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

Подождите немного. Документ загружается.

324 ADVANCED TECHNIQuES, AVOIDING MISTAKES

direct effects of the spouse variable will provide information about whether the differ-

ence between husbands and wives predicts the factors.

Reported in Table 11.10 are the correlations, standard deviations, and means for

the FOE indicators and marital adjustment indicators for the total sample of N = 206

cases (103 couples) in the Sabatelli and Bartle–Haring (2003) data set. Also reported in

Table 11.10 are the correlations (speciﬁcally, point-biserial correlations, r

) between the

spouse variable and each of the indicators calculated from the group means and standard

deviations (Kline, 2004, p. 114). With six observed variables altogether—including ﬁve

effect indicators and one cause indicator (spouse)—there are 6(7)/2 = 21 observations.

The MIMIC model of Figure 11.5 was ﬁtted to the covariance matrix assembled

from the correlations and standard deviations in Table 11.10 with the ML method of LIS-

REL 8.8. The factor loadings and measurement error variances were speciﬁed as ﬁxed

and equal to the corresponding values reported in Table 11.9, which assume equality of

husbands and wives across these parameters. The MIMIC model of Figure 11.5 does not

assume correlated errors between the father and mother indicators of the FOE factor.

This is because this correlation was estimated earlier to be negative for husbands but

positive for wives (Table 11.9). In the total sample, this error correlation may be about

zero. With these speciﬁcations, the MIMIC model has six free parameters, including

(1) three variances (of the spouse cause indicator and two-factor disturbances); (2) one

covariance between the factor disturbances; and (3) the two direct effects of the dichoto-

mous cause indicator on the factors, so df

= 21 – 6 = 15.

The analysis of the MIMIC model with ML estimation in LISREL converged to an

admissible solution. This occurred even though the data matrix is ill scaled (see Table

11.10)—rescale this matrix if you reproduce this analysis in a different SEM computer

taBle 11.10. Input data (Correlations, standard deviations) for analysis of

a MIMIC (Multiple Indicators and Multiple Causes) Model of Family-of-origin

experiences and Marital adjustment

Variables 1 2 3 4 5 6

Martial adjustment indicators

1. Problems

1.000

2. Intimacy

.700 1.000

Family-of-origin experiences indicators

3. Father

.253 .409 1.000

4. Mother

.247 .355 .659 1.000

5. Father-Mother

.301 .330 .622 .513 1.000

Predictor

6. Spouse

.097 .010 .139 .035 .145

1.000

158.663 138.177 84.497 85.943

83.025

.500

32.064 21.463 12.357 12.748 13.813

.500

Note. These data are derived from those summarized in Table 9.8; N = 206. Means are reported but not

analyzed.

Mean Structures and Latent Growth Models 325

tool and encounter a problem. Overall model ﬁt seems adequate based on the values of

selected ﬁt indexes:

(15) = 14.188, p = .511

RMSEA = 0 (0–.061); p

close-ﬁt H

= .890;

GFI = .977; CFI = 1.000; SRMR = .032

Estimates for the free model parameters except for the variance of the spouse cause indi-

cator, which is just the sample value (Table 11.10), are presented in Figure 11.5 in the

proper places. The unstandardized coefﬁcients (and standard errors) for direct effects of

the spouse cause indicator on the FOE factor and on the marital adjustment factor are,

respectively, 3.101 (1.626) and .684 (3.273). Because spouse is coded as 0 = husband and

1 = wife, these positive regression weights indicate higher predicted overall standings on

both factors for wives than husbands. The coefﬁcient for the direct effect of spouse on

marital adjustment is not statistically signiﬁcant at the .05 level because z = .684/3.273

= .209, p = .835. The standardized coefﬁcient for this path, .015, is also quite small. The

unstandardized coefﬁcient for the direct effect of spouse on FOE experiences is nearly

statistically signiﬁcant because z = 3.101/1.626 = 1.91, p = .057. The standardized coef-

ﬁcient for this path is .144. These results obtained in the total sample for the MIMIC

model are similar to those for the measurement model with structured means analyzed

earlier across both samples. To summarize, wives report somewhat better FOE experi-

ences than their husbands, but not clearly better marital adjustment. You can download

all LISREL and Mplus computer ﬁles for this analysis from the book’s website (p. 3). See

Kano (2001) for information about the speciﬁcation of MIMIC models in experimental

designs where groups are compared across latent variables instead of observed variables

only as in ANOVA or MANOVA.

suMMarY

Means are estimated in SEM by regressing exogenous or endogenous variables on a con-

stant that equals 1.0. It is usually not necessary to manually create a constant because

most full-featured SEM computer programs do so automatically when means are ana-

lyzed. The parameters of a mean structure include the means of the exogenous variables

and the intercepts of the endogenous variables. Means of endogenous variables are not

considered model parameters, but predicted means on these variables, calculated as

total effects of the constant, can be compared with the observed means. In order to be

identiﬁed, the number of parameters in a mean structure cannot exceed the number of

observed means. A latent growth model for longitudinal data is basically a structural

regression model with a mean structure. Each repeated measures variable is speciﬁed

as an indicator of at least two different factors—one representing the initial status and

the other, the rate of change. These factors are usually assumed to covary, which allows

326 ADVANCED TECHNIQuES, AVOIDING MISTAKES

for the possibility that rate of change is related to initial status. When estimating group

mean differences on factors, one way to identify the mean structure is to select one

sample as the reference group, constrain the factors’ means or intercepts to zero in this

group, but freely estimate these parameters in all other groups. The results indicate the

relative difference between each of these groups and the reference sample on the factor

mean or intercept. However, the hypothesis of at least partial measurement invariance

of the factor loadings and indicator intercepts should be tenable in order to reasonably

interpret group mean differences on factors. An alternative way to estimate group differ-

ences on latent variables is to analyze a MIMIC model with data from the total sample.

The factors in the MIMIC model are regressed on at least one cause indicator that rep-

resents group membership. An advantage of this approach is that there are no special

identiﬁcation requirements beyond those for a single-sample analysis.

reCoMMended readIngs

The Duncan et al. (1999) book on latent growth curve modeling is a seminal work, and the

more recent book by Bollen and Curran (2006) in this area covers newer topics and analysis

techniques. Green and Thompson (2006) contrast standard techniques for analyzing multiple

means, including MANOVA and discriminant analysis, with structured means modeling in SEM

including the analysis of MIMIC models.

Bollen, K. A., & Curran, P. J. (2006). Latent curve models: A structural equation perspective.

Hoboken, NJ: Wiley.

Duncan, T. E., Duncan, S. C., Strycker, L. A., Li, F., & Alpert, A. (1999). An introduction to

latent variable growth curve modeling: Concepts, issues, and applications. Mahwah, NJ:

Erlbaum.

Green, S. B., & Thompson, M. S. (2006). Structural equation modeling for conducting tests of

differences in multiple means. Psychosomatic Medicine, 68, 706–717.

327

Interaction Effects and Multilevel SEM

This chapter introduces two advanced techniques in SEM: (1) the estimation of interac-

tion effects of observed or latent variables and (2) multilevel SEM. Whole books have

been written about these topics, so there is no way to cover both of them in great

detail in a single chapter. Instead, I want to make you aware that these possibilities

exist in SEM and cite enough advanced works in each area so that you can learn

more through additional study. Each topic represents the expansion of SEM to other

horizons, some of which may help you to evaluate an even wider range of hypotheses

in the future. As the French chemist and microbiologist Louis Pasteur once said: Chance

only favors invention for minds that are prepared for discoveries by patient study and

persevering efforts. Keep this thought in mind as you read this chapter.

InteraCtIon eFFeCts oF oBserved varIaBles

Estimation of the interaction effects of continuous observed variables in SEM uses

the same method as in moderated multiple regression (MMR). This method involves

creating product terms that represent interaction effects. A product term is literally

the product of the scores from two different variables, such as XW = X × W. The same

method is used to estimate curvilinear relations (trends) except that product terms in

this case are created by exponentiation where the scores (base numbers) are raised to a

power, such as X

= X × X, which represents a quadratic trend. These terms are known

as power terms or polynomials. Estimation of the curvilinear effects of continuous

observed variables in multiple regression is not described here, but see Cohen, Cohen,

West, and Aiken (2003) for more information. There is also a supplemental reading

about the estimation of curvilinear effects in SEM that you can download from this

book’s website (see p. 3).

Consider the data in Table 12.1 where M

= 16.38. The multiple correlation of pre-

dictors X and W with the criterion Y is .183. This result summarizes the linear effects

328 ADVANCED TECHNIQuES, AVOIDING MISTAKES

of the predictors, but inspection of the scores in Table 12.1 indicates a more complex

pattern: the relation of X to Y is linear and positive for cases with lower scores on W, but

it is linear and negative for cases with higher scores on W. This conditional relation of

X to Y is illustrated in the scatterplot of Figure 12.1 where cases with scores < M

are

represented with closed circles and cases with scores > M

are represented with open

circles. Although it is not as apparent, there is a similar change in the direction of the

relation of W to Y: positive at higher levels of X, negative at lower levels. So W moderates

the relation of X to Y, just as X moderates the relation of W to Y. This describes interac-

tion, which is always symmetrical.

The product term XW in Table 12.1 represents the interaction effect just described

when entered in the regression equation along with X and W as predictors.

The multiple

correlation from this analysis is .910, much greater than the corresponding value with

just X and W in the equation (.183). The unstandardized regression equation is

1.768 .734 .108 3.118Y X W XW=+−−

(12.1)

The coefﬁcient for the product term in Equation 12.1, or −.108, estimates the interac-

tion effect controlling for both main effects. A method to interpret this effect is brieﬂy

explained (Cohen et al., 2003). First, rearrange the unstandardized regression equation

so that there is no product term. For example, the expression presented next

(1.768 .108 ) .734 3.118Y WX W=−+−

(12.2)

is algebraically equivalent to Equation 12.1 but has no product term. The regression

coefﬁcient for X in Equation 12.2 (or 1.768 – .108W) can be seen as one that depends on

the level of W.

Next, substitute in Equation 12.2 meaningful values of W and inspect the effect

taBle 12.1. data set for an Interaction

effect of Continuous observed variables

Predictors Criterion

X W XW Y

2 10 20 5

6 12 72 9

8 13 104 11

11 10 110 11

4 24 96 11

7 19 133 10

8 18 144 7

11 25 275 5

A common mistake is to omit the variables X and W when XW is entered into the equation, but XW by itself

does not represent an interaction effect.

Interaction Effects and Multilevel SEM 329

on the coefﬁcients for X. For instance, given M

= 16.38 and SD

= 6.02 for the data in

Table 12.1, scores on W that fall –2, –1, 0, +1, and +2 standard deviations away from the

mean are, respectively,

4.34, 10.36, 16.38, 22.40, and 28.42

Suppose that W = 22.40 (i.e., M

+ SD

). Plugging this value for W in Equation 12.2

generates the unstandardized regression equation presented next:

22.40

= –.651 X + 13.324

Presented in Table 12.2 are all simple regressions for predicting Y from X at each of

the ﬁve levels of W just deﬁned. The slopes of these conditional equations, or simple

slopes, progressively change from a negative slope of –1.301 for cases with scores on

W two standard deviations above the mean to a positive slope of 1.299 for cases with

scores on W two standard deviations below the mean. For cases with average scores

on W, the simple slope is practically zero (–.001), so X and Y are unrelated at this level

of W.

When the simple slope of a conditional regression line is divided by its standard

error (Preacher, Rucker, & Hayes, 2007, p. 192), the resulting ratio is in large samples

a z test of statistical signiﬁcance. A related concept is that of regions of signiﬁcance,

or a range of values on W for which the conditional regression of Y on X is statistically

signiﬁcant. Preacher et al. (2007) refer to conﬁdence intervals based on simple slopes

as conﬁdence bands, and these bands can be plotted in order to interpret interaction.

FIgure 12.1. Scatterplot for the data set in Table 12.1 for variables X and Y. Closed dots

indicate scores on W below the mean, and open dots indicate scores on W above the mean.

330 ADVANCED TECHNIQuES, AVOIDING MISTAKES

Speciﬁcally, a third variable W moderates the relation between X and Y where the conﬁ-

dence bands do not include slopes of zero. Preacher, Curran, and Bauer (2006) describe

computer tools for analyzing simple slopes, including some utilities freely available over

the Internet.

There are also freely available scripts for SAS/STAT for visualizing interac-

tions with three-dimensional spin plots.

Standardized regression coefﬁcients (beta weights) do not have the normal inter-

pretation for power terms that represent interaction effects. This is because, in most

cases, the product of z scores from two different variables, such as z

× z

, does not

equal the z score of the product of the corresponding unstandardized scores, or z

. As

noted by Whisman and McClelland (2005), multiple regression procedures in statistical

computer programs typically report beta weights for the data in Table 12.1 equivalent

to the model

= b

+ b

(12.3)

but the weight b

does not correctly estimate the standardized interaction effect. The

correct standardized model would include the term “b

,” but this term does not

appear in Equation 12.3. Whisman and McClelland (2005) offer this advice: It is best to

avoid standardized regression coefﬁcients in MMR, and instead one should focus on the

unstandardized regression coefﬁcients. Aiken and West (1991) describe how to obtain

correct beta weights in MMR.

The method for unstandardized variables just described can be extended to esti-

mate higher-order interactions. For example, the product term XW represents the lin-

ear × linear interaction of X and W in predicting Y. Such an interaction means that the

linear relation of X to Y changes uniformly over the levels of W (e.g., Table 12.2). The

product term XW

represents a linear × quadratic interaction, which means that the

linear relation of X to Y changes faster at higher (or lower) levels of W. Estimation of this

taBle 12.2 regression equations for Predicting Y from X

Conditional on the level of W for the data set in table 12.1

Level Score Regression equation

+2 SD 28.42

1.301 17.712YX=− +

+1 SD 22.40

.651 13.324YX=− +

Mean 16.38

.001 8.905YX=− +

−1 SD 10.36

.649 4.486YX=+

−2 SD 4.34

1.299 .068YX=+

www.people.ku.edu/~preacher/interact/index.html

www.ats.ucla.edu/stat/sas/faq/spplot/reg_int_cont.htm

Interaction Effects and Multilevel SEM 331

effect would require that the terms X, W, and W

(i.e., the quadratic trend) are in the

model along with XW

. It is also possible to estimate three-way or higher interactions.

For example, the product term XWU represents the three-way linear interaction among

these variables when all lower-order effects are also included in the equation. These

include terms for the linear effects (X, W, U) and all linear × linear interactions (XW,

XU, WU). A three-way linear interaction means that the linear × linear interaction of X

and W changes uniformly across the levels of U. Because interaction is symmetrical, the

same interpretation applies to each of the other two linear × linear interactions regard-

ing the corresponding third variable. Estimation of higher-order interactive (and cur-

vilinear) effects requires the analysis of numerous product terms, so very large samples

may be needed for adequate statistical power. See Dawson and Richter (2006) for more

information about the estimation of three-way interactions in MMR.

A problem that can occur when analyzing product terms is extreme collinearity.

This is because correlations between product terms and their constituent variables can

be so high that the analysis can fail or the results are unstable. One way to address

this problem is to mean-center the original variables before calculating product terms

based on them. Mean centering occurs when the average of a variable is adjusted to

zero (the mean is subtracted from every score), and centering tends to reduce—but

not typically to eliminate—correlations between product terms and constituent vari-

ables. An alternative is to create a residualized product term using the technique of

residual centering that is calculated controlling for the main effects and consequently

is uncorrelated with them (Lance, 1988; Little, Bovaird, & Widaman, 2006). A residu-

alized product term is created in two steps by ﬁrst regressing the product term on all

constituent main effect terms (e.g., XW scores are regressed on both X and W). The

residuals from the regression analysis just described are uncorrelated with the main

effects but still convey information about the interaction effect. In a second regression

analysis, the criterion is regressed on X, W, and the residualized XW term created in

the ﬁrst analysis.

Another complication is measurement error. Score reliabilities of product terms can

be lower than those of scores on the component variables. This in turn reduces both the

absolute coefﬁcient for the product term and the power of corresponding statistical tests

(Jaccard & Wan, 1995). Measurement error in the outcome variable that varies across

the levels of a predictor can bias the regression coefﬁcient for product terms that involve

that predictor (Baron & Kenny, 1986). One way to address these problems is to use pre-

dictor variables with excellent score reliabilities. Another is to estimate the interaction

effects of latent variables in the multiple-indicator (i.e., SEM) approach described later.

This method controls for score unreliability through the speciﬁcation of a measurement

model, just as in the technique of CFA or in the analysis of an SR model.

InteraCtIon eFFeCts In Path Models

The interactive effects of observed variables are represented in path models with the

appropriate product terms and all constituent variables. Consider the moderated

332 ADVANCED TECHNIQuES, AVOIDING MISTAKES

path analysis (MPA) model in Figure 12.2 where X, W, and the interaction XW are all

speciﬁed as direct causes of Y. These three exogenous variables are speciﬁed to covary,

but creation of a residualized product term would eliminate the paths X

XW and

XW in this model. The unstandardized coefﬁcients for the paths

X → Y, W → Y, and XW → Y

estimate, respectively, the linear effect of X, the linear effect of W, and the linear × lin-

ear interactive effect, each controlling for all other effects on Y. When means are not

analyzed in MPA, there are no intercepts. However, it is still possible to analyze sim-

ple slopes, regions of signiﬁcance, and conﬁdence bands for interaction effects in MPA

(Preacher et al., 2007). Keep the following points in mind:

1. Kenny (2009) reminds us that just as a mediational model is a causal model

(Chapter 7), so too is a model of moderation. This means that if the basic directionality

assumptions are incorrect, then the results may be of little value. For example, an inter-

action effect XW can be reversed if the direct effect between X and Y is reversed (i.e., Y

causes X instead of the opposite).

2. Kenny (2009) gives this example of how curvilinear effects can be confounded

with interactive effects. Suppose that X is income and Y is work motivation. The rela-

tion between these two variables is curvilinear such that their covaration is stronger at

lower levels of X. If variable W is age, then because younger workers earn less money,

the “interaction” between age and income effect could be found, such that the rela-

tion between income and motivation is stronger for younger workers. In order to avoid

confusing curvilinear and interactive effects, Edwards (2009) recommends the routine

inclusion of the power terms X

and W

whenever estimating coefﬁcients for the prod-

FIgure 12.2. Path analytic representation of an interactive effect of continuous observed

variables.

Interaction Effects and Multilevel SEM 333

uct term XW. That is, curvilinear testing cannot be disregarded when testing for interac-

tion.

3. Edwards (2009) reminds us that although product terms such as XW are repre-

sented as causal variables in MPA models (e.g., Figure 12.2), they actually have no causal

potency by themselves. This is because a product term does not represent a unique

entity apart from its component variables. Instead, it is a mathematical construction

that represents conditional or joint effects on the outcome variable (controlling for main

effects) that can be examined through inspection of simple slopes, regions of signiﬁ-

cance, and conﬁdence bands.

MedIatIon and ModeratIon together

It is also possible to represent both mediation (indirect effects) and moderation (interac-

tive effects) in the same structural model. Baron and Kenny (1986) described mediated

moderation, which involves the speciﬁcation that an interactive effect is mediated by

at least one other variable. Consider the MPA model in Figure 12.3, which represents

the hypothesis that the interaction of X and W is entirely mediated by the prior variable

M. In the analysis of a model basically identical to that of Figure 12.3, Lance (1988)

studied the relation of memory demand (X), complexity of social perception (W), and

their interaction effect (XW) on the overall accuracy of recall of the script of a lecture

(Y). The model also included a mediator, recollection of speciﬁc behaviors mentioned

in the script (M), through which the interaction effect was speciﬁed to inﬂuence overall

accuracy of recall. Lance’s (1988) results suggested that (1) the indirect effect of the

memory demand × cognitive complexity interaction was statistically signiﬁcant, but (2)

the direct effects of the individual component variables on overall accuracy were not

FIgure 12.3. Path model with mediated moderation.