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

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174 CORE TECHNIQuES

in turn are speciﬁed as the reciprocal causes of each other. For example, young women

may be more likely to leave school if they are pregnant, but leaving school could be a

risk factor for pregnancy.

With six observed variables in the model of Figure 7.2, there are 21 observations.

The total number of free parameters is 19, including six variances of exogenous variables,

three unanalyzed associations, and 10 direct effects, so df

= 2. This model satisﬁes the

order and rank conditions for the equation of every endogenous variable. (You should

verify these statements.) I used the ML method of EQS 6.1 to ﬁt the model of Figure 7.2

to the covariance matrix based on the data in Table 7.6. The program’s default start val-

ues were used. These warning messages were issued after the very ﬁrst iteration:

You have bad start values to begin with

Please provide better start values and re-run the job

Next, EQS recovered from this “stumble” and eventually went on to generate a con-

verged and an admissible solution. However, at other times the analysis of a nonrecur-

sive model may fail right away due to bad start values. The same thing can happen when

analyzing an complex structural equation model of any type with many observed and

latent variables.

When computer analysis is foiled by a start value problem, then it is up to you to

provide better initial estimates. In the present example, I followed the suggestions in

Appendix 7.A to generate start values for the reciprocal direct effects and disturbance

variances for the variables maternity age and mother education in Figure 7.2. Speciﬁ-

cally, a “typical” standardized effect size of .30 was assumed for the path from maternity

age to education, and a “smaller” standardized effect size of .10 was assumed for the

path from mother education to maternity age. Given the observed standard deviations

for the variables mother education and maternity age—respectively, 2.17 and 2.33 (Table

7.6)—start values were calculated as follows:

Maternity Age → Mother Education: .30 (2.17/2.33) = .28

Variance of D

: (1 – .10) 2.17

= .90 (4.71) = 4.24

Mother Education → Maternity Age: .10 (2.33/2.17) = .11

Variance of D

: (1 – .01) 2.33

= .99 (5.43) = 5.38

In a second analysis with EQS, the start values just calculated were speciﬁed in program

syntax. The second analysis terminated normally with no error messages, and this solu-

tion is admissible. The parameter estimates are not described here, but you can view

them in the output ﬁle. Both the EQS syntax ﬁle with start value speciﬁcations and the

output ﬁle for this analysis can be downloaded from this book’s website (p. 3). You can

also download LISREL ﬁles for the same analysis.

Some additional issues speciﬁc to the estimation of nonrecursive models are

described in the chapter appendices. These issues apply whether the structural model

consists of observed variables only (nonrecursive path model) or has factors (nonre-

Estimation 175

cursive structural regression model). Appendix 7.B deals with effect decomposition in

nonrecursive models and the assumption of equilibrium. Appendix 7.C is about the esti-

mation of corrected R

-type proportions of explained variance for endogenous variables

involved in feedback loops.

FIttIng Models to CorrelatIon MatrICes

Default ML estimation assumes the analysis of unstandardized variables. If the variables

are standardized, ML results may be inaccurate, including estimates of standard errors

and values of model ﬁt statistics. This can happen if a model is not scale invariant, which

means that its overall ﬁt to the data depends on whether the variables are standardized

or unstandardized. Whether or not a model is scale invariant is determined by a rather

complex combination of its characteristics, including how the factors are scaled and the

presence of equality constraints on certain parameter estimates (Cudeck, 1989). One

symptom of scale invariance when a correlation matrix is analyzed with default ML

estimation is the observation that some of the diagonal elements in the model-implied

correlation matrix do not equal 1.0.

There is a method for correctly ﬁtting a model to a correlation matrix instead

of a covariance matrix known as constrained estimation or constrained optimiza-

tion (Browne, 1982). This method involves the imposition of nonlinear constraints

on certain parameter estimation to guarantee that the model is scale invariant. These

constraints can be quite complicated to program manually (e.g., Steiger, 2002, p. 221),

and not all SEM computer tools support nonlinear constraints (LISREL, Mplus, Mx,

and TCALIS do). However, some SEM computer programs, including SEPATH and

RAMONA, allow constrained estimation to be performed automatically by selecting

an option. These automated methods accept as input either a raw data ﬁle or a cor-

relation matrix. The EQS and Mplus programs can also correctly analyze correlations,

but they require raw data ﬁles. There are at least three occasions for using constrained

estimation:

1. A researcher is conducting a secondary SEM analysis based on a source wherein

correlations are reported, but not standard deviations. The raw data are also not avail-

able.

2. There is a theoretical reason to impose equality constraints on standardized

estimates, such as when the standardized direct effects of different predictors on the

same outcome are presumed to be equal. When a covariance matrix is analyzed, equality

constraints are imposed in the unstandardized solution only.

3. A researcher wishes to report correct tests of statistical signiﬁcance for the stan-

dardized solution. This means that correct standard errors are needed for the standard-

ized estimates, too. Note that Mplus automatically reports correct standard errors for

standardized effects when the standardized solution is requested.

176 CORE TECHNIQuES

alternatIve estIMators

Standard ML estimation works ﬁne for 90% or more of the structural equation models

described in the literature. However, you should be aware of some alternative methods.

Some of these alternatives are options when the assumption of multivariate normality

is not tenable, and others are intended for noncontinuous outcome variables. In some

disciplines, such as education, categorical outcomes may be analyzed as often as con-

tinuous outcomes. The methods described next are generally iterative, simultaneous,

full information, and available in many SEM computer programs.

other normal theory Methods for Continuous outcomes

Two methods for endogenous variables with multivariate normal distributions include

generalized least squares (GLS) and unweighted least squares (ULS). The ULS

method is actually a type of OLS estimation that minimizes the sum of squared differ-

ences between sample and model-implied covariances. It can generate unbiased esti-

mates across random samples, but it is not as efﬁcient as ML estimation (Kaplan, 2009).

A drawback of the ULS method is its requirement that all observed variables have the

same scale. That is, this method is neither scale free nor scale invariant. A potential

advantage is that, unlike ML, the ULS method does not require a positive-deﬁnite cova-

riance matrix (Chapter 3). It is also robust concerning initial estimates (Wothke, 1993).

This means that ULS estimation could be used to generate start values for a second

analysis of the same model and data but with ML estimation.

The GLS method is a member of a larger family of methods known as fully weighted

least squares (WLS) estimation, and some other methods in this family can be used for

severely non-normal data. In contrast to ULS, the GLS estimator is both scale free and

scale invariant, and under the assumption of multivariate normality, the GLS and ML

methods are asymptotic. One potential advantage of GLS over ML estimation is that it

requires less computation time and computer memory. However, this potential advan-

tage is not as meaningful today, given fast processors and abundant memory in relatively

inexpensive personal computers. In general, ML estimation is preferred to both ULS and

GLS estimation.

Corrected normal theory Methods for Continuous but non-normal

outcomes

The results of computer simulation studies generally indicate that it is best not to ignore

the multivariate normality assumption of default ML estimation (e.g., Curran, West,

& Finch, 1997; Olsson, Foss, Troye, & Howell, 2000). For example, when endogenous

variables are continuous but have severely non-normal distributions:

1. Values of ML parameter estimates may be relatively accurate in large samples,

but their estimated standard errors tend to be too low, perhaps by as much as 25–50%,

Estimation 177

depending on the data and model. This results in rejection of the null hypothesis that

the corresponding population parameter is zero more often than is correct (Type I error

rate is inﬂated).

2. Values of statistical tests of model ﬁt tend to be too high. This results in rejection

of the null hypothesis that the model has perfect ﬁt in the population more often than

is correct. That is, true models tend to be rejected too often. The actual rate of this error

may be as high as 50% when the expected rate assuming normal distributions is 5%,

again depending on the data and model. The most widely reported model test statistic in

SEM, the model chi-square

, is described in the next chapter. Depending on the par-

ticular pattern and severity of nonnormality, the value of

may be too small, which

would favor the researcher’s model. In other words, model test statistics calculated using

normal theory methods when there is severe nonnormality are not trustworthy.

One option to avoid bias is to normalize the variables with transformations (Chap-

ter 3) and then analyze the transformed data with default ML estimation. Another option

for continuous but non-normal outcome variables is to use a corrected normal theory

method. This means to analyze the original data with a normal theory method, such

as ML, but use robust standard errors and corrected model test statistics. Robust

standard errors are estimates of standard errors that are supposedly robust against non-

normality. The best known example of corrected model test statistics is the Satorra–

Bentler statistic (Satorra & Bentler, 1994), which adjusts downward the value of

from standard ML estimation by an amount that reﬂects the degree of kurtosis. The

Satorra–Bentler statistic was originally associated with EQS but is now calculated by

other SEM computer programs. Results of computer simulation studies of the Satorra–

Bentler statistic are generally favorable (Chou & Bentler, 1995). Analysis of a raw data ﬁle

is required for a corrected normal theory method. Of the various methods for analyzing

continuous outcome variables with severely non-normal distributions described here, a

corrected normal theory method may be the most straightforward to apply (Finney &

DiStefano, 2006).

normal theory Methods with Bootstrapping for Continuous but

non-normal outcomes

Another option for analyzing continuous but severely non-normal endogenous variables

is to use a normal theory method (i.e., ML estimation) but with nonparametric boot-

strapping, which assumes only that the population and sample distributions have the

same shape. In a bootstrap approach, parameters, standard errors, and model test sta-

tistics are estimated with empirical sampling distributions from large numbers of gen-

erated samples (e.g., Figure 2.3). Results of a computer simulation study by Nevitt and

Hancock (2001) indicate that bootstrap estimates for a measurement model were gener-

ally less biased compared with those from standard ML estimation under conditions

of non-normality and for sample sizes of N ≥ 200. For N = 100, however, bootstrapped

estimates had relatively large standard errors, and many generated samples were unus-

178 CORE TECHNIQuES

able due to problems such as nonpositive deﬁnite covariance matrices. These problems

are consistent with the caution by Yung and Bentler (1996) that a small sample size will

not typically render accurate bootstrapped results.

elliptical and arbitrary distribution estimators for Continuous but

non-normal outcomes

Another option to analyze models with continuous but non-normal endogenous vari-

ables is to use a method that does not assume multivariate normality. For example, there

is a class of estimators based on elliptical distribution theory that requires only sym-

metrical distributions (Bentler & Dijkstra, 1985). These methods estimate the degree

of kurtosis in raw data. If all endogenous variables have a common degree of kurtosis,

positive or negative skew is allowed; otherwise, zero skew is assumed. Various elliptical

distribution estimators are available in EQS.

Another option known as arbitrary distribution function (ADF) estimation makes

no distributional assumptions for continuous variables (Browne, 1984). This is because

it estimates the degree of both skew and kurtosis in the raw data. The calculations for

the ADF estimator are complex in part because the method derives a relatively large

weight matrix that is applied to the covariance residuals as part of the ﬁt function to

be minimized. The number of rows or columns in this square weight matrix equals

the number of observations, or v (v + 1)/2 where v is the number of observed variables.

For a model with many observed variables, the size of this matrix can be so large that

it can be difﬁcult for the computer to derive the inverse of this matrix. For example,

if there are 15 observed variables, the dimensions of the ADF weight matrix would be

120 × 120, which would have a total of 120

= 14,400 elements. Also, calculations in

ADF estimation typically require very large sample sizes in order for the results to be

reasonably accurate. Relatively simple (i.e., uninteresting) models may require sample

sizes of 200–500, and thousands of cases may be required for more complex models.

These requirements are impractical for many researchers. The results of some computer

simulation studies indicate that ADF estimation yields overly optimistic values of ﬁt

statistics when the model is misspeciﬁed (Olsson et al., 2000).

options for analyzing dichotomous or ordered-Categorical

outcomes

Endogenous variables are not always continuous. The most obvious example is a binary

or dichotomous outcome, such as relapsed–not relapsed (Chapter 2). There are also

ordered-categorical (ordinal) variables with three or more levels that imply a rank order.

For example, the following item has a Likert scale that indicates degree of agreement:

I am happy with my life (0 = disagree, 1 = uncertain, 2 = agree)

Estimation 179

The numeric scale for this variable (0–2) can distinguish among only three levels of

agreement. It would be hard to argue that the numbers assigned to the three response

alternatives of this item make up a scale with equal intervals. Also, scores on variables

with so few levels cannot be normally distributed. Although there is no “golden rule”

concerning the minimum number of levels that is required before scores can be approxi-

mately normally distributed, a score range of at least 15 points or so may be required.

However, Likert scales with about 5–10 points may be favorable in terms of people’s

ability to reasonably discriminate between scale values (anchors). With more than 10 or

so scale points for individual items, respondents may choose arbitrarily between adja-

cent points. Suppose that research participants are asked to rate their degree of agree-

ment with some statement on a 25-point Likert scale. It would be difﬁcult to think of

25 distinct verbal labels for each point along the scale that would indicate progressively

increasing or decreasing levels of agreement. Even with fewer labels, participants may

struggle with trying to decide what is the difference between ratings of, say, 13 versus

14 or 23 versus 24. That is, it is not practical to somehow “force” a variable with a Likert

scale to become continuous by adding levels beyond 10 or so.

Results of some computer simulation studies indicate that results from standard

ML estimation may be inaccurate for models with dichotomous or ordinal endogenous

variables. These simulation studies generally assume a true population measurement

model with continuous indicators. Within generated samples, the indicators are cat-

egorized to approximate data from noncontinuous variables. Bernstein and Teng (1989)

found that when there is only a single factor in the population but the indicators have

few categories, one-factor measurement models tend to be rejected too often. That is,

categorization can spuriously suggest the presence of multiple factors. DiStefano (2002)

found that ML parameter estimates and their standard errors were both generally too

low when the data analyzed were from categorical indicators, and the degree of negative

bias was higher as distributions became increasingly non-normal. The message of the

studies just cited and others is that standard ML is not an appropriate method for ana-

lyzing ordered-categorical variables.

Two analytical options for ordinal outcome variables are outlined. The ﬁrst involves

the case where ordered-categorical outcomes are analyzed as “stand-alone” variables

that are not merged or combined across a set of similar variables. This approach requires

special estimators for this type of data (i.e., not ML) related to the WLS family. The

second option involves analyzing parcels. A parcel is a total score across a set of homo-

geneous items each with a Likert-type scale. Parcels are generally treated as continuous

variables. The score reliability of parcels (total scores) tends to be greater than that for

the individual items. If the distributions of all parcels are normal, then default ML esti-

mation could be used to analyze the data. Parcels are then typically speciﬁed as continu-

The PRELIS program of LISREL automatically classiﬁes a variable with less than 16 levels as ordinal, but

this default can be changed.

180 CORE TECHNIQUES

ous indicators of underlying latent variables in a measurement model, such as in a CFA

model or when analyzing a structural regression (SR) model (e.g., Figures 5.6, 5.8). But

parceling is controversial. The reasons why are outlined later in this chapter.

Special WLS Methods for Ordinal Outcomes

Muthén (e.g., 1984) describes an approach to estimating models with any combina-

tion of dichotomous, ordinal, or continuous outcome variables known as continuous/

categorical variable methodology (CVM). In CVM, bivariate associations among ob-

served variables are estimated with polychoric correlations, which assume that a normal,

continuous process underlies each observed variable (Flora & Curran, 2004). The model

is then estimated with a form of WLS, and values of corrected test statistics are provided.

In the CVM approach described by Muthén and Asparouhov (2002) that is imple-

mented in Mplus, each observed ordinal indicator is associated with an underlying

latent response variable, which is the underlying amount of a continuous and normally

distributed trait or characteristic that is required to respond in a certain category of

the corresponding observed ordinal item. When the observed indicator is dichotomous,

such as for items with a true–false response format, this amount, or threshold, is the

point on the latent response variable where one answer is given (e.g., true) when the

threshold is exceeded. It is also the point where the other response is given (e.g., false)

when the threshold is not exceeded (Brown, 2006). Dichotomous items have a single

threshold, but the number of thresholds for items with ≥ 3 response categories is the

number of categories minus one. Each latent response variable is in turn represented as

the continuous indicator of the underlying substantive factor that corresponds to a hypo-

thetical construct. The data matrix analyzed in this approach is an asymptotic correla-

tion matrix of the latent response variables. For dichotomous indicators, this estimated

matrix will be a tetrachoric correlation matrix; for items with at least three response

categories, the data matrix will be an estimated polychoric correlation matrix.

The arbitrary and elliptical estimators described earlier (e.g., ADF), which do not

assume normality, are also members of the WLS family of estimators.

The WLS estima-

tor can be applied to either continuous or ordinal outcomes because it does not assume

a particular distributional form. In general, WLS estimation is just as computationally

complex as ADF estimation, requires very large samples, and is subject to technical

problems in the analysis (e.g., Finney & DiStefano, 2006, pp. 281–288), such as the

failure of the computer to derive the inverse of the weight matrix. Muthén, du Toit, and

Spisic (1997) describe forms of robust WLS estimation that deal with problems of

using WLS when the sample size is not very large. These robust methods use somewhat

simpler matrix calculations compared with WLS estimation.

In Mplus, two of these robust estimators are designated as mean-adjusted weighted

least squares (WLSM) and mean- and variance-adjusted weighted least squares

The GLS estimator is also a member of this family, but it assumes multivariate normality.

Estimation 181

(WLSMV). The standard errors and parameter estimates from these two methods are

the same, but WLSMV does not calculate the model degrees of freedom in the stan-

dard way, and this method may be preferred when the number of observed variables

is relatively small. This method is also the default in Mplus when ordered-categorical

variables are analyzed. In computer simulation studies, the WLSMV method has gener-

ally performed well except when the sample size is only about N = 200 or distributions

on ordered-categorical variables are markedly skewed (Muthén et al., 1997). Results

of computer simulation studies by Flora and Curran (2004) also indicated generally

positive performance of robust WLS estimation methods in the analysis of measurement

models with ordinal indicators. In contrast, these authors observed technical problems

with WLS estimation when the sample size was even as large as N = 1,000 for larger

models with about 20 indicators.

Models with ordinal outcomes are analyzed in two steps in LISREL. First, the raw

data are submitted to the PRELIS program, which estimates polychoric correlations

among the observed variables. These correlations and other statistical information are

used to compute an asymptotic covariance matrix, which is then analyzed in LISREL

with WLS estimation. Another option in LISREL is diagonally weighted least squares

(DWLS) estimation, which is a mathematically simpler form of WLS estimation that

may be better when the sample size is not very large. See Jöreskog (2005) for examples.

The EQS program uses a two-stage method by Lee, Poon, and Bentler (1995) for

analyzing models with any combination of continuous or categorical endogenous vari-

ables. In the ﬁrst stage, a special form of ML estimation is used to estimate correlations

between the continuous latent variables presumed to underlie the observed variables. In

the second stage, an asymptotic covariance matrix is computed, and the model is ana-

lyzed with a method that in EQS is referred to as arbitrary generalized least squares

(AGLS) estimation, which is basically equivalent to WLS estimation (Finney & DiSte-

fano, 2006).

analyzing Items Parcels

Suppose that a questionnaire has 40 Likert scale items. Instead of analyzing all 40 items

as “stand-alone” outcome variables, a researcher partitions the items into two nonover-

lapping sets of 20 items each. The items within each set are presumed to be homoge-

neous; that is, they reﬂect a common domain. A total score is derived across the 20

items within each set, and the two resulting total scores, or parcels, are analyzed instead

of the 40 items. Because the total scores are continuous and normally distributed, the

researcher opts for standard ML estimation, which is easier to use than WLS estimators.

This is the basic rationale of parceling. But this technique is controversial because it

assumes that items within each parcel are known to measure a single construct, or are

unidimensional. This knowledge may come from familiarity with the item domain or

results of prior statistical analyses, such as an exploratory factor analysis.

Parceling is not recommended if unidimensionality cannot be assumed. Speciﬁ-

cally, parceling should not be part of an analysis aimed at determining whether a set of

182 CORE TECHNIQuES

items is unidimensional. This is because it is possible that parceling can mask a mul-

tidimensional factor structure in such a way that a seriously misspeciﬁed model may

nevertheless ﬁt the data reasonably well (Bandalos, 2002). There are also different ways

to parcel items, including random assignment of items to parcels and groupings of items

based on rational grounds (e.g., the items share similar content), and the choice can

affect the results. See Bandalos and Finney (2001) and Little, Cunningham, Shahar, and

Widamin (2002) for descriptions of the potential advantages and drawbacks of parcel-

ing in SEM. Williams and O’Boyle (2008) review human resource management research

using parcels.

a healthY PersPeCtIve on estIMatIon

The availability of so many different estimation methods can sometimes seem over-

whelming for newcomers to SEM. Loehlin (2004) cites the following proverb that may

describe this experience: a person with one watch always knows what time it is; a per-

son with two never does. It also doesn’t help that the same general estimator may be

referred to using different names in the documentation or syntax of different SEM com-

puter tools. Actually, the situation is not so bewildering because standard ML estima-

tion works just ﬁne for most types of structural equation models if the data have been

properly screened and distributions of continuous endogenous variables are reasonably

multivariate normal. But if the normality assumption is not tenable or if you are working

with outcome variables that are not continuous, you need alternative methods.

suMMarY

The method of maximum likelihood estimation is a normal theory, full-information

method that simultaneously analyzes all model equations in an iterative algorithm.

General statistical assumptions include independence of the scores, independence of

exogenous variables and residuals, multivariate normality, and correct speciﬁcation of

the model. Correct speciﬁcation of the model is especially important because of error

propagation, or the tendency for misspeciﬁcation in one part of the model to affect esti-

mates in other parts. Sometimes iterative estimation fails due to poor start values. When

this happens, it may be necessary to specify better initial estimates in order to “help”

the computer reach a converged solution. It can happen in estimation that the solution

contains illogical values, such as Heywood cases, which renders the solution inadmis-

sible. Thus, you should always carefully inspect the solution even if the computer output

contains no error or warning messages. When endogenous variables are continuous but

their distributions are severely non-normal, the most straightforward option is to use

a corrected normal theory method that generates robust standard errors and corrected

test statistics. When the endogenous variables are not continuous, then other estimation

methods, including forms of robust weighted least squares, should be applied.

Estimation 183