a five-step guide to conducting sem analysis in counseling research 2012
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Stephanie A. CrockettA Five-Step Guide to Conducting SEM Analysis in Counseling Research
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Outcome Research Design
A Five-Step Guide toConducting SEM Analysisin Counseling Research
Stephanie A. Crockett1
AbstractThe use of structural equation modeling (SEM), a second-generation multivariate analysis techniquethat determines the degree to which a theoretical model is supported by the sample data, is becom-ing increasingly popular in counseling research. SEM tests models that include both observed andlatent variables, allowing the counseling researcher to confirm the factor structure of a newly devel-oped or existing psychological instruments and to examine the plausibility of complex, theoreticalcounseling models. This article provides counseling researchers and practitioners with an overviewof SEM and presents five steps for conducting SEM analysis in counseling research.
Keywordsstructural equation modeling, counseling research
Submitted 2 October 2011. Revised 4 December 2011. Accepted 5 December 2011.
In recent decades, the field of counseling has
made increased efforts to empirically know
how and what works in client treatment and
to build a scientific foundation that substanti-
ates the efficacy of counseling practice in rela-
tion to client outcomes (Kaplan & Gladding,
2011; Ray et al., 2011). As we attend to what
works in counseling, it is critical that counsel-
ing researchers and practitioners employ clini-
cal interventions and assessments that are
grounded in empirically verified counseling
theories and constructs. The validation of com-
plex counseling theories and constructs
requires counseling researchers to employ
advanced statistical methods. Structural equa-
tion modeling (SEM) is one such advanced sta-
tistical method that allows for the testing of
multifaceted theories and constructs; and in the
social sciences, it is rapidly becoming the
favored method for determining the plausibility
of theoretical models (Martens, 2005; Quintana
& Maxwell, 1999; Schumacker & Lomax, 2010).
SEM is a collection of statistical techniques
that allows researchers to assess empirical
relationships among directly observed vari-
ables and underlying theoretical constructs
(i.e., latent variables; Raykov & Marcoulides,
2000). It is highly applicable within the field of
counseling as researchers often strive to validate
theoretical constructs and models. Specifically,
SEM can be used to confirm the factor structure
of a newly developed psychological instrument
1Department of Counseling, University of Oakland,
Rochester, MI, USA
Corresponding Author:
Stephanie A. Crockett, University of Oakland, 2200 N.
Squirrel Road, Rochester, MI 48309, USA
E-mail: [email protected]
Counseling Outcome Researchand Evaluation3(1) 30-47 The Author(s) 2012Reprints and permission:sagepub.com/journalsPermissions.navDOI: 10.1177/2150137811434142http://core.sagepub.com
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(Martens, 2005; Tomarken & Walker, 2005).
Counseling researchers may also wish to use
SEM to confirm the factor structure of an existing
psychological instrument with a new population
(Martens, 2005). SEM techniques can also be
employed to determine the plausibility of com-
plex, theoretical counseling models. Further,
counseling researchers can use SEM to com-
pare competing theoretical models in order to
determine which model is a better fit to the
empirical data (Chan, Lee, Lee, Kubota, &
Allen, 2007).
For these reasons, SEM is becoming increas-
ingly popular in counseling research. For exam-
ple, Bullock-Yowell, Peterson, Reardon,
Leierer, and Reed (2011) evaluated the cognitive
information process theory using SEM to deter-
mine whether career thoughts mediate the rela-
tionship between career/life stress and level of
career decidedness. Chao, Chu-Lien, and Sanjay
(2011) employed SEM techniques to examine the
role of ethnic identity, gender roles, and multicul-
tural training in college counselors multicultural
counseling competence. Cochran, Wang, Steven-
son, Johnson, and Crews (2010) sought to empiri-
cally verify Gottfredsons theory of
Circumscription and Compromise using SEM
to test the relationship between adolescent occu-
pational aspirations and midlife career success.
Villodas, Villodas, and Roesch (2011) examined
the factor structure of the Positive and Negative
Affect Schedule (PANAS) for a multiethnic sam-
ple of adolescents using a confirmatory factor
analysis (CFA). Tovar and Simon (2010)
employed a CFA to validate the factor structure
of the Sense of Belonging scales.
From the examples listed above, it is appar-
ent that SEM plays a vital role in the advance-
ment of counseling research and, as easy-to-use
SEM computer programs such as AMOS(Arbuckle & Wothke, 1999), become readily
accessible it can be expected that SEM will
be increasingly important in determining the
efficacy of counseling services and treatment.
While SEM is widely used in social science
research (Chan et al., 2007; Quintana &
Maxwell, 1999), to date no tutorial articles have
been published to assist counseling researchers
and practitioners in the step-by-step application
of SEM techniques. This article strives to famil-
iarize counseling researchers and practitioners
with the purpose and uses of SEM, as well as
provide an applied approach to conducting
SEM analysis. In particular, the article begins
with a general overview of SEM, including
key terms and definitions, a brief history of
SEM development, and the advantages and
limitations associated with the approach.
Readers will then learn how to conduct SEM
analysis in counseling research using a series
of five, applicable stages.
Overview of SEM
SEM is a second-generation multivariate analy-
sis technique that is used to determine the
extent to which an a priori theoretical model
is supported by the sample data (Raykov &
Marcoulides, 2000; Schumacker & Lomax,
2010). More specifically, SEM tests models
that specify how groups of variables define a
construct, as well as the relationships among
constructs. For example, consider a counseling
researcher who is interested in the impact of the
therapeutic working alliance, a construct that
cannot be directly measured, on the number
of counseling sessions a client attends. The
researcher could use SEM to determine
whether (a) variables such as agreement on
therapy tasks, agreement on therapy goals, and
the counselorclient emotional bond comprise
the construct therapeutic working alliance, and
(b) the therapeutic working alliance, as a
whole, is predictive of client number of coun-
seling sessions attended.
In essence, SEM uses hypothesis testing to
improve our understanding of the complex rela-
tionships that occur among observed variables
and latent constructs. Observed variables (i.e.,
indicator variables) are variables that can be
directly measured using tests, assessments, and
surveys, and are used to define a given latent
construct. Latent constructs cannot be directly
observed or measured and, as a result, must
be inferred from a set of observed variables.
In our example, agreement on therapy tasks,
agreement on therapy goals, and emotional
bond are observed variables that are directly
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measured by the Working Alliance Inventory
(WAI; Horvath & Greenberg, 1986). The
inventory yields three separate subscale scores
that are then used to make inferences regarding
the overall working alliance between the
counselor and client. Therefore, it can be said
that agreement on therapy tasks, agreement on
therapy goals, and the emotional bond between
the counselor and client define the latent con-
struct working alliance. The outcome variable,
number of counseling sessions attended, is also
an observed variable and, using SEM proce-
dures, a researcher can test the hypothesized
relationship between the observed outcome
variable and the latent predictor variable.
The Development of SEM
SEM was derived from the evolution of three
particular types of models: regression, path,
and confirmatory factor (Schumacker &
Lomax, 2010). The first step toward SEM
development was linear regression modeling.
Linear regression modeling is concerned with
observed variables only and attempts to predict
a dependent, observed variable from one or
more independent, observed variables. Regres-
sion models use a correlation coefficient and
least squares criterion to estimate the para-
meters of the model by minimizing the sum
of squared differences between observed and
predicted scores of the dependent variable. Path
analysis, another precursor to SEM, is also con-
cerned with observed variables, and predicts
relationships among observed variables by sol-
ving a series of concurrent regression equa-
tions. Path models permit the researcher to
test relationships among multiple independent
and dependent variables. Overall, path analysis
allows for the testing of more complex models
than linear regression analysis. The final model
that contributed to the development of SEM is
the confirmatory factor model. CFA assumes
that items on an inventory correlate with one
another and yield observed scores that measure
or define a construct. Confirmatory factor mod-
els seek to validate the existence of theoretical
constructs by empirically testing the relation-
ships between observed and latent variables.
SEM models combine path and factor analy-
tic models allowing for the incorporation of
both observed and latent variables into a model.
SEM procedures ultimately determine the plau-
sibility of a theoretical model by comparing the
estimated theoretical covariance matrixP
to
the observed covariance matrix S (i.e., the
matrix derived from the sample data; Schu-
macker & Lomax, 2010). Many SEM software
programs are currently available to researchers.
These include LISREL1, AMOS, EQS1,Mx, Mplus1, Ramona, and SEPATH1. Manyof the SEM software programs allow research-
ers to statistically analyze raw data and provide
procedures for managing missing data, outliers,
and variable transformations. Programs, such
as AMOS and LISREL1, offer researchersthe option to construct a path diagram that can
be translated by the software program into the
mathematical equations needed for analysis.
Advantages and Limitations of SEM Use
SEM techniques yield several advantages over
first-generation multivariate methods (Kline,
2010; Schumacker & Lomax, 2010). Most
importantly, SEM offers researchers an
enhanced understanding of the complex rela-
tionships that exist among theoretical con-
structs. As the counseling field continues to
explore increasingly complex phenomenon, the
theoretical models used to explain such phe-
nomenon are also increasing in complexity.
SEM techniques provide counseling research-
ers with a comprehensive method for specify-
ing and empirically testing the plausibility of
complex theoretical models (Kelloway, 1998).
SEM also allows for the simultaneous anal-
ysis of direct and indirect effects with multiple
exogenous and endogenous variables (Stage,
Carter, & Nora, 2004). A direct effect occurs
when the exogenous (i.e., independent) vari-
able influences an endogenous (i.e., dependent)
variable. An indirect effect, on the other hand,
occurs when the relationship between the exo-
genous and endogenous variable is mediated
by one or more intervening variables (Baron
& Kenny, 1986). While multiple regression
analysis can also be used to explore indirect
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relationships among variables (see e.g., Baron
& Kenny, 1986), it assumes that no measure-
ment error exists for the exogenous variables
(Raykov & Marcoulides, 2000). Such an
assumption rarely applies to actual practice.
Ignoring potential measurement error can
adversely impact the validity and reliability of
a study and, as a result, multiple regression
techniques may be highly susceptible to errors
in interpretation. SEM techniques, on the other
hand, overtly take into account the measure-
ment error in the models observed variables
(Schumacker & Lomax, 2010).
In addition, SEM affords counseling
researchers the ability to test increasingly com-
plex theoretical models. For example, SEM per-
mits the same variable to be interpreted as both
an exogenous and endogenous variable (Stage
et al., 2004), and allows for an interaction term
to be included in the theoretical model in order
to test main and interaction (i.e., moderator)
effects (Schumacker & Lomax, 2010). These
techniques can also be used to compare alterna-
tive theoretical models in order to assess the
relative fit of each model, which decreases the
high frequency of model misspecification found
in regression analysis (Skosireva, 2010).
Finally, SEM provides a path diagram, or visual
representation of the hypothesized relationships
among variables, that can be directly translated
into the mathematical equations needed for
analysis (Raykov & Marcoulides, 2000; Stage
et al., 2004).
While SEM has several advantages over tra-
ditional, first-generation multivariate methods,
there are limitations associated with using this
technique. Similar to other multivariate statisti-
cal techniques, SEM examines the correlations
among variables, but cannot establish causal
effects. As a result, the successful application
of SEM techniques relies on the researchers
theoretical knowledge of each variable (Stage
et al., 2004). SEM is also an inherently confir-
matory technique and is most advantageous
when the researcher has an a priori theoretical
model to test. It is not an exploratory technique
and is ill suited for exploring and identifying
relationships among variables (Kelloway,
1998, p. 7).
Steps for Conducting SEM Analysis
Prior to discussing the steps for conducting an
SEM analysis, counseling researchers should
be reminded that SEM is a correlational
research technique and, as a result, note the
analysis is impacted by measurement scales,
restriction of range, outliers, linearity, and non-
normality (Schumacker & Lomax, 2010).
Counseling researchers should take the time
to thoroughly screen the data, attending to out-
liers and missing data, as well as issues related
to linearity and normality before running SEM
analysis. The actual SEM analysis consists of a
series of five sequential steps: model specifica-
tion, model identification, model estimation,
model testing, and model modification (Bollen
& Long, 1993; see Table 1). The remainder of
this section discusses each of these steps at
length. To illustrate the application of SEM
procedures, an example theoretical model
based on a study by Crockett (2011) is used
throughout this section. The study examined
the impact of supervisor multicultural compe-
tence and the supervisory working alliance on
supervision outcomes in a sample of 221 coun-
seling trainees enrolled in masters and doc-
toral level counseling programs across the
United States.
Model Specification
Model specification is the first step of SEM
analysis and occurs prior to data collection and
analysis. It is often the most difficult step for
researchers as it involves the development of
a theoretical model using applicable, related
theory and research to determine variables of
interest and the relationships among them
(Cooley, 1978). It is critical that the hypothe-
sized theoretical model be grounded in and
derived from the extant literature. The
researcher must be able to provide plausible
explanations for relationships included in the
model and a rationale for the overall specifica-
tion of a model. The example theoretical
model attempts to specify the relationship
between supervisor multicultural competence
and supervisee outcomes. The model
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hypothesizes: (a) that supervisor multicultural
competence directly impacts supervisee coun-
seling self-efficacy (CSE) and (b) that supervi-
sor multicultural competence indirectly
impacts supervisee CSE through the supervi-
sory working alliance. That is, the supervisory
working alliance mediates the relationship
between the exogenous variable and endogen-
ous variables. The hypothesized relationships
among the variables are depicted as:
Supervisor Multicultural Competence
! Supervisee CSE Supervisor MulticulturalCompetence! Supervisory WorkingAlliance! Supervisee CSE:
Given that SEM models contain both observed
and latent variables, model specification is a
two-step building process (Anderson &
Gerbing, 1988). First, the measurement model
must be specified; this involves the identifica-
tion of observed variables that comprise each
of the models latent constructs. It is important
to note that the measurement model does not
specify directional relationships among the
latent variables. The measurement model in
the example includes three latent constructs.
The first latent variable, supervisory working
alliance, is estimated by the three observed
factors (i.e., task, goals, and bond subscales)
that comprise the underlying structure of the
WAI-Short Form (WAI-SF; Ladany, Mori, &
Mehr, 2007). The second latent variable,
supervisee CSE, is estimated by the five fac-
tors (i.e., microskills, counseling process, dif-
ficult client behaviors, cultural competence,
Table 1. Steps for Conducting SEM Analysis
SEM Step Description of Step
Model specification This step involves the specification of a theoretical model that utilizes applicable,related theory and research to determine the latent and observed variables ofinterest and the relationships among them. In particular, researcher must specify ameasurement and structural model. A path diagram can be constructed to visuallyrepresent the hypothesized relationships among variable in the theoretical model
Model identification This step helps the researcher to determine whether the specified model is capable ofproducing actual results that can be estimated in SEM analysis. Models must beindentified and able to generate a unique solution and parameter estimates.OBriens (1994) criteria can be used to establish whether a measurement model isidentified. To determine whether a structural model is indentified researchers canuse Bollens (1989) recursive rule and the t rule
Model estimation This step involves the use of an iterative procedure (i.e., fitting function) to generatethe theoretical covariance matrix
P, as well as minimize the differences between
the estimated theoretical covariance matrixP
and the observed covariance matrixS. Maximum likelihood (ML) and generalized least squares (GLS) are the mostcommonly used fitting functions
Model testing This step involves the analysis of both the measurement and structural models in orderto determine (a) the global fit of the entire model, and (b) the fit of individual modelparameters. Multiple indices of fit (i.e., absolute, comparative, and parsimonious)should be analyzed to determine the degree to which the theoretical model fits thesample data. The w2 difference test can also be used when working with nestedmodels to compare the plausibility of the theoretical model to viable alterativemodels. It should be noted that the measurement model must yield a good fit to thedata before the structural model can be analyzed
Model modification The final step involves using theory trimming or the addition of new parameters toattempt to improve the theoretical models fit to the data. Researchers should beadvised to model modification is an exploratory procedure and is based on thesample data instead of the extant literature. Respecified models will need to becross-validated with a new sample
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and counselor values/biases subscales) that
comprise the underlying structure of the Coun-
selor Self-Estimate Inventory (COSE; Larson
et al., 1992). The measurement model for the
example study can be expressed through a
series of eight equations wherein the error
term indicates the measurement error inherent
in the observed variable:
Tasks = function of supervisory
working alliance error;1
Goals = function of supervisory
working alliance error;2
Bond = function of supervisory
working alliance error;3
Microskills = function of supervisee
CSE error;4
Counseling process = function of
supervisee CSE error;5
Difficult client behaviors = function of
supervisee CSE error;6
Cultural competence = function of
supervisee CSE error;7
Counselor values/biases = function of
supervisee CSE error:8
If the latent constructs in the measurement
model are adequately measured by the observed
variables, then researchers can specify the struc-
tural model. The structural model specifies rela-
tionships among the latent variables in the
theoretical model. It is imperative that such rela-
tionships are indicated prior to model estimation
and testing as SEM is a confirmatory technique.
The structural model in the example study identi-
fies: (a) the hypothesized direct relationship
between the exogenous variable, supervisor mul-
ticultural competence, and the latent exogenous
variables, supervisee CSE; and (b) the hypothe-
sized indirect relationship between the
exogenous variable, supervisor multicultural
competence, and the latent exogenous variables,
supervisee CSE through the latent mediator vari-
able, supervisory working alliance. The struc-
tural model can also be illustrated through a
series of equations; because the model includes
a mediator variable three equations are
specified:
Supervisee CSE = structure coefficient1
Supervisor Multicultural Competence error;9
Supervisee CSE = structure coefficient2
Supervisory Working Alliance error;10
Supervisory Working Alliance = structure
coefficient3 Supervisor MulticulturalCompetence error:
11
The structural equations specify the estima-
tion of three structure coefficients (i.e., ele-
ments that comprise the estimated theoretical
covariance matrixP
). Each equation contains
a prediction error which specifies the degree
of variance in the latent endogenous variable
that is not accounted for by the other variables
in the equation (Schumacker & Lomax, 2010).
Finally, the equations specify the direction of
the predicted relationships.
The hypothesized relationships among
observed and latent variables in a theoretical
model can also be illustrated through a path
diagram (i.e., a graphical representation of the
theoretical model). Such diagrams use a series
of conventional symbols to depict the relation-
ships among model variables (see Figure 1). A
rectangle represents an observed variable,
whereas an oval denotes a latent variable. Uni-
directional arrows indicate a hypothesized
relationship in which one variable influences
another. These arrows are often referred to as
model paths. Bidirectional, curved arrows are
used to denote covariance between two inde-
pendent variables. Finally, the measurement
error for each observed, dependent variable
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is symbolized by a circle that includes an error
term and points toward the dependent vari-
able. Figure 2 provides an example path dia-
gram for the example model.
Model Identification
Model identification is a requirement for produc-
ing results that can be estimated in SEM analysis.
This step occurs prior to estimating model para-
meters (i.e., relationships among variables in the
model) and is concerned with whether a unique
solution to the model can be generated. For a
model to be considered identified, it must be
theoretically possible to establish a unique
estimate for each parameter (Kelloway, 1998;
Schumacker & Lomax, 2010) and is dependent
on the designation of model parameters as free
(i.e., a parameter that is unknown and needs to
be estimated), fixed (i.e., a parameter that is fixed
at a specific value, often a 0 or 1), or constrained
(i.e., a parameter that is unknown, but con-
strained to equal one or more other parameters).
For example, a theoretical model that exerts x y 20 has no sole solution; the value of x couldbe 10, 15, or 19. In order to find a unique solution
for x, the value of y must be fixed. If the value y is
fixed at 15 then x has to 5.
The measurement model must first be identi-
fied for the overall SEM to be identified. Accord-
ing to OBrien (1994), the measurement model is
most likely identified when: (a) there are two or
more latent variables, each with at least three
indicators that load on it, the errors of these indi-
cators are not correlated, and each indicator loads
on only one factor, or (b) there are two or more
latent variables, but there is a latent variable on
which only two indicators load, the errors of the
indicators are not correlated, each indicator loads
on only one factor, and the variances or covar-
iances between factors is zero. To increase the
likelihood of identification in the structural
model, a causal path from each latent variable
to a corresponding observed variable must be
fixed at zero. This one fixed, nonzero loading
is termed a reference variable and is often the
variable with the most reliable scores (Kline,
2010). CFA results (see section on model testing)
confirmed that the example measurement model
was identified as each latent variable had three or
more indicators that appropriately loaded on
each variable, the errors of the indicators were
not correlated, and each indicator in the model
only loaded on one factor. Additionally, the ref-
erence variable for each latent variable was iden-
tified in the CFA. The reference variable for
supervisory working alliance and supervisee
CSE was task and microskills, respectively.
Establishing that a structural model is iden-
tified can be extremely cumbersome and
involves highly complex mathematical calcula-
tions. As a result, Bollen (1989) outlined a
widely used set of rules for the identification
of structural models: the recursive rule and the
t rule. The recursive rule states that a structural
model should be recursive to be identified. A
structural model is recursive when all of the
relationships specified by the model are unidir-
ectional (i.e., two variables are not reciprocally
related; Schumacker & Lomax, 2010). To sat-
isfy the recursive rule: (a) the c matrix (i.e.,errors in the structural equations) of a structural
model must be diagonal, meaning that there are
no correlated errors in the endogenous vari-
ables, and (b) the b matrix must be able to be
Observed Variable
Latent Variable
Unidirectional, or recursive relationship
Nonrecursive relationship
Covariance among two independent variables
Measurement error for an observed variable
Figure 1. Hypothesized relationships amongobserved and latent variables in a theoretical modelcan be illustrated through a path diagram that uses aseries of conventional symbols to depict the rela-tionships among model variables.
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arranged so that all free elements are in the
lower triangle of the matrix, meaning that
no reciprocal relationships or feedback loops
exist among the endogenous variables (Bollen,
1989). A visual inspection of a models path
diagram, in conjunction with examining the cand b matrices found on the analysis outputof an SEM statistical software program allow
the researcher to determine whether the model
is recursive or not. An examination of Figure
2 indicates that the example model is recursive
as all relationships specified in the model
are unidirectional.
The t rule exerts that in an identified, recur-
sive model the number of parameters to be esti-
mated is less than the nonredundant (i.e., unique)
elements in the sample covariance matrix S
(i.e., the true model generated from the data).
Simply stated, the structural model must have
more known pieces of information than
unknown pieces in order to find unique solu-
tions. To determine whether this necessary con-
dition is met, the number of knowns (i.e., the
number of unique elements in the covariance
matrix of the structural model) is calculated
using p(p 1)/2, where p is equal to thenumber of observed variables. The number of
unknowns is equal to the number of free para-
meters to be estimated in the model (i.e., the
relationships between the exogenous and endo-
genous variables, relationships between the
endogenous variables, factor loadings, errors in
the equations, variance/covariance of the exo-
genous variables). In our example theoretical
model, there are nine observed variables; there-
fore, the number of unique elements in the cov-
ariance matrix (i.e., the number of knowns) is
45. The number of free parameters (i.e., the
number of unknowns) to be estimated in the
model is 9. Given that the number of unique ele-
ments in the covariance matrix exceeded the
number of free parameters in the model, the
model is said to be overidentified. SEM models
can also be underidentified or just-identified.
Underidentified models do not provide enough
information for the model parameters to be dis-
tinctively estimated and, as a result, fail to yield
a unique solution. Just-identified provide just
enough information for all of the model para-
meters to be uniquely estimated. Overidentified
and just-identified models are both considered to
be identified; however, an overidentified model
yields a number of possible solutions, whereas a
just-identified model produces only one solu-
tion. Given that the covariance matrix contains
many sources of error (e.g., sampling and mea-
surement error), researchers (Kelloway, 1998)
suggest that an overidentified model is ideal.
In an overidentified model, the goal of SEM is
to select the solution that comes closest to
explaining the observed data (Kelloway, 1998).
Underidentified models, such as x y 20, caneasily become identified by imposing additional
constraints on model parameters.
SupervisorMulticulturalCompetence
SupervisoryWorkingAlliance
CounselingSelf-Efficacy
TaskGoal
Bond
Microskill
Process
Difficult
Culture
Value
ee e
e
e
e
e
e
Figure 2. This path diagram depicts the hypothesized direct and indirect relationships among supervisor multi-cultural competence, the supervisory working alliance, and supervisee counseling self-efficacy as specified by theexample theoretical model.
Crockett 37
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Model Estimation
Model estimation, the third step of SEM analy-
sis, involves estimating the parameters of the
theoretical model in such a way that the theore-
tical parameter values yield a covariance
matrix as close as possible to the observed
covariance matrix S. SEM analysis programs
use an iterative procedure, often referred to as
a fitting function, to minimize the differences
between the estimated theoretical covariance
matrixP
and the observed covariance matrix
S. Specifically, the iterative procedure attempts
to improve the preliminary parameter estimates
with subsequent calculation cycles. The final
parameter estimates represent the best fit to
observed covariance matrix S.
Several fitting functions are available to
researchers (e.g., ordinary least squares [OLS],
generalized least squares [GLS], maximum like-
lihood [ML]). ML is the most widely used type
of estimation, followed by GLS (Kelloway,
1998). Although ML and GLS are comparable
to OLS estimation used in multiple regression,
they are slightly different from and several
advantages over OLS estimation. In particular,
ML and GLS are (a) not scale-dependent, (b)
allow dichotomous exogenous variables (Sko-
sireva, 2010), and (c) consistent and asymptoti-
cally efficient in large samples (Bollen, 1989;
Kelloway, 1998; Schumacker & Lomax,
2010). ML and GLS assume multivariate nor-
mality of dependent variables and, unlike OLS,
are full information techniques, meaning that
they estimate all model parameters simultane-
ously to produce a full estimation model. When
the assumption of multivariate normality is vio-
lated, researchers may use an asymptotically
distribution-free (ADF) estimator. ADF is not
dependent on the underlying distribution of the
data, but it does require a large sample size as the
estimator yields inaccurate chi-square (w2) statis-tics for smaller sample sizes (Mueller, 1996). For
more information on ADF, please see Raykov
and Widaman (1995). In the example model,
ML was employed by LISERL1 during theSEM analysis to minimize the differences
between the estimated theoretical covariance
matrixP
and the observed covariance matrix S.
Model Testing
As mentioned earlier SEM allows for the simul-
taneous analysis of direct and indirect relation-
ships among latent and observed variables;
however, many researchers (e.g., Anderson &
Gerbing, 1988; James, Mulaik, & Brett, 1982)
recommend a two-step approach to model test-
ing. In particular, James, Mulaik, and Brett
(1982) argued that model testing involved the
analysis of two conceptually distinct models: the
measurement model and the structural model.
The researcher must first determine whether the
proposed measurement model holds, ensuring
that the chosen observed indicators for a latent
construct actually measure the construct. If the
chosen indicators for a construct do not accu-
rately measure the construct, then the structural
model is meaningless (Joreskog & Sorbom,
1993). Accordingly, it is recommended that
researchers conduct a CFA of the measurement
model to determine whether the factor indicators
loaded on the latent variables in the direction
expected prior to testing the structural model.
A CFA of the example measurement model
was run prior to estimating the structural model
to ensure that all factors loaded on the latent
variables in the direction expected. Results
indicated an adequate fit of the CFA model,
w2(19) 44.72, p < .05; root mean squareerror of approximation (RMSEA) .07; com-parative fit index (CFI) .97; Parsimoniousnormed fit index (PNFI) .65, to the data (seeinformation on model fit). The standardized
parameter estimates were significant at the
p < .05 level and consistent with the specified
hypotheses, loading in the appropriate direc-
tion. The individual parameters comprising the
model were also analyzed. As predicted, the
latent variable supervisory working alliance
was significantly positively correlated with its
factor indicators: WAI-SF bond subscale (r .83, p < .05), WAI-SF task subscale (r .92,p < .05), and WAI-SF goal subscale (r .82,p < .05). The latent variable supervisor CSE
was also significantly positively correlated
with its factor indicators: COSE microskills
subscale (r .83, p < .05), COSE counselingprocess subscale (r .79, p < .05), COSE
38 Counseling Outcome Research and Evaluation 3(1)
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