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

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334 ADVANCED TECHNIQuES, AVOIDING MISTAKES

statistically signiﬁcant. These results are consistent with “pure” mediation of the effects

of memory demand, cognitive complexity, and their interaction on recall through recol-

lection of speciﬁc behaviors.

James and Brett (1984) described moderated mediation, also known as a condi-

tional indirect effect (Preacher et al., 2007). It is indicated when the strength of an indi-

rect effect varies across the levels of another variable. In the multiple-sample analysis of

a path model, this other variable is group membership. If the magnitude of an indirect

effect differs appreciably across the groups, there is evidence for moderated mediation.

If the moderator is instead a continuous variable W, then moderated mediation can be

represented in a moderated path model and estimated in a single sample. There is more

than one kind of moderated mediation. For example, the model in Figure 12.4(a) rep-

FIgure 12.4. Two examples of moderated mediation for the indirect effect X → M → Y. (a) Path

X → M depends on W. (b) Path M → Y depends on W.

Interaction Effects and Multilevel SEM 335

resents ﬁrst-stage moderation (Edwards & Lambert, 2007) where the ﬁrst path of the

indirect effect of X on Y through M, or X → M, depends on the fourth variable W. This

moderation effect is represented in Figure 12.4(a) by the regression of M on X, W, and

the product term XW. This model also represents the hypothesis that the ﬁrst path of the

indirect effect of W on Y through M, or W → M, depends on X.

The path model in Figure 12.4(b) represents second-stage moderation where the

second path of the indirect effect of X on Y through M, or M → Y, is moderated by W. In

this model, the interaction effect concerns variables W and M. The product term WM

is speciﬁed to covary with the component variable W and the disturbance variance for

the other component variable, M. The latter is required because mediators are always

endogenous, and endogenous variables are not permitted to have unanalyzed associa-

tions with other variables in structural equation models. Regressing the outcome vari-

able Y on X, M, W, and the product term WM

estimates whether the path M → Y depends

on W. Other forms of moderated mediation described by Edwards and Lambert (2007)

are as follows:

1. First- and second-stage moderation occurs when a fourth variable W moder-

ates both paths of the indirect effect X → M → Y. A variation is when one variable, such

as W, moderates the ﬁrst part of the indirect effect, or X → M, and a different variable,

such as Z, moderates the second path, or M → Y.

2. Conditional indirect effects can also involve the direct effect, or the path X → Y

in the examples to this point, and also the total effect of X on Y. For example, in direct

effect and ﬁrst-stage moderation, a fourth variable moderates both the direct effect

between X and Y and just the ﬁrst path of the indirect effect of X on Y, or X → M. In

direct effect and second-stage moderation, an external variable moderates both the

direct effect and the second path of the indirect effect, or M → Y. And in total effect

moderation, an external variable moderates both paths of the indirect effect and also

the direct effect.

Several recent works deal with the estimation of mediated moderation or moder-

ated mediation in MPA. The approach by Preacher et al. (2007) that features the analysis

of simple slopes, regions of signiﬁcance, and conﬁdence bands was mentioned earlier.

Edwards and Lambert (2007) describe an approach based on effect decomposition and

graphical plotting of conditional indirect effects. Fairchild and MacKinnon (2009) give

equations for standard errors of mediated moderation effects and moderated mediation

effects, as well as step-by-step suggestions for testing models with both kinds of effects

just mentioned. Hopwood (2007) does so for applications in the area of early interven-

tion research. For an example of the analysis of mediated moderation and moderated

mediation in the same path model, see Clapp and Beck (2009). These authors, studying

a sample of survivors of serious motor vehicle accidents, examined whether the indirect

effects of posttraumatic stress disorder on social support through attitudes about the

usefulness of social networks in coping with stress is moderated by childhood victim-

ization and elapsed time from the accident.

336 ADVANCED TECHNIQuES, AVOIDING MISTAKES

InteraCtIve eFFeCts oF latent varIaBles

In the indicant product approach of SEM, product terms are speciﬁed as multiple indi-

cators of latent product variables that represent curvilinear or interactive effects. We

will consider only the estimation of interactive effects of latent variables, but the same

basic principles apply to the estimation of curvilinear effects of latent variables. Suppose

that factor A has two indicators, X

and X

, and factor B has two indicators, W

and W

The reference variable for A is X

, and the reference variable for B is X

. Equations that

specify the measurement model for these indicators are:

AE+

BE+

AEλ+

BEλ+

The parameters of this measurement model include the loadings of X

and W

on their

respective factors (

), the variances of the four measurement errors, and the

variances and covariance of factors A and B.

The latent product variable AB represents the interactive effect of factors A and B

when they are analyzed together with A and B in the same structural model. Its indica-

tors are the four product indicators X

, X

, and X

. By taking the prod-

uct of the corresponding expressions in Equation 12.4 for the nonproduct indicators, the

equations of the measurement model for the product indicators are

1111

WXXW

AB AE BE E E+++

2 22112

W WWXXW

AB AE BE E Eλ + +λ +

2 21 221

X XW XXW

AB AE BE E Eλ +λ + +

22 22 22 22

XW XW WX XW

AB AE BE E Eλ λ +λ +λ +

These equations (12.5) show that the product indicators load on a total of eight addi-

tional latent product variables besides AB. For example, product indicator X

loads on

latent variables AB,

, and

. (The term

is the residual for X

All the factor loadings in the model deﬁned by Equation 12.5 are either the constant 1.0

or functions of

and

, the loadings of the nonproduct indicators X

and W

their respective factors (Equation 12.4). This means that no new factor loadings need to

be estimated for the product indicators.

The only other parameters of the measurement model for the product indicators

are the variances and covariances of the latent product variables implied by Equation

12.5. Assuming normal distributions for all nonproduct latent variables (Equation 12.4),

Kenny and Judd (1984) showed that (1) the covariances among the latent product vari-

ables and factors A and B are all zero; and (2) the variances of the latent product vari-

ables can be expressed as functions of the variances of the nonproduct latent variables

as follows:

(12.4)

(12.5)

Interaction Effects and Multilevel SEM 337

2222

,AB A B A B

σ =σσ+σ

1211

222

XW XW

EE EE

σ =σσ

222

BE B E

σ=σσ

1212

222

XW XW

EE EE

σ =σσ

222

BE B E

σ=σσ

2121

222

XWXW

EEEE

σ =σσ

222

AE A E

σ=σσ

2222

222

XWXW

EEEE

σ =σσ

222

AE A E

σ =σσ

where the term

,AB

represents the covariance between factors A and B. For example,

the variance of the latent product variable AB equals the product of the variances for

factors A and B plus their covariance. All variances of the other latent product variables

are related to the variances of the nonproduct latent variables. Thus, no new variances

need to be estimated, so the measurement model for the product indicators is theoreti-

cally identiﬁed.

Presented in Figure 12.5 is the whole SR model for the regression of Y on factors A,

B, and their interactive effect represented by AB. The measurement models for the non-

product indicators and the product indicators deﬁned by, respectively, Equations 12.4

and 12.5 are also represented in the ﬁgure. Among estimates for the structural model in

the ﬁgure, coefﬁcients for the paths

A → Y, B → Y, and AB → Y

estimate, respectively, the linear effect of factor A, the linear effect of factor B, and the lin-

ear × linear interaction of these latent variables, each controlling for the other effects.

estIMatIon WIth the kennY–Judd Method

Kenny and Judd (1984) were among the ﬁrst to describe a method for estimating struc-

tural equation models with product indicators. The Kenny–Judd method is generally

applied to observed variables in mean-deviation form (i.e., scores on nonproduct indi-

cators are centered before creating product indicators). It has two potential complica-

tions:

1. It requires the imposition of nonlinear constraints in order to estimate some

parameters of the measurement model for the product indicators (see Equations 12.4,

12.5). Not all SEM programs support nonlinear constraints; those that do include Mplus,

Mx, the TCALIS procedure of SAS/STAT, and LISREL.

Correctly programming all such

constraints can be tedious and error prone.

(12.6)

Nonlinear constraints must be speciﬁed in LISREL using its matrix-based programming language, not

SIMPLIS.

338 ADVANCED TECHNIQuES, AVOIDING MISTAKES

2. A product variable is not normally distributed even if each of its component

variables is normally distributed. For example, the Kenny–Judd method assumes that

factors A and B and the measurement errors for their nonproduct indicators in Figure

12.5 are normally distributed. But the products of these latent variables, such as AB, are

not normally distributed, which violates the normality requirement of default maxi-

mum likelihood (ML) estimation. Yang–Wallentin and Jöreskog (2001) demonstrate the

estimation of a model with product indicators using a corrected normal theory method

that can generate robust standard errors and corrected model test statistics. Also, mini-

mum sample sizes of 400–500 cases may be needed when estimating even relatively

small models, but the need for large samples is not speciﬁc to the Kenny–Judd method

per se.

FIgure 12.5. Model with interactive and main effects of factors A and B.

Interaction Effects and Multilevel SEM 339

Presented in Table 12.3 is a covariance matrix generated by Kenny and Judd (1984)

for a hypothetical sample of 500 cases. You can download from the website for this

book (see p. 3) the Mplus 5.2 syntax and data ﬁles that specify the nonlinear constraints

implied by Equations 12.4 and 12.5 and ﬁt the model in Figure 12.5 to the data in Table

12.3 using the Kenny–Judd method; the Mplus output ﬁle can be downloaded from the

site, too. The Mplus syntax ﬁle is annotated with comments that explain the nonlinear

constraints of this method. Because Kenny and Judd (1984) used a generalized least

squares (GLS) estimator in their original analysis of these data, I speciﬁed the same esti-

mator in this analysis with Mplus. The input data for this analysis are in matrix form, so

it is not possible to use a corrected normal theory method because such methods require

raw data ﬁles. You can also download from the book’s website Mplus computer ﬁles for

the analysis of a latent quadratic effect using the Kenny–Judd method with accompany-

ing text that explains this supplemental example.

With a total of nine observed variables (four nonproduct indicators, four prod-

uct indicators, and Y; see Figure 12.5), there are a total of 9(10)/2, or 45 observations

available for this analysis. There are a total of 13 free parameters, including (1) two

factor loadings (of X

and W

); (2) seven variances (of A, B, E

, E

, and D

)

and one covariance (A

B); and (3) three direct effects (of A, B, and AB on Y). There

are no free parameters for the measurement model of the product indicators, so df

= 45 – 13 = 32. The analysis in Mplus converged to an admissible solution. Values of

selected ﬁt statistics are reported next and generally indicate acceptable overall ﬁt. The

90% conﬁdence interval based on the RMSEA is reported in parentheses:

(32) = 41.989, p = . 111

RMSEA = .025 (0–.044), p

close-ﬁt H

= .988

CFI = .988; SRMR = .046

The Mplus-generated GLS parameter estimates for the model of Figure 12.5 are very

similar to those reported by Kenny and Judd (1984) in their original analysis. The main

taBle 12.3. Input data (Covariances) for analysis of a Model with an Interactive

effect of latent variables with the kenny–Judd Method

Variable 1 2 3 4 5 6 7 8 9

1. X

2.395

2. X

1.254 1.542

3. W

.445 .202 2.097

4. W

.231 .116 1.141 1.370

5. X

−.367 −.070 −.148 −.133 5.669

6. X

−.301 −.041 −.130 −.117 2.868 3.076

7. X

−.081 −.054 .038 .037 2.989 1.346 3.411

8. X

−.047 −.045 .039 −.043 1.341 1.392 1.719 1.960

9. Y

−.368 −.179 .402 .282 2.556 1.579 1.623 .971 2.174

Note. These data for a hypothetical sample are from Kenny and Judd (1984, p. 205); N = 500.

340 ADVANCED TECHNIQuES, AVOIDING MISTAKES

and interactive effects of factors A and B together explain 86.8% of the total variance in

Y. The unstandardized equation for predicting Y is

= –.169 A + .321 B + .699 AB

This prediction equation has no intercept because means were not analyzed. However,

you could use the same method described earlier to rearrange this unstandardized equa-

tion to (1) eliminate the product term and (2) generate simple regressions of, say, Y on

factor B where the slope for B varies as a function of factor A. Following these steps

will show that the relation between Y and B is positive for levels of A above the mean

(i.e., > 0) but negative for levels of A below the mean (< 0) for the model in Figure 12.5

and the data in Table 12.3.

alternatIve estIMatIon Methods

When using the Kenny–Judd method to estimate latent interaction effects, Jöreskog and

Yang (1996) recommend adding a mean structure to the model. (The basic Kenny–Judd

method has no mean structure.) They argue that because the means of the indicators

are functions of other parameters in the model, their intercepts should be added to the

model in order for the results to be more accurate. They also note that a single product

indicator is all that is needed for identiﬁcation. In contrast, the analysis of all possible

product indicators in the Kenny–Judd approach can make the model rather compli-

cated (e.g., Figure 12.5). As a compromise between analyzing a single-product indicator

and all possible product indicators, Marsh, Wen, and Hau (2006) recommend analyzing

matched-pair products in which information from the same indicator is not repeated.

For example, given indicators X

and X

of factor A and indicators W

and W

of factor

B, the pair of product indicators X

and X

is a set of matched-pair indicators

because no individual indicator appears twice in any product term. The pair X

and

is the other set of matched-pair indicators for this example.

Ping (1996) describes a two-step estimation method that does not require nonlinear

constraints, which means that it can be used with just about any SEM computer tool. It

requires essentially the same assumptions as the Kenny–Judd method. In the ﬁrst step

of Ping’s method, the model is analyzed without the product indicators. That is, only

the linear effects of latent variables in the structural model are estimated. One records

parameter estimates from this analysis and calculates the values of the parameters of

the measurement model for the product indicators implied in the Kenny–Judd method.

These values can be calculated either by hand or by using a set of templates for Micro-

soft Excel by R. Ping that can be freely downloaded over the Internet.

These calculated

values are then speciﬁed as ﬁxed parameters in the second step where all indicators,

www.wright.edu/~robert.ping/

Interaction Effects and Multilevel SEM 341

product and nonproduct, are analyzed together. Included in the results of the second

analysis are estimates of interaction effects of latent variables.

Bollen’s (1996) two-stage least squares (TSLS) method for latent variables is another

estimation option. This method requires at least one product indicator of a latent prod-

uct variable and a separate product indicator that is used as an instrumental variable. An

advantage of this method is that it does not assume normal distributions for the indica-

tors. Because it is not iterative, the TSLS method may also be less susceptible to techni-

cal problems in the analysis. A drawback is that because TSLS is a partial-information

technique, there is no statistical test (e.g.,

) of the overall ﬁt of the model to the data.

In a simulation study, Yang–Wallentin (2001) compared standard ML estimation and

Bollen’s (1996) TSLS method applied to the estimation of latent interaction effects. Nei-

ther method performed especially well for sample sizes of N < 400, especially TSLS. For

larger samples, differences in bias were generally negligible, but TSLS tended to under-

estimate standard errors even in large samples.

Wall and Amemiya (2001) describe the generalized appended product indicator

(GAPI) method for estimating latent curvilinear or interactive effects. As in the Kenny–

Judd method, products of observed variables are speciﬁed as indicators of latent product

terms, but the GAPI method does not assume that any of the variables are normally

distributed. Consequently, it is not assumed in the GAPI method that the latent prod-

uct variables are independent. Instead, these covariances are estimated as part of the

analysis. A mean structure is also added to the model. However, all other constraints of

the Kenny–Judd method, including the nonlinear constraints, are imposed in the GAPI

method. A disadvantage of this method is that its implementation in computer syntax

can be complicated (see Marsh et al., 2006).

Marsh, Wen, and Hau (2004) describe what is basically an unconstrained approach

to the estimation of latent interaction and quadratic effects that imposes no nonlinear

constraints and also does not assume multivariate normality. It features the speciﬁca-

tion of product indicators of latent curvilinear or interaction effects, but it imposes no

nonlinear constraints on estimates of the correspondence between product indicators

and latent product terms. The unconstrained approach is generally easier to implement

in computer syntax than the GAPI method (see Marsh et al., 2006). Results of com-

puter simulation studies by Marsh et al. (2004) generally support this method for large

samples and when normality assumptions are not met.

Klein and Moosbrugger’s (2000) latent moderated structural equations (LMS)

method uses a special form of ML estimation that assumes normal distributions for the

nonproduct variables but takes direct account of the degree of non-normality implied

by the latent product terms. This method adds a mean structure to the model, and it

uses a form of the expectation–maximization (EM) algorithm (Chapter 3) in estima-

tion. The LMS method directly analyzes raw data (there is no matrix input) from the

nonproduct indicators (e.g., X

, X

, W

, and W

in Figure 12.5) to estimate a latent inter-

action or curvilinear effect without creating any product indicators. Of all the methods

described here, the LMS method may be the most precise because it explicitly estimates

the form of nonnormality. The LMS method is computationally intensive, but Klein

342 ADVANCED TECHNIQuES, AVOIDING MISTAKES

and Muthén (2007) describe a simpler algorithm known as quasi-maximum likelihood

(QML) estimation that closely approximates results of the former. A version of the LMS/

QML method is incorporated in Mplus, along with special syntax for specifying latent

interaction or curvilinear effects. This syntax is very compact and much less complex

than the syntax required to implement the Kenny–Judd method and some other alter-

native methods just described. However, most traditional SEM ﬁt statistics, including

the model chi-square (

) and approximate ﬁt indexes, are not available in the Mplus

implementation of the LMS/QML method. Instead, the relative ﬁt of different models is

compared using the Akaike Information Criterion (AIC) (Chapter 8) or a related statistic

known as the Bayesian Information Criterion (BIC) (Raftery, 1995).

Little, Bovaird, and Widaman (2006) describe an extension of the method of residu-

alized centering for estimating the interactive or curvilinear effects of latent variables. In

this approach, the researcher creates every possible product indicator and then regresses

each product indicator on its own set of constituent nonproduct indicators. The residu-

als from the analysis represent interaction but are uncorrelated with the corresponding

set of nonproduct indicators. The residualized product indicators are then speciﬁed as

the indicators of a latent product factor that is uncorrelated with the corresponding

nonproduct latent factors (i.e., those that represent latent linear effects only). The only

other special parameterization in this approach is that error covariances are speciﬁed

between pairs of residualized product indicators based on common nonproduct indica-

tors (e.g., Little, Bovaird, & Widaman, 2006, p. 506). This method can be implemented

in basically any SEM computer tool (i.e., it does not require a speciﬁc software package),

and it relies on traditional ﬁt statistics in the assessment of model–data correspondence.

Based on computer simulation studies by Little, Bovaird, and Widaman (2006), their

residualized product indicator method generally yielded similar parameter estimates

compared with the LMS/QML method and also the Marsh et al. (2004) unconstrained

method used with mean centering.

No single method for estimating curvilinear or interactive effects of latent vari-

ables has so far emerged as the “best” approach, but this is an active research area. For

empirical examples, see Klein and Moosbrugger (2000), who applied the LMS method

in a sample of 304 middle-aged men to estimate the latent main and interactive effects

of ﬂexibility in goal adjustment and perceived physical ﬁtness on levels of complaining

about one’s mental or physical state. They found that high levels of perceived ﬁtness

neutralized the effects of goal ﬂexibility, but effects of goal ﬂexibility on complaining

were more substantial at lower levels of perceived ﬁtness. In a sample of 792 employees

in various commercial joint ventures, Song, Droge, Hanvanich, and Calantone (2005)

used the Kenny–Judd method to estimate the latent interactive effects of company

technological capabilities and marketing capabilities on marketing performance (sales,

proﬁts, etc.) They analyzed their moderation model across two different groups of

companies—those in areas where industry technology rapidly changes versus areas

where technological developments are minor. The results suggested that the interactive

effects of company technological and marketing resources on sales success depend on

industry context.

Interaction Effects and Multilevel SEM 343

ratIonale oF MultIlevel analYsIs

The term multilevel modeling (MLM)—also known as hierarchical linear modeling

and random coefﬁcient modeling, among other variations—refers to a family of sta-

tistical techniques for analyzing hierarchical (nested) data where (1) scores are clus-

tered into larger units and (2) scores within each level may not be independent. You

already know that repeated measures data are hierarchical in that multiple scores are

nested under the same person. Dependence among such scores is explicitly estimated in

various techniques for repeated measures data. For example, the error term in repeated

measures ANOVA takes account of score covariances across the levels of within-subject

factors. In SEM, the capability to specify an error covariance structure for repeated mea-

sures variables, such as when analyzing a latent growth model (Chapter 11), also takes

account of score dependencies.

Another situation for analyzing hierarchical data occurs in complex sampling

designs, in which the levels of at least one higher-order variable are selected prior to

sampling individual cases within each level. An example is the method of cluster sam-

pling. Suppose in a study of Grade 2 scholastic skills that a total of 100 public elementary

schools in a particular geographic region is randomly selected, and then every Grade 2

student in these schools is assessed. Here, students are clustered within schools. A vari-

ation is multistage sampling where only a portion of the students within each school

are randomly selected (e.g., 10%) for inclusion in the sample. In stratiﬁed sampling, a

population is divided into homogeneous, mutually exclusive subpopulations (strata),

such as by gender or ethic categories, and then cases within each stratum are randomly

selected. The resulting hierarchical data set may be representative on the variable(s)

selected for stratiﬁcation.

Scores clustered under a higher-level variable may not be independent. For instance,

siblings are affected by their common family situation. Score dependence in complex

samples means that the application of standard formulas for estimating standard errors

in a single-level analysis that assume independence (e.g., Equation 2.14) may not yield

correct results. Speciﬁcally, such formulas tend to underestimate sampling variance in

complex samples. Because standard errors are the denominators of basic statistical tests

(Chapter 2), underestimation tends to result in rejection of the null hypothesis too often

(inﬂation of Type I error). This is a second motivation for MLM: the correct estimation

of standard errors in a complex sampling design.

Increase of sampling error in complex samples compared with simple random

sampling of individual cases with no clusters is known as the design effect. It is

often estimated as the ratio of the variance of a statistic in a complex sample over

the variance of the same statistic in a simple random sample for the same number of

cases. For example, a design effect of 4.0 would mean that the variance is four times

greater in a complex sampling design than if the study were based on simple random

sampling. Another interpretation is that only ¼ or one-quarter as many cases would

be needed in a simple random sample to measure the same statistic instead of a com-

plex sample where the design effect is 4.0. Thus, higher estimates of the design effect