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A general approach torepresenting multifaceted

personality constructs:

Application to state

self ‐esteemRichard P. Bagozzi

a & Todd F. Heatherton

b

a School of Business Administration , University of Michigan , Ann Arbor, MI, 48109–1234b Harvard University ,

Published online: 03 Nov 2009.

To cite this article: Richard P. Bagozzi & Todd F. Heatherton (1994) A general

approach to representing multifaceted personality constructs: Application to state

self ‐

esteem, Structural Equation Modeling: A Multidisciplinary Journal, 1:1, 35-67,DOI: 10.1080/10705519409539961

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STRUCTURAL EQUATION MODELING, 1(1), 35-67

Copyright © 1994, Lawrence Erlbaum Asso ciate s, Inc.

A General Approach

to

Representing

Multifaceted Personality Constructs:

Application to State Self-Esteem

Richard

P.

Bagozzi

University of

Michigan

Todd F. Heatherton

Harvard University

This article proposes

a

framework

for

representing personality constructs

at

four levels of abstraction. The total aggregation model is the composite formed

by the sum of scores on all items in a scale. The partial aggregation model

treats separate dimensions

of a

personality construct

as

indicators

of a

single

latent variable, with each dimension being an aggregation of items. The partial

disaggregation m odel represents each dimension as a separate latent variable,

either freely correlated with

the

other dimensions

or

loading

on one or

more

than one higher order factor; the measures of the dimensions are multiple

indicators formed as aggregates of subsets of items. The total disaggregation

model also represents each dimension

as a

separate latent variable

but,

unlike

the partial disaggregation model, uses each item in the scale as an indicator of

its respective factor. Illustrations

of the

models

are

provided

on the

State

Self-Esteem Scale—including tests

of

psychometric properties, invariance,

and

generalizability.

It is often assumed in personality research that a single scale ought to

measure a single construct

(Briggs & Cheek, 1986, p. 109). Indeed, such a

point of view is at the heart of what is meant by

construct validity

(e.g.,

Campbell & Fiske, 1959; Cook & Campbell, 1979). Measures of a construct

are valid to the extent that they measure what they are supposed to measure.

As simple and reasonable as these assertions appear to be, it has proved

difficult to reach a consensus on the criteria for representing personality

Requests for reprints should be sent to Richard P. Bagozzi, School of Business Administra-

tion,

University of M ichigan, Ann Arbor, MI 48109-1234.

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3 6 BAGOZZI AN D HEATHERTON

constructs. Many personality constructs were originally conceived as uni-

dimensional but over time have come to reveal multifaceted components.

For example, self-monitoring was initially conceived along a continuum

from low to high (e.g ., Snyder, 1974,1979) but later showed multiple factors

underlying Snyder's 25-item scale (e.g., Briggs & Cheek, 1988; Briggs,

Cheek, & Buss, 1980; Gabrenya & Arkin, 1980; Hoyle & Lennox, 1991;

Lennox, 1988; Lennox & Wolfe, 1984; Miller & Thayer, 1989; cf. Snyder &

Gangestad, 1986). Similarly, self-esteem was conceived early on as a uni-

dimensional construct measured with 10 items (e.g., Rosenberg, 1965,

1979).

But recently, self-esteem or, more broadly, the self-concept has been

regarded as a multifaceted construct (e.g., Fleming & Courtney, 1984;

Marsh, Byrne, & Shavelson, 1988; Marsh & Shavelson, 1985; Shavelson &

Bolus, 1982; Shavelson & Marsh, 1986; for an early statement, see Gecas,

1971). One version of the Self Description Questionnaire, for example, has

seven dimensions and uses 66 items (e.g., Marsh, 1988).

Considerable uncertainty and disagreement exist about how to repre-

sent and measure many personality constructs. Part of the problem lies

with differences in interpretation between single-faceted and multifac-

eted conceptualizations. In a related vein, researchers often address a

personality construct from the perspective of only one of several possible

levels of abstraction but fail to point out the implications of doing so. For

instance, concepts can be represented on a single dimension on which

multiple dimensions have been collapsed or in some way aggregated;

likewise, items designed to measure a personality concept can be aggre-

gated or disaggregated, depending on one's purposes. As a consequence,

at least four possibilities exist when aggregation or disaggregation is

crossed with items and dimensions.

Carver (1989) offered two broad arguments in favor of multifaceted

constructs in personality research. First, Carver maintained that "individual

facets of a construct should sometimes predict dependent variables better

than should the broader construct (i.e., the overall index)" (p . 579). Indeed,

depending on the context, it is possible for some dimensions to positively

predict a criterion, others to be unrelated, and still others to be negatively

related. Lumping together items across dimensions in an index could mis-

leadingly yield a weighted average and obscure the differential contributions

of the dimensions. In support of this conclusion, Briggs and Cheek (1986)

presented findings showing that the total score and scores for the individual

dimensions of self-monitoring correlate at different levels and in different

patterns across various criteria. The second point made by Carver (1989) in

support of multifaceted constructs is based on the observation that the whole

may be greater than the sum of its parts. By this, Carver meant that "some

multifaceted constructs seem to be based on the assumption that the several

components interact with each other to produce the outcome effect of inter-

est" (p. 582). Obviously, summing items across subdimensions of a scale

would fail to account for such a possibility.

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MULTIFACETED

PERSONALITY CONSTRUCTS 3 7

Building on Carver's (1989) insights, Hull, Lehn, and Tedlie (1991)

proposed

an

approach

for

testing multifaceted personality constructs based

on structural equation modeling with latent variables (see also Marsh &

Hocevar, 1985). Hull et al. proposed two general models. One, a measure-

ment model, consists

of

testing

the

hypothesis that

an

underlying latent

variable accounts for variation among measures of subcomponents of a

personality scale. Hull

et al.

illustrated this model

on

measures

of

hardiness

and found that a single latent variable accounted for significant amounts of

variation in measures of five subdimensions: alienation from work, alien-

ation from self, security, powerlessness, and external locus of control. Hull

et al.'s second contribution was to model the effects of latent personality

constructs

on

criteria treated

as a

dependent variable.

The

advantages

of

this

approach over regression analysis

are

that measurement error

in the

predic-

tors

is

taken into account explicitly,

and the

effects

of the

latent personality

construct can be partitioned into a general effect shared by the sub-

dimensions and any additional specific effect due to one or more than one

subdimension. Hull

et al.

applied their predictive model

to two

contexts—

the effect of hardiness on depression and the effect of self-punitive attitudes

(i.e., self-criticism, high standards, overgeneralization)

on

depression.

One of our goals in the present article is to introduce a comprehensive

framework

for

representing personality constructs that brings issues

of con-

struct validity and levels of analysis into clearer focus. The framework

applies to personality concepts in which measures are related in additive,

linear ways

to one or

more than

one

dimension.

In the

Discussion section,

we

cover important classes of personality concepts in which the framework does

not apply. A second purpose is to illustrate models implied by the framework

on measures of the State Self-Esteem Scale (SSES), a scale recently intro-

duced by Heatherton and Polivy (1991). Heatherton and Polivy used classic

procedures (e.g., exploratory factor analysis; correlations)

to

analyze

the

psychometric properties

of

measures

and

placed emphasis

on the

total scale

as well as three dimensions: performance, social, and appearance. Based on

prior research, SSES items were chosen to measure these dimensions, and

the results of an exploratory factor analysis supported the scale and its

subdimensions (Heatherton

&

Polivy, 1991).

Our aim is to

develop

a

more

refined representation of the SSES based on alternative levels of abstraction

organized hierarchically and to test the alternatives. We discuss the relation

of our framework to previous approaches on which it builds.

GENER L FR MEWORK FOR REPRESENTING

PERSON LITY

CONSTRUCTS

Figure

1

illustrates

the

framework

and

uses

the

SSES

to

make

the

ideas more

concrete. The

total aggregation model

is the most abstract representation of

a scale.

The

personality construct

is

represented

as a

single composite made

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a. Total aggregation

Aggregation across dimensions and items:

Aggregation acro ss dimensions:

b.

artial

aggregation

Hierarchical

organization of components:

c.

artial disaggregation

First-order

model:

£p

r

m

L

Pn

I s

o

L S

P

z

a

r

d. Total disaggregation

First-order model:

Discrete components:

.

SSES

Second-order model:

SSES

Zp

m

2

Pn

L S

o

I S

P

£

a

r

I IXXXX XXXXXl

Second-order model:

FIGURE

General framework for representing personality constructs—application

to

the State Self-Esteem Scale (SSES).


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MULTIFACETED PERSONALITY CONSTRUCTS 3 9

up of the sum of items hypothesized to measure it. Following the convention

established in the literature, theoretical or latent variables are drawn as

circles or ellipses in Figure 1, and measurements are depicted as boxes. For

the total aggregation model, the relation between the SSES as a latent

variable and its measurement as the sum of items from the scale is taken to

reflect a 1:1 correspondence. No degree of correspondence is modeled, and

no attempt is made to account for m easurement error. Although a measure of

reliability might be computed based on the associations among measure-

ments (e.g., Cronbach alpha) and then used to correct for attentuation in

predictions of a criterion, no formal representation is provided of the mea-

surement properties of the scale, such as is possible with the other cases

illustrated in Figure 1 and as discussed here . The total aggregation model is

consistent with current conceptualizations of self-esteem as a "hypothetical

construct that is quantified ... as the sum of evaluations across salient

attributes of on e's self or personality" (Blascovich & Tomaka, 1991, p . 115).

Note that the total aggregation model constitutes an aggregation of both

dimensions and items. To examine the case in which dimensions are aggre-

gated and items disaggregated, a single-factor model can be investigated in

which all items load on the factor.

The main advantages of the total aggregation model are its simplicity and

ability to capture the essence of the underlying meaning of a personality

concept. However, the advantages accrue only to the extent that the mea-

sures share sufficient common variance. To the degree that measures share

common variance, the summation of items tends to smooth out random error

and permit the total aggregation model to function as a useful representation.

The primary disadvantage of the total aggregation model is that it fails to

represent the unique properties of subdimensions, if any, and obscures both

the differential dependence and the effects of subdimensions on other con-

structs of theoretical interest. The total aggregation model has its place in

molar assessments of behavior, but, for more fine-grained analyses, less

global representations are needed. The remaining models to be introduced

explore the representation of the aggregation and disaggregation of items

under the condition of when dimensions are hypothesized (i.e., when dimen-

sions are disaggregated).

The partial aggregation model shown in Figure 1 constitutes a more

molecular representation of a latent personality construct, yet it retains

the idea of a single underlying factor. Two cases are of interest. In the

first case, the dimensions of the construct are organized hierarchically as

indicators of an underlying factor. Three dimensions are shown, corre-

sponding to those proposed by Heathcrton and Polivy (1991) for the

SSES,

but it should be pointed out that other personality constructs with

more than three dimensions can be accommodated by such an approach,

depending on the theory at hand. For the Self-Description Questionnaire

(SDQ), for example, one might have as many as 12 dimensions (e.g.,

Marsh, Barnes, & Hocevar, 1985).



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To test the hierarchical representation of the partial aggregation model as

a null hypothesis, a confirmatory factor analysis (CFA) can be performed in

which composites of items for each dimension are treated as indicators of a

single factor. A failure to reject this model based on standard statistical

criteria suggests that each of the multiple indicators measures a single

underlying construct. The construct can then function as a predictor or

predicted variable in a structural equation model (SEM) while taking into

account measurement error in the measures of the construct. Hull et al.

(1991) advocated this approach in their study of hardiness, and Bagozzi and

Heatherton (1993) utilized this form of the partial aggregation model to

examine convergent and discriminant validity of measures of the SSES and

the Multiple Affect Adjective Check List.

If the partial aggregation model is rejected on the basis of standard

statistical criteria, then one of two conclusions can be drawn. Either two or

more factors underly the latent personality concept, or the measures of the

hypothesized dimensions of the latent personality construct are highly falli-

ble. The former possibility can be addressed by use of the partial disaggre-

gation model, which is discussed shortly. The latter possibility, of course,

requires that better measures be obtained.

A second kind of partial aggregation is possible and is labeled the discrete

components case in Figure 1. Here the dimensions of the personality con-

struct are not formally modeled as indicators of it but rather are treated as

separate subscales and are considered only loosely tied to the overall con-

struct. Two approaches might be taken with discrete components in the sense

shown in Figure lb. In the first, items particular to each component are

summed to form separate composites. This might be based on conceptual

criteria of shared meaning of items within components and of distinct mean-

ing of items across components, as well as empirical criteria derived ex post

facto from a factor analysis. Researchers claiming that the Self-Monitoring

Scale (SMS) is made up of acting, extraversion, and other-directed sub-

dimensions have basically taken this approach (e.g., Briggs & Cheek, 1986).

Likewise, Carver's (1989) treatment of multifaceted personality constructs

seems to rely on this interpretation, and Heatherton and Polivy's (1991) test

of the SSES rests on a discrete-components understanding. This discrete-

components approach is analogous to the total aggregation approach in that

each component is treated as a separate aggregate made up of the sum of

scores to items reflecting that component.

The other way to address discrete components is to model the relation

between each component and its measures with a separate CFA model. Each

latent variable is treated as a composite variable and the measures as indica-

tors.

The outcomes and conclusions already mentioned for the CFA of the

hierarchical organization of components also apply to each component so

treated in the discrete-components case.

The principal advantages of the hierarchical organization of components

in the partial aggregation model are that separate parameter estimates are



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derived representing the degree of correspondence between the latent per-

sonality construct and its subdimensions (i.e., the Xs in Figure lb ) and at the

same time estimates of m easurement error are provided (i.e., the 6s in Figure

lb) . This permits an assessment of the reliability of measures and the oppor-

tunity to correct for unreliability in predictions, as already noted. The main

disadvantage of this model is that the unique dimensions of the overall

personality construct, if any, are obscured.

A more fine-grained representation of multifaceted personality constructs

can be performed by use of the partial disaggrega tion model (see Figure 1 ).

Here each component or dimension is represented as a separate latent vari-

able indicated by composites of subscales. The first-order partial disaggrega-

tion model estimates the degree of correspondence between each component

and its respective measures, as well as the respective error variances. More-

over, separate estimates are provided for the correlations among dimensions,

which can be used to assess the degree of discrimination between dimen-

sions.

These correlations are corrected for attentuation as a consequence of

standard estimation procedures. We have shown two measures per dimension

in the partial disaggregation model—each formed as composites of items

from the subscales—but it should be emphasized that more measures can be

specified, depending on the number of items available per dimension. Guide-

lines for forming composites are discussed in the Method section. The

first-order partial disaggregation model assumes that the dimensions are

distinct—that is, the measures of the separate dimensions are presumed to

achieve discriminant validity between dimensions. The degree of discrimi-

nant validity is inversely proportional to the magnitude of the correlations

among the first-order factors (shown as (j>s in Figure lc ) . Strong discrim inant

validity will be achieved when the 4>s are nonsignificant or small; weak

discriminant validity will be achieved when the

s

are high but less than 1.00

by an amount greater than twice the standard error (SE) of the estim ate of .

Lack of discrim ination in a strict statistical sense occurs when the 4>s are

within 2 SE of 1.00.

The second-order partial disaggregation model treats the components as

first-order factors and introduces a second-order factor explaining variation

in the first-order factors. The second-order factor can be thought of as an

abstract representation of the overall personality construct. The second-

order model partitions variation in measures into three components—ran-

dom error, measure specificity, and common variance. The first-order model

only partitions variation into that attributable to error and the components;

random error is confounded with measure specificity (Marsh & Hocevar,

1988).

One interpretation of the second-order partial disaggregation model is the

following. To the extent that the first-order factors include common vari-

ance, the second-order factor captures the shared variance across factors. A

single second-order factor suggests that the dimensions measure the same

hierarchical concept, except for random error and measure specificity. It



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should be noted that more than one higher order factor might be appropriate,

depending on the theoretical context and personality concept.

The principal advantage of the first-order partial disaggregation model is

that it is possible to specify and test for the existence of multiple dimensions

of a personality construct. Useful diagnostics and information are provided

on properties of the dimensions and their measures. One of the first studies

to use a first-order partial disaggregation model to test a theory of personal-

ity was Marsh and Hocevar's (1985) investigation of the SDQ. Hull et al.

(1991) examined a somewhat similar model for self-punitive attitudes as part

of a predictive model of depression. Because structural equation methods are

based on full-information procedures, a measurement model embedded

within a predictive model might differ from a measurement model treated in

isolation, such as the partial disaggregation model.

The main advantage of the second-order, partial disaggregation model is

that hypotheses can be tested about the hierarchical structure of a personality

construct (e.g., Marsh & Hocevar, 1985). In addition, both versions of the

partial disaggregation model permit interesting inquires into discriminant

validity among dimensions of a personality construct. The first-order model

provides estimates of the degree of association among the dimensions with

the measures corrected for attentuation. Discriminant validity occurs to the

extent that the factors are distinct. However, it should be remembered that

random error and measure specificity are confounded. The second-order

model estimates measure specificity for the indicators of each dimension.

Yet a trade-off can be seen through the two models. Discriminant validity in

a strong sense will be achieved when dimensions are uncorrelated or w eakly

correlated . Here a higher order model is not meaningful because the dimen-

sions do not share common variance. Discriminant validity will be achieved

in a weak sense when they are highly correlated, but significantly less than

1.00. But to the extent that a single second-order factor represents the data,

discriminant validity among dimensions may be less defensible, and the

dimensions should be interpreted as subcomponents of a higher order organ-

izing concept. Of course, depending on the theory, it is possible to have

multiple higher order factors and achieve discriminant validity across these.

Conclusions about discriminant validity may be ambiguous when drawing

the line between "high" and "moderate" correlations.

The final model we consider is the total

disaggregation

model (see Figure

1).

In this model, each dimension of a multifaceted construct is modeled as

a distinct latent variable, as with the partial disaggregation model, but,

unlike the partial disaggregation model, which uses composites based on

subscales as indicators of the latent factors, each individual item from the

personality scale is used to operationalize its respective hypothesized dimen-

sion. This yields what might be termed an "atomistic" level of analysis to

contrast it with the "molar" total aggregation model and the more "molecu-

lar" partial aggregation and partial disaggregation models. In principal, the

total disaggregation model provides the most detailed level of analysis of a



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personality construct because psychometric properties are provided for each

individual item. In practice, however, the total disaggregation model can be

unwieldy because of likely high levels of random error in typical items and

the many parameters that must be estimated. As the number of items per

factor increases and sample sizes increase, it is likely that many total disag-

gregation models may fail to fit the data satisfactorily. As a consequence, the

total disaggregation model is likely to be applied successfully, if at all, only

when about four or five measures per factor or fewer are used. Nevertheless,

even with more than five measures per factor, the total disaggregation model

can

be

useful

in

scale development, item analysis,

and

modeling

of

method

effects.

Hoyle and Lennox (1991) provided an illustration of the total disaggrega-

tion model applied

to the SMS.

None

of

their models

fit

very well, however,

on the basis of contemporary goodness-of-fit tests. Under some conditions,

correlated uniquenesses may be used to investigate method biases and other

sources of model misspecification, but this should be done cautiously and

should be guided by theoretical considerations (e.g., Marsh, 1989). It is also

possible that method factors can be introduced corresponding to item rever-

sals or other systematic format or wording patterns.

THE PRESENT STUDY

As an illustration of the general framework for representing personality

constructs,

the

hierarchical organization

of

responses

to the

SSES

was

exam-

ined in two samples. The first objective was to investigate the reasonable-

ness of the part ia l aggregat ion, part ia l disaggregat ion, and total

disaggregation models for the data at hand. We focused on goodness-of-fit

tests,

reliabilities

of

measures, discrimination among components,

and

parti-

tioning of variance into useful components. Second, an indication of the

generalizability of the partial aggregation and partial disaggregation models

for the SSES was explored. This was done by examining the replicability of

the models and key parameters across the samples. Last, a comparison was

made by gender to see if the structure of the SSES is equally applicable to

women and men.

METHOD

Subjects

Subjects in Sample 1 were 102 undergraduate volunteers from the St. George

campus of the University of Toronto—72 women and 30 men ranging in age

from 18 to 43 years (M = 22.0 years, SD = 5.2 years). Subjects in Sample 2

were 428 undergraduate volunteers from Erindale College of the University

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4 4 BAGOZZI AND HEATHERTON

of Toronto—284 women and 144 men ranging in age from 17 to 57 years

(M

= 20.3 years, SD - 4.3 years).

SSES

The SSES is a 20-item scale developed by Heatherton and Polivy (1991) and

based on items from the Fleming and Courtney (1984) and Pliner, Chaiken,

and Flett (1990) modifications of the Janis-Field Feelings of Inadequacy

Scale (Janis & Field, 1959). Items were selected to reflect acute as opposed

to typical levels of self-esteem and to include performance, social, and

appearance subdimensions, which a subsequent factor analysis supported

(Heatherton & Polivy, 1991). To further emphasize current feelings of self-

esteem, the following instructions were used:

This is a questionnaire designed to measure what you are thinking at

this moment. There is, of course, no right answer for any statement.

The best answer is what you feel is true of yourself at this moment. Be

sure to answer all of the items, even if you are not certain of the best

answer. Again, answer these questions as they are true for you RIGHT

NOW.

Responses are recorded on a 5-point scale with points labeled not at all (1),

a little bit

(2),

somewhat

(3),

very much

(4), and

extremely

(5). The 20-item

SSES can be found in the Appendix.

Procedure

Subjects in Sample 1 completed the SSES and several other scales not

relevant to the current study. Subjects responded to the items while seated in

a quiet room and were tested individually. Subjects in Sample 2 completed

the SSES and other scales not pertinent to the present study and did so in a

single mass-testing session.

Statistical

Criteria

In this article, the models examined can be fit and the hypotheses can be

tested using structural equations methods (Bentler, 1989; Bollen, 1989;

Joreskog & Sorbom, 1989). These procedures permit the representation of

latent variables corresponding to the overall SSES, as well as its dimensions

or components, and facilitate the investigation of the hierarchical arrange-

ments presented in Figure 1.


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Assessment of overall model fit. The degree of correspondence be-

tween any particular model and the data can be assessed with the use of

several measures. The chi-square goodness-of-fit test indicates the discrep-

ancy between a hypothesized model and data. Significant values of the

chi-square goodness-of-fit test indicate that the data and model deviate in a

fundamental way and that the model should be rejected. A nonsignificant

chi-square goodness-of-fit test with

p

a .05, say (e.g., Bentler, 1989), sug-

gests that a model is a reasonable representation of the data.

Because the chi-square test is sensitive to sample size and can lead to

a rejection of a model differing in a trivial way from the data for large

sample sizes—and conversely can result in the acceptance of a model

with important differences from the data for small sample sizes—it is

prudent also to examine other measures of fit. McDonald and Marsh

(1990) proposed the relative noncentrality index (RNI) in this regard (see

also the comparative fit index proposed by Bentler, 1990, which is similar

to the RNI). The RNI is a refinement of the normed fit index (Bentler &

Bo nett, 1980), which measures the amount of variance accounted for by a

model in a practical sense. Like the normed fit index, the RNI takes on

values from 0 to 1, inclusive, but is not a statistic because its sampling

distribution is unknown. Unlike the normed fit index, the RNI corrects for

small sample biases and has been shown to perform better than many

other indices (e.g., Bentler, 1990; McDonald & Marsh, 1990). Following

the rule-of-thumb suggested by Bentler and Bonett (1980), RNI values

grea ter than or equal to about .90 are taken to indicate a satisfactory fit

from a practical standpoint.

Tests of hypotheses. Chi-square difference tests are used to test

hypotheses. One set of hypotheses concerns the equality of factor loadings

and of error variances for measures of a single factor, such as shown in

Figure lb for the partial aggregation model. In general, with two measures

of a factor, the model will be underidentified (the variance-covariance

matrix for measures is not sufficient to provide enough information to

estimate parameters); with three measures, the model is exactly identified;

with four or more measures, it is overidentified. An exactly identified

model provides enough information to estimate all parameters, but, be-

cause df = 0 for this model, the chi-square test indicates a perfect fit. To

test how well the single-factor model with three measures fits the data

versus the most general alternative—that the variance-covariance matrix

is any positive definite matrix—we can constrain all three factor loadings

(or selected subsets of two) to be equal. This yields a chi square with 2 df

(1 d/will result for the case in which a pair of loadings is constrained to

be equal). The chi square in this case gives a test of whether the three

components of the SSES each load on a single factor and equally reflect

that factor.

D o w n l o a d e d b y [

C e n t r a l M i c h i g a n U n i v e

r s i t y ] a t 0 7 : 2 7 0 5 J a n u a r y 2 0 1 5


14/35


Further, one might want to test whether the error variances are equal for

the single-factor model in Figure l b . This can be accomplished by constrain-

ing both the factor loadings and the error variances to be equal and compar-

ing the chi square for this model to the one in which only the factor loadings

are constrained to be equal. The difference in chi-square values with degrees

of freedom equal to the difference in degrees of freedom for the two mod-

els—in this case,

df =

4 - 2 = 2—gives a test of the reasonableness of

assuming equal error variances. Note further that this hypotheses and the

ones considered later must be performed on the variance-covariance matrix

(e.g., Cudeck, 1989). For multiple-factor models with two or more measures

per factor, such as presented in Figures lc and Id, the models will be

overidentified.

Another set of hypotheses to address is whether the SSES—as reflected in

any or all levels depicted in Figure 1—generalizes to other samples. We can

test whether the same factor structure and parameters for a particular model

apply to two or more samples. In other words, we are interested in the degree

of replicability of the SSES. One way to do this is to compare models with

identical structures—in which key parameters are fixed to be equal across

samples—to models with no equality constraints. This can be done in a

sequential manner. For example, the model in Figure lb might begin with a

test of whether the Xs are invariant across samples; next, one could test

whether both the Xs and 0ss are invariant. Likewise, for the models in Figure

lc , one might test the following: for the first-order, three-factor model,

whether the Xs are invariant, then the Xs and 6as, and finally the Xs, 6&s, and

s assesses whether the SSES compo-

nents covary equally. The additional test of invariance in vs and in ips

investigates whether the sources of common and specific variance, re-

spectively, are equal across samples. The aforementioned sequences of

hypotheses constitute increasingly demanding tests of the generalizabil-

ity of the SSES. These tests should be performed only after it is demon-

strated that the variance-covariance matrices differ across samples and

that the same factors underlie the data (e.g., Cole & Maxwell, 1985;

Joreskog & Sorbom, 1989).

The final tests of hypotheses concern the appropriateness of the SSES for

men and women. The models for the representation of the SSES shown in

Figure 1 can be compared for men and women, and sequences of hypotheses

can be tested for key param eters, as already described for the tests performed

on different samples (cf. Byrne & Shavelson, 1987).



r s i t y ] a t 0 7 : 2 7 0 5 J a n u a r y 2 0 1 5


15/35


Examination of parameters.

Further insights into the fits of models

and tests of hypotheses can be obtained by inspection of parameter esti-

mates.

Reliabilities can be computed for measures of the overall SSES and

for the individual components by use of the following formula:

in which

X\

refers to the ith factor loading on the factor corresponding to

either the overall SSES in the partial aggregation model (i.e., 2=) or the

individual components for the first-order partial disaggregation model (i.e.,

=p,

| s ,

£A),

and 6ei stands for the error variances of the measures of compo-

nents in either the partial aggregation model (i.e., Zpj, Zsk, Zai) or the partial

disaggregation model (i.e., 2p

m

, 2p

n

, 2s

0

,Zs

p

, 2a

q

, Za

r

). An indication of the

degree of discrimination among the components of the SSES can be garnered

by inspection of the correlations among factors (i.e., the s) in the first-

order, three-factor partial disaggregation model. Information on the parti-

tioning of variance in measures of the SSES components can be obtained by

looking at the parameter estimates for the second-order, three-factor partial

disaggregation model. Three types of variance are of interest: random error,

measure specificity (i.e., variance shared by measures of the same factor),

and common variance (i.e., variance shared by all measures).

Specification

of

Measures

for the Partial Disaggregation

Model

The specification of operationalizations for the total aggregation, partial

aggregation, and total disaggregation models shown in Figure 1 is straight-

forward. For the total aggregation model, the SSES is operationalized as the

sum of items in the total scale. For the partial aggregation model shown in

Figure 1, each dimension is operationalized as the sum of items hypothesized

to measure that dimension.

For the total disaggregation model, each item is treated as a measure of its

respective SSES dimension. In practice, particularly with large numbers of

items, the total disaggregation model may be impractical in the sense that the

pattern of responses to items will deviate significantly from the hypothesized

structure, and correlated residuals may be needed to achieve satisfactory fits

for the model. However, correlated residuals are frequently difficult to

interpret, too many distort the meaning of the hypothesized structure, and, in

any case, one should have a theoretical rationale to permit an interpretation.

In our experience, when more than about five items per factor are treated as

individual measures of factors in a multifactor CFA, it is difficult to achieve

a satisfactorily fitting model that is interpretable in an unambiguous sense.



y 2 0 1 5


16/35

4 8 BAGOZZI

AND

HEATHERTON

Nevertheless,

as

mentioned earlier,

the

total disaggregation model

has its

place

and can be

useful

in

item analyses

and

investigation

of

method effects

due

to

item reversals

or

other systematic biases.

Several alternatives

can be

identified

for the

operationalization

of a

partial disaggregation model. When

the

number

of

items

per

dimension

is

relatively small—say, as many as five to seven items, it seems prudent to

form two composites for each dimension in which each composite is a

sum of items. In fact, this was the strategy chosen for the partial d isaggre-

gation model

of the

SSES examined

in

this study. When nine

or

more

items exist

per

dimension

in a

scale,

it is

feasible

to

form three

or

more

composites

as

indicators

for

each dimension.

For

their analysis

of the

SDQ, Marsh

and

Hocevar (1985) used four com posite indicators

for

each

of seven dimensions.

RESULTS

Tests

of

Basic

Models for Organization of the

S S E S

Total aggregation model. When

the

SSES

is

treated

as the sum of 20

items, the

Cronbach alpha reliabilities

are .91 for

Sample

1 and .92 for

Sample 2. From one perspective, the total aggregation model constitutes an

aggregation

of

both dimensions

and

items.

As a

point

of

comparison,

it is

interesting

to

examine

the

case

in

which dimensions remain aggregated

but

items

are

disaggregated. This

is the

model

in

which

all

items load

on a

single

factor.

For

Sample

1, the

model fits quite poorly, X

2

(17O,

Ni =

102)

=

514.44,

p

<

.001,

RNI = .64.

L ikewise,

for

Sample

2, the fit is

very poor, x

2

(170,

N2

= 428) =

1395.16,

p


17/35

TABLE

1

Findings for

Partial

Aggregation Model of State Self-Esteem-Hierarchical Organization of Components for Samples 1 and 2

Sample

1

Sample 2

b

M odel Goodness

of

Fi t

Test

of

Hypotheses'

Key Parameter Estimates*

Goodness of Fit

Test

of

Hypotheses Key Parameter Estimates

6

M,: Null , N = 102) = 88.66,

p - .00

NA

NA

= 428) = 421.20,

p = .00

NA

NA

M

2

: Baseline

,

N =

102) = 0.00 ,

p = 1.00

NA

X, = .81

(.10),

e

tl

=

\

3

= .66 (.10), 0

S3

=

.34

(.10)

.44

(.10)

.56 (.10)

^ ( 0 ,

N

= 428) = 0.00,

p = 1.00

NA

X, = .79

(.05),

0

S1

= .37 (.05)

X

2

= .78 (.05), 0

82

= .39 (.05)

X

3

= .72 (.05),


18/35


eter estimates, based on the correlation matrix as input, are presented for

ease of interpretation, in which it can be seen that factor loadings are high

and error variances are low for Sample 1. Using the formula for reliability

presented in the Method section, we find that the SSES is reliable in Sample

1 (p = .79). Model M3 hypothesizes that the factor loadings for the three

SSES dimensions are equal. For this and each subsequent hypothesis, the

covariance matrix is used as input to LISREL (Cudeck, 1989). Although the

chi-square test is significant for M3, X

2

(2, N = 102) = 7.72, p - .02, the RNI

of .93 suggests that the model fits satisfactorily as a practical matter. Never-

theless, a comparison of models M3 and M2 shows that one must reject the

hypothesis that the factor loadings are equal, x d(2) = 7.72,

p <

.02. Model

M4 introduces the additional restriction that the error variances of the mea-

sures of performance, social, and appearance are equal. Although the chi-

square value for M4 is significant, x (4, N = 102) = 12.15, p ~ .02, the model

fits w ell in a practical sense (RNI = .90). A comparison of M4 and M3 reveals

that we cannot reject the hypothesis that the error variances are equal, x d(2)

= 4.43, /?>.12.

The findings for Sample 1 so far suggest that the partial disaggregation

model fits the data well but that the factor loadings, although high and

significant in each case, are not equal. Actually, the test of equality of factor

loadings (and error variances) is a demanding one in the sense that tau-

equivalency or parallel forms are not necessary in order to achieve meaning-

ful scales and that the congeneric test, implied by the model in which no

equality restrictions are placed on factor loadings and error variances, pro-

vides sufficient information on the properties of a scale . Occasionally, a

researcher may, given a rejection of the hypothesis of equal factor loadings

(and/or error variances), want to test whether subsets of loadings (and/or

error variances) are equal. This was done for illustrative purposes, and the

results are summarized under models Ms and M6 in Table 1. Model Ms,

which hypothesizes that the factor loadings for performance and social are

equal, fits well, x

2

( l , N = 102) = .07, p ~ .78, RNI = 1.00. A comparison of

M5 and M2 shows that we cannot reject the hypothesis of equal factor

loadings, x

2

d(l) = 07, p > .78. Model M6 also fits well, x

2

(2 , N

=

102) = .53 ,

p .77, RNI = 1.00. A comparison of M6 and Ms indicates that we cannot

reject the hypothesis of equal error variances, x

2

d( l) = .46, p > .50.

Looking next at the findings for Sample 2, we see that factor loadings

under the baseline model are high and error variances are low. The reliability

of the m easures for the SSES is p = .81. The results for M3 point to a

satisfactory fit, x

2

(2, N= 428) = 9.67, p - .01 , RNI = .98, but the hypothesis

of equal factor loadings must be rejected, x d(2) = 9.67, p < .01 . Likewise,

although M4 fits reasonably well, x (4, N= 428) = 25.24,p - .00, RNI = .95 ,

we must reject the hypothesis of equal error variances, x

2

d(2) = 15.57, p <

.001. The test of equal factor loadings for performance and social reveals

that we cannot reject this hypothesis, x

2

d( l) = 3.46,

p

> .06, but the hypoth-



y 2 0 1 5


19/35

MU L T IFA CE TE D PE RSO N A L IT Y CO N ST RU CT S 5 1

esis of equal error variances for performance and social must be rejected,

X

2

d(l) = 14-65,

p


20/35

ro

TABLE

2

Findings for Partial Disaggregation Model of State Self-Esteem

—

First-Order Three-Factor Model for Samples 1 and 2

K ey

Parameter Estimates*

Factor Loading

Error

Factor Correlation

Model

M,: Null ;

Goodness

of

F it

^(15,

N

= 102) = 331.32,

p

=

.00

P er fo rm an ce S oc ia l A pp ea ra nc e

Sample

l

b

Variance Performance Social Appearance

M

2

: Full x*(6,

N

= 102) = 4.32,

P = .63,

RNI = 1.00

.87

(.08)

.91 (.08)

.00

.00

.00

.00

.00

.00

.97 (.09)

.74 (.09)

.00

.00

.00

.00

.00

.00

.66 (.10)

.95 (.10)

.25

(.06)

.17 (.06)

.08 (.09)

.46 (.08)

.57 (.10)

.10 (.13)

1.00

.72 (.07)

.63 (.08)

1.00

.64 (.08)

1.00

Sample

2

C

M,: Null

x

2

d 5 ,

N

= 428) =

1384.82,

p= .00

M

2

: Full

N = 428) = 25.41,

p = .00,

RN I = .99

.83 (.04)

.86 (.04)

.00

.00

.00

.00

.00

.00

.91 (.04)

.78 (.04)

.00

.00

.00

.00

.00

.00

.70 (.05)

.98

(.04)

.31 (.04)

.27 (.04)

.18 (.04)

.39 (.04)

.51

(.04)

.03

(.05)

1.00

.74 (.03)

.67 (.04)

1.00

.70 (.04)

1.00

•"Standard errors in parentheses.

h

N

= 102.

e

N

= 428.

D o w n l o a

d e d b y [ C e n t r a l M i c h i g a n U n i v e r s i t y ]

a t 0 7 : 2 7 0 5 J a n u a r y 2 0 1 5


21/35

TABLE

3

Findings for Partial Disaggregation

Model

of State Self-Esteem —Second-Order

Three-Factor Model

for Samples 1 and 2

Goodness

M odel of Fit

Full ^ (6 ) = 4 .32,

P =

.63,

RNI = 1.00

Full

x*(6)

= 25.41,

p

= .00,

RNI = .99

First-Order

Factor Loading

Performance

1.00°

1.05 (.10)

.00

.0 0

.00

.0 0

1.00

c

1.02 (.06)

.0 0

.00

.0 0

.00

Social

.00

.00

1.00

c

.76 (.10)

.0 0

.0 0

.0 0

.00

1.00

.86 (.05)

.00

.00

Appearance

.00

.00

.00

.00

1.00

c

1.44 (.26)

.00

.0 0

.00

.0 0

1.00

1.40 (.10)

Key Param eter Estimates*

Second-Order Error in

Factor Loading First-Order Factors

7 ;

12 7 J

* / * 2 * i

Sample l

b

.73 (.10) .82 (.09) .49 (.10) .21 (.08) .26 (.12) .19 (.06)

Sample 2

d

.70 (.05) .79 (.04) .56 (.05) .20 (.04) .19 (.04) .18 (.03)

Variance Decomposition

Random

Error

.25 (.06)

.17 (.06)

.07 (.09)

.46 (.08)

.57 (.10)

.10 (.13)

.30 (.04)

.27 (.04)

.18 (.04)

.39 (.04)

.51 (.04)

.03 (.05)

Specific

.21

.23

.2 6

.15

.19

.39

.2 0

.21

.19

.14

.18

.35

Common

.53

.59

.67

.3 9

.24

.5 0

.49

.51

.6 2

.46

.31

.61

CO

D o w n l o a d e d b y [ C e n t r a l M i c h i g a n U n i v e r s i t y ]

a t 0 7 : 2 7 0 5 J a n u a r y 2 0 1 5


22/35

5 4

BAGOZZI

AND HEATHERTON

TABLE 4

Factor Loadings

and

Correlations Among Factors

for Total

Disaggregation

Model

tem

Number

I

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

Note.

•N. =

Performance

.63 (.10)

.00

.00

.61 (.10)

.62 (.10)

.00

.00

.00

.73

(.09)

.00

.00

.00

.00

.64 (.09)

.00

.00

.00

.75 (.09)

.70 (.09)

.00

1.00

.64

(.08)

.64 (.07)

Standard

error:

102.

h

N

2

- 428

Sample

1*

Social

.00

.67

(.09)

.00

.00

.00

.00

.00

.65 (.09)

.00

.64 (.09)

.00

.00

.81 (.09)

.00

.55 (.10)

.00

.74

(.09)

.00

.00

.66 (.09)

ppearance

.00

.00

.49 (.10)

.00

.00

.52 (.10)

.56 (.10)

.0 0

.00

.00

.88 (.08)

.85 (.08)

.00

.0 0

.00

.62 (.09)

.00

.00

.00

.00

Performance

.74

(.04)

.00

.00

.61 (.05)

.49 (.05)

.00

.00

.0 0

.68 (.04)

.00

.00

.00

.00

.63

(.05)

.00

.00

.00

.70 (.04)

.69 (.04)

.00

Factor

Correlations

1.00

.54 (.08)

1.00

s

in

parentheses.

1.00

.74

(.03)

.72 (.03)

Sample

2

b

Social

.00

.61

(.05)

.00

.0 0

.00

.0 0

.00

.68 (.04)

.00

.73 (.04)

.00

.00

.73 (.04)

.00

.72 (.05)

.00

.63

(.05)

.00

.00

.69 (.04)

1.00

.69

(.03)

Appearance

.00

.00

.71 (.04)

.00

.00

.52 (.05)

.60 (.05)

.00

.00

.00

.80 (.04)

.80 (.04)

.00

.00

.00

.72

(.04)

.00

.00

.00

.00

1.00

cant in both samples; only Item 7 failed to reach significance. Nevertheless,

the models do not fit well, and thus a second method factor was tried with

the non-reverse-coded items loading on it. This, too, resulted in a significant

improvement in fit for Sample 1, x

2

(146,

N\

= 102) = 227.25, RNI = .92, and

X

2

d(8)

=

81.29,

p <

.001;

and

Sample

2,

X

2

(146,

Nz = 428) =

396.10,

RNI =

.94, and x

2

d(8) = 237.90, p


23/35


pies 1 and 2 showed that x

2

(6,

Ni

= 102,

Nz

= 428) = 8.95,

p

- .18. Therefore,

we cannot reject the hypothesis that the measures are equivalent between the

two samples.

Partial disaggregation model. The test of the equality of variance-

covariance matrices for Samples 1 and 2 showed that x (21, Ni = 102, N2 =

428) = 26.06, p - .20. Hence, we cannot reject the hypothesis that the

measures are equivalent between the two samples.

In sum, the variance-covariance matrices among measures of the SSES

are quite similar across the samples when we examine (a) measures of the

components treated as separate aggregates but loading on a single factor (the

partial aggregation model) and (b) measures of the components treated as

subaggregates loading on three factors (the partial disaggregation model). As

a consequence, it is not meaningful to test for differences in factor structure

and parameter estimates across the samples. It should be noted that the

analyses to be reported for gender differences show that the variance-covar-

iance matrices do in fact differ between men and women in Sample 2, and we

are thus able to explore more fully hypotheses related to equality of factor

structures for the partial aggregation and partial disaggregation models.

Because the results for the total disaggregation model revealed that the

single- and three-factor models fit poorly, no tests of generalizability were

performed on these models.

Tests

of Generality of SSES cross Gender

Examination of basic models for organization of the SSES. Table

5 summarizes the findings for the partial aggregation model for women and

men from Sample 2 (Sample 1 is too small to permit a comparison of men

and women). Looking first at the results for women, we see that factor

loadings are high and error variances are low. The reliability of measures is

p = .80. All models fit well as a practical matter, as shown by the RNIs.

However, tests of equality of parameters show that neither factor loadings,

X

2

d(2) = 11.11, p> .001, nor error variances, x

2

d(2) = 9.72, p

<

.01, are equal

for this single-factor model. Further, tests for the subset of factor loadings

and error variances corresponding to performance and social indicate that

neither pair is equal, x

2

d(l) = 6.28,

p <

.01, and X

2

d(l) = 9.53,

p <

.001,

respectively. Looking next at the findings for men, we see that factor load-

ings are high and error variances are low. The reliability of measures is p =

.84.

All models fit well on the basis of RNIs, and, indeed, three of four fit

well on the basis of the chi-square test. Likewise, one cannot reject the

hypothesis of equal factor loadings, x

2

d(2) = 3.34, p > .19. The hypothesis of

equal error variances, however, must be rejected, x d(2) = 9.24,p


24/35

Findings

Model

M,: Null

M

2

: Baseline

M

3

:

X, = X

2

= X

3

M

4

t

Xi = X2

=

X3,

hi - hi

=

hi

M $ : X] = X

2

M $ : X| = X2,

for Partial Aggregation

Goodness of Fit

X * Q, N = 284) = 267.11,

p - .00

^ ( 0 , N = 284) = 0.00,

p = 1.00

X * ( 2 ,

N= 284) = 11.11,

P = .00, :

RNI = .97

X

2

(4,

N = 284) = 20.83,

, P = .00,

RNI = .94

X*(1,N= 284) = 6.28,

P ~

.01,

RNI = .98

^ ( 2 , N = 284) = 15.81,

p = .00,

RNI = .95

Model of

Women

Test of

Hypotheses

NA

NA

M

3

- M

2

d 2 =

11.11,

p < .001

M

4

-M

3

xj 2 = 9.72,

p

< .01

M

3

- M

2

x3(0

= 6.28,

p < .01

M

s

-M

s

x3 D

= 9.53,

P < .001

TABLE 5

State Self-Esteem-Hierarchical Organization of

a

Key Parameter Estimates

6

NA

X, = .77 (.06), 0

M

= .41 (.06)

X

2

= .81 (.06), 0,

2

= .34 (.06)

X

3

= .69 (.06), 0,3 = .52 (.06)

X, = 3.83 (.20), 9,, = 8.70(1.14)

Xj = 3.83 (.20), 9,

2

= 14.72 (1.57)

X

3

= 3.83 (.20), 9

M

= 11.61 (1.34)

X, = 3.88 (.21), 0,, = 11.56 (.69)

X

2

= 3.88 (.21), 0,

2

= 11.56 (.69)

X

3

= 3.88 (.21), 0

M

= 11.56 (.69)

X, = 4.05 (.23), 0,, = 7.66 (1.21)

X

2

= 4.05 (.23), 0,

2

= 14.14 (1.57)

X

3

= 3.36 (.29), 0,3 = 12.84(1.41)

X, = 4.12 (.24), 0,, = 10.90 (.92)

X

2

= 4.12 (.24), B

a

= 10.90 (.92)

X

3

= 3.42 (.29), 0,3 = 12.40 (1.39)

Goodness of Fit

X* (3 , N = 144) = 169.98

p = .00

^ ( 0 , N

= 144) = 0.00,

p = 1.00

X

2

(2,

N = 144) = 3.34,

P = .19,

RNI = .99

^ ( 4 ,

A' = 144) = 12.58,

p =

.01,

RNI = .95

X^l,

N

= 144) = .62,

p = .43,

RNI = 1.00

^ ( 2 , N = 144) = 2.86,

P ~ -24,

RNI = .99

Components

M en

b

Test

of

Hypotheses'

NA

NA

M

3

- M

2

x3(2) = 3.34,

p > .19

M

4

- M

3

x2(2) = 9.24,

p < .01

M

5

- M

2

x3(0

= -62,

p

> .43

M

6

- M,

xiO

= 2.24,

p >

.15

for Women andMen

Key Parameter Estimates

1

NA

X, = .80 (.08), 9,,

X

2

= .79 (.08), 9

U

X

3

= .80 (.08), fl,

3

X, = 3.98 (.28), 0,,

X

2

= 3.98 (.28), 0

M

X

3

= 3.98 (.28), S

a

X, = 4.07 (.29), 0

}

i

X

2

= 4.07 (.29), 0j

2

X

3

= 4.07 (.29), 0

H

X, = 4.27 (.34), 0,,

X

2

= 4.27 (.34), 0

M

X

3

= 3.63 (.35), 0,,

X, = 4.29 (.34), 0,,

X

2

= 4.29 (.34), 0

M

X

3

= 3.64 (.35), 0,3

= .36 (.07)

= .38 (.07)

= .36 (.07)

= 10.06(1.64)

= 13.97 (2.05)

= 6.32(1.31)

= 9.94 (.83)

= 9.94 (.83)

= 9.94 (.83)

= 8.94(1.72)

= 13.02 (2.05)

= 7.51 (1.43)

= 10.98(1.30)

= 10.98(1.30)

= 7.43(1.42)

JV = 284.

b

N = 144.

C

NA = not applicable. Standard errors in parentheses.

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25/35


for the tests of subsets of factor loadings and error variances, one cannot

reject the hypothesis of equal factor loadings for performance and social,

X d(l) = .62,

p

> .43, or equal error variances for the measures of these two

dimensions, x

2

d(l) = 2.24,

p >

.15.

Table 6 presents the results for the partial disaggregation model for the

first-order, three-factor case. The model fits the data well for women and

men, based on the values for the RNIs. For women and m en, factor loadings

are high and error variances are low. The reliabilities for m easures of perfor-

mance, social, and appearance are, respectively, p

P

= .80, p

s

= .84, and p

a

=

.83 for women and p

p

= .89, p

s

= .84, and p

a

= ,84 for men. The correlations

among the performance, social, and appearance factors are high and yet are

significantly less than 1.00. Table 7 shows the findings for the partial disag-

gregation model for the second-order, three-factor case. The model fits w ell

in both samples, factor loadings are high, error variances and specific vari-

ance are generally low, and common variance ranges from moderate to high

for the most part.

Tests of equ ality of models across wom en and men.

For the par-

tial aggregation m odel, the test of the equality of variance-covariance matri-

ces for women and men revealed that x (6, JVi = 284,

N2

= 144) = 8.09,

p -

.23. Thus, we cannot reject the hypothesis that the measures are equivalent

for women and men.

Table 8 summarizes the findings for the tests of equality of the partial

disaggregation model (first-order, three-factor case) for women and men.

The test of the equality of variance-covariance matrices shows that x (21,

Ni =

284, JV2 = 144) = 34.03,

p

~ .03, and therefore we must reject the

hypothesis that the measures are equivalent (see M i in Table 8) . The second

row in Table 8 reveals that the factor pattern is the same for women and

men—that is, the same three factors exist. Next it can be seen that the

hypothesis of equal factor loadings cannot be rejected, x

2

d(6) = 8.62, p > .21

(see the third row in Table 8). Likewise, we cannot reject the hypothesis that

the error variances are equal, x

2

d(6) = 12.62, p > .05 . Last, the hypothesis of

equal covariances among the factors of performance, social, and appearance

cannot be rejected, x

2

d(3) = 6.40,

p >

.09.

Table 9 presents the results for the tests of equality of the partial disaggre-

gation model (second-order, three-factor case) for women and men. Unlike

the first-order model in which the disturbance terms confound random error

with measure specificity, these analyses provide separate estimates of ran-

dom error and measure specificity. The tests of equal covariance matrices

and factor patterns are the same as for the first-order model. The test of equal

factor loadings shows that we cannot reject this hypothesis, x

2

d(3) = 5.47, p

> .15 (see the third row in Table 9). The hypothesis of equal error variances

for m easures, however, must be rejected, x d(6) = 14.32,

p <

.03. We cannot

reject the hypothesis of equal error variances for the first-order factors—



r s i t y ] a t 0 7 : 2 7 0 5 J a n u a r y 2 0 1 5


26/35

00

TABLE 6

Findings

for Partial Disaggregation Model of State Self-Esteem - First-Order Three-Factor Model for Women and Men

Model

M,: Null

M

2

: Full

M,: Null

M

2

: Full

Goodness of Fi t

X*(1S , N = 284) = 870.78,

p = .00

X

i

{6,N=

284) =

21.33,

p

=

.00,

R N I = .98

X ^IS, N

= 144) = 549.72,

p

-

.00

X*(6 ,

N

= 144) = 8.64,

p = .20,

R NI = 1.00

Performance

.79

(.06)

.84

(.05)

.0 0

.0 0

.00

.00

.92 (.07)

.87 (.07)

.0 0

.0 0

.0 0

.0 0

Factor Loading

Social

Key Parameter Estimates'

1

Appearance

Women*

.0 0

.0 0

.91 (.05)

.78 (.05)

.0 0

.0 0

.0 0

.0 0

.0 0

.0 0

.7 2 (.06)

.95 (.05)

Men'

.0 0

.0 0

.88 (.07)

.8 3 (.07)

.0 0

.0 0

.0 0

.0 0

.0 0

.0 0

.68

(.07)

.9 9 (.08)

Error

Variance

.3 7

(.05)

.2 9 (.05)

.17 (.04)

.3 9 (.04)

.4 9 (.05)

.1 0 (.06)

.15 (.05)

.2 4 (.05)

.2 2 (.06)

.3 2 (.06)

.53

(.07)

.01 (.08)

Factor Correlation

Performance

1.00

.7 6 (.04)

.66 (.05)

1.00

.73 (.05)

.71 (.05)

Social Appearance

1.00

.7 0 (.04) 1.00

1.00

.77 (.05) 1.00

Standard errors in parentheses.

b

N

= 284.

N

= 144.



27/35

TABLE 7

Findings for Partial Disaggregation Model of State Self-Esteem —Second-Order Three-Factor Model for Women and Men

Goodness

M odel of Fit

Fu ll x*(6) = 21.33,

p = .00,

RNI = .98

Full

x*(6)

= 8.64,

p = .20,

RNI = 1.00

First-Order

Factor Loading

Performance

1.00

c

1.06 (.08)

.0 0

.0 0

.00

.00

1.00

c

.94 (.08)

.0 0

.00

.0 0

.0 0

Social

.00

.00

1.00

c

.86 (.06)

.00

.00

.0 0

.00

1.00

C

.94 (.08)

.0 0

.00

Appearance

.00

.0 0

.0 0

.0 0

LOO*

1.33

( . 1 2 )

.0 0

.00

.0 0

.00

1.00

C

1.45

( . 1 7 )

K ey Parameter Estimates'

Second-Order Error in

Factor Loading First Order Factors

7/ y

2

yj * ; * ; * 3

Women

0

.67 (.06) .81 (.06) .56 (.06) .18 (.04) .17 (.06) .20 (.04)

M en

6

.75 (.08) .78 (.08) .59 (.08) .28 (.07) .17 (.06) .12 (.04)

Variance Decomposition

Random

Error

.37 (.05)

.29 (.05)

.17 (.04)

.39 (.04)

.49 (.05)

.10 (.06)

.15 (.05)

.24 (.05)

.22 (.06)

.32 (.06)

.53 (.07)

.01 (.08)

Specific

.18

.2 0

.17

.13

.20

.35

.28

.25

.17

.15

.1 2

.25

Common

.45

.50

.6 6

.49

.31

.55

.56

.5 0

.61

.54

.35

.73

CO



28/35

TABLE 8

Findings for Partial Disaggregation Model of State Self-Esteem -

Multiple-Group Analysis for

First-Order

Three-Factor

Model:

Tes ts

of

Invariance

for

Women

and Men

Model

Mi. Equal

covariance

matrices

M

2

: Equal

factor

pattern

M

3

: X invaiiant

M

4

: X invariant,

Q_s invaiiant

M

5

: X invaiiant,

81

invaiiant,

t> inv aiia nt

G oodness of F it

^ ( 2 1, Af, = 284,

N

2

= 144) = 34.03,

p = .03

1?(\2, N, =

284,

N

2

= 144) = 29.97,

p = .00

X^IS, Nt = 28 4, AT

2

= 144) = 38.59,

p = .00

^ ( 2 4 ,

N

t

= 284, A^ = 144) = 51 .21,

p = .00

X

2

(27, N

t

= 284, N

2

= 144) = 57.61,

p = .00

Test of H ypotheses

M , - M

2

x2(6) = 8.62,

p > .21

M

4

- M

3

j^(6) = 12.62,

p > .05

M

3

- M

4

x2(3) = 6.40,

p > .09

TABLE

9

Findings for Partial Disaggregation

Model

of State Sel f -Esteem-

Multiple-Group Analysis of Second-Order, Three-Factor

Model:

Tests

of

Invariance

for

Women

and Men

Model

M,. Equal

covariance

matrices

M

2

: Equal

factor

pattern

M

3

: X invaiiant

M

4

: X invaiiant,

6

e

invaiiant

M

5

: X invariant,

8_ e invariant,

• invaiiant

Goodness of Fit

3^(21,

N,

= 2 84, A/i = 144) = 34.0 3,

p = .03

T?(\2, N,

= 28 4, Aij = 144) = 29.97,

p = .00

^ ( 1 5 , N, =s 28 4, AT

2

= 144) = 35.4 4,

p = .00

^ ( 2 1 , N, =

284, AT

2

= 144) = 49.76,

p = .00

^ ( 2 4 , N

t

= 284, N

2

= 144) = 56.21,

p = .00

Test of H ypotheses

M

3

-

P >

M

4

-

Xd(6) =

P <

M

5

-

P >

M

2

5.47,

.15

M

3

14.32,

.03

M

4

6.45,

.09

M

6

: X invariant,

8 e invariant,

* invariant,

T invaiiant

, AT, = 28 4,

N

2

= 144) = 57.61,

p = .00

M

6

- M

5

= 1-40,

p > .46

D o w n l o a d e d b y

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MULTIFACETED PERSONALITY CONSTR UCTS 6 1

which corresponds to measure specificity, x d(3) = 6.45, p > .09. Last, the

hypothesis of equal factor loadings for the regression of the first-order

factors on the second-order factor—which corresponds to common variance

of the measures—cannot be rejected, x

2

d(3) = 1.40, p > .46.

DISCUSSION

S S E S

Overall, the findings show that the SSES can be represented in a psychomet-

rically sound way through both the partial aggregation model (treated as a

hierarchical organization of components under a single SSES latent variable)

and the partial disaggregation model (treated as either a first-order or sec-

ond-order model of three latent variables corresponding to the dimensions of

the SSES). The total disaggregation model of the SSES also showed that the

scale can be represented as three factors corresponding to performance,

social, and appearance dimensions. However, a satisfactory fit was achieved

only after the introduction of method factors. These results complement the

findings of Heatherton and Polivy (1991), who confined analyses to the total

aggregation model and the partial aggregation model, in which the latter

dealt with each dimension of the SSES as a discrete component measured

without error.

The partial aggregation model provided a cogent representation of the

SSES at a high level of abstraction. In this sense, it supports the interpreta-

tion and use of the SSES as a multifaceted composite. Unlike the traditional

total score approach, which fails both to provide a precise representation of

the SSES as a latent construct and to model measurement error explicitly, the

partial aggregation model captures the hierarchical organization of the SSES

as a singular, general factor with three dimensions and supplies information

on the amount of trait and error variance in the dimensions. The results

showed that the measures of the SSES under the partial aggregation model

are highly reliable and replicate across the two independent samples in the

sense of demonstrating equality of variance-covariance matrices. Further,

the examination of the partial aggregation model by gender revealed that

men and women also exhibited identical variance-covariance matrices for

the composite measures of the SSES. The findings for the across-sample and

across-gender analyses suggest that the structure of responses to the SSES is

quite generalizable for the partial aggregation model.

The partial disaggregation model yielded insights into the SSES at a more

fine-grained and less abstract level of analysis than the aggregate models.

This model captured the unique dimensions of the SSES and demonstrated

that each dimension is reliably measured and achieves discrimination from

the others in a moderate sense. Yet, the findings revealed that significant

amounts of common variance exist across the measures of the three dimen-


r s i t y ] a t 0 7 : 2 7 0 5 J a n u a r y 2 0 1 5

8/16/2019 A General Approac

a general approach to representing multifaceted

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