Use of structural equation modeling in operations management research: Looking back and forward

www.elsevier.com/locate/dsw

Journal of Operations Management 24 (2006) 148–169

Use of structural equation modeling in operations

management research: Looking back and forward§

Rachna Shah *, Susan Meyer Goldstein 1

Operations and Management Science Department, Carlson School of Management,

321, 19th Avenue South, University of Minnesota, Minneapolis, MN 55455, USA

Received 10 October 2003; received in revised form 28 March 2005; accepted 3 May 2005

Available online 5 July 2005

Abstract

This paper reviews applications of structural equation modeling (SEM) in four major Operations Management journals

(Management Science, Journal of Operations Management, Decision Sciences, and Journal of Production and Operations

Management Society) and provides guidelines for improving the use of SEM in operations management (OM) research. We

review 93 articles from the earliest application of SEM in these journals in 1984 through August 2003. We document and assess

these published applications and identify methodological issues gleaned from the SEM literature. The implications of

overlooking fundamental assumptions of SEM and ignoring serious methodological issues are presented along with guidelines

for improving future applications of SEM in OM research. We find that while SEM is a valuable tool for testing and advancing

OM theory, OM researchers need to pay greater attention to these highlighted issues to take full advantage of its potential.

# 2005 Elsevier B.V. All rights reserved.

Keywords: Empirical research methods; Structural equation modeling; Operations management

1. Introduction

Structural equation modeling as a method for

measuring relationships among latent variables has

been around since early in the 20th century originating

in Sewall Wright’s 1916 work (Bollen, 1989). Despite

a slow but steady increase in its use, it was not until the

monograph by Bagozzi in 1980 that the technique was

§ Note: List of reviewed articles is available upon request from the

authors.

* Corresponding author. Tel.: +1 612 624 4432.

E-mail addresses: [email protected] (R. Shah),

[email protected] (S.M. Goldstein).1 Tel.: +1 612 626 0271.

0272-6963/$ – see front matter # 2005 Elsevier B.V. All rights reserved

doi:10.1016/j.jom.2005.05.001

brought to the attention of a much wider audience of

marketing and consumer behavior researchers. While

Operations Management (OM) researchers were slow

to use this new statistical approach, structural equation

modeling (SEM) has more recently become one of the

preferred data analysis methods among empirical OM

researchers, and articles that employ SEM as the

primary data analytic tool now routinely appear in

major OM journals.

Despite its regular and frequent application in the

OM literature, there are few guidelines for the

application of SEM and even fewer standards that

researchers adhere to in conducting analyses and

presenting and interpreting results, resulting in a large

.

R. Shah, S.M. Goldstein / Journal of Operations Management 24 (2006) 148–169 149

variance across articles that use SEM. To the best of

our knowledge, there are no reviews of the applica-

tions of SEM in the OM literature, while there are

regular reviews in other research areas that use this

technique. For instance, focused reviews have

appeared periodically in psychology (Hershberger,

2003), marketing (Baumgartner and Homburg, 1996),

MIS (Chin and Todd, 1995; Gefen et al., 2000),

strategic management (Shook et al., 2004), logistics

(Garver and Mentzer, 1999), and organizational

research (Medsker et al., 1994). These reviews have

revealed vast discrepancies and serious flaws in the use

of SEM. Steiger (2001) notes that even SEM textbooks

ignore many important issues, suggesting that

researchers may not have sufficient guidance to use

SEM appropriately.

Due to the complexities involved in using SEM and

problems uncovered in its use in other fields, a review

specific to OM literature seems timely and warranted.

Our objectives in conducting this review are three-

fold. First, we characterize published OM research in

terms of relevant criteria such as software used,

sample size, parameters estimated, purpose for using

SEM (e.g. measurement model development, struc-

tural model evaluation), and fit measures used. In

using SEM, researchers have to make subjective

choices on complex elements that are highly inter-

dependent in order to align research objectives with

analytical requirements. Therefore, our second objec-

tive is to highlight these interdependencies, identify

problem areas, and discuss their implications. Third,

we provide guidelines to improve analysis and

reporting of SEM applications. Our goal is to promote

improved usage of SEM, standardize terminology, and

help prevent some common pitfalls in future OM

research.

2. Overview of structural equation modeling

To provide a basis for subsequent discussion, we

present a brief overview of structural equation

modeling along with two special cases frequently

used in the OM literature. The overview is intended to

be a brief synopsis rather than a comprehensive

detailing of mathematical model specification. There

are a number of books (Maruyama, 1998; Bollen,

1989) and articles dealing with mathematical speci-

fication (Anderson and Gerbing, 1988), key assump-

tions underlying model specification (Bagozzi and Yi,

1988; Fornell, 1983), and other methodological issues

of evaluation and fit (MacCallum, 1986; MacCallum

et al., 1992).

At the outset, we point to a distinction in the use of

two terms that are often used interchangeably in OM:

covariance structure modeling (CSM) and structural

equation modeling (SEM). CSM represents a general

class of models that include ARMA (autoregressive

and moving average) time series models, multi-

plicative models for multi-faceted data, circumplex

models, as well as all SEM models (Long, 1983).

Thus, SEM models are a subset of CSM models. We

restrict the current review to SEM models because

other types of CSM models are rarely used in OM

research.

Structural equation modeling is a technique to

specify, estimate, and evaluate models of linear

relationships among a set of observed variables in

terms of a generally smaller number of unobserved

variables (see Appendix A for detail). SEM models

consist of observed variables (also called manifest or

measured, MV for short) and unobserved variables

(also called underlying or latent, LV for short) that can

be independent (exogenous) or dependent (endogen-

ous) in nature. LVs are hypothetical constructs that

cannot be directly measured, and in SEM are typically

represented by multiple MVs that serve as indicators

of the underlying constructs. The SEM model is an a

priori hypothesis about a pattern of linear relationships

among a set of observed and unobserved variables.

The objective in using SEM is to determine whether

the a priori model is valid, rather than to ‘find’ a

suitable model (Gefen et al., 2000).

Path analysis and confirmatory factor analysis are

two special cases of SEM that are regularly used in

OM. Path analysis (PA) models specify patterns of

directional and non-directional relationships among

MVs. The only LVs in such models are error terms

(Hair et al., 1998). Thus, PA provides for the testing of

structural relationships amongMVs when theMVs are

of primary interest or whenmultiple indicators for LVs

are not available. Confirmatory factor analysis (CFA)

requires that LVs and their associated MVs be

specified before analyzing the data. This is accom-

plished by restricting the MVs to load on specific LVs

and by designating which LVs are allowed to correlate.

R. Shah, S.M. Goldstein / Journal of Operations Management 24 (2006) 148–169150

Fig. 1. Illustrations of PA, CFA, and SEM models.

A CFA model allows for directional influences

between LVs and their MVs and (only) non-

directional (correlational) relationships between

LVs. Long (1983) provides a detailed (mathematical)

treatment of each of these techniques. Fig. 1 shows

graphical illustrations of SEM, PA and CFA models.

Throughout this paper, we use the term SEM to refer to

all three model types (SEM, PA, CFA) and note any

exceptions to this.

3. Review of published SEM research

Our review focuses on empirical applications of

SEM which include: (1) CFA models alone, such as

in measurement or validation research; (2) PA

models (provided they are estimated using software

which allows latent variable modeling); and (3) SEM

models that combine both measurement and struc-

tural components. We exclude theoretical papers,


papers using simulation, conventional exploratory

factor analysis (EFA), structural models estimated by

regression models (e.g. models estimated by two

stage least squares), and partial least squares (PLS)

models. EFA models are not included because the

measurement model is not specified a priori (MVs

are not restricted to load on a specific LV and a MV

can load on multiple LVs),1 whereas in SEM, the

model is explicitly defined a priori. The main

objective of regression and PLS models is prediction

of variance explanation in the dependent variable(s)

compared to theory development and testing in the

form of structural relationships (i.e. parameter

estimation) in SEM. This philosophical distinction

between these approaches is critical in deciding

whether to use PLS or SEM (Anderson and Gerbing,

1988). In addition, because assumptions underlying

PLS and regression are less constraining than

SEM, the problems and concerns in conducting

these analyses are significantly different. Therefore,

we do not include regression and PLS models in our

review.

3.1. Journal selection

We considered all OM journals that are recognized

as publishing high quality and relevant empirical OM

research. Recently, Barman et al. (2001) ranked

Management Science (MS), Operations Research

(OR), Journal of Operations Management (JOM),

Decision Sciences (DS), and Journal of Production

and Operations Management Society (POMS) as the

top OM journals in terms of quality. In the past decade,

several additional reviews have examined the quality

and/or relevance of OM journals and have consistently

ranked these journals in the top tier (Vokurka, 1996;

Goh et al., 1997; Soteriou et al., 1998; Malhotra and

Grover, 1998). We do not include OR in our review as

its mission does not include publishing empirical

research.We selected MS, JOM, DS, and POMS as the

journals most representative of high quality and

relevant empirical research in OM. In our review, we

include articles from these four journals that meet our

methodology criteria and do not exclude articles due

to topic of research.

1 Target rotation, rarely used in OM research, is an instance of

EFA in which the model is specified a priori.

3.2. Time horizon and article selection

Rather than use specific search terms for selecting

articles, we manually checked each article of the

reviewed journals. Although more time consuming,

the manual search gave us more control and better

coverage than a ‘‘keyword’’ based search because

there is no widely accepted terminology for research

methods in OM to conduct such a search. In selecting

an appropriate time horizon, we started with the most

recent issue of each journal available until August

2003 and moved backwards in time. Using this

approach, we reviewed all published issues of JOM

from 1982 (Volume 1, Number 1) to 2003 (Volume 21,

Number 4) and POM from 1992 (Volume 1, Number

1) to 2003 (Volume 12, Number 1). For MS and DS,

we moved backward in time until we no longer found

applications of SEM. The earliest application of SEM

in DS was found in 1984 (Volume 15, Number 2) and

the most recent issue reviewed is Volume 34, Number

1 (2003). The incidence of SEM in MS began in 1987

(Volume 34, Number 6) and we reviewed all issues

through Volume 49, Number 8 (2003). The earliest

publication in these two journals corresponds with our

knowledge of the field and seems to have face validity

as such because it coincides with the general

timeframe when SEM was beginning to gain attention

of the wider audience in other literature streams.

In total, we found 93 research articles that satisfied

our selection criteria. Fig. 2 shows the number of

articles stacked by journal for the years we reviewed.

This figure is very informative: overall, it is clear that

the number of SEM articles has increased significantly

over the past 20 years in the four journals individually

Fig. 2. Number of articles by journal and year.


and cumulatively. To assess the growth trend in the use

of SEM, we regress the number of articles on an index

of year of publication (beginning with 1984). We use

both linear and quadratic effects of time in the

regression model.

The regression model is significant (F2,17 = 39.93,

p = 0.000) and indicates that 82% of the variance in

the number of SEM publications is explained by the

linear and quadratic effects of time. Further, the linear

trend is not significant (t = �0.850, p = 0.41), whereas

the quadratic effect is significant (t = 2.94, p = .009).

So the use of SEM has not grown linearly as a function

of time, rather it has accelerated over time. In contrast,

the use of SEM in marketing and psychology grew

steadily over time and there is no indication of its

accelerated use in more recent years (Baumgartner

and Homburg, 1996; Hershberger, 2003).

There are several software programs available for

conducting SEM analysis, and each has idiosyncra-

sies and fundamental requirements for conducting

analysis. In our database, 19.6% of the articles did

not report the software used. Of the articles that

reported the software, LISREL accounted for 48.3%

followed by EQS (18.9%), SAS (9.1%), AMOS

(2.8%), RAMONA (0.7%) and SPSS (0.7%).

LISREL was the first software developed to solve

structural equation models and seems to have

capitalized on its first mover advantage not only in

psychology (MacCallum and Austin, 2000) and

marketing (Baumgartner and Homburg, 1996) but

also in OM.

3.3. Unit of analysis

In our review we found that multiple models were

sometimes presented in one article. Therefore, the unit

of analysis from this point forward (unless specified

otherwise) is the actual applications (one or more

models for each article). A single model is included in

our data set in the following situations: (1) when a

single model is proposed and evaluated using a single

sample; (2) when multiple alternative or nested

models are evaluated using a single sample, only

the final model is included in our analysis; (3) when a

single model is evaluated with either multiple samples

or by splitting a sample, only the model tested with the

verification sample is included in our analysis. Thus,

in these three cases, each article contributed only one

model to the analysis. When more than one model is

evaluated (using single, multiple, or split samples)

each distinct model is included in our analysis. In this

situation, each article contributed more than one

model to the analysis. A total of 143 models were

drawn from the 93 research articles, thus the overall

sample size for the remainder of the paper is 143. Of

the 143 models, we could not determine the method

used for four models. Of the remaining 139 models, 26

are PAs, 38 are CFAs, and 75 are SEMs. There are a

small number of articles that reported models that

never achieved adequate fit (by the authors’ descrip-

tions), and while we include these articles in our

review, the fit measures are omitted from our analysis

to avoid inclusion of data related to models with

inadequate fit.

4. Critical issues in the application of SEM

There are many important issues to consider when

using SEM, whether for evaluating a measurement

model or examining the fit of structural relationships,

separately or simultaneously. Our discussion of issues

is organized into three groups: (1) issues to consider or

address prior to analysis are categorized under the

‘‘pre-analysis’’ stage; (2) issues and concerns to

address during analysis; and (3) issues related to the

post-analysis stage, which includes issues related to

evaluation, interpretation and presentation of results.

Decisions made at each stage are highly interdepen-

dent and significantly impact the quality of results, and

we cross-reference and discuss these interdependen-

cies whenever possible.

4.1. Issues related to pre-analysis stage

Issues related to the pre-analysis stage need to be

considered prior to conducting SEM analysis and

include conceptual issues, sample size issues, mea-

surement model specification, latent model specifica-

tion, and degrees of freedom issues. A summary of

pre-analysis data from the reviewed OM studies is

presented in Table 1.

4.1.1. Conceptual issues

An underlying assumption of SEM analysis is that

the items or indicators used to measure a LV are


Table 1

Issues related to pre-analysis stage

Path analysis

models

Confirmatory

factor analysis

models

Structural

equation

models

All modelsa

Number of models revieweda 26 38 75 143

Sample size

Median 125.0 141.0 202.0 176.0

Mean 251.2 245.4 246.4 243.3

Range (18, 2338) (63, 902) (52, 840) (16, 2338)

Number of parameters estimated

Median 10.0 31.0 34.0 26.0

Mean 11.3 38.3 37.5 31.9

Range (2, 34) (8, 98) (11, 101) (2, 101)

Sample size/parameters estimated

Median 9.6 6.2 5.6 6.4

Mean 33.5 8.8 7.4 13.2

Range (2.9, 389.7) (2.3, 36.1) (1.6, 25.4) (1.6, 389.7)

Number of manifest variables

Median 6.0 12.5 12.0 11.0

Mean 6.3 13.5 16.3 14.0

Range (3, 10) (4, 32) (5, 80) (3, 80)

Number of latent variables

Median Not relevant 3.0 4.0 4.0

Mean Not relevant 3.66 4.7 4.4

Range Not relevant (1, 10) (1, 12) (1, 12)

Manifest variables/latent variable

Median Not relevant 4.0 3.3 3.6

Mean Not relevant 5.2 4.1 4.5

Range Not relevant (1.3, 16.0) (1.3, 9.0) (1.3, 16.0)

Number of single indicator

latent variablesbNot relevant Reported for 1 model Reported for 25 models Reported for 28 models

Correlated measurement

errors (CMEs)

1 model unknownc 11 models (28.9%) 8 models (10.7%),

4 models unknownc19 models (13.3%),

6 models unknownc

Theoretical justification

for CMEs

Not relevant 0 (0% of CFA models

with CMEs)

4 (50% of SEM models

with CMEs)

4 (21% of all models

with CMEs)

Recursiveness 127 (88.8%) recursive; 13 (9.1%) nonrecursive; not reported or could not be determined from model

description for 3 (2.1%) models

Evidence of model identification Reported by 3.8% Reported by 26.3% Reported by 5.3% Reported by 10.5%

Degrees of freedom (d.f.)

Median 4.5 62.0 52.5 48.0

Mean 4.6 90.1 124.5 99.7

Range (1, 11) (5, 367) (4, 690) (1, 690)

Proportion reporting 53.8% 52.6% 88.0% 71.3%a The type of analysis performed could not be determined for 4 of 143 models published in 93 articles.b The number of latent variables modeled using a single measured variable (i.e. single indicator).c Presence of CMEs could not be determined due to inadequate model description.


reflective (i.e. caused by the same underlying LV) in

nature. Yet researchers frequently apply SEM to

formative indicators. Formative (also called causal)

indicators are measures that form or cause the creation

of a LV (MacCallum and Browne, 1993; Bollen,

1989). An example of formative measures is the

amount of beer, wine and hard liquor consumed to

indicate level of mental inebriation (Chin, 1998). It

can be hardly argued that mental inebriation causes the

amount of beer, wine and hard liquor consumption. On

the contrary, the amount of each type of alcoholic

beverage affects the level of mental inebriation.

Formative indicators do not need to be highly

correlated or have high internal consistency (Bollen,

1989). In this example, an increase in beer consump-

tion does not imply an increase in wine or hard liquor

consumption. Measurement of formative indicators

requires an index (as opposed to developing a scale

when using reflective indicators), and can be modeled

using SEM, but requires additional constraints

(Bollen, 1989; MacCallum and Browne, 1993). Using

SEM without additional constraints makes the

resulting estimates invalid (Fornell et al., 1991) and

the model statistically unidentified (Bollen and

Lennox, 1991).

Another underlying assumption for SEM is that the

theoretical relationships hypothesized in the models

being tested represent actual relationships in the

studied population. SEM assesses how closely the

observed data correspond to the expected patterns

and requires that relationships represented by the

model are well established and amenable to accurate

measurement in the population. SEM is not recom-

mended for exploratory research when the measure-

ment structure is not well defined or when the theory

that underlies patterns of relationships among LVs is

not well established (Brannick, 1995; Hurley et al.,

1997).

Thus, researchers need to carefully consider: (1)

type of items, (2) state of underlying theory, and (3)

stage of development of measurement instrument,

prior to using SEM. For formative measurement items,

researchers should consider alternative techniques

such as SEM using formative indicators (MacCallum

and Browne, 1993) and components-based approaches

such as partial least squares (Cohen et al., 1990).

When the underlying theory or the measurement

structure is not well developed, simpler data analytic

techniques such as EFA and regression analysis may

be more appropriate (Hurley et al., 1997).

4.1.2. Sample size issues

Adequacy of sample size has a significant impact

on the reliability of parameter estimates, model fit,

and statistical power. Using a simulation experiment

to examine the effect of varying sample size to

parameter estimate ratios, Jackson (2003) reports that

smaller sample sizes are generally characterized by

parameter estimates with low reliability, greater

bias in x2 and RMSEA fit statistics, and greater

uncertainty in future replication. How large a sample

should be for SEM is deceptively difficult to

determine because it is dependent upon several

characteristics such as number of MVs per LV

(MacCallum et al., 1996), degree of multivariate

normality (West et al., 1995), and estimation method

(Tanaka, 1987). Suggested approaches for determin-

ing sample size include establishing a minimum (e.g.,

200), having a certain number of observations per

MV, having a certain number of observations per

parameters estimated (Bentler and Chou, 1987;

Bollen, 1989; Marsh et al., 1988), and through

conducting power analysis (MacCallum et al., 1996).

While the first two approaches are simply rules

of thumbs, the latter two have been studied

extensively.

Table 1 reports the results of analysis of SEM

applications in the OM literature related to sample size

and number of parameters estimated. The smallest

sample sizes for PA (n = 18), CFA (n = 63), and SEM

(n = 52) are significantly smaller than established

guidelines for models with even minimal complexity

(MacCallum et al., 1996; Marsh et al., 1988).

Additionally, 67.9% of all models have ratios of

sample size to parameters estimates of less than 10:1

and 35.7% of models have ratios of less than 5:1. The

lower end of both sample size and sample size to

parameter estimate ratios are significantly smaller in

the reviewed OM research than those studied by

Jackson (2003), indicating that the OM literature may

be highly susceptible to the negative outcomes

reported in his study.

Statistical power (i.e. the ability to detect and

reject a poor model) is critical to SEM analysis

because, in contrast to traditional hypothesis testing,

the goal in SEM analysis is to produce a non-


significant result between sample data and the

implied covariance matrix derived from model

parameter estimates. Yet, a non-significant result

may also be due to a lack of ability (i.e. power) to

detect model misspecification. Few studies in our

review mentioned power and none estimated power

explicitly. Therefore, we employed MacCallum et al.

(1996), who define minimum sample size as a

function of degrees of freedom that is needed for

adequate power (0.80) to detect close model fit, to

assess the power of models in our sample. (We were

not able to assess power for 41 of 143 models due to

insufficient information.) Our analysis indicates that

37% of the models have adequate power and 63% do

not. These proportions are consistent with similar

analyses in psychology (MacCallum and Austin,

2000), MIS (Chin and Todd, 1995), and strategy

(Shook et al., 2004), and have not changed since

1960 (Sedlmeier and Gigenrenzer, 1989). We

recommend that future researchers use MacCallum

et al. (1996) to calculate the minimum sample size

needed to ensure adequate statistical power.

4.1.3. Degrees of freedom and model

identification

Degrees of freedom are calculated as follows:

d:f: ¼ ð1=2Þf pð pþ 1Þg � q, where p is the number

of MVs, ð1=2Þf pð pþ 1Þg is the number of equations

(or alternately, the number of distinct elements in the

input matrix ‘‘S’’), and q is the effective number of

free (unknown) parameters to be estimated minus the

number of implied variances. As the formula

indicates, degrees of freedom is a function of model

specification in terms of the number of equations and

the effective number of free parameters that need to be

estimated.

When the effective number of free parameters is

exactly equal to the number of equations (that is, the

degrees of freedom are zero), the model is said to be

‘‘just-identified’’ or ‘‘saturated’’. Just-identified mod-

els provide an exact solution for parameters (i.e. point

estimates with no confidence intervals). When the

effective number of free parameters is greater than the

number of equations (degrees of freedom are less than

zero), the model is ‘‘under-identified’’ and sufficient

information is not available to uniquely estimate the

parameters. Under-identified models may not con-

verge during model estimation, and when they do, the

parameter estimates they provide are not reliable and

overall fit statistics cannot be interpreted (Rigdon,

1995). For models in which there are fewer unknowns

than equations (degrees of freedom are one or greater),

the model is ‘‘over-identified’’. An over-identified

model is highly desirable because more than one

equation is used to estimate at least some of the

parameters, significantly enhancing reliability of the

estimate (Bollen, 1989).

Model identification is a complex issue and while

non-negative degrees of freedom is a necessary

condition, additional conditions such as establishing

a scale for each LV are frequently required (for a

detailed discourse on sufficiency conditions, see Long,

1983; Bollen, 1989). In our review, degrees of freedom

were not reported for 41 (28.7%) models (see Table 1).

We recalculated the degrees of freedom independently

for each reviewed model to assess discrepancies

between the reported and our calculated degrees of

freedom. Wewere not able to reproduce the degrees of

freedom for 18 applications based on authors’

descriptions of their models. This lack of reproduci-

bility may be due in part to poor model description or

to correlated errors in the measurement or latent

variable models that are not stated in the text. We also

examined whether the issue of identification was

explicitly addressed for each model. One author

reported that the estimated model was not identified

and only 10.5% mentioned anything about model

identification. Perhaps the issue of identification was

considered implicitly because many software pro-

grams provide a warning message if a model is not

identified.

Model identification has a significant impact on

parameter estimates: in an unidentified model, more

than one set of parameter estimates could generate the

observed data and a researcher has no way to choose

among the various solutions because each is equally

valid (or invalid, if you wish). Degrees of freedom are

critically linked to the minimum sample size required

for adequate model fit; the greater the degrees of

freedom, the smaller the sample size needed for a

given level of model fit (MacCallum et al., 1996).

Calculating and reporting the degrees of freedom are

fundamental to understanding the specified model, its

identification, and its fit. Thus, we recommend that

degrees of freedom and model identification should be

reported for every tested model.


4.1.4. Measurement model specification

4.1.4.1. Number of items (MVs) per LV. It is generally

accepted that multiple MVs should measure each LV

but the number of MVs that should be used is less

clear. A ratio of fewer than three MVs per LV is of

concern because the model is statistically unidentified

in the absence of additional constraints (Long, 1983).

A large number of MVs per LV is advantageous as it

helps to compensate for a small sample (Marsh et al.,

1988) but disadvantageous as it means more para-

meters to estimate, requiring a larger sample size for

adequate power. A large number of MVs per LV also

makes it difficult to parsimoniously represent the

measurement structure constituting the set of MVs

(Anderson and Gerbing, 1984). In cases where a large

number of MVs are needed to represent a LV, Bagozzi

and Heatherton (1994) suggest four methods to reduce

the number of MVs per LV. In our review, 24% of CFA

models (9 of 38) and 39% of SEM models (29 of 75)

had a MV:LV ratio of less than 3. Generally, these

applications did not explicitly discuss identification

issues or additional constraints. The number of MVs

per LV characteristic is not applicable to PA models.

4.1.4.2. Single indicator constructs. We identified

LVs represented by a single indicator in 2.6% of CFA

models and 33.3% of SEM models in our sample (not

applicable to PA models). The low occurrence of

single indicator variables for CFA is not surprising

because the central objective of CFA is construct

measurement. However, the relatively high occurrence

of single indicator constructs in SEM models is

troublesome because single indicators ignore mea-

surement reliability, one of the challenges SEM is

designed to circumvent (Bentler and Chou, 1987). The

single indicator issue is also tied to model identifica-

tion as discussed above. Single indicators are only

sufficient when one measure perfectly represents a

concept, a rare situation, or when measurement

reliability is not an issue. Generally, single MVs

should be modeled as MVs rather than LVs.

4.1.4.3. Correlated measurement errors. Measure-

ment errors should sometimes be modeled as

correlated, for instance, in a longitudinal study when

the same item is measured at two points in time

(Bollen, 1989 p. 232). The statistical effect of

correlated error terms is the same as double loading,

but the substantive meaning is significantly different.

Double loading implies that each MV is affected by

two underlying LVs. Fundamental to LV unidimen-

sionality is that eachMV load on one LVwith loadings

on all other LVs restricted to zero. Because adding

correlated measurement errors to SEM models nearly

always improves model fit, they are often used post

hoc without improving the substantive interpretation

of the model (Fornell, 1983; Gerbing and Anderson,

1984) and making reliability estimates ambiguous

(Bollen, 1989 p. 222).

To the best of our knowledge, our sample contains

no instances of double loading MVs but we found a

number of models with correlated measurement

errors: 3.8% of PA, 28.9% of CFA, and 10.7% of

SEMmodels. We read the text of each article carefully

to determine whether the authors provided any

theoretical justification for using correlated errors or

whether they were introduced simply to improve

model fit. In more than half of the applications, no

justification was provided. Correlated measurement

errors should be used only when warranted on

theoretical or methodological grounds (Fornell,

1983) and their statistical and substantive impact

should be explicitly discussed.

4.1.5. Latent model specification

4.1.5.1. Recursive/non-recursive models. Models are

non-recursive when they contain reciprocal causation,

feedback loops, or correlated error terms (Bollen,

1989, p. 83). In such models, the matrix representing

latent exogenous variables (B; seeAppendixA formore

detail) has non-zero elements both above and below the

diagonal. If B is lower triangular and the errors in

equations are uncorrelated, then the model is called

recursive (Hair et al., 1998). Non-recursive models

require additional restrictions for the model to be

identified, for the stability of estimated reciprocal

effects, and for the interpretation of measures of

variation accounted for in the endogenous variables (for

a more detailed treatment of non-recursive models, see

Long, 1983; Teel et al., 1986). In our review, we

examined each application for recursive and non-

recursive models due to either simultaneous effects or

correlated errors in equations.Whilewe did not observe

any instances of simultaneous effects, we found that in

9.1% of the models, either the authors defined their

model as non-recursiveor a careful readingof the article


led to such a conclusion. However, even when authors

explicitly stated that they were testing a non-recursive

model, we saw little if any explanation of issues such as

model identification in the text. We recommend that if

non-recursive models are specified, additional restric-

tions and implications for model identification are

explicitly stated in the paper.

4.2. Issues related to data analysis

Data analysis issues comprise examining sample

data for distributional characteristics and generating

an input matrix. Distributional characteristics of the

data impact researchers’ choices of estimation

method, and the type of input matrix impacts the

selection of software used for analysis.

4.2.1. Data screening

Data screening is critical to prepare data for SEM

analysis (Hair et al., 1998). Screening through

exploratory data analysis includes investigating for

missing data, influential outliers, and distributional

characteristics. Significant missing data result in

convergence failures, biased parameter estimates,

and inflated fit indices (Brown, 1994; Muthen et al.,

1987). Influential outliers are linked to normality and

skewness issues with MVs. Assessing data normality

(along with skewness and kurtosis) is important

because many model estimation methods are based on

an assumption of normality. Non-normal data may

result in inflated goodness of fit statistics and

underestimated standard errors (MacCallum et al.,

1992), although these effects are lessened with larger

sample sizes (Lei and Lomax, 2005).

In our review, only a handful of applications

discussed missing data. In the psychology literature,

listwise deletion, pairwise deletion, data imputation

and full information maximum likelihood (FIML)

methods are commonly used to manage missing data

(Marsh, 1998). Results from Monte Carlo simulation

examining the performance of these four methods

indicate the superiority of FIML, leading to the lowest

rate of convergence failures, least bias in parameter

estimates, and lowest inflation in goodness of fit

statistics (Enders and Bandalos, 2001; Brown, 1994).

FIML method is currently available in LISREL

(version 8.50 and above), SYSTAT (RAMONA) and

AMOS.

We found that for 26.6% of applications, normality

was discussed qualitatively in the text of the reviewed

articles. Estimation methods such as maximum

likelihood ratio and generalized least square assume

normality, although some non-normality can be

accommodated (Hu and Bentler, 1998; Lei and

Lomax, 2005). Weighted least square, ordinary least

square, and asymptotically distribution free estimation

methods do not require normality. Additionally, ‘‘ML,

Robust’’ in EQS software adjusts model fit and

parameter estimates for non-normality. Finally,

researchers can transform non-normal data, although

serious problems have been noted with data transfor-

mation (cf. Satorra, 2001). We suggest that some

discussion of data screening methods be included

generally, and normality be discussed specifically in

relation to the choice of estimation method.

4.2.2. Type of input matrix

While raw data can be used as input for SEM

analysis, a covariance (S) or correlation (R) matrix is

generally used. In our review of the OM literature, no

papers report using raw data, 30.8% report using S,

and 25.2% report using R (44.1% of applications did

not report the type of matrix used to conduct analysis).

Seven of 44 applications using S and 25 of 36

applications using R provide the input matrix in the

paper. Not providing the input matrix makes it

impossible to replicate the results reported by the

author(s).

While conventional estimation methods in SEM are

based on statistical distribution theory that is

appropriate for S but not for R, there are interpreta-

tional advantages to using R: if MVs are standardized

and the model is fit to R, then parameter estimates can

be interpreted in terms of standardized variables.

However, it is not correct to fit a model to R while

treating R as if it were a covariance matrix. Cudeck

(1989) conducted exhaustive analysis on the implica-

tions of treating R as if it were S and concludes that the

consequences depend on the properties of the model

being fitted: standard errors, confidence intervals and

test statistics for the parameter estimates are incorrect

in all cases. In some cases, parameter estimates and

values of fit indices are also incorrect.

Software programs commonly used to conduct

SEM deal with this issue in different ways. Correct

estimation of a correlation matrix can be done in


LISREL (Joreskog and Sorbom, 1996) but requires the

user to introduce specific parameter constraints.

Although not widely used in OM, RAMONA (Browne

and Mels, 1998), EQS (Bentler, 1989) and SEPATH

(Steiger, 1999) automatically provide correct estima-

tion with a correlation matrix. Currently, AMOS

cannot analyze correlation matrices. In our review, we

found 24 instances where authors reported using a

correlation matrix with LISREL (out of 69 models run

with LISREL) but most did not mention the necessary

additional constraints. We found one instance of using

AMOS with a correlation matrix.

Given the lack of awareness among users about the

treatment of R versus S by various software programs,

we direct readers’ attention to a test devised by

MacCallum and Austin (2000) to help users determine

whether a particular SEM program provides correct

estimation of a model fit to a correlation matrix.

Otherwise, it is preferable to fit models to covariance

matrices, thus insuring correct results.

4.2.3. Estimation methods

A variety of estimation methods such as maximum

likelihood ratio (ML), generalized least square (GLS),

weighted and unweighted least square (WLS andULS),

asymptotically distribution free (ADF), and ordinary

least square (OLS) are available. Their use depends

upon the distributional properties of theMVs, and each

has computational advantages and disadvantages

relative to the others. For instance, ML assumes data

are univariate andmultivariate normal and requires that

the input data matrix be positive definite, but it is

Table 2a

Issues related to data analysis for structural model

Number of models reporting (n = 143)

x2 107

x2, p-value 76

GFI 84

AGFI 59

RMR (or RMSR) 51

RMSEA 51

NFI 49

NNFI (or TLI) 62

CFI 73

IFI (or BL89) 16

Normed x2 (x2/d.f.) reported 52

Normed x2 calculated 98b

a One model reported RMR = 145.4; this data point omitted as an outb Data not available to calculate others.

relatively unbiased under moderate violations of

normality (Bollen, 1989). GLS assumes normality

but does not impose the restriction of a positive definite

input matrix. ADF has few distributional assumptions

but requires very large sample sizes for accurate

estimates. OLS, the simplest method, has no distribu-

tional assumptions and is computationally the most

robust, but it is scale invariant and does not provide fit

indices or standard errors for estimates.

Forty-eight percent of applications in our review

did not report the estimation method used. Of the

applications that reported the estimation method, a

majority (68.9%) used ML. Estimation method, data

normality, sample size, and model specification are

inextricably linked and must be considered simulta-

neously by the researcher. We suggest that authors

explicitly state the estimation method used and link it

to the properties of the observed variables.

4.3. Issues related to post-analysis

Post-analysis issues include evaluating the solution

achieved from model estimation, model fit, and

respecification of the model. Reports of these data

from the studied sample are summarized in Tables 2a

and 2b.

4.3.1. Evaluation of solution

We have organized our discussion of evaluation of

solutions into overall model fit, measurement model

fit, and structural model fit. To focus solely on the

overall fit of the model while overlooking important

Proportion reporting (%) Results: mean; median Range

74.8 204.0; 64.2 (0.0, 1270.0)

53.1 0.21; 0.13 (0.0, 0.94)

58.7 0.93; 0.94 (0.75, 0.99)

41.3 0.89; 0.90 (0.63, 0.97)

35.7 0.052; 0.050 (0.01, 0.14)a

35.7 0.058; 0.060 (0.00, 0.13)

34.3 0.91; 0.92 (0.72, 0.99)

43.4 0.95; 0.95 (0.73, 1.07)

51.0 0.96; 0.96 (0.88, 1.00)

11.2 0.94; 0.95 (0.88, 0.98)

36.4 1.82; 1.59 (0.02, 4.80)

68.5 2.17; 1.62 (0.01, 21.71)

lier relative to other reported RMRs.


Table 2b

Issues related to data analysis for measurement model

Number of

models reporting

(n = 143)

Proportion

reporting

(%)

Reliability assessment 123 86.0

Unidimensionality assessment 94 65.7

Discriminant validity addressed 99 69.2

Validity issues addressed

(R2; variance explained)

76 53.1

Path coefficients

(confidence intervals)

138 (3) 96.5 (2.1)

Path t-statistics (standard errors) 90 (21) 62.9 (14.7)

Residual information/analysis

provided

19 13.3

Specification search conducted

for model respecification

20 14.0

Modification indices used

for model respecification

21 14.7

Alternative models compared 29 20.3

Inconsistency between described

and tested models

31 21.7

Cross-validation sample used 22 15.4

Split sample approach used 27 18.9

information about parameters is a common error that

we encountered in our review. A model with good

overall fit but yielding nonsensical parameter esti-

mates is not a useful model.

4.3.1.1. Overall model fit. Assessing a model’s fit is

one of the more complicated aspects of SEM because,

unlike traditional statistical methods, it relies on non-

significance. Historically, the most popular index used

to assess the overall goodness of fit has been the x2-

statistic, although its conclusions regarding model

significance are generally ignored. The x2-statistic is

inherently biased when the sample size is large but is

dependent on distributional assumptions associated

with large samples. Additionally, a x2-test offers a

dichotomous decision strategy (accept /reject) for

assessing the adequacy of fit implied by a statistical

decision rule (Bollen, 1989). In light of these issues,

numerous alternative fit indices have been developed

to quantify the degree of fit along a continuum (see

Joreskog, 1993; Tanaka, 1993; Bollen, 1989, pp. 256–

289; Mulaik et al., 1989 for comprehensive reviews).

Fit indices are commonly distinguished as either

absolute or incremental (Bollen, 1989). In general,

absolute fit indices indicate the degree to which the

hypothesized model reproduces the sample data, and

incremental fit indices measure the proportional

improvement in fit when the hypothesized model is

compared with a restricted, nested baseline model (Hu

and Bentler, 1998).

Absolute measures of fit: The most basicmeasure of

absolute fit is the x2-statistic. Other commonly used

measures include root mean square error of approx-

imation (RMSEA), root mean square residual (RMR or

SRMR), goodness-of-fit index (GFI) and adjusted

goodness of fit (AGFI). GFI and AGFI increase as

goodness of fit increases and are bounded above by

1.00, while RMSEA and RMR decrease as goodness of

fit increases and are bounded below by zero (Browne

and Cudeck, 1989). Ninety-four percent of the

applications we reviewed report at least one of these

measures (Table 2a). Although the frequency of use and

the magnitude of each of these measures are similar to

those reported in marketing by Baumgartner and

Homburg (1996), the ranges in our sample are much

wider indicating greater variability in empirical OM

research. The variability may be an indication of more

complex models and/or a less established theory base.

Incremental fit measures: Incremental fit measures

compare the model under study to two reference

models: (1) a worst case or null model, and (2) an ideal

model that perfectly represents the modeled phenom-

ena in the studied population. While there are many

incremental fit indices, some of the most popular are

normed fit index (NFI), non-normed fit index (NNFI or

TLI), comparative fit index (CFI) and incremental fit

index (IFI or BL89). Sixty-nine percent of the

reviewed studies report at least one of the four

measures (Table 2a). An additional fit index that is

frequently used is the normed x2 which is reported for

36.4% of models. Because the x2-statistic by itself is

beset with problems, the ratio of x2 to degrees of

freedom (x2/d.f.) is informative because it corrects for

model size. Additionally, we calculated the normed x2

for all models that reported x2 and either reported

degrees of freedom or enough model specification

information to allow us to ascertain the degrees of

freedom (68.5% of all applications) and found a

median of 1.62 (range 0.01, 21.71). Small values of

normed x2 (<1.0) can indicate an over-fitted model

and higher values (>3.0–5.0) can indicate an under-

parameterized model (Joreskog, 1969).

A brief summary of the effects on fit indices of

small samples, normality violations, model misspe-


cification, and estimation method are reported in

Table 3. An ongoing debate about superiority or even

appropriateness of one index over another makes the

issue of selecting which to use in assessing fit very

complex. For instance, Hu and Bentler (1998) advise

against using GFI and AGFI because they are

significantly influenced by sample size and are

insufficiently sensitive to model misspecification.

Most fit indices are influenced by sample size and

should not be interpreted independently of sample

size (Hu and Bentler, 1998; Marsh et al., 1988).

Therefore, no consistent criteria (i.e. cut-offs) can be

defined to apply in all (or most) instances (Marsh

et al., 1988).

Until definitive fit indices are developed, research-

ers should report multiple measures of fit so reviewers

and readers have the opportunity to evaluate the

underlying fit of the data to the model from multiple

Table 3

Influence of sample and estimation characteristics on model fit indices

Small sample (n) biasa Violations of

normalitybM

Absolute

x2 Bias establishedf

GFI Poor for small ne

can be usedfProblematic with

ADFeM

id

AGFI Poor for small ne,f Problematic with

ADFeM

id

RMR (or SRMR) ML preferred for

small neM

RMSEA Tends to over reject

modeleM

Incremental

NFI Poor for small ne S

id

NNFI (or TLI) Best index for small nf

tends to over reject

modele

M

CFI ML preferred for

small neM

IFI (or BL89) ML preferred for

small neM

Normed x2 Bias establishedf

a While all fit indexes listed suffer small sample bias (approximatelyb Most normality violations have insignificant effects on fit indexes, ec Identifying model misspecification is a positive characteristic; fit indexd The following estimation methods investigated: maximum likelih

distribution free (ADF)e,f.e Hu and Bentler (1998).f Marsh et al. (1988).

perspectives. x2 should be reported with its corre-

sponding degrees of freedom in order to be insightful.

RMR and RMSEA, two measures that reflect the

residual differences between the input and implied

(reproduced) matrices, indicate how well matrix

covariance terms are predicted by the tested model.

RMR in particular performs well under many

conditions (Hu and Bentler, 1998; Marsh et al.,

1988). Researchers might also report a summary of

standardized (correlation) residuals because when

most or all are ‘‘quite small’’ relative to correlations in

the tested sample (Browne et al., 2002, p. 418), they

indicate good model fit (Bollen, 1989, p. 258).

4.3.1.2. Measurement model fit. Measurement model

fit can be evaluated in two ways: first, by assessing

constructs’ reliability and convergent and discriminant

odel misspecificationc Estimation

method effectdGeneral comments

No preference

isspec’s not

entified by ADFeML preferred Use of index not

recommendede

isspec’s not

entified by ADFeML preferred Use of index not

recommendede

isspec’s identified ML preferred Recommended for all

analysese

isspec’s identified No preference Use with ADF not

recommendede

ome misspec’s

entified

ML preferred Use of index not

recommendede

isspec’s identified ML preferred



No preference

n < 250), we consolidate findings by leading researchers.

xcept those noted.

es that do not identify misspecification are considered poor choices.

ood ratio (ML), generalized least square (GLS), asymptotically


validity, and second, by examining the individual path

(parameter) estimates (Bollen, 1989).

Various indices of reliability can be computed to

summarize how well LVs are measured by their MVs

individually or jointly (individual item reliability,

composite reliability, and average variance extracted;

cf. Bagozzi and Yi, 1988; Fornell and Larcker, 1981).

Our initial attempt to report reliability measures used

by the authors proved difficult due to the diversity of

methods used. Therefore, we limit our review to

whether authors report at least one of the various

measures. Overall, 86.0% of the applications describe

some form of reliability assessment. We recommend

that authors report at least one measure of construct

reliability based on estimated model parameters (e.g.

composite reliability or average variance extracted)

(Bollen, 1989).

Cronbach alpha is an inferior measure of reliability

because in most cases it is only a lower bound on

reliability (Bollen, 1989). In our review we found that

Cronbach alpha was frequently presented as proof to

establish unidimensionality. It is not sufficient for this

purpose because a scale may not be unidimensional

even if it has high reliability (Gerbing and Anderson,

1984). Our review also examined how published

research dealt with the issue of discriminant validity.

We found that 69.2% of all applications included

evidence of discriminant validity. Our review indicates

that despite a lack of standardization in the reported

measures, most published research in OM includes

some measure of reliability, unidimensionality and

validity.

Another way to assess measurement model fit is to

evaluate path estimates. In evaluating path estimates,

sign (positive or negative), strength, and significance

should be aligned with theory. The magnitude of

standard errors associated with path estimates should

be small; a large standard error indicates an unstable

parameter estimate that is subject to sampling error.

Although recommended but rarely used in practice,

the 90% confidence interval (CI) around each path

estimate is very useful (Browne and Cudeck, 1993).

The CI provides an explicit indication of the degree of

parameter estimate precision. Additionally, the sta-

tistical significance of path estimates can be inferred

from the 90% CI: if the 90% CI includes zero, then the

path estimate is not significantly different from zero

(at a = 0.05). Overall, confidence intervals are very

informative and we recommend their use in future

studies. In our review, we found that 96.5% of the

applications report path coefficients, 62.9% provide t

statistics, 14.7% provide standard errors, and 2.1%

report confidence intervals.

4.3.1.3. Structural model fit. In SEM models, the

latent variable model represents the structural model

fit, and generally, the hypotheses of interest. In PA

models that do not have LVs, the hypotheses of interest

are generally represented by the paths between MVs.

Like measurement model fit, the sign, magnitude and

statistical significance of the structural path coeffi-

cients are examined in testing the hypotheses.

Researchers should recognize the important distinc-

tion between variance fit (explained variance in

endogenous variables as measured by R2 for each

structural equation) and covariance fit (overall good-

ness of fit, such as that tested by a x2-test). Authors

emphasize covariance fit a great deal more than

variance fit; in our review, 53.1% of the models

presented evidence of the variance fit compared to 96%

that presented at least one index of overall fit. It is

important to distinguish between these two types of fit

because a model might fit well but not explain a

significant amount of variation in endogenous variables

or conversely, fit poorly and explain a large amount of

variation in endogenous variables (Fornell, 1983).

In summary, we suggest that fit indices should

not be regarded as measures of usefulness of a model.

They each contain some information aboutmodel fit but

none about model plausibility (Browne and Cudeck,

1993). Rather than establishing that fit indices meet

arbitrarily established cut-offs, future research should

report a variety of absolute and incremental fit indices

for measurement, structural, and overall models and

include a discussion of interpretation of fit indices

relative to the study design.We foundmany instances in

which authors conclude that a particular model had

better fit than alternativemodels based on comparing fit

indices. While some fit indices can be useful for such

comparisons, most commonly employed fit indices

cannot be compared acrossmodels in thismanner (e.g. a

model with a lower RMSEA does not indicate better fit

than a model with a higher RMSEA). For nested

alternate models, x2 difference test or Target Coeffi-

cient can be used (Marsh and Hocevar, 1985). For

alternate models that are not nested, parsimony fit


measures such as Parsimonious NFI, Parsimonious

GFI, Akaike information criterion (AIC) and normed

x2 can be used (Hair et al., 1998).

4.3.2. Model respecification

Although no model fits the real world exactly, a

desirable outcome in SEM analysis is to show that a

hypothesized model provides a good approximation of

real world phenomena, as represented by an observed

set of data. When an initial model of interest does not

satisfy this objective, researchers often alter the model

to improve its fit to the data. Modification of a

hypothesized model to improve its parsimony and/or

fit to the data is termed a ‘‘specification search’’

(Leamer, 1978; Long, 1983). A specification search is

designed to identify and eliminate errors from the

original specification of the hypothesized model.

Joreskog and Sorbom (1996) describe three

strategies in model specification (and evaluation):

(1) strictly confirmatory, where a single a priori model

is studied; (2) model generation, where an initial

model is fit to data and then modified (frequently with

the use of modification indices) until it fits adequately;

and (3) alternative models, where multiple a priori

models are studied. Although not improper, the

‘‘strictly confirmatory’’ approach is highly restrictive

and does not leave the researcher any latitude if the

model does not work. The model generation approach

is troublesome because of the potential for abuse,

results that lack validity (MacCallum, 1986), and high

susceptibility to capitalization on chance (MacCallum

et al., 1992). Simulation work by MacCallum (1990)

and Homburg and Dobartz (1992) indicates that only

half of specification searches (even with correct

restrictions and large samples) are successful in

recovering the correct underlying model.

In our review, 28.7% (41 of 143) of the applications

reported making post hoc changes to respecify the

model. We also examined the published articles for

inconsistency between the model that was tested

versus the model described in the text. In 31 out of 143

cases we found such inconsistency, where we could

not match the described model with the tested model.

We suspect that in many cases, authors made post hoc

changes (perhaps to improve model fit), but those

changes were not well described. We found only

20.3% of the models were tested using alternate

models. We recommend that researchers compare

alternate a priori models (either nested or unnested) to

uncover the model that the observed data support best

rather than use specification searches (Browne and

Cudeck, 1989). Such practices may have a lower

probability of identifying models with great fit, but

they increase the alignment of modeling results with

our existing knowledge and theories. Leading journals

must show a willingness to publish poor fitting models

for such advancement of knowledge and theory.

5. Presentation and interpretation of results

We encountered many difficulties related to

presentation and interpretation of models, methods,

analysis, and results in our review. In a majority of

articles, we had difficulty determining either the

complete model (e.g. correlated measurement errors)

or the complete set of MVs. Whether the model was fit

to a correlation or covariance matrix could not be

ascertained for nearly half of the models, and reporting

of fit results was incomplete in a majority of models.

In addition, issues of causation in cross-sectional

designs, generalizability, and confirmation bias also

raise concerns and are discussed in detail below.

5.1. Causality

Each of the applications we reviewed used a cross-

sectional research design. The debate over whether

concurrent measurement of variables can be used to

infer causality is vibrant but unresolved (Gollob and

Reichardt, 1991; Gollob and Reichardt, 1987;

MacCallum and Austin, 2000). One point of agree-

ment is that causal interpretation must be based on the

theoretical grounding of and empirical support for a

model (Pearl, 2000). In light of this ongoing debate,

we suggest that OM researchers describe the theory

they are testing and its expected manifested results as

clearly as possible prior to conducting analysis.

5.2. Generalizability

‘‘Generalizability of findings’’ refers to the applic-

ability of findings from one study with a finite, often

small sample to a population (or other populations).

Findings from single studies are subject to limitations

due to sample or selection effects and their impact on


the conclusions that can be drawn. In our review, such

limitationswere seldomacknowledged and resultswere

usually interpreted and discussed as if they were

expansively generalizable. Sample and selection effects

are controlled (but not eliminated) by identifying a

specific population and from it selecting a sample that is

appropriate to the objectives of the study. Rather than

identifying a specific population, the articles we

reviewed focused predominantly on describing their

samples. However, a structural equation model is a

hypothesis about the structure of relationships among

MVs and LVs in a specific population, and this

population should be explicitly identified.

Another aspect of generalizability involves repli-

cating the results of a study in a different sample from

the same population. We found that 15.4% of the

reviewed applications used cross-validation and

18.9% used a split sample approach. Given the

difficulty in obtaining responses from multiple

samples from a given population, the expected

cross-validation index (ECVI), an index computed

from a single sample, can indicate how well a solution

obtained in one sample is likely to fit an independent

sample from the same population (Browne and

Cudeck, 1989; Cudeck and Browne, 1983).

Selecting the most appropriate set of measurement

items to represent the domain of underlying LVs is

critical when using SEM. However, there are few

standardized instruments for LVs, making progress in

empirical OM research slow and difficult. Appropriate

operationalization of LVs is as critical as their repeated

use: repetition helps to establish validity and

reliability. (For a detailed discussion and guidelines

on the selection effects related to good indicators, see

Little et al., 1999; forOMmeasurement scales, seeRoth

andSchroeder, inpress.)Achallengingissueariseswhen

researchersareunable tovalidatepreviouslyusedscales.

In such situations, we suggest a two-pronged strategy.

First, a priori the researcher must examine the

assumptions employed in developing the previous

scales and state their impact on replication. Second,

upon failure to replicate with validity, the researcher

must use an exploratory means to develop modified

scales to be validated by future researchers. However,

this respecified model should not be given the status

of a hypothesizedmodel andwould need to bevalidated

in the future with another sample from the same

population.

5.3. Confirmation bias

Confirmation bias is defined as a prejudice in favor

of the evaluated model (Greenwald et al., 1986). Our

review suggests that OM researchers (not unlike

researchers in other fields) are highly susceptible to

confirmation bias. Researchers evaluate a single

model, give an overly positive evaluation of model

fit, and are reluctant to consider alternative explana-

tions of data. An associated problem in this context is

the existence of equivalent models, alternative models

that are indistinct from the original model in terms of

goodness of fit to the data but with a distinct

substantive meaning in terms of the underlying theory

(MacCallum et al., 1993). In a study of 53 published

applications in psychology, MacCallum et al. (1993)

showed that equivalent models exist routinely in large

numbers and are universally ignored by researchers. In

order to mitigate problems related to confirmation

bias, we recommend that OM researchers generate

multiple alternate, equivalent models a priori and if

one or more of these models cannot be eliminated due

to theoretical reasons or poor fit, to explicitly discuss

the alternate explanation(s) underlying the data rather

than confirming and presenting results from one

definitive model (MacCallum et al., 1993).

6. Discussion and conclusion

SEM has rapidly become an important and widely

used research tool in the OM literature. Its attractive-

ness to OM researchers can be attributed to two

factors. From CFA, SEM draws upon the notion of

unobserved or latent variables, and from PA, SEM

adopts the notion of modeling direct and indirect

relationships. These advantages, combined with the

availability of ever more user-friendly software, make

it likely that SEM will enjoy widespread use in the

future. We have provided both a review of the OM

literature employing SEM as well as discussion and

guidelines for improving its future use. Table 4

contains a summary of some of the most important

issues discussed here, their implications, and recom-

mendations for resolving these challenges. Below, we

briefly discuss these issues.

As researchers, we should ensure that SEM is the

correct method for examining the research question at


Table 4

Implications and recommendations for select SEM issues

Implications Recommendations

Formative (causal indicators) Bollen (1989), MacCallum and

Browne (1993):

Model as causal indicators

(MacCallum and Browne, 1993)

Without additional constraints, the model

is generally unidentified

Report appropriate conditions and modeling issues

Poorly developed or weak

relationships

Hurley et al. (1997):

More likely to result in a poor fitting

model requiring specification searches and

post hoc model respecification

Use alternative methods that demand less

rigorous model specification such as EFA

and Regression Analysis (Hurley et al., 1997)

Violating multivariate

normality

MacCallum et al. (1992):

Inflated goodness of fit

statistics; Underestimated

standard errors

Use estimation methods that adjust for violation

such as ‘‘ML, Robust’’ available in EQS;

Use estimation methods that do not assume

multivariate normality such as GLS, ADF

Correlation matrix

as input data

LISREL is inappropriate without

additional constraints (Cudeck, 1989):

Type of input matrix and software must be reported

RAMONA in SYSTAT (Browne and Mels, 1998),

EQS (Bentler, 1989), SEPATH (Steiger, 1999)

can be used LISREL can be used with additional

constraints (LISREL 8.50)

AMOS cannot be used

Standard errors, confidence intervals and

test statistics for parameter estimates are

incorrect in all cases

Parameter estimates and fit indices are

incorrect in some cases

Small sample size MacCallum et al. (1996), Marsh et al. (1988),

Hu and Bentler (1998):

Conduct and report statistical power

Simpler models (fewer parameters estimated, higher

degrees of freedom) are associated with higher

power (MacCallum et al., 1996)

Use fit indices that are less biased to small sample size

such as NNFI; avoid fit indices that are more biased,

such as x2, GFI and NFI (Hu and Bentler, 1998)

Associated with lower power, ceteris paribus

Parameter estimates have lower reliability

Fit indices are overestimated

Few degrees of freedom (d.f.) MacCallum et al. (1996): Report degrees of freedom

Conduct and report statistical power

Simpler models (fewer parameters estimated, higher

degrees of freedom) are associated with higher

power (MacCallum et al., 1996)

Associated with lower power, ceteris paribus

Parameter estimates have lower reliability

Fit indices are overestimated

Model identification d.f. = 0, results are not generalizable Desirable condition (d.f. > 0)

Assess and report model identification

Explicitly discuss implication of unidentified models

on generalizability of results

d.f. < 0, model cannot be estimated unless

some parameters are fixed or held constant

Number of MVs/LV To provide adequate representation of

content domain, need sufficient MVs/LV

Have at least three MVs per LV for CFA/SEM

(Rigdon, 1995)

One MV per LV May not provide adequate representation of

content domain Poor reliability and

validity because error variance cannot be

estimated (Maruyama, 1998)

Model as MV (not LV)

Single MV can be modeled as LV only when MV is the

perfect representation of the LV; specific conditions must

be imposed for identification purposes (LISREL 8.50)

Model is generally unidentified

Correlated measurement

errors

Gerbing and Anderson (1984): Report correlated errors Justify their theoretical

validity a priori

Discuss the impact on measurement and structural

parameter estimates and model fit

Alters measurement and structural

parameter estimates

Almost always improves model fit

Changes the substantive meaning of the model

Non-recursive models Without additional constraints the model

is unidentified

Explicitly report model is non-recursive and its cause

Add constraints and report their impact (Long, 1983)


hand. When theory development is at a nascent stage

and patterns of relationships among LVs are relatively

weak, SEM should be used with caution so that model

confirmation and theory testing do not degenerate into

extensive model respecification. Likewise, it is

important that we use appropriate measurement

methods and understand the distinction between

formative and reflective variables.

Determining minimum sample size is, in part,

dependent upon the number of parameter estimates in

the hypothesized model. But emerging research in this

area indicates that the relationship between sample

size and number of parameter estimates is complex

and dependent upon MV characteristics (MacCallum

et al., 2001). Likewise, guidelines on degrees of

freedom and model identification are not simple or

straightforward. Researchers must be cognizant of

these issues and we recommend that all studies discuss

them explicitly. As the powerful capabilities of SEM

derive partly from its highly restrictive simplifying

assumptions, it is important that assumptions such as

normality and skewness are carefully assessed prior to

generating an input matrix and conducting analysis.

With regard to model estimation, researchers

should recognize that parameter estimates are not

fixed values, but rather depend upon the estimation

method. For instance, parameter estimates obtained by

using maximum likelihood ratio are different from

those obtained using ordinary least squares (Browne

and Arminger, 1995). Further, in evaluating model fit,

the correspondence between the hypothesized model

and the observed data should be assessed using a

variety of absolute and incremental fit indices for

measurement, structural, and overall models. In

addition to path coefficients, confidence intervals

and standard errors should be assessed.

Rather than hypothesizing a single model, multiple

alternate models should be evaluated when possible,

and research results shouldbe cross validated using split

or multiple samples. Given the very real possibility of

alternate, equivalent models, researchers should be

cautious in over-interpreting results. Because no model

represents the real world exactly, we must be more

forthright about the ‘‘imperfection’’ inherent in any

model and acknowledge the literal implausibility of the

model more explicitly (MacCallum, 2003).

One of the most poignant observations in conduct-

ing this study was the inconsistency in the published

reporting of results and, in numerous instances, our

inability to reconstruct the tested model based on the

description in the text and the reported degrees of

freedom. These issues can be resolved by attention to

published guidelines for presenting results of SEM (e.g.

Hoyle and Panter, 1995). To assist both during the

review process and in building a cumulative tradition in

the OM field, sufficient information needs to be

provided to understand (1) the population from which

the data samplewas obtained, (2) the distribution of the

data, (3) the hypothesized measurement and structural

models, and (4) statistical results to corroborate the

subsequent interpretation and conclusions.

We recommend that every published application of

SEM provide a clear and complete specification of the

model(s) and variables, preferably in the form of a

graphical figure, including the measurement model

linking LVs to MVs, the structural model connecting

LVs, and specification of which parameters are being

estimated and which are fixed. It is helpful to identify

specific research hypotheses on the graphical figure,

both to clarify the model and to reduce the text needed

to describe them. In addition to including a statement

about the type of input data matrix, software and

estimation method used, we recommend the input

matrix be included in paper for future replications and

meta-analytical research studies, but we recognize this

is an editorial decision subject to space constraints. In

terms of statistical results, we suggest researchers

include multiple measures of fit and criteria for

evaluating fit along with parameter estimates, and

associated confidence intervals and standard errors.

Finally, interpretation of results should be guided by

an understanding that models are imperfect and cannot

be made to be exactly correct.

We can enrich our knowledge by reviewing the use

of SEM in more mature research fields such as

psychology and marketing, including methodological

advances. Some advances worthy of mention are

validation studies using the multi-trait multi-method

(MTMM) matrix method (cf. Cudeck, 1988; Wida-

man, 1985), measurement invariance (Widaman and

Reise, 1997), and using categorical (Muthen, 1983) or

experimental data (Russell et al., 1998).

Our review of published SEM applications in the

OM literature suggests that while reporting has

improved over time, we need to pay attention to

methodological issues in using SEM. Like any


statistical technique or tool, it is important that SEM

be used prudently if researchers want to take full

advantage of its potential. SEM is a useful tool to

represent multidimensional unobservable constructs

and simultaneously examine structural relationships

that are not well captured by traditional research

methods (Gefen et al., 2000, p. 6). In the future,

utilizing the guidelines presented here will improve

the use of SEM in OM research, and thus, improve our

collective understanding of OM theory and practice.

Acknowledgements

We thank Michael Browne and Sriram Thirumalai

for helpful comments on this paper. We also thank

Carlos Rodriguez for assistance with article screening

and data coding.

Appendix A. Mathematical specification of

structural equation modeling

A structural equation model can be defined as a

hypothesis of a specific pattern of relations among a

set of measured variables (MVs) and latent variables

(LVs). The three equations presented below are

fundamental to SEM. Eq. (1) represents the directional

influences of the exogenous LVs (j) on their indicators

(x). Eq. (2) represents the directional influences of the

endogenous LVs (h) on their indicators (y). Thus,

Eqs. (1) and (2) link the observed (manifest) variables

to unobserved (latent) variables through a factor

analytic model and constitute the ‘‘measurement’’

portion of the model. Eq. (3) represents the

endogenous LVs (h) as linear functions of other

exogenous LVs (j) and endogenous LVs plus residual

terms (z). Thus, Eq. (3) specifies relationships between

LVs through a structural equation model and

constitutes the ‘‘structural’’ portion of the model

x ¼ Lxjþ d (1)

y ¼ Lyhþ e (2)

h ¼ Bhþ Gjþ z (3)

where x is the measures of exogenous manifest vari-

ables, Lx the effect of exogenous LVs on their MVs

(matrix), d the error of measurement in exogenous

manifest variables, y the measures of endogenous

manifest variables, Ly the effect of endogenous LVs

on their MVs (matrix), e the error of measurement in

endogenous manifest variables, j the latent exogenous

constructs, h the latent endogenous constructs, G the

effect of exogenous constructs on endogenous con-

structs (matrix), B the effect of endogenous constructs

on each of the other endogenous constructs (matrix)

and z is the errors in equations or residuals.

It is also necessary to define the following

covariance matrices:

(a) f
= E(jj0) is a covariance matrix for the exogen-
ous LVs.

(b) u
d = E(dd0) is a covariance matrix for the
measurement errors in the exogenous MVs.

(c) u
e = E(ee0) is a covariance matrix for the measure-
ment errors in the endogenous MVs.

(d) c
= E(zz0) is a covariance matrix for the errors in
equation for the endogenous LVs.

Given this mathematical representation, it can be

shown that the population covariance matrix for the

MVs is a function of eight parameter matrices:Lx,Ly,

G, B, f, ud, ue andc. Thus, given a hypothesized model

in terms of fixed and free parameters of the eight-

parameter matrices, and given a sample covariance

matrix for the MVs, one can solve for estimates of the

free parameters of the model. The most common

approach for fitting the model to data is to obtain

maximum likelihood estimates of parameters, and an

accompanying likelihood ratio x2-test of the null

hypothesis that the model holds in the population.

The notation above uses SEM as developed by

Joreskog (1974) and represented in LISREL (Joreskog

and Sorbom, 1996).

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