds) Structural Equation Modeling, Concepts, Issues, and Applications, Sage, Newbury Park, CA. Joreskog, K.G. (1973) Analysis of covariance structures. In Multivariate AnalysisÐIII, P. Krishnaiah (eds), Academic Press, New York, pp. 263±85. Joreskog, K.G. (1977) Structural equation models in the social sciences Speci®cation, estimation and testing. In Applications of Statistics, P. Krishnaiah (eds), North-Holland Publishing Co., Amsterdam, pp. 265±87. Joreskog, K.G. (1981) Analysis of covariance structures. Scandinavian Journal of Statistics, 8, 65± 92. Joreskog, K.G. and Sorbom, D. (1982) Recent developments in structural equation modeling. Journal of Marketing Research, 19, 404±16. Joreskog, K.G. and Sorbom, D. (1996a) PRELISTM2 User's Reference Guide, Scienti®c Software International. Joreskog, K.G. and Sorbom, D. (1996b) LISREL18 User's Reference Guide, Scienti®c Software International. Long, E.R., MacDonald, D.D., Smith, S.L., and Calder, F.D. (1995) Incidence of adverse biological effects within ranges of chemical concentrations in marine and estuarine sediments. Environmental Management, 19, 81±97. Malaeb, Z.A. (1997) A SAS1 code to correct for non-normality and non-constant variance in regression and ANOVA models using the Box-Cox method of power transformation. Environmental Monitoring and Assessment, 47, 255±73. Marcoulides, G.A. and Schumacker, R.E. (1996) (eds) Advanced Structural Equation Modeling, Issues and Techniques, Lawrence Erlbaum Associates, Mahwah, New Jersey. Stevens, D.L. Jr. (1994) Implementation of a National Monitoring Program. Journal of Environmental Management, 42, 1±29. Summers, J.K., Paul, J.F., and Robertson, A. (1995) Monitoring the Ecological Condition of Estuaries in the United States. Toxicological and Environmental Chemistry, 49, 93±108. Wright, S. (1934) The method of path coef®cients. Annals of Mathematical Statistics, 5, 161±215.
Biographical sketches Ziad A. Malaeb was the senior statistician at the USGS/National Wetlands Research Center, Gulf Breeze Project Of®ce. He is currently the lead statistician at the US
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Department of Defense/US Army Safety Center at Fort Rucker, Alabama. He received his bachelor and master of science degrees in mathematics from Lamar University, his master of science degree in statistics from Texas A&M University, and completed all Ph.D. course work requirements in statistics at the University of Southwestern Louisiana. For the past ®ve years, Malaeb has been the senior statistical consultant for the U.S Environmental Protection Agency's Environmental Monitoring and Assessment ProgramÐEstuaries (EMAP-E) in Gulf Breeze, Florida. His primary responsibilities are to ensure that EMAPE data are collected and analyzed according to its probability-based design. Malaeb also compared EMAP-E data from two different coastal regions, the Louisianian and Virginian Provinces, using structural equation modeling. J. Kevin Summers has been conducting research in systems ecology and monitoring of estuarine ecosystems for the past twenty years. His contributions have primarily been in the areas of developing methods and tools for analysis including the development of categorical time series analysis to examine the indirect effects of climate and anthropogenic effects on marine/estuarine ®sh populations, the development of indices of estuarine condition, and his contribution efforts to develop indices of coastal eutrophication, estuarine integrity, and environmental substainability He presently works with the U.S. Environmental Protection Agency's Of®ce of Research and Development and he is the chief of the Coastal Ecology Branch of the Gulf Ecology Division. In addition, he is the technical Director of the Coastal Resources element of the Environmental Monitoring and Assessment Program. Dr. Summers is presently working with the states of Florida, Alabama, and Mississippi to develop integrated, comprehensive monitoring programs with those states. The primary goal of this effort is to develop a Gulf of Mexico region-wide monitoring program that is embedded within the each of the Gulf States. Bruce H. Pugesek holds bachelor's degrees in zoology and environmental biology from the University of Montana, a master of science degree from the University of Wyoming in zoology and physiology, and a Ph.D in biology from Bowling Green State University. Bruce was an assistant professor of biology at Purdue University before accepting a threeyear NIH postdoctoral fellowship to train in statistics at the Pennsylvania State University. For the past 8 years he has held the position of research statistician at the National Wetlands Research Center. Dr. Pugesek is currently at the USGS/Northern Rocky Mountain Research Center in Bozeman, Montana.
