Kline R.B. Principles and Practice of Structural Equation Modeling

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4 CONCEPTS AND TOOLS

you can still download and open on your own computer the EQS output ﬁle for a par-

ticular analysis and review the results. This is because all of the computer ﬁles on this

book’s website are plain-text (ASCII) ﬁles that require nothing more than a basic text

editor, such as Notepad in Microsoft Windows, to view their contents. Even if you are

using an SEM computer tool other than EQS, LISREL, or Mplus, it is still worthwhile to

review the computer ﬁles on the site. This is because (1) common principles about pro-

gramming apply across different SEM computer tools, and (2) it can be helpful to view

the same analysis from somewhat different perspectives. Some of the exercises for this

book involve extensions of the original analyses for these examples, so there are plenty

of opportunities for practice with real data sets. Suggested answers for all exercises are

presented at the end of the book.

PedagogICal aPProaCh

You may be reading this book while participating in a course or workshop on SEM. This

context offers the potential advantages of the structure and support available in a class-

room setting, but formal coursework is not the only way to learn about SEM. Another

is self-study, a method through which many researchers learn about what is, for them,

a new statistical technique. (This is how I ﬁrst learned about SEM, not in classes.) I

assume that most readers are relative newcomers to SEM or that they already have some

knowledge of the area, but wish to hone their skills. Consequently, I will speak to you

(through my author’s voice) as one researcher to another, not as a statistician to the quan-

titatively untutored. For example, the instructional language of statisticians is matrix

algebra, which can convey a lot of information in a relatively small amount of space, but

you must already be familiar with linear algebra to decode the message. There are other,

more advanced works about SEM that emphasize matrix representations (Bollen, 1989;

Kaplan, 2009; Mulaik, 2009), and these works can be consulted by those interested in

such presentations (i.e., when you are ready). Instead, fundamental concepts about SEM

are presented here using the language of researchers: words and ﬁgures, not matrix alge-

bra. I will not shelter you from some of the more technical aspects of SEM, but I aim to

cover requisite concepts in an accessible way that supports continued learning.

You may be relieved to know that you are not at a disadvantage if at present you

have no experience using an SEM computer tool. This is because the presentation in

this book is not based on the symbolism or syntax associated with a particular software

package. A number of books are linked to speciﬁc SEM computer tools, including

Byrne (2006, 2009, 2010) for, respectively, EQS, Amos, and Mplus.•

Blunch (2008) for Amos.•

Diamantopoulos and Siguaw (2000), Hayduk (1996), and Kelloway (1998) for •

L I S R E L .

Mueller (1996) for both LISREL and EQS.•

Introduction 5

Software-centric books can be invaluable for users of a particular computer tool, but

perhaps less so for others. Instead, essential principles of SEM that users of any com-

puter tool must understand are emphasized here. In this way, this book is more like a

guide to writing style and composition than a handbook about how to use a particular

word processor. Besides, becoming proﬁcient with a particular software package is just

a matter of practice. But without strong concept knowledge, the output one gets from

a computer tool for statistical analyses—including SEM—may be meaningless or, even

worse, misleading.

As with other statistical techniques, there is no gold standard for notation in SEM.

Although the symbol set associated with the original syntax of LISREL is probably the

most widely used in advanced works about SEM, it features a profusion of subscripted

lowercase Greek letters (e.g.,

) for individual model parameters, uppercase

Greek letters for parameter matrices (e.g.,

Φ, Λ

), and two-letter acronyms for parameter

matrices (e.g., TE for theta–epsilon) or matrix forms (e.g., DI for diagonal) that can be

confusing to follow unless you have memorized the entire system. Instead, this book

uses a minimum number of alphabetic characters to represent various aspects of SEM

such as observed versus latent variables.

Learning to use a new set of statistical techniques is like making a journey through

a strange land. Such a journey requires a substantial commitment of time, patience, and

a willingness to tolerate the frustration of initial uncertainty and inevitable trial and

error. But this is one journey you do not have to make alone. Think of this book as a

travel atlas or even as someone to counsel you about language and customs, what to see

and what to avoid, and what lies just over the horizon. I hope that the combination of

a conceptually based approach, numerous examples, and the occasional bit of practical

advice presented in this book will help to make this statistical journey a little easier,

maybe even enjoyable. (Imagine that!)

GettinG Ready to LeaRn about SeM

Listed next are suggestions about the best way to prepare yourself for learning about

SEM. I offer these suggestions in the spirit of giving you a healthy perspective at the

beginning of this journey, one that empowers your sense of being a researcher.

Know your area

Strong familiarity with the theoretical and empirical literature in your research area is

the single most important thing you need for SEM. This is because everything, from the

speciﬁcation of your initial model to modiﬁcation of that model in subsequent reanaly-

ses to interpretation of the results, must be guided by your domain knowledge. So you

need ﬁrst and foremost to be a researcher, not a statistician or computer geek. This is true

for most kinds of statistical analysis, in that the value of the product (numerical results)

depends on the quality of the ideas (your hypotheses) on which the analysis is based.

6 CONCEPTS AND TOOLS

Otherwise, that familiar expression about computer analysis, “garbage in, garbage out,”

applies.

know Your Measures

Kühnel (2001) reminds us that learning about SEM has the by-product that newcomers

must deal with fundamental issues of measurement. Speciﬁcally, the analysis of measures

with strong psychometric characteristics, such as good score reliability and validity, is

essential in SEM. For example, it is impossible to analyze a structural equation model

with latent variables that represent hypothetical constructs without thinking about how

to measure those constructs. When you have just a single measure of a construct, then

it is especially critical for this single indicator to have good psychometric properties.

Likewise, the analysis of measures with deﬁcient psychometric characteristics could

bias the results. Unfortunately, measurement theory is too often neglected nowadays in

undergraduate and graduate degree programs in psychology (Frederich, Buday, & Kerr,

2000) and related areas, but SEM requires strong knowledge in this area. Some crucial

measurement-related concepts are considered in Chapter 3.

review Fundamental statistical Concepts and techniques

Before learning about SEM, you should have a good understanding of (1) principles of

multiple correlation/regression,

(2) the correct interpretation of results from statistical

tests, and (3) data screening techniques. These topics are reviewed in the next two chap-

ters, but it may help to know now why they are so important. Some kinds of statistical

results in SEM are interpreted exactly as regression coefﬁcients in multiple regression

(MR). Values of these coefﬁcients are corrected for the presence of correlated predictors

in SEM just as they are in MR. The potential for bias due to the omission of a predictor

that is correlated with others in the equation is basically the same in SEM and MR. The

technique of MR plays an important role in data screening. There are many statistical

tests in SEM, and their correct interpretation is essential. So with strong knowledge of

these topics, you are better prepared to learn about SEM.

use the Best research Computer in the World

Which is the human brain; speciﬁcally—yours. At the end of the analysis in SEM—or any

type of statistical analysis—it is you as the researcher who must evaluate the degree of

support for the hypotheses, explain any unexpected ﬁndings, relate the results to those

of previous studies, and reﬂect on implications of the ﬁndings for future research. These

are all matters of judgment. A statistician or computer geek could help you to select

appropriate statistical tools, but not with the rest without your domain knowledge. As

The simpler term multiple regression is used from this point.

Introduction 7

aptly put by Pedhazur and Schmelkin (1991), “no amount of technical proﬁciency will

do you any good, if you do not think” (p. 2).

get a Computer tool for seM

Obviously, you need a computer tool to conduct the analysis. In SEM, many choices

of computer tools are now available. Some of these include EQS, LISREL, and Mplus,

but there are still more, including Amos, CALIS/TCALIS of SAS/STAT, Mx, RAMONA

of SYSTAT, and SEPATH of STATISTICA. There are freely available student versions

of Amos, LISREL, and Mplus, and student versions are great for honing basic skills.

However, student versions are typically limited in terms of the number of variables

that can be analyzed, so they are not generally suitable for more complex analyses.

However, Mx can analyze a wide range of structural equation models, and it is freely

available over the Internet. All the SEM computer tools just mentioned, and others, are

described in Chapter 4. The website for this book (p. 3) has links to home pages for

SEM computer tools.

Join the Community

An electronic mail network called SEMNET operates over the Internet and is dedicated

to SEM.

It serves as an open forum for discussion and debate about the whole range of

issues associated with SEM. It also provides a place to ask questions about analyses or

about more general issues, including philosophical ones (e.g., the nature of causality).

Members of SEMNET come from different disciplines, and they range from newcom-

ers to seasoned veterans. Many works of the latter are cited in this book. (I subscribe

to SEMNET, too.) Sometimes the discussion gets, ah, lively (sparks can ﬂy), but this is

the nature of scientiﬁc discourse. Whether you participate as a “lurker” (someone who

mainly reads posts) or as an active poster, SEMNET offers opportunities to learn some-

thing new. There is even a theme song for SEM, the hilarious Ballad of the Casual Modeler

(Rogosa, 1988). I think that you might enjoy listening to it, too.

CharaCterIstICs oF seM

The term structural equation modeling (SEM) does not designate a single statistical

technique but instead refers to a family of related procedures. Other terms such as cova-

riance structure analysis, covariance structure modeling, or analysis of covariance

structures are also used in the literature to classify these techniques together under a

single label. These terms are essentially interchangeable, but only the ﬁrst will be used

www2.gsu.edu/~mkteer/semnet.html

www.stanford.edu/class/ed260/ballad.mp3

8 CONCEPTS AND TOOLS

throughout this book. Another term that you may have heard is causal modeling, which

is a somewhat dated expression ﬁrst associated with the SEM technique of path analysis.

For reasons elaborated later, the results of an SEM analysis cannot generally be taken

as evidence for causation. Wilkinson and the Task Force on Statistical Inference (1999)

were even more blunt when they noted that use of SEM computer tools “rarely yields

any results that have any interpretation as causal effects” (p. 600). Some newcomers to

SEM have unrealistic expectations in this regard. They may see SEM as a kind of magical

technique that allows one to discern causal relations in the absence of experimental or

even quasi-experimental designs. Unfortunately, no statistical technique, SEM or other-

wise, can somehow “prove” causality in nonexperimental designs. The correct and real-

istic interpretation of results from SEM analyses is emphasized throughout this book.

Summarized next are the characteristics of most applications of SEM.

a Priori does not Mean exclusively Conﬁrmatory

Computer tools for SEM require you to provide a lot of information about things such

as which variables are assumed to affect other variables and the directionalities of these

effects. These a priori speciﬁcations reﬂect your hypotheses, and in total they make up

the model to be analyzed. In this sense, SEM can be viewed as conﬁrmatory. That is,

your model is a given at the start of the analysis, and one of the main questions to be

answered is whether it is supported by the data. But as often happens, the data may be

inconsistent with your model, which means that you must either abandon your model or

modify the hypotheses on which it is based. In a strictly conﬁrmatory application, the

researcher has a single model that is accepted or rejected based on its correspondence

to the data (Jöreskog, 1993), and that’s it. However, on few occasions will the scope of

model testing be so narrow.

A second, somewhat less restrictive context concerns the testing of alternative

models, and it refers to situations in which more than one a priori model is available

(Jöreskog, 1993). This context requires sufﬁcient theoretical or empirical bases to spec-

ify more than one model; the particular model with acceptable correspondence to the

data may be retained, but the rest will be rejected.

A third context, that of model generation, is probably the most common and occurs

when an initial model does not ﬁt the data and is subsequently modiﬁed by the researcher.

The altered model is then tested again with the same data (Jöreskog, 1993). The goal of

this process is to “discover” a model with three properties: It makes theoretical sense, it

is reasonably parsimonious, and its correspondence to the data is acceptably close.

explicit distinction between observed and latent variables

There are two broad classes of variables in SEM, observed and latent. The observed class

represents your data—that is, variables for which you have collected scores and entered

in a data ﬁle. Another term for observed variables is manifest variables. Observed vari-

ables can be categorical, ordinal, or continuous, but all latent variables in SEM are con-

Introduction 9

tinuous. There are other statistical techniques for analyzing models with categorical

latent variables, but SEM deals with continuous latent variables only.

Latent variables in SEM generally correspond to hypothetical constructs or fac-

tors, which are explanatory variables presumed to reﬂect a continuum that is not directly

observable. An example is the construct of intelligence. There is no single, deﬁnitive mea-

sure of intelligence. Instead, researchers use different types of observed variables, such

as tasks of verbal reasoning or memory capacity, to assess various facets of intelligence.

Latent variables in SEM can represent a wide range of phenomena. For example, con-

structs about attributes of people (e.g., intelligence, neuroticism), higher-level units of

analysis (e.g., groups, geographic regions), or measures, such as method effects (e.g.,

self-report, observational), can all be represented as latent variables in SEM.

An observed variable used as an indirect measure of a construct is referred to as an

indicator. The explicit distinction between factors and indicators in SEM allows one to

test a wide variety of hypotheses about measurement. Suppose that a researcher believes

that variables X

, X

, and X

tap some common domain that is distinct from the one

assessed by X

and X

. In SEM, it is relatively easy to specify a model where X

–X

are

the indicators of one factor and X

–X

are indicators of a different factor. If the ﬁt of

the model just described to the data is poor, then this measurement hypothesis would

be rejected. The ability to analyze both observed and latent variables distinguishes

SEM from some more standard statistical techniques, such as the analysis of variance

(ANOVA) and MR, which analyze observed variables only.

Another class of variables in SEM corresponds to residual or error terms, which can

be associated with either observed variables or factors speciﬁed as outcome (dependent)

variables. In the case of indicators, a residual term represents variance unexplained

by the factor that the corresponding indicator is supposed to measure. Part of this

unexplained variance is due to random measurement error, or score unreliability.

The

explicit representation of measurement error is a special characteristic of SEM. This is

not to say that SEM can compensate for gross psychometric ﬂaws—no technique can—

but this property lends a more realistic quality to an analysis. Some more standard sta-

tistical techniques make unrealistic assumptions in this area. For example, it is assumed

in MR that all predictor variables are measured without error. In diagrams of structural

equation variables, residual terms may be represented using the same symbols as for

substantive latent variables. This is because error variance must be estimated, given the

whole model and the data; thus in this sense error variance is not directly observable in

the raw data. Also, residual terms are explicitly represented in the syntax or diagrams

of some SEM computer tools as latent variables. Even if they are not, error variance is

estimated in basically all SEM analyses, and estimates about the degree of residual vari-

ance often have interpretive import.

As already mentioned, it is possible in SEM to analyze substantive latent variables

The other part of unexplained variance is systematic (i.e., reliable) but unrelated to the underlying

construct. Another term for this part of residual variance is speciﬁc variance.

10 CONCEPTS AND TOOLS

or observed variables (or any combination of the two) as outcome variables. For such

variables, each will typically have an error term that represents variance unexplained

by their predictors. It is also possible to specify either observed or latent variables (or

any combination of the two) as predictors in structural equation models. This capability

permits great ﬂexibility in the types of hypotheses that can be tested in SEM. I should

say now that models in SEM do not necessarily have to have substantive latent variables

at all. (Most structural equation models have error terms represented as latent variables,

however.) That is, the evaluation of models that concern effects only among observed

variables is certainly possible in SEM. This describes the technique of path analysis, a

member of the SEM family.

Covariances always, but Means Can Be analyzed, too

The basic statistic of SEM is the covariance, which is deﬁned for two continuous observed

variables X and Y as follows:

XY XY X Y

cov r SD SD=

(1.1)

where r

is the Pearson correlation and SD

and SD

are their standard deviations. A

covariance represents the strength of the association between X and Y and their vari-

abilities, albeit with a single number. Because the covariance is an unstandardized sta-

tistic, its value has no upper or lower bound. For example, covariances of, say, –1,003.26

or 13.58 are possible. In any event, cov

conveys more information than r

, which says

something about association in a standardized metric only.

To say that the covariance is the basic statistic of SEM means that the analysis

has two main goals: (1) to understand patterns of covariances among a set of observed

variables and (2) to explain as much of their variance as possible with the researcher’s

model. The part of a structural equation model that represents hypotheses about vari-

ances and covariances is the covariance structure. The next several chapters outline

the rationale of analyzing covariance structures, but essentially all models in SEM have

a covariance structure.

Some researchers, especially those who use ANOVA as their main analytical tool,

have the impression that SEM is concerned solely with covariances. However, this view

is too narrow because means can also be analyzed in SEM, too. But what really distin-

guishes the analysis of means in SEM is that means of latent variables can be estimated.

In contrast, ANOVA is concerned with means of observed variables only. It is also pos-

sible in SEM to analyze effects traditionally associated with ANOVA, including between-

group and within-group (e.g., repeated measures) mean contrasts. For example, in SEM

one can estimate the magnitude of group mean differences on latent variables, some-

thing that is not really feasible in ANOVA.

When means are analyzed along with covariances in SEM, the model has both a

covariance structure and a mean structure, and the mean structure often represents

the estimation of factor means. Means are not analyzed in most SEM analyses—that is,

Introduction 11

a mean structure is not required—but the option to do so provides additional ﬂexibility.

For example, sometimes we are interested in estimating factors by analyzing covari-

ances among the observed variables, but also want to test whether means on these latent

variables are equal across different groups, such as boys versus girls. In this case, both

covariances and means would be analyzed in SEM. At other times, however, we are not

interested in means on the latent variables. Instead, we are concerned only with factor

covariances, and focus solely on what are the latent variables or factors, based on analy-

sis of the covariances among the observed variables. In the second case just mentioned,

we may only want to know how many factors underlie the scores on the observed vari-

ables. But in the ﬁrst case, we may be interested in both questions—that is, how many

factor underlie the indicators, and whether boys and girls have different means on each

of these factors.

seM Can Be applied to experimental data, too

Another too narrow view of SEM is that it is appropriate only for data from nonexperi-

mental designs. The heavy emphasis on covariances in the SEM literature may be at the

root of this perception, but the discussion to this point should suggest that this belief

is without foundation. For example, between-group comparisons in SEM could involve

experimental conditions to which cases are randomly assigned. In this context, the

application of SEM could be used to estimate group differences on latent variables that

are hypothesized to correspond to the observed outcome measures in a particular way.

Techniques in SEM can also be used in studies that have a mix of experimental and

nonexperimental features, as would occur if cases with various physical disorders were

randomly assigned to receive particular kinds of medications.

seM requires large samples

Attempts have been made to adapt SEM techniques to accommodate smaller sample

sizes (e.g., Nevitt & Hancock, 2004), but it is still generally true that SEM is a large-

sample technique. Implications of this property are considered throughout the book, but

I can say now that some kinds of statistical estimates in SEM, such as standard errors,

may not be accurate when the sample size is not large. The likelihood of technical prob-

lems in the analysis is greater, too.

Because sample size is such an important issue, let us now consider the bottom-line

question: What is a “large enough” sample size in SEM? It is difﬁcult to give a single

answer because several factors affect sample size requirements. For example, the analy-

sis of a complex model generally requires more cases than that of a simpler model. This

is because more complex models have more parameters than simpler models. More

precise deﬁnitions of parameters are given later in this volume, but for now you can

Bruce Thompson, personal communication, April 22, 2008.

12 CONCEPTS AND TOOLS

view them as hypothesized effects that require statistical estimates based on your data.

Models with more parameters require more estimates, so larger samples are necessary

in order for the results to be reasonably stable. The type of estimation algorithm used in

the analysis affects sample size requirements, too. There is more than one type of esti-

mation method in SEM, and some types need very large samples because of assumptions

they make (or do not make) about the data. Another factor involves the distributional

characteristics of the data. In general, smaller sample sizes are needed when the distri-

butions of continuous outcome variables are all normal in shape and their associations

with one another are all linear.

A useful rule of thumb concerning the relation between sample size and model

complexity that also has some empirical support was referred to by Jackson (2003) as

the N:q rule. This rule is applicable when the estimation method used is maximum

likelihood (ML), which is by far the method used most often in SEM. Indeed, ML is the

default method in most SEM computer tools. Properties of ML estimation are described

in Chapter 7, but it is no exaggeration to describe this method as the motor of SEM. (You

are the driver.) In ML estimation, Jackson (2003) suggested that researchers think about

minimum sample size in terms of the ratio of cases (N) to the number of model parame-

ters that require statistical estimates (q). An ideal sample size-to-parameters ratio would

be 20:1. For example, if a total of q = 10 model parameters require statistical estimates,

then an ideal minimum sample size would be 20 × 10, or N = 200. Less ideal would be an

N:q ratio of 10:1, which for the example just given for q = 10 would be a minimal sample

size of 10 × 10, or N = 100. As the N:q ratio decreases below 10:1 (e.g., N = 50, q = 10 for

a 5:1 ratio), so does the trustworthiness of the results.

It also helps to think about recommended sample size in more absolute terms. A

“typical” sample size in studies where SEM is used is about 200 cases. This number

corresponds to the approximate median sample size in surveys of published articles in

which SEM results are reported. These include an earlier review by Breckler (1990) of 72

articles in personality and social psychology journals and a more recent review by Shah

and Goldstein (2006) of 93 articles in management science journals. However, a sample

size of 200 cases may be too small when analyzing a complex model, using an estima-

tion method other than ML, or distributions are severely non-normal. With < 100 cases,

almost any type of SEM may be untenable unless a very simple model is evaluated. Such

simple models may be so bare-bones as to be uninteresting. Barrett (2007) suggested

that reviewers of journal submissions routinely reject for publication any SEM analysis

where N < 200 unless the population studied is restricted in size. This recommendation

is not standard practice, but it highlights the fact that analyzing small samples in SEM is

problematic. One of these problems is low statistical power. I will show you in Chapter

8 how to estimate power in SEM.

less emphasis on statistical tests

A great many effects can be tested for statistical signiﬁcance in SEM, ranging from things

such as the variance of a single variable up to entire models evaluated across multiple

samples. There are four reasons, however, why the results of statistical tests may be less

Introduction 13

relevant than other types of techniques, including ANOVA and MR. First, SEM allows

the evaluation of entire models, which brings a higher-level perspective to the analysis.

Statistical tests of individual effects represented in models may be of interest, but at

some point you must make a decision about the whole model: Should it be rejected?—

modiﬁed?—if so, how? Thus, there is a sense in SEM that the view of the entire landscape

(the whole model) has precedence over that of speciﬁc details (individual effects).

The second reason statistical tests play a smaller role in SEM concerns the general

requirement for large sample sizes discussed earlier. With most statistical tests, it is pos-

sible to have results that are “highly signiﬁcant” (e.g., p < .0001) but trivial in absolute

magnitude when the sample size is large. By the same token, virtually all effects that are

not nil will be statistically signiﬁcant in a sufﬁciently large sample. In fact, if the sample

size is large, then a statistically signiﬁcant result just basically conﬁrms a large sample

(Thompson, 1992), which is a tautology, or a needless repetition of the same sense in

different words.

The third reason is that statistical signiﬁcance (i.e., p values) for effects of latent

variables is estimated by the computer, but this estimate could change if, say, a different

estimation algorithm is used or sometimes even across different computer tools for the

same analysis and data. Differences in estimated p values across different software pack-

ages are usually not great, but small differences in p can affect hypothesis testing (e.g.,

p = .053 vs. p = .047 for the same effect when testing at the .05 level).

The fourth reason is not speciﬁc to SEM, but concerns most kinds of statistical

analyses in the behavioral sciences: We should in general be more concerned with esti-

mating the sizes or magnitudes of effects (i.e., effect sizes) than with the outcome of

statistical tests (e.g., Kline, 2004). Also, SEM gives better estimates of effect size than

traditional techniques for observed variables, including MR and ANOVA. Suggestions

for the conduct of statistical signiﬁcance testing in SEM will be discussed at various

points throughout the book.

seM and the general linear Model

You may know that ANOVA is just a special (restricted) case of MR. The two techniques

are based on the same underlying mathematical model that belongs to a larger family

known as the general linear model (GLM). The multivariate techniques of MANOVA

(i.e., multivariate ANOVA) and canonical correlation, among others, are also part of the

GLM. The whole of the GLM can be seen as just a restricted case of SEM (Fan, 1997). So

learning about SEM really means extending your repertoire of data analysis skills to the

next level, one that offers even more ﬂexibility than the GLM.

WIdesPread enthusIasM, But WIth a CautIonarY tale

It cannot be denied that SEM is increasingly “popular” among researchers in many dif-

ferent disciplines. This has become evident by the growing numbers of computer tools

for SEM, formal courses at the graduate level, continuing-education workshops, and