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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]

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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

Crockett 31


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

32 Counseling Outcome Research and Evaluation 3(1)


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

Crockett 33


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



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


Bond = function of supervisory


Microskills = function of supervisee

CSE error;4

Counseling process = function of

supervisee CSE error;5

Difficult client behaviors = function of


Cultural competence = function of


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

Crockett 35


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.



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


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



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A Five-Step Guide to Conducting SEM Analysis in Counseling Research 2012