18067_Chapter_32

8/2/2019 18067_Chapter_32

1/23

488

32BEST PRACTICES IN STRUCTURALEQUATION MODELING

RA LPH O. MUELLER

GREGORY R. HA NCOCK

S

tructural equation modeling (SEM) hasevolved into a mature and popularmethodology to investigate theory-derived

structural/causal hypotheses. Indeed, with thecontinued development of SEM software pack-ages such as AMOS (Arbuckle, 2007), EQS(Bentler, 2006), LISREL (Jreskog & Srbom,2006), and Mplus (Muthn & Muthn, 2006),SEM has become the preeminent multivariatemethod of data analysis (Hershberger, 2003,pp. 4344).Yet, we believe that many practitionersstill have little, if any, formal SEM background,potentially leading to misapplications and pub-lications of questionable utility. Drawing on ourown experiences as authors and reviewers of

SEM studies, as well as on existing guides forreporting SEM results (e.g., Boomsma, 2000;Hoyle & Panter, 1995; McDonald & Ho, 2002),we offer a collection of best practices guidelinesto those analysts and authors who contemplate

using SEM to help answer their substantiveresearch questions. Throughout, we assume thatreaders have at least some familiarity with thegoals and language of SEM as covered in anyintroductory textbook (e.g., Byrne, 1998, 2001,2006; Kline, 2005; Loehlin, 2004; Mueller, 1996;Schumacker & Lomax, 2004). For those desiringeven more in-depth or advanced knowledge, werecommend Bollen (1989), Kaplan (2000), orHancock and Mueller (2006).

SETTING THE STAGE

The foundations of SEM are rooted in classical

measured variable path analysis (e.g., Wright,1918) and confirmatory factor analysis (e.g.,Jreskog, 1966, 1967). From a purely statisticalperspective, traditional data analytical techniquessuch as the analysis of variance, the analysis of

Authors Note: During the writing of this chapter, the first author was on sabbatical leave from The George Washington

University and was partially supported by its Center for the Study of Language and Education and the Institute for Education

Studies, both in the Graduate School of Education and Human Development. While on leave, he was visiting professor in the

Department of Measurement, Statistics and Evaluation (EDMS) at the University of Maryland, College Park, and visiting

scholar in its Center for Integrated Latent Variable Research (CILVR). He thanks the EDMS and CILVR faculty and staff for

their hospitality, generosity, and collegiality. Portions of this chapter were adapted from a presentation by the authors at the

2004 meeting of the American Educational Research Association in San Diego.

32-Osborne (Best)-45409.qxd 10/9/2007 1:00 PM Page 488

8/2/2019 18067_Chapter_32

2/23

Structural Equation Modeling 489

covariance, multiple linear regression, canonicalcorrelation, and exploratory factor analysisaswell as measured variable path and confirmatoryfactor analysiscan be regarded as special cases

of SEM. However, classical path- and factor-ana-lytic techniques have historically emphasized anexplicit link to a theoretically conceptualizedunderlying causal model and hence are moststrongly identified with the more general SEMframework. Simply put, SEM defines a set ofdata analysis tools that allows for the testing oftheoretically derived and a priori specified causalhypotheses.

Many contemporary treatments introduceSEM not just as a statistical technique but as a

processinvolving several stages: (a) initial model

conceptualization, (b) parameter identificationand estimation, (c) data-model fit assessment,and (d) potential model modification. As anystudy using SEM should address these fourstages (e.g., Mueller, 1997), we provide briefdescriptions here and subsequently use them asa framework for our best practices analysis illus-trations and publication guidelines.

Initial Model Conceptualization

The first stage of any SEM analysis should

consist of developing a thorough understandingof, and justification for, the underlying theory ortheories that gave rise to the particular model(s)being investigated. In most of the traditionaland typical SEM applications, the operational-ized theories assume one of three forms:

A measured variable path analysis(MVPA)

model: hypothesized structural/causal

relations among directly measured variables;

the four-stage SEM process applied to MVPA

models was illustrated in, for example,

Hancock & Mueller, 2004.

A confirmatory factor analysis(CFA) model:

structural/causal relations between

unobserved latent factors and their measured

indicators; the four-stage SEM process applied

to CFA models was illustrated in, for example,

Mueller & Hancock, 2001.

A latent variable path analysis(LVPA) model:

structural/causal relations among latent

factors. This type of SEM model is the focus

in this chapter and constitutes a combination

of the previous two. A distinction is made

between the structural and the measurement

portions of the model: While the former is

concerned with causal relations among latent

constructs and typically is the focus in LVPA

studies, the latter specifies how these

constructs are modeled using measuredindicator variables (i.e., a CFA model).

More complex models (e.g., multisample,latent means, latent growth, multilevel, or mix-ture models) with their own specific recommen-dations certainly exist but are beyond thepresent scope. Regardless of model type, how-ever, a lack of consonance between model andunderlying theory will have negative repercus-sions for the entire SEM process. Hence, metic-ulous attention to theoretical detail cannot be

overemphasized.

Parameter Identification and Estimation

A models hypothesized structural and non-structural relations can be expressed as popula-tion parameters that convey both magnitudeand sign of those relations. Before sample esti-mates of these parameters can be obtained, eachparameterand hence the whole modelmustbe shown to be identified; that is, it must be pos-sible to express each parameter as a function of

the variances and covariances of the measuredvariables. Even though this is difficult and cum-bersome to demonstrate, fortunately, the identi-fication status of a model can often be assessedby comparing the total number of parametersto be estimated, t, with the number of unique(co)variances of measured variables,

where p is the total number of measured vari-ables in the model. When t > u (i.e., whenattempting to estimate more parameters thanthere are unique variances and covariances), themodel is underidentified, and estimation ofsome (if not all) parameters is impossible. Onthe other hand, t u is a necessary but not suf-ficient condition for identification, and usuallyparameter estimation can commence: t = uimplies that the model is justidentified, whilet < u implies that it is overidentified (providedthat indeed all parameters are identified and anylatent variables in the system have been assignedan appropriate metric; see Note 4).

u =pp + 1

2,


8/2/2019 18067_Chapter_32

3/23

490 BEST ADVANCED PRACTICES IN QUANTITATIVE METHODS

SEM software packages offer a variety ofparameter estimation techniques for modelswhose identification can be established. Themost popular estimation method (and the

default in most SEM software packages) is max-imum likelihood(ML), an iterative large-sampletechnique that assumes underlying multivariatenormality. Alternative techniques exist (e.g.,generalized least squares [GLS], asymptoticallydistribution free [ADF; Browne, 1984], androbust estimators [Satorra & Bentler, 1994]),some of which do not depend on a particularunderlying distribution of the data, but still, thevast majority of substantive studies use ML.

Data-Model Fit Assessment

A central issue addressed by SEM is how toassess the fit between observed data and thehypothesized model, ideally operationalized asan evaluation of the degree of discrepancybetween the true population covariance matrixand that implied by the models structural andnonstructural parameters. As the populationparameter values are seldom known, the differ-ence between an observed, sample-based covari-ance matrix and that implied by parameterestimates must serve to approximate the popula-tion discrepancy. For a justidentified model, theobserved data will fit the model perfectly: Thesystem of equations expressing each model param-eter as a function of the observed (co)variances isuniquely solvable; thus, the sample estimate ofthe model-implied covariance matrix will, bydefault, equal the sample estimate of the popula-tion covariance matrix. However, if a model isoveridentified, it is unlikely that these two matri-ces are equal as the system of equations (express-ing model parameters as functions of observedvariances and covariances) is solvable in morethan a single way.

Abiding by a general desire for parsimony,overidentified models tend to be of more sub-stantive interest than justidentified ones becausethey represent simpler potential explanations ofthe observed associations. While data-model fitfor such models was initially conceived as a for-mal statistical test of the discrepancy between thetrue and model-implied covariance matrices (achi-square test with df = ut; Jreskog, 1966,1967), such a test now is often viewed as overlystrict given its power to detect even trivial devia-tions of a proposed model from reality. Hence,

many alternative assessment strategies have

emerged (for a now classic review, see Tanaka,1993) and continue to be developed. Data-modelfit indices for such assessment can be categorizedroughly into three broad classes (with recom-

mended indices in italics):

Absolute indices evaluate the overall dis-crepancy between observed and implied covari-ance matrices; fit improves as more parametersare added to the model and degrees of freedomdecrease: for example, the standardized rootmean square residual(SRMR), the chi-square test(recommended to be reported mostly for its his-torical significance), and the goodness-of-fitindex (GFI).

Parsimonious indices evaluate the overalldiscrepancy between observed and impliedcovariance matrices while taking into account amodels complexity; fit improves as more param-eters are added to the model, as long as thoseparameters are making a useful contribution: forexample, the root mean square error of approxima-tion (RMSEA) with its associated confidenceinterval, the Akaike information criterion (AIC)for fit comparisons across nonnested models, andthe adjusted goodness-of-fit index (AGFI).

Incremental indicesassess absolute or parsi-

monious fit relative to a baseline model, usuallythe null model (a model that specifies no relationsamong measured variables): for example, thecomparative fit index(CFI), the normed fit index(NFI), and the nonnormed fit index (NNFI).

If, after considering several indices, data-model fit is deemed acceptable (and judged bestcompared to competing models, if applicable),the model is retained as tenable, and individualparameters may be interpreted. If, however, evi-dence suggests unacceptable data-model fit, the

next and often final stage in the SEM processis considered: modifying the model to improvefit in hopes of also improving the models corre-spondence to reality.

Potential Model Modification

In a strict sense, any hypothesized modelis, at best, only an approximation to reality; theremaining question is one of degree of that mis-specification. With regard to external specificationerrorswhen irrelevant variables were included inthe model or substantively important ones were left

outremediation can only occur by respecifying


8/2/2019 18067_Chapter_32

4/23


the model based on more relevant theory. On theother hand, internal specification errorswhenunimportant paths among variables were includedor when important paths were omittedcan

potentially be diagnosed and remedied using Waldstatistics(predicted increase in chi-square if a previ-ously estimated parameter were fixed to someknown value, e.g., zero) and Lagrange multiplierstatistics (also referred to as modification indices;estimated decrease in chi-square if a previouslyfixed parameter were to be estimated). As thesetests recommendations are directly motivated bythe data and not by theoretical considerations, anyresulting respecifications must be viewed asexploratory in nature and might not lead to amodel that resembles reality any more closely than

the one(s) initially conceptualized.

BEST PRACTICES IN SEM DATAANALYSIS: A SET OF ILLUSTRATIONS

Using the four-stage SEM process as a frame-work, we turn to an illustration of best practicesin the most common type of SEM analyses. Wechose to focus on a set of hypothesized modelsinvolving structural/causal relations amonglatent factors (i.e., LVPA models) to demonstrateour preference for using a two-phase approach(i.e., a measurement phase followed by a struc-tural phase) over a single-phase, all-in-oneanalysis. We conclude this section by illustratingthe statistical comparison of hierarchicallyrelated or nestedmodels (occurring, for example,when one models parameters are a proper sub-set of another models parameters) and address-ing the disattenuation (i.e., purification andstrengthening) of structural parameter esti-mates obtained from an LVPA when comparedwith those obtained from an analysis of the

same overall structure but one that uses mea-sured variables only.

Suppose an educational researcher is inter-ested in investigating the structural effects ofgirls reading and mathematics self-concept(Read-SC and Math-SC, respectively) on math-ematics proficiency (Math-Prof), as potentiallymediated by task-goal orientation (Task-Goal).More specifically, the investigator might havestrong theoretical reasons to believe that at leastone of three scenarios is tenable: In Model 1(Figure 32.1a), it is hypothesized that the effects

of Read-SC and Math-SC on Math-Prof are

both completely mediated by Task-Goal. InModel 2 (Figure 32.1b), only the effect of Read-SC on Math-Prof is completely mediated byTask-Goal, while Math-SC affects Math-Prof

not only indirectly via Task-Goal but alsodirectly without other intervening variables.Finally, in Model 3 (Figure 32.1c), Read-SC andMath-SC are thought to affect Math-Profdirectly as well as indirectly via Task-Goal. Toillustrate the testing of the tenability of thesethree competing models, multivariate normaldata on three indicator variables for each of thefour constructs were simulated for a sample ofn = 1,000 ninth-grade girls. Table 32.1 describesthe 12 indicator variables in more detail, whileTable 32.2 contains relevant summary statistics.1

At this point, it is possible and might seementirely appropriate to address the research ques-tions implied by the hypothesized models througha series of multiple linear regression (MLR) analy-ses. For example, for Model 2 in Figure 32.1b, twoseparate regressions could be conducted: (1) Anappropriate surrogate measure of Math-Profcould be regressed on proxy variables for Math-SC and Task-Goal, and (2) a suitable indicator ofTask-Goal could be regressed on proxies for Read-SC and Math-SC. If the researcher would chooseitems ReadSC3, MathSC3, TG1, and Proc from

Table 32.1 as surrogates for their respective con-structs, MLR results would indicate that eventhough all hypothesized effects are statistically sig-nificantly different from zero, only small amountsof variance in the dependent variables TG1 andProc are explained by their respective predictorvariables (R2TG1 = 0.034, R

2Proc = 0.26; see Table

32.6 for the unstandardized and standardizedregression coefficients obtained from the twoMLR analyses2). As we will show through thecourse of the illustrations below, an appropriatelyconducted LVPA of the models in Figure 32.1 andthe data in Table 32.2 will greatly enhance the util-ity of the data to extract more meaningful resultsthat address the researchers key questions.

SEM Notation

As the three alternative structural modelsdepicted in Figure 32.1 are at the theoretical/latent construct level, we followed commonpractice and enclosed the four factors of Read-SC, Math-SC, Task-Goal, and Math-Prof inellipses/circles. On the other hand, a glanceahead at the operationalized model in Figure 32.2

reveals that the now included measured variables


8/2/2019 18067_Chapter_32

5/23


Read-SC

(F1)

(a) Model 1

Task-Goal

(F3)

Math-Prof

(F4)

Math-SC

(F2)

cF1

cF2

cD3 cD4

bF3F1

bF4F3

bF3F2

cF2F1

D3 D4

Read-SC

(F1)

Task-Goal

(F3)

Math-Prof

(F4)

Math-SC

(F2)

cF1

cF2

cD3 cD4

bF3F1

bF4F3

bF4F2bF3F2

cF2F1

D3 D4

(b) Model 2

Read-SC

(F1)

Task-Goal

(F3)

Math-Prof

(F4)

Math-SC

(F2)

cF1

cF2

cD3 cD4

bF3F1

bF4F3

bF4F2

bF4F1

bF3F2

cF2F1

D3 D4

(c) Model 3

Figure 32.1 The theoretical models.


8/2/2019 18067_Chapter_32

6/23

(items RSC1 to RSC3, MSC1 to MSC3, TG1 toTG3, Math, Prob, and Proc) are enclosed in rec-

tangles/squares. Using the Bentler-WeeksVFED labeling convention (V for measuredVariable/item, F for latent Factor/construct, Efor Error/measured variable residual, D forDisturbance/latent factor residual), the latentand measured variables in the current modelsare labeled F1 through F4 and V1 through V12,respectively. The hypothesized presence orabsence of relations between variables in themodel is indicated by the presence or absence ofarrows in the corresponding path diagram:One-headed arrows signify direct structural or

causal effects hypothesized from one variable toanother, while two-headed arrows denote

hypothesized covariation and variation withoutstructural specificity. For example, for Model 1

in Figure 32.1a, note (a) the hypothesizedcovariance between Read-SC and Math-SC andthe constructs depicted variances (two-headedarrows connect the factors to each other and tothemselves, given that a variables variance canbe thought of as a covariance of the variablewith itself), (b) the hypothesized structuraleffects of these two factors on Task-Goal (one-headed arrows lead from both to Task-Goal),but (c) the absence of such hypothesized directeffects on Math-Prof (there are no one-headedarrows directly leading from Read-SC and

Math-SC to Math-Prof; the former two con-structs are hypothesized to affect the latter only


Table 32.1 Indicator Variable/Item Description

Construct Variable Label Item Scores

Read-SC (F1)

RSC1 (V1) Compared to others my age, 1 (false) to 6 (true)

I am good at reading.

RSC2 (V2) I get good grades in reading.

RSC3 (V3) Work in reading class is easy for me.

Math-SC (F2)

MSC1 (V4) Compared to others my age, 1 (false) to 6 (true)

I am good at math.

MSC2 (V5) I get good grades in math.

MSC3 (V6) Work in math class is easy for me.

Task-Goal (F3)

TG1 (V7) I like school work that Ill learn 1 (false) to 6 (true)

from, even if I make a lot of mistakes.

TG2 (V8) An important reason why I do

my school work is because I like

to learn new things.

TG3 (V9) I like school work best when

it really makes me think.

Math-Prof (F4)

Math (V10) Mathematics subtest scores

of the Stanford AchievementTest 9

Prob (V11) Problem Solving subtest

scores of the Stanford

Achievement Test 9

Proc (V12) Procedure subtest scores

of the Stanford Achievement

Test 9


8/2/2019 18067_Chapter_32

7/23


indirectly, mediated by Task-Goal). Finally,because variation in dependent variables usuallyis not fully explainable by the amount of variationor covariation in their specified causes, eachdependent variable has an associated residualterm. For example, in the operationalized modelin Figure 32.2, D3 and D4 denote the predictionerrors associated with the latent factors F3(Task-Goal) and F4 (Math-Prof), while E1through E3 indicate the residuals associated

with the measured indicator variables (V1 toV3) of the latent construct Read-SC.

For purposes of labeling structural and non-structural parameters associated with the connec-tions between measured and/or latent variables in apath diagram, we used the abcsystem3 (Hancock &Mueller, 2006, pp. 46). Structural effects from onevariable (measured or latent) to another are labeledbto,from, with the subscripts indicating the to andfromvariables (e.g.,in Figure 32.2,bF3F1 indicates the path

Table 32.2 Correlations and Standard Deviations of Simulated Data

READSC1 READSC2 READSC3 MATHSC1 MATHSC2 MATHSC3

(V1) (V2) (V3) (V4) (V5) (V6)

READSC1 1.000

READSC2 0.499 1.000

READSC3 0.398 0.483 1.000

MATHSC1 0.206 0.148 0.123 1.000

MATHSC2 0.150 0.244 0.095 0.668 1.000

MATHSC3 0.121 0.091 0.308 0.633 0.641 1.000

GOALS1 0.141 0.150 0.123 0.140 0.143 0.167

GOALS2 0.123 0.151 0.134 0.163 0.180 0.145

GOALS3 0.161 0.199 0.160 0.147 0.151 0.158

SATMATH 0.049 0.007 0.003 0.556 0.539 0.521SATPROB 0.031 0.009 0.023 0.544 0.505 0.472

SATPROC 0.025 0.029 0.006 0.513 0.483 0.480

SD 1.273 1.353 1.285 1.396 1.308 1.300

GOALS1 GOALS2 GOALS3 SATMATH SATPROB SATPROC

(V7) (V8) (V9) (V10) (V11) (V12)

READSC1

READSC2

READSC3

MATHSC1MATHSC2

MATHSC3

GOALS1 1.000

GOALS2 0.499 1.000

GOALS3 0.433 0.514 1.000

SATMATH 0.345 0.385 0.337 1.000

SATPROB 0.304 0.359 0.281 0.738 1.000

SATPROC 0.259 0.330 0.279 0.714 0.645 1.000

SD 1.334 1.277 1.265 37.087 37.325 45.098


8/2/2019 18067_Chapter_32

8/23


to F3 from F1, and bV2F1 denotes the path/factorloading to item V2 from factor F1). On the otherhand, variances and covariances are labeled by theletter c(e.g., in Figure 32.2, cF1 denotes the varianceof the latent construct F1, while cF2F1 represents thecovariance between the factors F2 and F1).

All-in-One SEM AnalysisGenerally Not Recommended

Although we generally do not recommendthe analytic strategy outlined in this section, itnevertheless will prove pedagogically instruc-tive and will motivate arguments in later sec-tions. With a hypothesized structure amonglatent constructs in place and associated mea-sured indicators selected, Model 1 in Figure 32.1acan be operationalized as illustrated in Figure32.2. This path diagram implies a set of 14structural equations, one for each dependentvariable: two equations from the structural por-tion of the model (i.e., the part that specifies thecausal structure among latent constructs) and12 equations from the measurement portion of

the model (i.e., the part that links each of theindicator variables with the designated latentconstructs). Table 32.3 lists all 14 structuralequations and their associated endogenous(dependent) and exogenous (independent)variables that together specify the model inFigure 32.2 (variables are assumed to be mean-centered, thus eliminating the need for inter-cept terms; items V1, V4, V7, and V10 are usedas reference variables for their respective fac-tors, and thus their factor loadings are notfreeto be estimated butfixedto 1.0; also see Note 4).

Though it might seem that the statisticalestimation of the unknown coefficients in thestructural equations (the b and c parameters)should be the focus at this stage of the analysis,a prior assessment of the data-model fit is moreessential as it allows for an overall judgmentabout whether the data fit the structure ashypothesized (indeed, should evidence materi-alize that the data do not fit the model, inter-pretations of individual parameter estimatesmight be useless). As can be verified from thepath diagram in Figure 32.2 by counting one- and

cE7

cE1

cE2

cE3

cE4

cE5

cE6

cE8 cE9 cE10 cE11 cE12

cD3 cD4

bV3F1 cF1

cF2

Proc

(V12)

Prob

(V11)Math

(V10)

TG3

(V9)

TG2

(V8)

TG1

(V7)

E7 E8 E9 E10 E11 E12

bV6F2

bV8F3

bV9F3 bV12F3

bF4F3

bF3F2

bF3F1

cF2F1

bV11F4

D3

E4

E1

E2

E3

E5

E6

D4

Math-Prof

(F4)

Task-Goal

(F3)

1

1

1

1

Math-SC

(F2)

Read-SC

(F1)

RSC3

(V3)

RSC2

(V2)

RSC1

(V1)

MSC1

(V4)

MSC2

(V5)

MSC3

(V6)

1 1 1 1 1 1

1

1

1

1

1

1

1 1

bV1F1

bV5F2

Figure 32.2 Initially operationalized Model 1.


8/2/2019 18067_Chapter_32

9/23


Table 32.3 Structural Equations Implied by the Path Diagram in Figure 32.2

Structural Portion

Endogenous Variable Structural Equations Exogenous Variables a

Task-Goal (F3) F3 = bF3F1 F1 + bF3F2 F2 + D3 Read-SC (F1)

Math-SC (F2)

Math-Prof (F4) F4 = bF4F3 F3 + D4 Task-Goal (F3)

Measurement Portion

Endogenous Variable Structural Equations Exogenous Variables b

RSC1 (V1) V1 = (1)F1 + E1 Read-SC (F1)

RSC2 (V2) V2 = bV2F1 F1 + E2

RSC3 (V3) V3 = bV3F1 F1 + E3

MSC1 (V4) V4 = (1)F2 + E4 Math-SC (F2)MSC2 (V5) V5 = bV5F2 F2 + E5

MSC3 (V6) V6 = bV6F2 F2 + E6

TG1 (V7) V7 = (1)F3 + E7 Task-Goal (F3)

TG2 (V8) V8 = bV8F3 F3 + E8

TG3 (V9) V9 = bV9F3 F3 + E9

Math (V10) V10 = (1)F4 + E10 Math-Prof (F4)

Prob (V11) V11 = bV11F4 F4 + E11

Proc (V12) V12 = bV12F4 F4 + E12

a. Residuals D, though technically independent, are not listed.b. Residuals E, though technically independent, are not listed.

two-headed arrows labeled with bor csymbols,the model contains t= 28 parameters to be esti-mated:4 two variances of the independent latentconstructs and one covariance between them,two variances of residuals associated with thetwo dependent latent constructs, three pathcoefficients relating the latent constructs, eightfactor loadings, and 12 variances of residualsassociated with the measured variables. Further-

more, the 12 measured variables in the modelproduce u = 12 (12 + 1)/2 = 78 unique variancesand covariances; the model is overidentified (t=28 < u = 78), and it is likely that some degree ofdata-model misfit exists (i.e., the observedcovariance matrix will likely differ, to somedegree, from that implied by the model). Toassess the degree of data-model misfit, variousfit indices can be obtained and then should becompared against established cutoff criteria(e.g., those empirically derived by Hu & Bentler,1999, and listed here in Table 32.4). Though

here LISREL 8.8 (Jreskog & Srbom, 2006)

was employed, running any of the availableSEM software packages will verify the followingdata-model fit results for the data in Table 32.2and the model in Figure 32.2 (because the dataare assumed multivariate normal, the maxi-mum likelihood estimation method was used):2 = 3624.59 (df= u t= 50,p < .001), SRMR =0.13, RMSEA = 0.20 with CI90: (0.19, 0.20), andCFI = 0.55.

As is evident from comparing these resultswith the desired values in Table 32.4, the currentdata do not fit the proposed model; thus, it isnot appropriate to interpret any individual para-meter estimates as, on the whole, the model inFigure 32.2 should be rejected based on the cur-rent data. Now the researcher is faced with thequestion of what went wrong: (a) Is the sourceof the data-model misfit indeed primarily a flawin the underlying structural theory (Figure 32.1a),(b) can the misfit be attributed to misspecifica-tions in the measurement portion of the model

with the hypothesized structure among latent


8/2/2019 18067_Chapter_32

10/23

constructs actually having been specified cor-rectly, or (c) do misspecifications exist in boththe measurement and structural portions of themodel? To help address these questions and pre-vent potential confusion about the source ofobserved data-model misfit, we do not recom-

mend that researchers conduct SEM analyses byinitially analyzing the structural and measure-ment portions of their model simultaneously,as was done here. Instead, analysts are urged tofollow a two-phase analysis process, as describednext.

Two-Phase SEM AnalysisRecommended

Usually, the primary reason for conceptualiz-ing LVPA models is to investigate the tenabilityof theoretical causal structures among latent

variables. The main motivation for recommend-ing a two-phase process over an all-in-oneapproach is to initially separate a model into itsmeasurement and structural portions so thatmisspecifications in the former, if present, canbe realized and addressed first, before the struc-ture among latent constructs is assessed.5 Thisapproach will simplify the identification ofsources of data-model misfit and might also aidin the prevention of nonconvergence problemswith SEM software (i.e., when the iterative esti-mation algorithm cannot converge upon a

viable solution for parameter estimates).

Consider the path diagram in Figure 32.3a. Itis similar to the one depicted in Figure 32.2 as itinvolves the same measured and latent variablesbut differs in two important ways: Not only are

Read-SC and Math-SC now explicitly connectedto Math-Prof, but all structural links amonglatent variables have been changed to nonstruc-tural relations (note in Figure 32.3a the two-headed arrows between all latent constructs thatare now labeled with csymbols). That is, latentconstructs are allowed to freely covary withoutan explicit causal structure among them. Inshort, Figure 32.3a represents a CFA model ofthe latent factors Read-SC, Math-SC, Task-Goal,and Math-Prof, using the measured variables inTable 32.1 as their respective effect indicators.6

Measurement Phase. An analysis of the CFAmodel in Figure 32.3a constitutes the beginningof the measurement phaseof the proposed two-phase analysis process and produced the follow-ing data-model fit results: 2 = 3137.16 (df= 48,

p < .001), SRMR = 0.062, RMSEA = 0.18 withCI90: (0.17, 0.18), and CFI = 0.61. These valuessignify a slight improvement over fit results forthe model in Figure 32.2. To the experiencedmodeler, this improvement was predictablegiven that the model in Figure 32.2 is more

restrictive than, and a special case of, the CFAmodel in Figure 32.3a (with the paths fromRead-SC and Math-SC to Math-Prof fixed tozero); that is, the former model is nestedwithinthe latter,a topic more fully discussed in the nextsection. Irrespective of this minor improvement,however, why did the data-model fit remainunsatisfactory (as judged by the criteria listed inTable 32.4)? Beginning to analyze and addressthis misfit constitutes a move toward the fourthand final phase in the general SEM process,potential post hoc model modification.

First, reconsider the list of items in Table 32.1.While all variables certainly seem to belong tothe latent factors they were selected to indicate,note that for the reading and mathematics self-concept factors, corresponding items are identicalexcept for one word: The word readingin itemsRSC1 through RSC3 was replaced by the wordmath to obtain items MSC1 through MSC3. Thus,it seems plausible that individuals responses tocorresponding reading and mathematics self-con-cept items are influenced by some of the same orrelated causes. In fact, the model in Figure 32.3a

explicitly posits that two such related causes are


Table 32.4 Target Values for Selected Fit Indicesto Retain a Model by Class

Index Class

Incremental Absolute Parsimonious

NFI 0.90

NNFI 0.95 GFI 0.90 AGFI 0.90

CFI 0.95 SRMR 0.08 RMSEA 0.06

Joint Criteria

NNFI, CFI 0.96 and SRMR 0.09

SRMR 0.09 and RMSEA 0.06

SOURCE: Partially taken from Hu and Bentler (1999).

NOTE: CFI = comparative fit index; NFI = normed fit index;

NNFI = nonnormed fit index; GFI = goodness-of-fit index;

AGFI = adjusted goodness-of-fit index; RMSEA = root mean

square error of approximation; SRMR = standardized root

mean square residual.


8/2/2019 18067_Chapter_32

11/23


the latent constructs Read-SC and Math-SC.However,as the specification of residual terms (E)indicates, responses to items are influenced bycauses other than the hypothesized latent con-

structs. Those other, unspecified causes could alsobe associated. Thus, for example, the residualterms E1 and E4 might covary to some degree,particularly since both are associated with itemsthat differ by just one word. Based on similar the-oretical reasoning, a nonzero covariance mightexist between E2 and E5 and also between E3 andE6. In sum, it seems theoretically justifiable tomodify the CFA model in Figure 32.3a to allowresidual terms of corresponding reading andmathematics self-concept items to freely covary, asshown in Figure 32.3b. In fact, with enough fore-

sight, these covariances probably should havebeen included in the initially hypothesized model.

Second, as part of the analysis of the initial CFAmodel in Figure 32.3a, Lagrange multiplier (LM)statistics may be consulted for empirically basedmodel modification suggestions. These statisticsestimate the potential improvement in data-modelfit (as measured by the estimated decrease in chi-square) if a previously fixed parameter were to beestimated. Here, the three largest LM statistics were652.0, 567.8, and 541.7, associated with the fixedparameters cE5E2, cE4E1, and cE6E3, respectively.

Compared with the overall chi-square value of3137.16, these estimated chi-square decreases seemsubstantial,7 foreshadowing a statistically signifi-cant improvement in data-model fit. Indeed, afterrespecifying the model accordingly (i.e., freeingcE5E2, cE4E1,and cE6E3; see Figure 32.3b) and reanalyz-ing the data, fit results for the modified CFA modelimproved dramatically: 2 = 108.04 (df= 45,p

18067_Chapter_32

Documents

Transcript of 18067_Chapter_32