The Interface Between Theory and Data in 

Structural Equation Models 

By James B. Grace and Kenneth A. Bollen 

Microhabitat

η4

Diversity

η5

Litter

η3

Herbaceous

η2

lake, x1

impound, x2

swale, x3

vhit1, y1

vhit2, y2

herbl, y3

herbc, y4

wlitr, y5

litrd, y6

litrc, y7

rich, y8

0

.26

.75

.98

.92

.64

.90

.81 -.47

.95

1.13

.44

ns

ns

1.0 1.0

.80

1.00

.59

.77

0

Macrohabitat

η1

Microhabitat

η4

Diversity

η5

Litter

η3

Herbaceous

η2

lake, x1

impound, x2

swale, x3

vhit1, y1

vhit2, y2

herbl, y3

herbc, y4

wlitr, y5

litrd, y6

litrc, y7

rich, y8

0

.26

.75

.98

.92

.64

.90

.81 -.47

.95

1.13

.44

ns

ns

1.0 1.0

.80

1.00

.59

.77

0

Macrohabitat

η1

Open-File Report 2006-1363

U.S. Department of the Interior 

U.S. Geological Survey 
U.S. Department of the Interior 

DIRK KEMPTHORNE, Secretary 

U.S. Geological Survey 

Mark D. Myers, Director 

U.S. Geological Survey, Reston, Virginia 2006 

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Suggested citation: 

Grace, J.B., and Bollen, K.A., 2006, The interface between theory and data in structural 

equation models: U. S. Geological Survey Open-File Report 2006-1363, 33 p. 

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Although this report is in the public domain, permission must be secured from the individual 

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2
The Interface Between Theory and Data in 

Structural Equation Models 

By James B. Grace1 and Kenneth A. Bollen2

 

 1 U.S. Geological Survey, National Wetlands Research Center, 700 Cajundome Blvd., Lafayette, LA 70506. 

 2 Department of Sociology, CB 3210 Hamilton Hall, University of North Carolina, Chapel Hill, NC 27599. 

Abstract 

Structural equation modeling (SEM) holds the 

promise of providing natural scientists the capacity to 

evaluate complex multivariate hypotheses about 

ecological systems. Building on its predecessors, path 

analysis and factor analysis, SEM allows for the 

incorporation of both observed and unobserved 

(latent) variables into theoretically based probabilistic 

models. In this paper we discuss the interface 

between theory and data in SEM and the use of an 

additional variable type, the composite, for 

representing general concepts. In simple terms, 

composite variables specify the influences of 

collections of other variables and can be helpful in 

modeling general relationships of the sort commonly 

of interest to ecologists. While long recognized as a 

potentially important element of SEM, composite 

variables have received very limited use, in part 

because of a lack of theoretical consideration, but 

also because of difficulties that arise in parameter 

estimation when using conventional solution 

procedures. In this paper we present a framework for 

discussing composites and demonstrate how the use 

of partially reduced form models can help to 

overcome some of the parameter estimation and 

evaluation problems associated with models 

containing composites. Diagnostic procedures for 

evaluating the most appropriate and effective use of 

composites are illustrated with an example from the 

ecological literature. It is argued that an ability to 

incorporate composite variables into structural 

equation models may be particularly valuable in the 

study of natural systems, where concepts are 

frequently multifaceted and the influences of suites of 

variables are often of interest. 

Introduction 

The application of path analysis to problems in 

the natural sciences has a long history (Wright, 1921, 

1968). Only recently has the capacity for evaluating 

multivariate hypotheses using the more 

comprehensive approach known as structural 

equation modeling (SEM) attracted the interest of 

ecologists (Johnson and others, 1991; Mitchell, 1992; 

Pugesek and Tomer, 1996; Shipley, 2000; Pugesek 

and others, 2003; Grace 2006). At present, there 

appears to be a rapidly growing interest in SEM in 

the natural sciences. Reasons that SEM may prove to 

be of utility to researchers studying natural systems 

include its capacity for representing and testing 

multivariate hypotheses and its flexibility as a 

statistical modeling framework. Several of the recent 

suggestions for how to advance ecological sciences 

given by Belovsky and others (2004), such as a better 

integration of theory and empirical evidence and the 

need to study multiple causes simultaneously, are 

enhanced by SEM. It can be argued that the 

traditional univariate hypothesis testing framework 

limits our ability to study interacting systems and that 

SEM may provide some relief from this limitation. 

Along with the potential utility of SEM, however, are 

certain challenges to matching models to theories. 

A key feature of SEM is the capacity to include 

latent (unobserved) variables along with observed 

ones in models. The development of an appropriate 

structural equation model (SE model) is driven by the 

nature of the theoretical constructs or ideas that 

represent the mechanisms of interest, as well as the 

variables measured and the nature of the data. Often, 

general concepts are represented in SE models by 

using latent variables. In such models, latent 

variables are often used to represent underlying 

3
causes, while observed variables represent the effects 

(manifestations) of the latent factors. However, a 

growing body of literature indicates that in many 

cases, constructs are more appropriately modeled in a 

fundamentally different way, with observed variables 

representing a collection of causes rather than 

manifestations of effects. Frequently, when 

constructs and the available indicators (indicators are 

observed variables related to a construct) possess this 

sort of relationship, the constructs are more 

appropriately modeled by using composite variables, 

which differ in their properties and interpretations 

from conventional latent variables. 

As early as 1964, Blalock pointed out that the 

classic representation of latent variables influencing 

their indicators is not always appropriate. In 

particular, indicator variables might be determinants 

of the construct rather than effects of it. Blalock 

(1964) referred to the former, classic indicators as 

“effect” indicators since the indicators are effects of 

the latent variable. The latter indicators are “causal” 

indicators since these indicators are modeled as 

causes instead of effects. Periodically, practitioners 

of SEM have continued to state the need for careful 

consideration of the best way to represent theoretical 

constructs in models. Bollen (1984) cautioned that 

effect indicators should have “internal consistency” 

(i.e., they depend on a single entity and, therefore, 

would be expected to be correlated and joint 

responsive) while causal indicators need not. 

A number of authors have continued to discuss 

the specification of constructs in SE models (Bollen 

and Lennox, 1991; MacCallum and Browne, 1993), 

and there has been an increased interest in the subject 

emerging in recent years (Bollen and Ting, 2000; 

Diamantopoulous and Winklhofer, 2001; Edwards, 

2001; Fayers and Hand 2002; Jarvis and others, 2003; 

Williams and others, 2003). This literature has 

provided a number of useful insights, diagnostic 

procedures, and approaches to the question of how to 

treat causal indicators. A related issue is the treatment 

of constructs composed entirely by causal indicators, 

which may invoke the researcher to use composite 

variables to represent relationships. A clear, 

comprehensive, and satisfying treatment of 

composites has just begun to emerge (Grace and 

Bollen in press). In this report we expand on the main 

issues associated with using composites in models 

discussed by Grace and Bollen (in press) and provide 

both diagnostic and solution procedures. Further, we 

use an ecological example to demonstrate how 

composites can be used to represent constructs in SE 

models that match up with general theories. 

Framework and Terminology 

A consideration of the subject of composites 

requires that we have an adequate framework and 

sufficient terminology to express clearly the relevant 

issues. The theoretical background for the analysis 

plays an important part in how we proceed. It is 

typical in SEM applications that the investigators 

wish to evaluate some a priori hypotheses. Indeed, 

one of the dominant features of SEM is its capacity 

for leading to the rejection of multivariate hypotheses 

based on comparisons between data and model 

expectations (e.g., using chi-square goodness of fit 

tests based on likelihood ratios). It is also 

characteristic of SEM practice that the analysis is 

based on evaluating an a priori model rather than a 

null hypothesis. Thus, we expect to give a high 

priority to theoretical knowledge in both the 

processes of constructing and interpreting our SE 

models. 

One way to begin the modeling process is by 

defining the relevant theoretical concepts (also 

known as “constructs”). The meaning that a person 

assigns to a construct is given in its theoretical 

definition. A theoretical definition explains in as 

simple and precise terms as possible the meaning of a 

construct (Bollen, 1989, p. 180), as well as the 

number of distinct components or aspects of the 

construct. The “construct model” represents the 

hypothesized relationships among constructs based 

on known or expected underlying mechanisms. 

Figure 1 provides an overall framework for 

representing the relationships between general 

theories, construct models, and SE models. The 

construct model represents the researchers’ 

theoretical ideas in a form that facilitates the 

construction of the specific SE model(s) by 

expressing the general, expected statistical 

associations. Thus, the construct model can be seen 

as a useful bridge between the abstractions of theory 

and the realities of data modeling. In the development 

of SE models that are meant to relate to general 

theories, there is the need to consider carefully the 

relationships between the available data and the 

constructs of interest. This paper is largely about 

these relationships between construct models and SE 

models and the roles that composite variables might 

play. 

The specification of SE models has typically 

involved the use of both observed and latent 

variables. Here we define a “latent variable” as one 

that represents a factor which we believe to exist and 

be relevant to our analysis but which is unmeasured 

(Bollen, 2002). Latent variables are often used in SE 

models to represent such things as the true value of a 

4
The MxL block shown in figure 2 represents the 

situation in which manifest variables have causal 

influences on a latent (unmeasured) variable. In this 

case, our figure implies that the latent variable does 

indeed exist independent from our data despite the 

fact that we do not have measures of it. Graphically, 

this property of independent existence for the latent 

variable is represented by the presence of an error 

variance ζ1, which implies that while the xs have 

causal effects on η1, they do not completely 

determine it. 

parameter, an underlying cause, or an important piece 

of missing (unmeasured) information. Latent 

variables play a special role in SEM because they 

represent a kind of bridge between observed data and 

theoretical generalization. As an example of how 

latent variables might be used, if a construct is 

envisioned to have distinct dimensions, then each 

dimension might be represented by a single latent 

variable. In such a case, a two-dimensional construct 

would be represented by two latent variables in the 

SE model. 

“Composites” are another type of variable that 

can be specified in SE models, and they represent 

collections of causes. Composites can reflect the 

effects of sets of either manifest or latent variables. A 

composite can, thus, be referred to as either a 

“composite of manifest variables” or a “composite of 

latent variables.”3

 We also recognize that the issue of 

how to represent a construct deals with the 

interrelations between a latent or composite variable 

and the associated variables that give it meaning. To 

aid our discussion, we define a “block” as the basic 

unit to consider in an SE model and illustrate several 

types of blocks in figures 2 and 3. The reader should 

also be aware that we use the terms “observed” and 

“manifest” interchangeably to refer to the directly 

measured variables. The term “indicator” refers to an 

observed or manifest variable that is linked to either a 

latent or composite variable. 

In contrast to the MxL block is the MxC block, 

in which we have a composite summarizing the 

collective influences of x1 - x3. Since the error 

variance is specified to be 0 for a composite, this 

condition signifies that it is completely determined by 

its causes. We can distinguish two kinds of 

composites; one is the “fixed composite”. In this 

type, the loadings from causes are specified a priori 

to have particular values by way of definition. An 

ecologically relevant example would be the 

“importance value,” which is defined as the sum of 

the relative density, relative abundance, and relative 

frequency (usually for a species within a 

community). A second type of composite can be 

referred to as a “statistical composite”, which is the 

type considered in this paper. A statistical composite 

represents the collective effects of a set of causes on 

some response variable(s). In simplistic terms, the 

statistical composite is similar to a multiple 

regression predictor and usually represents some 

weighted combination of causal influences that 

maximizes variance explanation in one or more 

response variables (not shown in fig. 2). 

In Figure 2 we see that given a set of three 

manifest variables, x1 - x3, there are various ways 

they can be interpreted (and thus represented) in an 

SE model depending on how the data relate to the 

theoretical construct. The most common 

representation is the LxM block, where directed 

paths run from latent to manifest variables. In this 

block type, the latent variable is postulated to cause, 

at least in part, the correlations/covariances among 

the manifest variables. A great deal has been written 

about the properties of LxM blocks and they form 

the backbone of latent variable modeling (Bollen, 

1989). Some of the expected properties of blocks of 

this type will be considered later in the paper, along 

with diagnostics that help evaluate whether either 

theory or empirical relations are consistent with this 

representation of a construct. An alternative 

terminology that has been used for this case is to 

refer to a “latent variable with effect indicators.” We 

note that there is a long history of the use of the 

opposing terms “effect indicators” (as in the LxM 

block in figure 2 and “causal indicators” (as in the 

MxL block in figure 2) and include that usage in our 

discussion. 

Figure 3 presents more complex examples of 

blocks where relationships between latent variables 

are considered. The LxL block represents effects of 

latent causes on a latent response, where all latent 

variables have effect indicators. This is another 

standard block type found in models discussed in 

most introductory SEM treatments. What is 

represented by this block structure is that the causal 

mechanisms involve direct relations among latent 

factors and that we can understand the covariances 

among a set of x and y variables based on that 

mechanism. Also implied is that the effect indicators 

associated with each latent variable (e.g., x1 - x3) 

serve as multiple measures for that latent factor. 

Because the flow of causation is from the latent 

factors to the indicators, error terms are used to 

represent how well the indicators are predicted by the 

model, while the latent factors are presumed to be 

free from measurement error. 

 

3

 It is even possible to have a composite that is a 

mixture of observed and latent variables. 

5
General Theories

Empirical

Observations

Theoretical

Explorations

Construct Models

Structural Equation

Model

General Theories

Empirical

Observations

Theoretical

Explorations

Construct Models

Structural Equation

Model

Figure 1. Overview of the relationships between general theories, construct models, and structural equation models. 

Note that what are referred to here as general theories are sometimes referred to as hypotheses, for example, in the 

case of the "intermediate disturbance hypothesis", or models, as in the "dynamic equilibrium model". Construct 

models refer to hypothesized relations among theoretical constructs and take a form that defines how particular 

structural equation models relate to general theoretical ideas. The structural equation model itself is the specific 

formulation of the equations used to describe relations among the included variables. 
ξ1

x1

x2

x3

δ1

δ2

δ3

L->M block

x1

x2

x3

ζ1

η1

M->L block

x1

x2

x3

0

M->C block

ξ1 C1

x1

x2

x3

δ1

δ2

δ3

L->M block

x1

x2

x3

ζ1

η1

M->L block

x1

x2

x3

0

M->C block

C1

Figure 2. Representation of blocks specifying relationships between manifest (observed) indicators (in boxes) and 

latent variables (in circles) or composite variables (in hexagons). “L” refers to a latent variable or unmeasured cause, 

“M” refers to manifest variables, and “C” refers to a composite, which is a direct product of some set of causes and 

has zero error variance. 

 

ξ1

x1

x2

x3

η1 y1

δ1

ε1

δ2

δ3

ζ1

L->L block

ξ2

x4

x5

x6

δ4

δ5

δ6

ξ1

x1

x2

x3

η1

δ1

δ2

δ3

ζ1

L->L(no M) block

ξ2

x4

x5

x6

δ4

δ5

δ6

ξ1

x1

x2

x3

δ1

δ2

δ3

0

L->C block

ξ2

x4

x5

x6

δ4

δ5

δ6

η1

ξ1

x1

x2

x3

η1 y1

δ1

ε1

δ2

δ3

ζ1

L->L block

ξ2

x4

x5

x6

δ4

δ5

δ6

ξ1

x1

x2

x3

η1

δ1

δ2

δ3

ζ1

L->L(no M) block

ξ2

x4

x5

x6

δ4

δ5

δ6

ξ1

x1

x2

x3

δ1

δ2

δ3

0

L->C block

ξ2

x4

x5

x6

δ4

δ5

δ6

η1

Figure 3. Representation of blocks specifying relationships between latent causes and latent or composite variables. 

Block terminology is as described in the caption of figure 2 except that here in this figure composites of latent 

variables are in circles, though still with zero error variance.

7
The LxL(no M) block in figure 3 represents 

causal influences from latent variables that have 

effect indicators to a latent variable for which there is 

no effect indicator. This block type resembles the 

MxL block in figure 2, except for the fact that latent 

causes replace the manifest ones. Similarly, the LxC 

block in figure 3 is analogous to the MxC block in 

figure 2. The relevant situations for these 

representations are illustrated in the example 

application below. 

An Ecological Example: 

Colonization of Forest Patches by 

Understory Herbs 

Rather than continue to elaborate on constructs, 

blocks, and composites in the abstract, the remaining 

development of ideas (as well as the development of 

formal notation), will be conducted in the context of 

a specific example (hereafter referred to as Example 

1). The reader should keep in mind that while not 

representing an exhaustive treatment of the subject, 

the example chosen and procedures discussed in 

conjunction are broadly applicable to the general 

issues associated with representing constructs in SE 

models. 

Background 

In a recent paper, Verheyen and others (2003) 

addressed the influences of factors on the ability of 

herbaceous plant species to recolonize forest stands 

in a fragmented landscape. Their work can be viewed 

as an effort to evaluate a dichotomy of preexisting 

theories. On the one hand, island biogeographic 

theory suggests that successful colonization will be 

limited by factors such as the age of an island (in this 

case, a forest stand is analogous to an island) and the 

distance to a source population. On the other hand, 

the local conditions in a forest stand (such as soil 

conditions and the abundance of competitors) are 

frequently believed to be the ultimate limiting factors 

for successful colonization. The relative importance 

of effects is what is of interest in this case. 

In the case of Example 1, Verheyen and others 

(2003) examined colonization success and related it 

to forest stand attributes in a nature reserve in 

Belgium that has had a long history of agricultural 

use and forest fragmentation. In this landscape, much 

of the original forest cover was gradually removed, 

and in places, discrete stands of trees were replanted, 

creating a mosaic of ancient and contemporary 

stands. Land-use history of the 360-ha preserve was 

reconstructed by using maps dating back into the 

1700s and aerial photographs dating back to the 

1940s. Using the historical records, the age of each 

forest stand in the landscape was determined. 

Extensive surveys of forest herb species throughout 

the entire preserve were conducted in 1999, along 

with characterizations of forest understory cover and 

soil conditions in individual stands. Together, the 

sources of data allowed Verheyen and others (2003) 

to estimate the number of colonizations by each herb 

species in contemporary forest stands and the 

distance to the nearest population in the ancient 

stands, which was presumed to be a possible source 

for colonization. A total of 180 forest stands, with 

ages from 1 to 195 years old, were in the final data 

set. All species that were included in the analyses are 

obligate forest understory perennials. 

Verheyen and others’ (2003) initial model of 

constructs and the relations among them are shown in 

figure 4. As represented here, the authors were 

interested in differentiating between the effects of 

two exogenous (external) influences, landscape 

properties and soil conditions, on competitor 

abundance and ultimately colonization success (the 

latter two of which are considered endogenous or 

internal to the model). The authors used a two-stage 

analysis to evaluate the model in figure 4, first 

deriving composite indices, followed by a path 

analysis of relations among constructed indices using 

SEM. Here we wish to provide a more formal 

illustration of how their constructs and the relations 

to associated indicators can be evaluated. We also 

wish to discuss issues related to the solution of SEMs 

containing composites. We use a selective subset of 

the data analyzed by Verheyen and others (2003), and 

our purpose is not to confirm or contradict the 

biological conclusions of the original analysis but 

rather to illustrate statistical methods. Data 

characteristics were considered in this analysis, and 

corrective actions (e.g., transformations) were taken 

where needed, though the particulars are not 

discussed further here. 

In Example 1, our starting point is the initial 

conceptual (construct) model summarizing the core 

theoretical questions posed (fig. 4). We start by 

considering the various ways we might address the 

hypotheses implied by figure 4. To address this, we 

must consider the available indicators and then the 

linkages between indicators and constructs. In this 

study, Verheyen and others (2003) measured eight 

variables in the process of developing indicators for 

the four constructs of interest. Two 

8
Soil

Conditions

Landscape

Properties

Competitor

Abundance

Colonization

Success

Soil

Conditions

Landscape

Properties

Competitor

Abundance

Colonization

Success

Figure 4. Initial conceptual model showing the proposed relations among constructs (in dashed rounded rectangles) 

relating herb colonization success to the characteristics of forest stands (from Verheyen and others 2003).

distance

stand age

texture

moisture

pH

col. freq.

cover

?

?

Soil

Conditions

Landscape

Properties

Competitor

Abundance

Colonization

Success

distance

stand age

texture

moisture

pH

col. freq.

cover

?

?

Soil

Conditions

Landscape

Properties

Competitor

Abundance

Colonization

Success

Figure 5. Associations between indicator and constructs for Example 1. 
indicators of landscape conditions were measured, 

the estimated age of each forest stand and the 

distance from each reforested target stand to the 

nearest source patch. Three indicators of soil 

conditions were measured, including soil texture, soil 

moisture, and soil pH. Two indicators of competitor 

abundance were measured, the cover of herbaceous 

competitors and the abundance of understory shrubs. 

For the herb species whose colonization will be 

considered in this example application (Lamium 

galeobdolon), shrub abundance was not significantly 

related to colonization success (though herbaceous 

cover was). So, to simplify our example, we only use 

a single indicator for competitor abundance, which is 

herbaceous cover. The proportion of sampling points 

where a species was found in a forest stand serves as 

the single indicator of colonization success. 

Figure 5 shows which measured indicators are 

associated with particular constructs based on the 

theoretical reasoning of the authors. In this initial 

representation, we make no attempt to express the 

structure of the blocks, therefore, constructs are 

related to observed indicators with nondirectional 

lines rather than directional arrows. Error terms are 

specified for Competitors and Colonization, while the 

diagram is ambiguous as to whether or not Soil and 

Landscape possess error terms because the directions 

of influences are not given. 

Possible Model Structures 

There are several possibilities for how the 

relations in figure 5 might be developed into SE 

models. The primary determining factor for deciding 

which of the possibilities is most appropriate is the 

nature of the causal relations. There are, in fact, a 

great many possibilities that could be considered, 

though here we will only discuss a few key ones of 

contrasting type. One of these possibilities is that the 

most appropriate model is a classic latent variable 

model of the sort shown in figure 6. Here we see that 

the block structure used to represent relations 

between indicators and individual latent variables is 

the LxM block type. In this model, which will be 

referred to as Model A, two of the latent variables, 

Competitors and Colonization, are associated with 

one effect indicator each. As appropriate for such 

relationships, arrows are directed from the causes 

(the latent variables) to the indicators. The two other 

latent variables in this model, Soil and Landscape, 

are associated with multiple effect indicators. Causal 

relations are represented among latent variables, and 

direct paths between manifest variables are not 

specified. 

A second possibility is that the multiple 

indicators associated with Soil and Landscape are 

causal indicators. This type of structure is illustrated 

in Model B (fig. 7). Here, the arrows pointing from 

pH, moisture, and texture to Soil represent the 

presumption that the measured soil properties jointly 

determine the soil's influences rather than vice versa. 

A similar logic applies to distance and patch age, 

which are presumed to contribute to Landscape in 

this model. For the sake of simplicity, we assume in 

this case that the composites Soil and Landscape are 

completely defined by their indicators. Thus, the 

multi-indicator blocks are of the MxC type, and both 

Soil and Landscape are represented as observed 

composites with zero error variances. 

If we assume that the causal indicators in Model 

B are themselves imperfect measures, which is very 

reasonable in this case, then we might represent the 

situation by using latent variables for the true values 

of pH, moisture, texture, distance, stand age, 

competitors, and colonization. Given the available 

data in this example, each of these latent variables 

would be matched with a single indicator, being the 

observed manifestations of those properties. In this 

paper we will demonstrate the principles of 

composites while using the simpler case where the 

indicators x1 - x5 are assumed to be measured without 

error. It should be understood by the reader that the 

method of composites can be extended to the case 

where latent variables are used to represent the true 

values of all our measured properties. 

Our discussion of the range of possible ways to 

represent the relations in Figure 5 would be 

incomplete without considering another alternative, 

which we will refer to as Model C (fig. 8). In this 

model, our two multidimensional constructs, soil 

conditions and landscape properties, are represented 

simply as groups of variables. 

Formal Representations 

To formalize our discussion of the relationships 

embodied in our models, we present the three 

characteristic equations of the LISREL model 

(Bollen, 1989); 

, (1) 

, (2) 

and 

, (3) 

where x and y are vectors of observed indicators of 

exogenous (predictor) and endogenous (response) 

10
Soil

ξ1

Landscape

ξ2

Competitors

η1

Colonization

η2

distance, x4

stand age, x5

texture, x3

moisture, x2

pH, x1

col. freq., y2

cover, y1

Soil

ξ1

Landscape

ξ2

Competitors

η1

Colonization

η2

distance, x4

stand age, x5

texture, x3

moisture, x2

pH, x1

col. freq., y2

cover, y1

Figure 6. Model A. All observed (manifest) variables (in boxes) are represented as “effect indicators” associated 

with latent variables (in ovals). Thus, all blocks are of the LxM type. “col. freq” refers to colonization frequency. 

Competitors

η3

Colonization

η4

distance, x4

stand age, x5

texture, x3

moisture, x2

pH, x1

col. freq., y2

cover, y1

Landscape

η2

Soil

η1

0

0

Competitors

η3

Colonization

η4

distance, x4

stand age, x5

texture, x3

moisture, x2

pH, x1

col. freq., y2

cover, y1

Landscape

η2

Soil

η1

0

0

Figure 7. Model B. Soil conditions and landscape properties are represented as components of MxL blocks, with 

the associated manifest variables (x1

 - x5) representing causal indicators. “col. freq” refers to colonization 

frequency. 

11
distance, x4

stand age, x5

texture, x3

moisture, x2

pH, x1

Soil Conditions

Landscape Properties

Competitors

η1

Colonization

η2

col. freq., y2

cover, y1

distance, x4

stand age, x5

texture, x3

moisture, x2

pH, x1

Soil Conditions

Landscape Properties

Competitors

η1

Colonization

η2

col. freq., y2

cover, y1

Figure 8. Model C. A partially reduced form of Model B omitting explicit consideration of collective soil and 

landscape effects. 

12
latent variables, ξ and η are vectors containing the 

individual exogenous and endogenous latent 

variables, Λx and Λy are vectors of coefficients 

relating indicators to latent variables, B and Γ are 

coefficient matrices for effects of endogenous and 

exogenous latent variables on endogenous latent 

variables, δ and ε are vectors of measurement errors 

for x and y, and ζ is a vector of errors for the η

variables. For exogenous latent variables, ξ, their 

variances are represented by the diagonal elements of 

the matrix Φ while the off-diagonal elements of the 

matrix are the covariances. The variances of the error 

terms for the endogenous latent variables (ζ) are 

contained within the diagonal elements of the Ψ

matrix, while the off-diagonal elements of that matrix 

represent any covariances among errors (normally 

assumed to be zero). Error variables (in δ and ε) are 

expected to be uncorrelated with ξ and η. However, it 

is possible to model correlations among errors, 

typically represented by nonzero off-diagonal 

elements in the matrices representing cross products 

among measurement errors, θδ and θε. 

This notation is sufficiently general that we can 

use it to express the various models with causal 

indicators and composites that we discussed 

previously. We can incorporate the causal indicators 

in a model such as Model B by defining a separate ξ

variable for each causal indicator and setting its 

factor loading to 1.0 and its error to zero. For 

instance, if x1 to x3 are causal indicators, we would 

have 

. (4) 

as the part of the measurement model that 

corresponds to the causal indicators. This 

representation explicitly shows that we are assuming 

no measurement error in the xs. 

Composites are included by specifying the 

coefficients in Γ that correspond to the variables that 

make-up the composite. So if the causal indicators 

from equation 4 form a composite and η1 is the 

composite, we would estimate or fix the values in the 

first three columns of the first row of Γ to create the 

η1 composite. 

Estimation of Composites 

Previous authors have typically estimated 

composites with nonzero error variances (e.g., 

MacCallum and Browne, 1993). In a sense, such 

composites are actually types of latent variables in 

that they are not fully measured by the data. As a 

practical matter, for such “latent composites” in 

blocks, such as MxL and LxL(no M), we usually 

have insufficient information to estimate the true 

variance for the latent variables since they do not 

have at least one effect indicator. In these situations, 

we cannot estimate the model as originally 

formulated and we must look for alternatives. One 

possibility is to replace MxL with MxC and 

LxL(no M) with LxC, where C refers to a 

composite variable. In the case of MxC, the block 

describes a composite of manifest variables, while in 

the case of LxC, the block describes a composite of 

latent variables. Such composites are feasible 

elements of models, being both estimable and 

straightforward to interpret. This contrasts with the 

problematic characteristics of composites having 

non-zero errors (MacCallum and Browne, 1980). 

Composites are, in all cases that we describe, 

considered endogenous (η) variables but they are 

endogenous variables completely determined by the 

variables that influence them. If a composite does not 

fully represent the construct with which it is 

associated, we must recognize that it is an imperfect 

measure of that construct. With regard to the example 

of forest herb colonization, if stand age and distance 

from source constitute the predominant landscape 

features of importance for the model, then a 

composite formed from their effects will have a very 

general meaning. If stand age and distance from 

source are just two of many unique landscape 

properties of importance to herb colonization, our 

composite will simply represent their joint effects and 

will be a poor representation of a general landscape 

effect. Typically, unless an effort has been made to 

sample the many facets of a construct, any composite 

derived from the measures of that construct should be 

understood to represent the collective effects of its 

components, regardless of the label placed on that 

construct. In such a case, an error estimate of zero is 

appropriate. Despite their limitations, composites can 

be useful as a means of representing certain types of 

constructs, facilitating general conclusions, and 

managing model complexity. 

33

22

11

ξ

ξ

ξ

=

=

=

x

x

x

Evaluating the Different Model 

Possibilities - Theoretical 

Considerations 

Both theoretical information and empirical 

information contribute to an evaluation of the 

suitability of model architecture for a given situation. 

13
As stated earlier, there are a great number of 

possibilities for how a set of manifest variables may 

be connected. In the example under consideration 

here involving herb colonization of forest stands, 

initial guidance comes from the theory specified in 

figure 4 and the associations in figure 5. We begin 

our evaluation of the applicability of Models A-C in 

this case by considering the causal relations between 

latent variables and indicators. 

Borrowing, with modification, from Jarvis and 

others (2003), we recognize certain conceptual 

criteria for deciding whether a block should be 

modeled in the traditional LxM format (e.g., Model 

A) or whether MxC blocks are more appropriate 

(e.g., Model B). Three kinds of questions can be 

considered to gauge the theoretical basis for forming 

models such as Model A versus Model B: (1) What is 

the direction of causality? (2) Are the indicators in a 

single block interchangeable? (3) Is there an 

expectation that indicators in a block should covary 

(e.g., because they are jointly controlled by an 

underlying factor)? 

For the first question, we must begin by asking 

whether Soil has causal influences on pH, moisture, 

and texture. In other words, we are asking whether 

the variation among stands in soil conditions is such 

that pH, moisture, and texture are controlled by a 

common difference among soils. In such a case, we 

view that causation flows from the general construct 

to the indicators. Alternatively, if it is the case that 

pH, moisture, and texture behave independently from 

one another, then causation flows from the indicators 

to the latent variable. In this study, Verheyen and 

others (2003) felt that causation flows from the 

indicators and could be represented by a MxC block. 

A similar determination was made for the latent 

variable Landscape which is envisioned by the 

authors to be the result of largely distinct influences 

by certain landscape properties. 

The second question to be considered is whether 

the indicators in a block are interchangeable; if so, 

then they constitute redundant measures and are 

likely to represent effect indicators consistent with a 

LxM block. Stated in another way, we might ask if 

dropping one of the indicators in a block alters the 

meaning of the construct. For both Soil and 

Landscape, the indicators in the separate blocks 

would seem to be unique based on the Verheyen and 

others’ (2003) model specification. For the construct 

Landscape, an MxC block specification would 

appear to be indicated based on this criterion. The 

age of a patch is conceptually quite distinct from its 

distance from a source population. If data for the 

distance to source populations was absent, the 

construct would reduce to solely an age dimension. 

For Soil, there are precedents for specifying common 

soil factors of the LxM type (Grace, 2003; Weiher 

and others, 2004). In this case, Verheyen and others 

(2003) felt that the individual indicators (pH, texture, 

and moisture) reflect separate dimensions of Soil and 

are best represented by an MxC block structure. 

The third question is whether the indicators are 

expected to covary. Addressing this question serves 

as an additional way of evaluating the flow of 

causation in a block. If indicators are under common 

causal control by a latent factor, then when that latent 

factor varies, the indicators should all likewise vary. 

Such a situation would imply that an LxM block 

would be an appropriate means of modeling the 

situation. On the other hand, if causation flows from 

the indicators to the latent variable, there is no basis 

for expecting correlations among indicators since 

their causes are not specified. We should, therefore, 

recognize that a correlation among indicators does 

not inform us as to the direction of causal flow, 

though a lack of correlation among indicators would 

contraindicate the prospect that a block should be of 

the LxM form. For the herb colonization example, 

Verheyen and others’ (2003) presentation did not 

produce any expectations of whether indicators in 

multi-indicator blocks would correlate or not. So, 

again, by this criteria, it would seem that Soil and 

Landscape are constructs best represented as MxC 

blocks given the available indicators. 

Collectively, it would seem that the expectations 

of Verheyen and others (2003) as reflected by the 

three questions would lean towards a specification of 

the multi-indicator blocks as MxC for both Soil and 

Landscape. This assessment does not guarantee that 

this is the correct causal structure nor does it override 

the possibility that empirical characteristics of the 

data might imply a different block structure. What is 

most important to realize is that there are logical tests 

for developing, justifying, and interpreting block 

structure and that automatically presuming a 

conventional model structure (LxM) is not justified. 

Neither is it advisable to simply rely on data 

properties to make the determination as to whether a 

construct is best represented by a particular structure. 

Evaluating the Different Model 

Possibilities - Empirical Evaluations of 

Blocks 

It is often not known with absolute confidence 

what the causal flow is for a set of manifest variables. 

This condition is especially common for applications 

in the natural sciences where latent variables are only 

beginning to receive attention. In SEM practice, 

logical and statistical approaches to evaluating the 

appropriateness of model structure exist (see 

14
numerous illustrations in Bollen, 1989). We will 

address overall model fit later. For now, our 

considerations relate to the empirical characteristics 

of individual blocks. Ultimately these are of great 

importance, for no matter what logic one brings to a 

model, if the indicators in a block do not behave like 

effect indicators, then it will be unprofitable to model 

them as such. Reviews of SEM studies in certain 

disciplines (e.g., Jarvis and others, 2003) suggest that 

researchers commonly misjudge the empirical 

support for the causal direction specified in their 

models. The greatest culprit is the assumption that 

blocks are of the LxM form without seriously 

considering the use of MxL and LxL(no M) blocks 

(or MxC and LxC). In the example being discussed 

here, Verheyen and others (2003) assumed the form 

(MxC) for their multi-indicator blocks. It seems, 

however, that the logical justification for this 

assumption is stronger for the Landscape construct 

and less definitive for the Soil construct. In this 

section, we consider some of the empirical properties 

that inform us as to whether either block has 

properties consistent with LxM form. 

Table 1 presents correlations for colonization 

frequency and other manifest variables (note that the 

covariances, the basis for SEM analysis, can be 

reconstructed from the correlations and standard 

deviations) for the species Lamium galeobdolon. As 

Bollen and Ting (2000) have described, there is an 

expectation that a set of effect indicators associated 

with a single latent variable in a LxM block will be 

correlated, as implied by equation 1. For such a set of 

indicators, their degree of correlation will depend on 

the strength of the common influence of the latent 

cause relative to their total variances. So, for Model 

A, we would expect conspicuous correlations among 

soil pH, moisture, and texture because of the joint 

influence of Soil on those specific factors. We would 

have similar expectations for a strong correlation 

between distance and patch age. Again, the degree of 

correlation expected would depend on the relative 

importance of the individual errors for each indicator, 

though for a reliable set of indicators, correlations 

should be at least moderately high. In contrast, for a 

set of causal indicators associated with a single latent 

variable in a single MxC block, there is no basis for 

expecting any particular correlation pattern. None of 

the equations that apply to Model B imply common 

causal influence on sets of causal indicators. So, a set 

of causal indicators either may or may not 

intercorrelate in such a case, since our equations do 

not describe their interrelationships, except that they 

are classified as being of common interest. 

Inspection of table 1 reveals a correlation of 

moderate magnitude between age and distance of 

-0.5934. In addition, the correlations between these 

variables and the other variables in the model are 

approximately of the same magnitude. Thus, based 

on a crude inspection of correlations in the matrix, 

we are unable to rule out the possibility that either a 

LxM or MxC block structure would be consistent 

with the empirical characteristics of the indicators 

related to Landscape. 

The correlations between pH, moisture, and 

texture are 0.0265, 0.1324, and 0.5767 respectively. 

The low magnitude of correlations between pH and 

the other indicators in the block suggests that these 

three soil properties would not be likely to represent 

redundant measures of the sort normally found in 

LxM blocks. A method for formally evaluating 

causal versus effect indicators has been proposed by 

Bollen and Ting (2000) based on a simultaneous 

analysis of vanishing tetrads in models.4

Correlations/covariances among a set of truly 

redundant indicators in a LxM block should possess 

the mathematical property of vanishing tetrads, with 

a tetrad being the difference between the products of 

pairs of covariances among four random variables. It 

can be said in this case that the pattern of correlations 

among pH, moisture, and texture does not appear to 

be consistent with such a block structure. To increase 

our confidence in this assessment, we applied Bollen 

and Ting’s (2000) vanishing tetrad test to more 

formally compare Model A to Model B. Models B 

and C do not imply any vanishing tetrads whereas 

Model A does. We used the SAS macro for vanishing 

tetrads provided in Hipp and others (2005) and found 

that the fit of Model A is significantly worse than is 

the fit of Model B or C (p < 0.05). 

Comparing Results for Different Models 

Results for Model A 

The data presented in table 1 are sufficient to 

provide for an evaluation of all models discussed in 

this paper. Analyses were conducted by using the 

program Mplus (Muthén and Muthén, 2004). We 

used maximum likelihood estimation in all analyses. 

For the purposes of this illustration, we assume a 

modest, nonzero degree of measurement error for the 

 

4

 Spirtes and others, 1993 have done important work 

in exploratory tetrad analysis, providing methods that 

can help to generate plausible model structures for a 

data set. Here we are interested in confirmatory tetrad 

analysis where we are testing specific hypotheses by 

using a simultaneous test statistic developed in 

Bollen (1990). 

15
single-indicator measures (cover and colonization 

frequency) of 10 %. 

The specification of a nonzero percentage of 

measurement error deserves discussion. Here we 

specify 10 % measurement error primarily to 

illustrate a common use for single-indicator latent 

variables. It is widely recognized among statisticians 

that the assumption of perfect measurement (zero 

measurement error) is frequently unjustified and a 

common source of bias for model parameters. In 

SEM practice, there exist both a readily available 

means of adjusting for known measurement error (the 

specification of measurement error estimates using 

latent variables) and a tendency to make such 

adjustments. When single-indicator latent variables 

are specified, one must specify the most reasonable 

estimate for the degree of measurement error. Often, 

though not always, this is some nonzero percent of 

observed variance. It is best if this estimate comes 

from empirical evidence, such as from replicate 

samples or a calibration data set. However, 

sometimes an estimate is made based on subjective 

reasoning, which, although approximate, is viewed as 

less arbitrary than assuming zero error and likely to 

produce a lower degree of bias in parameters than 

achieved from a zero error estimate. We should say 

that the specification of measurement error in this 

example is not a requirement for the use of 

composites, nor does it play a major role in our 

presentation. That said, those who choose to specify 

nonzero measurement error in SE models should be 

aware of the impacts on model results and their 

stability (Bollen 1989). 

The fitting of data to Model A (fig. 6) resulted in 

poor fit based on model chi-square statistics, error 

variances for indicators, and examination of the 

residual covariance matrix. A chi-square of 45.20 

was obtained for the model, with 10 degrees of 

freedom. The associated probability of good fit 

between data and model was found to be less than 

0.00005. Note that since Model A is saturated with 

regard to the structural model (i.e., all paths between 

latent variables are estimated), the inflated chi-square 

does not result from unspecified relations among 

latent variables but instead resides in the 

measurement model. In other words, in this example 

the lack of model fit can be attributed to 

inappropriate relations between latent and manifest 

variables. Standardized parameter estimates (for 

alternatives to the use of conventionally standardized 

parameters, see Grace and Bollen, 2005) for Model A 

are shown in figure 9. Here it can be seen that the 

standardized error variance for pH is 0.99, indicating 

that the model provides no explanation of the 

variance of this variable. Finally, residuals indicate 

that in general, the model does a poor job of 

resolving the covariances among indicators. 

A modification of Model A was evaluated to 

determine the degree to which the lack of fit for pH 

contributed to poor model fit. In this modified model, 

only moisture and texture were used as indicators for 

Soil, while pH was specified to be a single indicator 

for an additional exogenous latent variable, True pH. 

Again, 10 % of the variance of observed pH was 

specified as being measurement error. A chi-square 

of 34.57 with 7 degrees of freedom was obtained for 

this model. The associated probability of good fit 

remained less than 0.00005. Thus, this result 

indicates that the observed lack of fit between model 

and data is spread throughout the measurement model 

and not solely due to a lack of correlation between 

pH and the other soil variables. 

A Digression on Parameter Identification 

in Models with Composites 

It is essential that we consider certain 

complications associated with the estimation of 

models containing composites if we are to overcome 

some of the limitations inherent in earlier 

presentations. These complications have specifically 

to do with being able to avoid problems of model 

underidentification. Model B (fig. 7), because it 

includes composite variables, has a somewhat 

different set of parameters that must be estimated 

compared to a similar model without composites 

(e.g., Model C, fig. 8). As mentioned earlier, since 

composites are defined by having zero error 

variances, the identification of the error variances for 

composites is not an issue. For Model B, associated 

with the composite variables are four paths that 

represent the effects of the composites on the latent 

variables. In this particular case, the relationships 

between the exogenous manifest variables (xs) and 

the composites, •1 and •2, are reduced to single 

effects to each composite from each x. So, for our 

example, the 10 potential paths from the individual x

variables to Competitors and Colonization in Model 

C are replaced in Model B with 5 paths from the x

variables to the two composites plus 4 paths from the 

composites to Competitors and Colonization. Despite 

the net gain in degrees of freedom from this 

substitution, problems remain with parameter 

identification. Ultimately, there is a general problem 

that arises when attempting to identify all paths 

leading to, as well as flowing out from, a composite. 

This problem is similar to the routinely encountered 

problem associated with latent variables with effect 

16
Table 1. Correlations among variables and the standard deviations. "col. freq" refers to colonization frequency. 

col. freq. distance age texture moisture pH cover 

col. freq. 1.0 

distance -0.5785 1.0 

age 0.6424 -0.5934 1.0 

texture -0.2553 0.1844 -0.3146 1.0 

moisture -0.3369 0.4604 -0.3462 0.5767 1.0 

pH -0.0073 -0.0465 -0.0976 0.1324 0.0265 1.0 

cover -0.3423 0.2070 -0.4062 0.3189 0.2952 0.2394 1.0 

std. dev. 0.1867 0.8207 0.3390 0.4588 1.1359 0.7507 0.3017 

17
Soil

ξ1

Landscape

ξ2

Competitors

η1

Colonization

η2

distance, x4

stand age, x5

texture, x3

moisture, x2

pH, x1

col. freq., y2

cover, y1

.83

.69

.57

.73

-.81

.24

.31

-.87

.76

.30

.95

.95

0.1

0.1

.99

.37

.57

.47

.34

Soil

ξ1

Landscape

ξ2

Competitors

η1

Colonization

η2

distance, x4

stand age, x5

texture, x3

moisture, x2

pH, x1

col. freq., y2

cover, y1

.83

.69

.57

.73

-.81

.24

.31

-.87

.76

.30

.95

.95

0.1

0.1

.99

.37

.57

.47

.34

Figure 9. Standardized parameter estimates for Model A. Chi-square for model fit was 45.20 with 10 degrees of 

freedom (sample size = 180) and a p-value < 0.00005, indicating very poor fit of data to model expectations. 

Competitors

η3

Colonization

η4

distance, x4

stand age, x5

texture, x3

moisture, x2

pH, x1

col. freq., y2

cover, y1

.65

.40

-.70

.33

.30

-.68

.75

.46

.95

.95

0.1

0.1

Landscape

η2

Soil

η1

0

0

Competitors

η3

Colonization

η4

distance, x4

stand age, x5

texture, x3

moisture, x2

pH, x1

col. freq., y2

cover, y1

.65

.40

-.70

.33

.30

-.68

.75

.46

.95

.95

0.1

0.1

Landscape

η2

Soil

η1

0

0

Figure 10. Standardized parameter estimates for Model B. Model chi-square was 6.888 with 3 degrees of freedom 

(p = 0.075). 

18
indicators where the scale of the latent variable needs 

to be established. Both of these problems can be 

solved in the same fashion, by specifying a single 

incoming or outgoing relationship so as to establish 

the scale of measurement. 

Despite the fact that parameter identification 

issues can be resolved by specification of select 

parameters, an issue still remains as to the statistical 

significance of the specified pathways. In Model B 

(fig. 7), we set the unstandardized coefficients for the 

paths from pH to Soil and from distance to 

Landscape to 1.0 to establish the scale of 

measurement for the composites (note that this is not 

shown in later results because only the standardized 

parameters are presented). This procedure ignores the 

question of whether the parameter values for these 

paths are detectably different from zero; specifically, 

does pH have a significant (nonzero) contribution to 

Soil, and similarly, does distance have a nonzero 

contribution to Landscape? One approach to 

evaluating fixed paths from xs to composites is to use 

a reduced form model such as Model C (fig. 8), in 

which the composites are omitted and the direct 

effects of xs on Competitors and Colonization are 

tested. For our example, the reduced form equations 

that apply to Figure 8 are 

15154143132121111 η γ γ γ γ ++++= γ xxxxx + ζ 

(5) 

and 

η γ γ γ γ ++++= γ xxxxx 5254243232221212 +

2121ηβ + ζ (6). 

For models with more than a single path flowing 

out from the composites, as is the case in Model B, 

an evaluation of Model C only provides an 

approximate answer to the question of whether the 

parameters in Model B are significant. The reason the 

reason is only approximate is because the gammas 

relating xs to Competitors must be proportional to 

those relating the same xs to Colonization. In other 

words, it is assumed that the weightings specifying 

the contributions of pH, moisture, and texture to a 

composite of their effects on Competitors are the 

same as the weightings for the effects of those same 

variables on Colonization. This means that for the 

reduced form equations (5 and 6), •11 - •15 should be 

proportional to •21 - •25. Otherwise, a single 

coefficient will not, for example, adequately 

represent the paths from •1 to •3 and •4, in Model B 

(fig. 7). As we show below, it is important to evaluate 

such relations as part of the assessment process. 

Further Results 

Results for Model B 

Maximum likelihood estimation yielded a chisquare of 6.888 with 3 degrees of freedom and an 

associated probability of 0.075. Examination of 

residuals and the chi-square results indicate adequate 

fit of the data to Model B, though there were some 

slight indications of residual relations between 

distance and stand age and •3 and •4. In the 

presentation of results in figure 10, all numerical 

estimates shown are associated with significant p

values for the relevant unstandardized parameters, 

except for the paths from pH to Soil and distance to 

Landscape, which were set to fixed values to 

establish the scales for the composites. 

Conspicuous in the results for Model B (fig. 10) 

is that neither moisture nor texture contributed 

significantly to the composite variable Soil. Also, 

Soil had a significant effect on Competitors but not 

on Colonization. In contrast, the relationship between 

stand age and Landscape was significant, as were 

both paths from Landscape to Competitors and 

Colonization. Examination of the residual variances 

for Competitors and Colonization shows that 25 % 

and 54 % of the variance in these response variables 

were explained (R2

 = 1 - the standardized error 

variance). 

Results for Model C 

It is instructive to evaluate Model C, which 

effectively represents what Bollen and Davis (1994) 

referred to as a “partially reduced form” of Model B 

in which composites are omitted. In Model C, 

separate effects of the x variables on Competitors and 

Colonization are estimated, and the individual x

variables are associated with constructs nominally. 

Since this model is saturated, maximum likelihood 

estimation yields a chi-square of 0 due to the fact that 

there are no degrees of freedom. In a side analysis, 

the stability of results was tested by deleting 

nonsignificant paths, yielding a chi-square of 9.773 

with 6 degrees of freedom (p = 0.1344). This side 

analysis indicated that the results for Model C are 

stable. 

Results for Model C (fig. 11) show that only four 

of the paths from x variables to Competitors and 

Colonization were significant. Variances explained 

for Competitors and Colonization in the saturated 

19
model were 30 % and 57 % respectively. Again, for 

the purpose of comparing among models, the 

saturated model results will be used even though 

nonsignificant paths are included. Aside from 

providing us with a comparison to Model B, Model C 

also provides a representation of our constructs from 

figure 4, except that in this case, the dimensions of 

soil conditions and landscape properties are 

represented separately. 

A Further Model to Consider – Model D 

Above it was mentioned that Model B presumes 

a single set of coefficients defines composite effects 

on more than one response variable. This is akin to 

saying that the slopes of the regression relations of xs 

on •3

 would be proportional to the effects of xs on •4. 

If this condition holds, then Model B is our best 

model for the situation. On the other hand, if this 

condition does not hold because relations are not 

proportional, our model will underestimate the 

variance explanation for Competitors and 

Colonization. It can be easily seen that, when 

gammas in equations 5 and 6 are not proportional, the 

composites in Model B will be insufficiently relevant 

to accurately predict both Competitors and 

Colonization accurately. To evaluate this possibility, 

we consider one additional model, Model D (fig. 12). 

In this model, separate composites are derived to 

estimate effects on Competitors and Colonization. 

Results for Model D were as for Model C in that 

when all paths are included there are no degrees of 

freedom for testing model fit. However, deletion of 

nonsignificant paths allowed for model testing, and 

again, no unspecified paths were indicated. 

Comparisons Among Models 

The exercise of comparing different models as 

contrasting representations of the construct relations 

in Figure 4 is meant not only to illustrate some of the 

variety of modeling possibilities but also to introduce 

a number of important concepts and procedures. 

Most of the presentations of SEM in introductory 

textbooks would likely lead beginning users to 

consider Model A to represent the construct relations 

in figure 4. In fact, several review papers have 

suggested that researchers frequently fail to consider 

the full range of possibilities for model development 

(Jarvis and others, 2003; Williams and others, 2003). 

Models that contain causal indicators and composites 

are rather rare compared to those in which constructs 

are exclusively represented by effect indicators in 

LxM blocks. In Example 1, the bulk of the 

theoretical information and the great majority of the 

empirical analyses indicate that Model A is not an 

appropriate representation of the situation. 

Conceptually, the constructs were not envisioned as 

unidimensional underlying causes but rather, as 

collections of influencing factors. Further, the 

available indicators do not represent a suite of effect 

indicators as would be appropriate for an LxM 

block. The inadequacy of fit of data to Model A is 

clearly indicated by the very low probability 

associated with the chi-square test. Thus, we can 

reject this model as inadequate. Considerations of 

modifications such as splitting out pH from the Soil 

block, as well as other attempts at small 

modifications, did not lead to the discovery of an 

adequate model akin to Model A either. 

The remaining comparisons of most interest are 

those between Models B, C, and D. All three of these 

models had adequate fit to the data, and the 

differences among them can be viewed as tradeoffs 

between model generality and model accuracy 

(Levins 1968). Model B is the most conceptually 

general of the three (though it is more statistically 

restrictive), representing the collective effects of soil 

conditions and landscape properties with a single 

composite for each. As described in the above 

discussion, for Model B to be appropriate, the 

composites estimated would need to be near optimal 

for representing effects on both Competitors and 

Colonization simultaneously. At the other end of the 

spectrum, Model C is the most accurate and least 

conceptually general of the three models. 

Compared to Model C, which has no composites, 

Model D represents an advance in generality, having 

an individual composite for each effect of the 

multidimensional constructs. As would be expected 

for such a case, the error variances for Competitors 

and Colonization are virtually identical for Models C 

and D. Thus, Models C and D are both equally 

accurate with regard to variance explanation, but they 

differ in the ways in which effects are represented. 

Overall, Model C provides a more succinct 

representation of the individual effects of x variables 

on Competitors and Colonization. All paths are tested 

for significance, and coefficients are readily 

interpretable as individual effects. In comparison, 

Model D provides a superior representation of 

collective effects, with single path coefficients 

representing the effects of suites of variables. 

An important comparison to make between 

Models B and D has to do with the magnitudes of the 

error variances for the response variables. It has been 

recognized previously that, when composites with 

multiple effects on other variables are not 

20
distance, x4

stand age, x5

texture, x3

moisture, x2

pH, x1

Soil Conditions

Landscape Properties

Competitors

η1

Colonization

η2

col. freq., y2

cover, y1

.72

.47

.19

.47

-.37

-.31

.95

.95

distance, x4

stand age, x5

texture, x3

moisture, x2

pH, x1

Soil Conditions

Landscape Properties

Competitors

η1

Colonization

η2

col. freq., y2

cover, y1

.72

.47

.19

.47

-.37

-.31

.95

.95

Figure 11. Select standardized parameter estimates for Model C. Here composites are omitted, thus, Model C 

represents a partially reduced form of Model B. 

Competitors

η5

Colonization

η6 distance, x4

stand age, x5

texture, x3

moisture, x2

pH, x1

col. freq., y2

cover, y1

.32

-.33

-.67

.60

.48

-.64

.72

.46

0

0

0

0

1.12

.95

.95

Soil->Com

η1

Soil->Col

η2

Land->Com

η3

Land->Col

η4

Competitors

η5

Colonization

η6 distance, x4

stand age, x5

texture, x3

moisture, x2

pH, x1

col. freq., y2

cover, y1

.32

-.33

-.67

.60

.48

-.64

.72

.46

0

0

0

0

1.12

.95

.95

Soil->Com

η1

Soil->Col

η2

Land->Com

η3

Land->Col

η4

Figure 12. Standardized parameter estimates for Model D. Here four composites (recognized by the zero error 

variances) are included, one for each response variable. 
appropriate, there is a conspicuous loss of variance 

explanation for the response variables. In this case, 

error variances are 0.75 and 0.46 for Competitors and 

Colonization, respectively, in Model B. In contrast, in 

Model D, the values are 0.72 and 0.46. A test of the 

adequacy of Model B is whether its error variances 

are significantly greater than those for the less 

general case where composites have single effects. 

For such a test, the t-value for a test of difference 

between the error variances for Competitors must be 

obtained. For Model B, the unstandardized residual 

variance for Competitors is 0.059 + 0.007 (standard 

error), while for Model D the value is 0.057 + 0.007, 

yielding a t-value of 0.14 and an associated 

probability of no difference approaching 0.9. For 

Colonization, residual error variances were virtually 

identical (0.013 + 0.002) for Models B and D. Thus, 

there is no indication that Model D is superior to 

Model B in variance explanation for the response 

variables. Further, the paths from composites to the 

response variables are also very similar between 

models. Collectively, this evidence supports the 

conclusion that Model B is superior to Model D 

overall because of its greater conceptual generality 

and that Model B is our best model (of those 

examined) for this example and these data. 

Composites in More Complex 

Situations 

In the example above, only the exogenous 

observed variables were related to composites. 

Situations can exist where endogenous constructs are 

best represented by composites. In this case, all the 

above material applies, but there are some additional 

complications to model structure that apply. An 

example of the evaluation of such a case is given in 

this section, dealing with the associations between 

anuran species richness and habitat conditions. In this 

example, it can be seen that the principles presented 

here can be readily extended to include a broader 

range of model types than the one considered above. 

Example Involving Endogenous 

and Exogenous Composites: 

Examination of Habitat 

Associations for Anurans 

Background 

Anurans are a group of amphibians, comprising 

frogs and toads, that are of high conservation concern 

because of worldwide population declines. Example 

2 uses data from a study by Lichtenberg and others 

(2006) in which 25 wetlands in the lower Mississippi 

River alluvial valley were examined for chorusing 

anuran species and associated habitat characteristics. 

One major goal of this study was to understand the 

habitat characteristics associated with diversity 

“hotspots,” places where a variety of species are 

abundant. Figure 13 provides an illustration of the 

construct relations of interest in Example 2. Stated in 

general terms, the goal of their analysis of anuran 

diversity was to understand the degree to which it 

depends on the type of habitat (lake, impoundment, 

etc.) versus the particular conditions (vegetation, 

litter, etc.) within that habitat. Specifically, the 

investigators were interested in the question of 

whether correlations between high diversity and 

particular wetland types can be explained by the 

microhabitat conditions within the wetlands. Such 

information could prove useful in making decisions 

about types of habitats to protect and the conditions 

to maintain. 

In the study by Lichtenberg and others (2006), 

several types of wetlands were examined. These 

could be classified as being of one of four types – 

lakes, impoundments, swales, and riverine areas. 

Specific microhabitat conditions were assessed at 

each site, and these included a number of vegetation 

and topographic features. Anuran diversity was 

assessed by using nighttime surveys of chorusing 

individuals during several seasons of the year. 

Chorus surveys during the year were used to produce 

an estimate of the species using a site. 

While macrohabitat types were clearly defined a 

priori, microhabitat conditions were sampled in a 

more exploratory fashion. A wide range of 

characteristics of the vegetation, including 

herbaceous cover, vegetation density at different 

vertical positions, woody cover, tree heights, canopy 

cover, as well as litter cover and depth (by type) were 

22
measured. Also measured were hydrologic features at 

the site, including the area of open water and the 

mean and maximum water depths. 

Example 1 provided a detailed consideration of 

how constructs may be represented and both 

theoretical and empirical criteria for arriving at 

decisions about block structure. In Example 2, our 

emphasis is more on the question of how to model a 

situation where an endogenous variable (microhabitat 

conditions) has multiple indicators and may involve 

composites. This question was not addressed in 

Example 1, where multiple indicators existed only for 

the exogenous constructs. 

In this second example, we begin our analysis 

with the construct labeled Macrohabitat Type. Since 

our measure is nominal and multilevel (whether a site 

is classified as lake, impoundment, swale, or 

riverine), it immediately suggests the need to model 

this construct by using a set of dummy variables 

representing the possible macrohabitat types. We 

assume for the sake of simplicity that the 

classification of individual sites as to habitat type was 

correct. Therefore, our presumption is that the 

construct Macrohabitat Type can be modeled by 

using the MxC block type. 

The construct labeled Microhabitat Conditions is 

one where the specific details of how the measured 

variables are related was not known a priori. For this 

reason, Lichtenberg and others (unpub. data) 

performed an exploratory factor analysis to see if the 

correlations among the many measured variables 

might suggest the operation of a smaller number of 

latent factors. Here we will not go into the details of 

that analysis but only say that the result was the 

recognition by the authors of two factors of 

importance to anuran diversity, the abundance of 

herbaceous vegetation and the abundance of leaf 

litter. Based on the conceptualization of Microhabitat 

Conditions by Lichtenberg and others (2006), it is 

clear that the indicators could represent a collection 

of factors that affect anurans based on theoretical 

grounds. Thus, we begin with the expectation of a 

block structure of LxC, with two latent variables, 

Herbaceous Vegetation and Litter, contributing to the 

construct. A total of seven indicators of the two latent 

variables were included in their final model (see 

below). 

Lichtenberg and others (2006) discuss certain 

issues of measurement regarding the construct 

Anuran Diversity. It is widely held that there are 

several causes of measurement error for wildlife 

populations and communities. In addition to the usual 

matter of sampling, varying detectability can 

contribute to error. Lichtenberg and others (2006) 

addressed the issue of detectability to some degree by 

using the total number of species recorded across 

samplings instead of the mean. Nonetheless, error in 

assessing the true number of species at each site is 

likely significant and, while no estimate of this error 

exists, we again use an arbitrary estimate of 10 % of 

the total variance. 

Possible Model Structures

In the first example we spent some time 

discussing and illustrating criteria for evaluating the 

cause and effect relations among variables and how 

these influence model structure. In this second 

example, we forego a detailed discussion of such 

considerations and, instead, focus on the possible 

ways to model a case where composites have 

directional effects on other composites. 

A model that logically follows from the 

construct relations in figure 13 is shown in figure 14. 

We refer to this model as Model E to avoid confusion 

with the models discussed in conjunction with the 

first example. Based on the information presented 

above, we represent Macrohabitat by using the block 

structure MxC. The Microhabitat construct is 

represented in this case as an LxC block containing 

two latent variables, both with multiple indicators. 

Diversity is represented by an LxM block with a 

single indicator. Consistent with their declarations as 

composites, the error variances for Macrohabitat and 

Microhabitat are set to zero. 

Of the models we consider here, Model E 

represents the one that is most abstract. At the same 

time, Model E is based on the greatest number of 

assumptions. It is presumed in this model that the 

influences of Macrohabitat (•1

) on Microhabitat (•2) 

can be summarized by a single coefficient, •21, 

despite the fact that •2 is actually a composite of 

latent effect of Herbaceous and Litter effects on 

Diversity rather than the microhabitat conditions 

themselves. Stated in other terms, in Model E the 

covariances between x1 - x3 and y1 - y7 must be 

resolved by their joint relations to a linear predictor 

that depends on •3. These are fairly critical 

assumptions in that their failure likely means an 

underestimate of Macrohabitat effects on 

Microhabitat and possibly unresolved covariances 

among manifest variables. 

An alternative formulation that is perhaps more 

biologically meaningful, Model F (fig. 15), represents 

the modeling implications of relaxing the 

assumptions just discussed for Model E. Here we are 

allowed to consider effects of Macrohabitat on the 

two dimensions of Microhabitat, Herbaceous and 

Litter. Since these two dimensions are independently 

23
Macrohabitat

Type

Anuran

Diversity

Microhabitat

Type

Macrohabitat

Type

Anuran

Diversity

Microhabitat

Type

Figure 13. Construct model relating macrohabitat type and microhabitat conditions to anuran diversity. 

Microhabitat

η2

Diversity

η3

Litter

ξ2

Herbaceous

ξ1

lake, x1

impound, x2

swale, x3

vhit1, y1

vhit2, y2

herbl, y3

herbc, y4

wlitr, y5

litrd, y6

litrc, y7

rich, y8

0

0

Macrohabitat

η1

β21

Microhabitat

η2

Diversity

η3

Litter

ξ2

Herbaceous

ξ1

lake, x1

impound, x2

swale, x3

vhit1, y1

vhit2, y2

herbl, y3

herbc, y4

wlitr, y5

litrd, y6

litrc, y7

rich, y8

0

0

Macrohabitat

η1

β21

Figure 14. Model E, which shows one of the possible ways that indicators could be related to the construct model 

shown in Figure 13. Note that by omission, the riverine macrohabitat condition represents the baseline against which 

other macrohabitats are compared. “rich” refers to number of anurans at a site, “imp” refers to impoundments, 

“vhit2” and “vhit1” are measures of vegetation density, “herbl” and “herbc” are measures of dead and live 

herbaceous vegetation, “wlitr” is woody litter, “litrd” is litter depth, and “litc” is litter cover. 

24
Macrohabitat

ηc1

Microhabitat

ηc2

Diversity

ηe3

Litter

ηe2

Herbaceous

ηe1

lake, x1

impound, x2

swale, x3

vhit1, y1

vhit2, y2

herbl, y3

herbc, y4

wlitr, y5

litrd, y6

litrc, y7

rich, y8

0

0

Macrohabitat

ηc1

Microhabitat

ηc2

Diversity

ηe3

Litter

ηe2

Herbaceous

ηe1

lake, x1

impound, x2

swale, x3

vhit1, y1

vhit2, y2

herbl, y3

herbc, y4

wlitr, y5

litrd, y6

litrc, y7

rich, y8

0

0

Figure 15. Model F, which differs from Model E by allowing separate coefficients (represented by separate paths) 

to convey the effects of Macrohabitat on the Herbaceous and Litter dimensions of Microhabitat (see caption for fig. 

14 for abbreviations). 

Microhabitat

η6

Diversity

η7

Litter

η5

Herbaceous

η4

lake, x1

impound, x2

swale, x3

vhit1, y1

vhit2, y2

herbl, y3

herbc, y4

wlitr, y5

litrd, y6

litrc, y7

rich, y8

0

0

0

0

Macro->Div

η1

Macro->Lit

η2

Macro->Herb

η3

Microhabitat

η6

Diversity

η7

Litter

η5

Herbaceous

η4

lake, x1

impound, x2

swale, x3

vhit1, y1

vhit2, y2

herbl, y3

herbc, y4

wlitr, y5

litrd, y6

litrc, y7

rich, y8

0

0

0

0

Macro->Div

η1

Macro->Lit

η2

Macro->Herb

η3

Figure 16. Model G, which differs from Model F by using separate composites (•1

 - •3) to convey the effects of 

individual Macrohabitat types on •4, •5, and •7 (see caption for fig. 14 for abbreviations). 

25
Diversity

η3

Litter

η2

Herbaceous

η1

lake, x1

impound, x2

swale, x3

vhit1, y1

vhit2, y2

herbl, y3

herbc, y4

wlitr, y5

litrd, y6

litrc, y7

rich, y8

Macrohabitat

Microhabitat

Diversity

η3

Litter

η2

Herbaceous

η1

lake, x1

impound, x2

swale, x3

vhit1, y1

vhit2, y2

herbl, y3

herbc, y4

wlitr, y5

litrd, y6

litrc, y7

rich, y8

Macrohabitat

Microhabitat

Diversity

η3

Litter

η2

Herbaceous

η1

lake, x1

impound, x2

swale, x3

vhit1, y1

vhit2, y2

herbl, y3

herbc, y4

wlitr, y5

litrd, y6

litrc, y7

rich, y8

Diversity

η3

Litter

η2

Herbaceous

η1

lake, x1

impound, x2

swale, x3

vhit1, y1

vhit2, y2

herbl, y3

herbc, y4

wlitr, y5

litrd, y6

litrc, y7

rich, y8

Macrohabitat

Microhabitat

Figure 17. Model H, partially reduced form model representing effects of Macrohabitat and Microhabitat nominally 

through grouping variables (see caption for fig. 15 for abbreviations). 

Table 2. Correlations among variables related to anuran richness and their standard deviations. “rich” refers to 

number of anurans at a site, “imp” refers to impoundments, “vhit2” and “vhit1” are measures of vegetation density, 

“herbl” and “herbc” are measures of dead and live herbaceous vegetation, “wlitr” is woody litter, “litrd” is litter 

depth, and “litc” is litter cover. "sd" refers to standard deviation. 

rich lake imp swale vhit2 vhit1 herbl herbc wlitr litrd litrc 

rich 1.0 

lake .696 1.0 

imp -.167 -.355 1.0 

swale -.431 -.659 -.253 1.0 

vhit2 .372 .167 .111 -.099 1.0 

vhit1 .222 -.156 .552 -.118 .653 1.0 

herbl .060 -.252 .562 -.009 .581 .825 1.0 

herbc .091 -.087 .419 -.132 .437 .745 .756 1.0 

wlitr .509 .430 -.284 -.099 -.051 -.290 -.395 -.396 1.0 

litrd .238 .146 -.433 .383 .027 -.097 -.180 -.281 .419 1.0 

litrc .219 .194 -.442 .273 -.118 -.414 -.509 -.580 .568 .762 1.0 

sd 2.170 0.510 0.332 0.476 0.512 1.482 0.173 0.122 0.100 0.122 0.148 

26
estimated in LxM blocks, the interpretative meaning 

of the paths from Macrohabitat to each dimension is 

clear and also independent of other relationships in 

the model. In contrast, since the composite 

Microhabitat is defined as a linear predictor of 

Diversity, it depends on the covariances between 

three latent variables, •2

, •3, and •5. If any of the 

covariances among these variables changes, the 

meaning of •4

 changes and, thus, the meaning of a 

direct path from Macrohabitat to Microhabitat (as in 

Model E). In Model F, the effect of Macrohabitat on 

Microhabitat can be summarized by calculating the 

total effect of •1

 on •4 , which can be summarized by 

the equation 

41 4221 βββββ 4331 ∗+∗= , (7) 

where •41 now refers to a calculated total effect. 

 Certain restrictive assumptions that warrant 

discussion remain in Model F. In particular, implied 

is the assumption that a single combination of 

coefficients from macrohabitat types (x1

 - x3) to the 

Macrohabitat composite (•1

) simultaneously 

summarizes the individual effects of x1

 - x3 on •2, •3, 

and •5

. We feel that it is always wise to check such 

an assumption (as was done in Example 1). Here, this 

assumption is explicitly removed in Model G (fig. 

16), where one composite representing Macrohabitat 

effects is replaced by three (MacroxHerb, 

MacroxLit, MacroxDiv). 

Also as was done in Example 1, we offer for 

comparison a partially reduced form model, Model H 

(fig. 17), in which composites are omitted and 

constructs are represented nominally. While Model H 

offers a less complete modeling of the construct 

relations (e.g., there are no path coefficients derived 

for interactions among the major constructs), it 

permits a complete evaluation of the significance of 

all individual effects, thus complementing the other 

models. 

Comparisons Among Models 

Sample correlations and standard deviations are 

presented in table 2 for the manifest variables 

considered in this example. As stated earlier, we 

forego an empirical evaluation of the relations 

between manifest variables and either composites or 

latent variables. We also bypass any discussion of the 

bivariate relations in Table 2 and move directly into a 

consideration of model results. 

 As was brought out in our discussion of 

Example 1, the solution and evaluation of general 

models containing composite variables often depend 

on information from more specific models. For this 

reason, we will begin with the model containing the 

least number of assumptions (and thereby, our least 

general model), Model H. We then will proceed 

progressively through models G, F, and E to 

determine whether more general models result in a 

substantial loss of information. 

Results for Model H 

The results for Model H are given in figure 18. 

For the purposes of this evaluation (and in contrast to 

our practice in Example 1), nonsignificant effects of 

macrohabitat types on endogenous variables were 

dropped from the final model. Despite the small 

sample size, results were stable, and the fit between 

model expectations and data was acceptable. As these 

results show, impoundments had significantly higher 

levels of herbaceous vegetation than did other 

macrohabitat types. Litter accumulation, in contrast, 

was substantially higher in lakes and swales than in 

impoundments and riverine habitats (recall that the 

riverine variable was omitted and, therefore, serves as 

the baseline condition). Anuran diversity was found 

to be higher in lakes than in all other habitat types. 

Results for Model G 

Model G provides for a single path from each 

composite to replace the multiple paths that would 

otherwise connect one construct with another. Aside 

from that, the models are very similar. Results from 

the estimation of Model G are presented in Figure 19. 

A comparison of results from Models H and G show 

numerous similarities, as well as a few differences. 

Model fit parameters are identical for the two models. 

Also, variance explanation for endogenous latent 

variables is the same, with R2

s of 0.33, 0.39, and 0.78 

for •4

, •5, and •7, respectively. Loadings in LxM 

blocks are the same for both models, as are outward 

paths from composites possessing single causes (•1

x •7 and •3 x •4) in comparison to the equivalent 

effects in Model H (x1 x •3 and x2 x •1). 

Composites with multiple causes in Model G yielded 

parameters not found in Model H, such as those 

associated with •6 x •7 (Microhabitat to Diversity) 

and •2 x •5 (MacroxLit to Litter). The path 

coefficients associated with these paths represent 

standardized collective effects of the composites’ 

causes on the response variables involved. Heise 

(1972) referred to these coefficients as “sheath” 

coefficients to designate the fact that they represent a 

collection of causes. 

27
Diversity

η3

Litter

η2

Herbaceous

η1

lake, x1

impound, x2

swale, x3

vhit1, y1

vhit2, y2

herbl, y3

herbc, y4

wlitr, y5

litrd, y6

litrc, y7

rich, y8

Macrohabitat

Microhabitat

.76

.60

.79

.98

.92

.63

.91

.82 -.30

.95

.50 .37

.57

.73

.78

.28

.67 .61

Diversity

η3

Litter

η2

Herbaceous

η1

lake, x1

impound, x2

swale, x3

vhit1, y1

vhit2, y2

herbl, y3

herbc, y4

wlitr, y5

litrd, y6

litrc, y7

rich, y8

Macrohabitat

Microhabitat

.76

.60

.79

.98

.92

.63

.91

.82 -.30

.95

.50 .37

.57

.73

.78

.28

.67 .61

Figure 18. Results obtained for Model H, showing standardized values for path coefficients and the error variances 

of latent endogenous variables. Correlations among xs and errors for ys are not shown for simplicity. Nonsignificant 

effects of macrohabitat types were dropped from the final model, which possessed a chi-square of 50.42 with 39 

degrees of freedom and a p of 0.104 (see caption for fig. 15 for abbreviations). 

Microhabitat

η6

Diversity

η7

Litter

η5

Herbaceous

η4

lake, x1

impound, x2

swale, x3

vhit1, y1

vhit2, y2

herbl, y3

herbc, y4

wlitr, y5

litrd, y6

litrc, y7

rich, y8

0

.28

.76

.60

.79

.98

.92

.63

.91

.82 -.30

.95

1.18

.42

.57

.62

.67 .61

.87

1.17

1.25

1.00

1.00

0

Macro->Div

η1

0

Macro->Lit

η2

0

Macrohabitat

η3

Microhabitat

η6

Diversity

η7

Litter

η5

Herbaceous

η4

lake, x1

impound, x2

swale, x3

vhit1, y1

vhit2, y2

herbl, y3

herbc, y4

wlitr, y5

litrd, y6

litrc, y7

rich, y8

0

.28

.76

.60

.79

.98

.92

.63

.91

.82 -.30

.95

1.18

.42

.57

.62

.67 .61

.87

1.17

1.25

1.00

1.00

0

Macro->Div

η1

0

Macro->Lit

η2

0

Macrohabitat

η3

Figure 19. Results obtained for Model G, showing standardized values for path coefficients and the error variances 

of latent endogenous variables. Correlations among xs and errors for ys are not shown for simplicity. Nonsignificant 

effects of macrohabitat types were dropped from the final model, which possessed a chi-square of 50.42 with 39 

degrees of freedom and a p of 0.104, precisely as for Model H (see caption for fig. 15 for abbreviations). 
Microhabitat

η4

Diversity

η5

Litter

η3

Herbaceous

η2

lake, x1

impound, x2

swale, x3

vhit1, y1

vhit2, y2

herbl, y3

herbc, y4

wlitr, y5

litrd, y6

litrc, y7

rich, y8

0

.26

.75

.98

.92

.64

.90

.81 -.47

.95

1.13

.44

ns

ns

1.0 1.0

.80

1.00

.59

.77

0

Macrohabitat

η1

Microhabitat

η4

Diversity

η5

Litter

η3

Herbaceous

η2

lake, x1

impound, x2

swale, x3

vhit1, y1

vhit2, y2

herbl, y3

herbc, y4

wlitr, y5

litrd, y6

litrc, y7

rich, y8

0

.26

.75

.98

.92

.64

.90

.81 -.47

.95

1.13

.44

ns

ns

1.0 1.0

.80

1.00

.59

.77

0

Macrohabitat

η1

Figure 20. Results obtained for Model F, showing standardized values for path coefficients and the error variances 

of latent endogenous variables. Correlations among xs and errors for ys are not shown for simplicity. Chi-square for 

this model was 30.03 with 24 degrees of freedom and a p of 0.1837 (see caption for fig. 15 for abbreviations). 

29
The most conspicuous differences between 

Models H and G reside with the paths from causes to 

composites. For example, in Model H, the effects of 

Herbaceous and Litter on Diversity are represented 

by two path coefficients (0.50 and 0.37), while in 

Model G, the same effects are represented by three 

paths, two from Herbaceous and Litter to 

Microhabitat (1.18 and 0.87) and one 

fromMicrohabitat to Diversity (0.42). Upon first 

examination, the paths from Herbaceous and Litter to 

Microhabitat appear unusually inflated, particularly 

since Herbaceous and Litter are only modestly 

correlated (-0.30). The equivalency of relationships 

in Models H and G can be made clear by realizing 

that the total effects of Herbaceous and Litter on 

Diversity are simply compound paths in Model G 

(e.g., the effect of •4

 on •7 = 1.18 times 0.42 = 0.50). 

The same holds true for all effects involving 

composites (e.g., the effects of x1

 and x3 on •5). 

Results for Model F 

Model F, as described above, is a more general 

statement than Model G and presumes that a single 

composite is sufficient to represent the effects of 

Macrohabitat types on •2 , •3, and •5 . Results from 

maximum likelihood estimation of Model F included 

a chi-square of model fit of 66.12 with 38 degrees of 

freedom and an associated p-value of 0.0031, 

indicating poor fit. It was found that when estimated 

by using a single composite for Macrohabitat effects, 

substantial residual correlations between individual 

macrohabitat types and the indicators for •2 , •3, and 

•5 were not resolved. These results suggested that a 

single composite representing the effects of 

macrohabitat types on Herbaceous, Litter, and 

Diversity was inconsistent with the data. However, 

neither impoundment or swale indicators contributed 

significantly to the composite in this case; therefore, 

it was possible to simplify the model by dropping 

these indicators from the model. With these 

indicators deleted, model fit improved to 30.03 with 

24 degrees of freedom and a p-value of 0.1837, 

indicating acceptable fit. The results presented in 

figure 20 are based on these results. 

The version of Model F presented in figure 20, 

while representing a model that fits the data, fails to 

explain significant variation in either •2 or •3. We 

might choose to select this model to represent the 

direct and indirect effects of Macrohabitat because it 

addresses the central question of interest in 

Lichtenberg and others' (2006) study. While it 

succeeds at that more narrow objective, Model F fails 

to provide an explanation for macrohabitat effects on 

both diversity and microhabitat. The clear reason for 

that failure is that macrohabitat effects on 

microhabitat conditions are quite different from those 

on diversity, and that a single composite would not 

be able to represent both adequately. 

A Consideration of Model E 

Our most abstract model, Model E, represents 

direct effects of Macrohabitat on Microhabitat rather 

than on the dimensions of Microhabitat (Herbaceous 

and Litter). We found that a proper estimation of this 

model using a simultaneous solution procedure could 

not be achieved. The fundamental difficulty is that a 

zero error variance is specified for Microhabitat in 

order to permit it to represent the effects of 

Herbaceous and Litter on Diversity. It is not possible 

to estimate values for Microhabitat, a composite, 

based on the effects of Macrohabitat when the latter 

is also a composite. Thus, with Microhabitat 

represented as a composite with zero error variance, 

Model E is unresolvable without resorting to the 

derivation of a composite index for Microhabitat 

(such an approach would require a piecewise solution 

procedure, which we do not prefer). This condition 

serves as an additional argument in favor of modeling 

approaches in which exogenous influences are 

measured against the individual dimensions of 

endogenous multidimensional constructs (e.g., 

Models G and F) rather than the composite of their 

combined effects (in this case, on Diversity). With 

such an approach, effects on the endogenous 

composite can be estimated by using compound 

paths. 

 In conclusion, the principles presented by Grace 

and Bollen (in press) can be extended to models that 

contain endogenous composites. Some additional 

complexities arise with regard to model structure. In 

particular, the effects of composites representing 

exogenous indicators on composites representing 

endogenous indicators will typically need to be 

viewed as being indirect, through effects on the 

endogenous indicators themselves. This requirement 

poses no real problems for the inclusion of 

composites in such models, and our analysis is meant 

primarily to make clear the logic whereby effects in 

such models are considered. 

Conclusions and 

Recommendations 

Ecological theories often deal with theoretical 

concepts that are heterogeneous. In fact, it can be 

stated that one of the major factors that limit our 

ability to generalize in ecology is that we seek to 

generalize across variable entities of interest, such as 

communities and ecosystems (Grace, 2006). 

30
Composites have, we believe, great potential to 

facilitate our ability to create models that are 

empirically meaningful and also of theoretical 

relevance. Without composites, models that consider 

substantial complexity and/or seek to address 

relations among a large number of variables will have 

the tendency to be highly specific, possessing a 

separate set of effects from each of a suite of 

intercorrelated indicators. Such specificity will limit 

the utility of SEM for effectively addressing the 

general questions often asked by ecologists. 

Composites represent something of a compromise, 

but one that often matches the characteristics of our 

theoretical constructs reasonably well. In the 

examples in this paper, using composites allows us to 

emphasize the interpretation of only 4 pathways from 

our two main predictor constructs instead of the 10 

pathways that would require emphasis without the 

use of composites. 

It should be pointed out that some have argued 

that because composites have data-specific properties 

defined by the relationships between predictors and 

responses, their meaning is vague, their effects less 

general, and they should be avoided (see discussion 

in Edwards 2001). Others, ourselves included, feel 

that the benefits associated with using composites 

outweigh their limitations. To some degree, the 

differences in opinion between these two 

perspectives may depend on the context and the 

emphasis of the investigator relative to the elements 

in figure 1. For those who wish to maximize their 

ability to relate statistical findings to some general 

theory, composites may provide a very useful device 

(e.g., Laughlin and Grace, 2006). For other 

investigators, who either possess theories that make 

very specific predictions or who do not wish to 

generalize far from the data, composites may be less 

valuable. 

It should be clear from the examples considered 

in this paper that the standard latent variable model is 

not always adequate for representing constructs and 

their interrelations. A consideration of composite 

variables contributes balance to the modeling process 

by providing options and encouraging a thorough 

evaluation of constructs and the measures at hand. 

While the need for composite variable specification 

has been pointed out several times (Heise, 1972; 

Bollen and Lennox, 1991; MacCallum and Brown, 

1993), the inclusion of composites in SEMs remains 

rare, while the misspecification of models using only 

LxM blocks is common (Jarvis and others, 2003). 

At present, much of the literature on composite 

variables is based on the premise that composites 

should have two or more effects on other variables 

(outward directed paths) and that composites with 

single effects should not be included in models 

(MacCallum and Browne, 1993; Edwards, 2001; 

Jarvis and others, 2003; Williams and others, 2003). 

This recommendation has been offered as a 

necessary, though insufficient, requirement for the 

identification of composite error variances. In our 

first example, we found such a model (Model C) to 

be appropriate and satisfactory; in many cases models 

of this sort will not be adequate, and those with single 

effects (e.g., Model E) will be required. We argue 

that (1) including composites that have only single 

outward directed paths can be quite valuable, even 

though the covariances for a set of data can be 

represented without them, and (2) confining the use 

of composites to cases where there are multiple 

outward directed pathways is ill advised because it 

can lead the researcher into developing and testing 

models that fail to match the relations in the data. 

A key factor that has limited the use of 

composites has been the aforementioned problem of 

parameter identification for models with single 

effects (as well as some models with multiple 

effects). In this paper, we show that this problem can 

be resolved by reference to a partially reduced form 

model (of the type represented by Model C) in which 

specific effects can be evaluated, in conjunction with 

the specification of a scale for the composite. Further, 

we recommend that, for models that incorporate 

composites, reduced form models such as Model C 

and single-effect models such as Model D should be 

routinely examined as part of the evaluation process. 

When this approach is taken, a more complete 

consideration of construct representation can be 

achieved. 

As illustrated through our considerations of 

Example 2, representing endogenous constructs by 

using composites poses additional theoretical and 

practical issues. Models of the most general type, 

which posit direct effects of exogenous variables on 

endogenous composites, while of interest, may be 

difficult to solve and also to interpret. Models in 

which effects on endogenous constructs are modeled 

through influences on their dimensions prove to be 

more tractable, as well as more generally 

interpretable. Again, as a general strategy, comparing 

results with those from models with fewer restrictions 

contributes to the evaluation process. 

Modern SEM using latent variables has largely 

developed through applications to the social and 

economic sciences, though it is now being widely 

used in a broad range of disciplines. Part of the 

emphasis on common factor-type models (multiindicator, latent variable models) certainly comes 

from the availability of redundant measures, such as 

batteries of questionnaire items or exam questions 

that can be used to estimate underlying causes. Even 

in these disciplines, it seems that data and constructs 

31
are such that composites could be widely used to 

represent relationships more properly. In the natural 

sciences, redundant measures are less common, a fact 

that has contributed to an emphasis on path analysis 

using only observed variables and encouraged little 

use of latent variables. We expect that multi-indicator 

latent variables of the LxM block type will find wide 

usage in the natural sciences over time. However, we 

anticipate that a significant need will develop for 

considerations of composite variables to represent 

collections of effects. Composites, when used with 

care, provide a useful way of summarizing groups of 

effects that match our theories about construct 

relations. 

Acknowledgments 

We wish to thank K. Verheyen, G.R. 

Guntenspergen, B. Biesbrouck, and M. Hermy for 

use of their data (with permission) in Example 1. We 

also thank J.S. Lichtenberg, S.L. King, and S. Walls 

for use of their unpublished data in Example 2. We 

are grateful to Timothy Gregoire and three 

anonymous reviewers for constructive comments that 

aided with the revision of the manuscript. Thanks 

also to Bill Shipley for helpful comments on an 

earlier version of this manuscript and to Connie 

Herndon, Victoria Jenkins, and Darren Johnson for 

additional reviews. 

References Cited 

Belovsky, G.E., Botkin, D.B., Crowl, T.A., 

Cummins, K.W., Franklin, J.F., Hunter, M.L., 

Joern, A., Lindemayer, D.B., MacMahon, J.A., 

Margules, C.R., and Scott, M.J., 2004, Ten 

suggestions to strengthen the science of ecology: 

Bioscience, v. 54, p. 345—351. 

Blalock, H.M., 1964, Causal inferences in 

nonexperimental research: Chapel Hill, 

University of North Carolina Press. 

Bollen, K.A., 1984, Multiple indicators: internal 

consistency or no necessary relationship. Quality 

and Quantity, v. 18, p. 377—385. 

Bollen, K.A., 1989, Structural equations with latent 

Variables: New York, John Wiley & Sons. 

Bollen, K.A., and Lennox, R., 1991, Conventional 

wisdom on measurement: a structural equation 

perspective. Psychological Bulletin, v. 110, p. 

305—314. 

Bollen, K.A., and Ting, K., 2000, A tetrad test for 

causal indicators: Psychological Methods, v. 5, 

p. 3—22. 

Diamantopoulous, A., and Winklhofer, H.M., 2001, 

Index construction with formative indicators: an 

alternative to scale development: Journal of 

Marketing Research, v. 38, p. 269—277. 

Edwards, J.R., 2001, Multidimensional constructs in 

organizational behavior research: an integrative 

analytical framework: Organizational Research 

Methods, v. 4, p. 144—192. 

Fayers, P.M., and Hand, D.J., 2002, Causal variables, 

indicator variables and measurement scales: an 

example from quality of life. Journal of the 

Royal Statistical Society: Series A, v. 165, p. 

233—261. 

Grace, J.B., 2003, Examining the relationship 

between environmental variables and ordination 

axes using latent variables and structural 

equation modeling. chap. 7 (Pages 171—193), in

Pugesek, B.H., Tomer, A., and von Eye, A., eds., 

Structural equation modeling, Cambridge, UK, 

Cambridge University Press, p. 171—193. 

Grace, J.B., 2006, Structural equation modeling and 

natural systems. Cambridge, UK, Cambridge 

University Press. 

Grace, J.B., and Bollen, K.A., 2005, Interpreting the 

results from multiple regression and structural 

equation models: Bulletin of the Ecological 

Society of America, v. 86, p. 283—295. 

Grace, J.B., and Bollen, K.A., in press, Representing 

general theoretical concepts in structural 

equation models: the role of composite variables: 

Environmental and Ecological Statistics, v. 0, p. 

000—000. 

Heise, D.R., 1972, Employing nominal variables, 

induced variables, and block variables in path 

analyses: Sociological Methods & Research, v. 

1, p. 147—173. 

Hipp, J., Bauer, D.J., and Bollen, K.A., 2005, 

Conducting tetrad tests of model fit and 

contrasts of tetrad-nested models: a new SAS 

macro: Structural Equation Modeling, v. 12, p. 

76—93. 

Jarvis, C.B., MacKenzie, S.B., and Podsakoff, P.M., 

2003, A critical review of construct indicators 

and measurement model misspecification in 

marketing and consumer research: Journal of 

Consumer Research, v. 30, p. 199—218. 

32
Johnson, M.L., Huggins, D.G., and deNoyelles, F., 

Jr., 1991, Ecosystem modeling with LISREL: a 

new approach for measuring direct and indirect 

effects: Ecological Applications, v. 1, p. 383—

398. 

Laughlin, D.C. and Grace, J.B., 2006, A multivariate 

model of plant species richness in forested 

systems: old-growth montane forests with a long 

history of fire. Oikos, v. 114, p. 60—70. 

Levins, R., 1968, Evolution in changing 

environments, Princeton, Princeton University 

Press. 

Lichtenberg, J.S., King, S.L., Grace, J.B., and Walls, 

S., 2006, Habitat associations of chorusing 

anurans in the lower Mississippi river alluvial 

valley: Wetlands, v. 26, p. 000—000. 

MacCallum, R.C., and Browne, M.W., 1993, The use 

of causal indicators in covariance structure 

models: some practical issues: Psychological 

Bulletin, v. 114, p. 533—541. 

Mitchell, R.J., 1992, Testing evolutionary and 

ecological hypotheses using path analysis and 

structural equation modelling: Functional 

Ecology, v. 6, p. 123—129. 

Muthén, L.K., and Muthén, B.O., 2004, Mplus user’s 

guide. (3rd ed.): Los Angeles, Muthén and 

Muthén. 

Pugesek, B.H., and Tomer, A., 1996, The Bumpus 

house sparrow data: a reanalysis using structural 

equation models: Evolutionary Ecology, v. 10, p. 

387—404. 

Pugesek, B.H., Tomer, A., and von Eye, A., 2003, 

Structural equation modeling: Cambridge: UK, 

Cambridge University Press. 

Shipley, B., 2000. Cause and correlation in biology: 

Cambridge, UK, Cambridge University Press. 

Spirtes, P., Glymour, C., and Scheines, R., 1993, 

Causation, prediction, and search: New York, 

Springer-Verlag. 

Verheyen, K., Guntenspergen, G.R., Biesbrouck, B., 

and Hermy, M., 2003, An integrated analysis of 

the effects of past land use on forest herb 

colonization at the landscape scale: Journal of 

Ecology, v. 91, p. 731—742. 

Weiher, E., Forbes, S., Schauwecker, T., and Grace, 

J.B., 2004, Multivariate control of plant species 

richness in a blackland prairie: Oikos, v. 106, p. 

151—157. 

Williams, L.J., Edwards, J.R., and Vandenberg, R.J., 

2003, Recent advances in causal modeling 

methods for organizational and management 

research. Journal of Management, v. 29, p. 

903—936. 

Wright, S., 1921, Correlation and causation: Journal 

of Agricultural Research, v. 10, p. 557—585. 

Wright, S., 1968. Evolution and the genetics of 

populations, Vol. 1: Genetic and Biometric 

Foundations, Chicago, University of Chicago 

Press. 

 

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