STRUCTURAL EQUATION MODELING, 12(3), 391–410
Copyright © 2005, Lawrence Erlbaum Associates, Inc.
Imperfect or Perfect Dynamic
Bipolarity? The Case of Antonymous
Affective Judgments
Stéphane Vautier
Université de Toulouse Le Mirail
Rolf Steyer
University of Jena
Saïd Jmel and Eric Raufaste
Université de Toulouse Le Mirail
How is affective change rated with positive adjectives such as good related to change
rated with negative adjectives such as bad? Two nested perfect and imperfect forms
of dynamic bipolarity are defined using latent change structural equation models
based on tetrads of items. Perfect bipolarity means that latent change scores correlate
–1. Meaningful structural equation modeling (SEM) analyses of self-rated affect
may require analyzing polychoric correlations, if self-ratings are collected using ordered categories. The models were applied to 6 4-wave datasets from Steyer and
Riedl (2004). Results suggest that perfect bipolarity is generally compatible with valence self-ratings, whereas imperfect bipolarity is compatible with tension and energy self-ratings. Methodological and substantive limits of the approach are discussed.
In the psychology of core affect (Russell, 2003; Russell & Feldman Barrett, 1999),
positive and negative activation (PA and NA, respectively) have been thought of as
“affective state dimensions” (Watson, Clark, & Tellegen, 1988, p. 1063), and more
recently “as reflecting two basic biobehavioral systems of activation” (Watson,
Wiese, Vaidya, & Tellegen, 1999, p. 827), “systems that mediate goal-directed apRequests for reprints should be sent to Stéphane Vautier, Maison de la Recherche, CERPP,
Université de Toulouse Le Mirail, 5 allées A. Machado, F–31058 cedex 9, France. E-mail: vautier@
univ-tlse2.fr
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proach and withdrawal behaviors” (pp. 829–830). Thus, are the biobehavioral systems functionally synchronized or not (see also Cacioppo & Berntson, 1994;
Cacioppo, Gardner, & Berntson, 1999)? Usual findings based on factor analyses of
self-ratings collected with Watson et al.’s (1988) PANAS scales suggest weakly and
negatively correlated factors, a finding interpreted as supportive of asynchronism in
PA and NA. However, weakly uncorrelated factors are compatible with a more subtle
form of synchronism: The two systems can be functionally related in such a way that
a positive change in one system co-occurs with a negative change in the other system,
and vice versa. Cacioppo and Berntson (1994) called that reciprocity, and Green,
Solovey, and Truax (1999) used the term dynamic bipolarity, in opposition to static
bipolarity. A general issue relates to the reasons why systems would work together or
not (e.g., Reich, Zautra, & Davis, 2003). Thus, considering some self-ratings gathered at two (or more) occasions, the idea of dynamic bipolarity can be elaborated further using a specific structural equation modeling (SEM) approach.
Vautier and Raufaste (2003) investigated dynamic bipolarity of PA and NA using an SEM approach based on the concept of true intraindividual change
(McArdle, 2001; Raykov, 1999; Steyer, Eid, & Schwenkmezger, 1997; Steyer,
Partchev, & Shanahan, 2000). They estimated a moderate and negative correlation
between the latent change PA scores and the latent change NA scores, a finding
that emphasizes the idea of separable underlying affective functions.
A substantive goal of this study was to extend the modeling of dynamic bipolarity of PA and NA scores to the case of affective judgments based on antonymous
items, which were studied using a static approach to bipolarity by Steyer and Riedl
(2004). Antonymous affective judgments are self-ratings gathered using items diametrically opposed on the affective circumplex (e.g., Larsen, McGraw, &
Cacioppo, 2001; Russell, 2003; Russell & Carroll, 1999; Watson et al., 1999).
Thus, the linguistic relation of antonymous items may be geometrically expressed
by univariate bipolarity, which in turn suggests that a given change on one pole
should be associated with a change of opposite magnitude on the other pole.
True-change models are not yet widely used in empirical studies. Hence, a
methodological goal was to delineate some conceptual and technical problems associated with their application.
DYNAMIC BIPOLARITY
Before examining the methodological aspects of modeling dynamic bipolarity
through SEM, we introduce the concept of dynamic bipolarity by contrasting it
with the concept of static bipolarity. A psychological phenomenon may be rated
with positively (Y+) or negatively (Y–) formulated items. Static bipolarity means
that in a given sample, the two ratings, Y+ and Y–, correlate negatively. Perfect static
bipolarity means that ratings correlate –1. However, even when measurement er-
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rors are statistically controlled for, perfect static bipolarity is hard to observe. Russell and Carroll (1999) addressed the issue of multiple interpretation of the response formats, and showed that several understandings are possible, which means
the Pearson correlation between positive and negative ratings may be misleading.
Moreover, Schimmack (2001) underscored the potential existence of response
styles, which could bias the latent correlation between ratings Y+ and Y–.
Steyer and Riedl (2004) presented a multiwave SEM approach based on latent
state–trait (LST) theory (see Steyer, Ferring, & Schmitt, 1992; Steyer, Schmitt, &
Eid, 1999, for details about LST theory). Using this method, static bipolarity of antonymous ratings can be analyzed while controlling for temporally stable individual
differences. State factors pertaining to ratings Y+ are decomposed into a trait factor,
common to all waves, and wave-specific state residuals. Ratings Y– are decomposed
this way as well. Thus, within each wave, state residuals related to ratings Y+ can be
linearly related to state residuals corresponding to ratings Y–. Furthermore, the correlation between the relevant state residuals can be estimated. If the state residuals are
linearly dependent, they correlate perfectly (and negatively), which corresponds to
perfect static bipolarity, temporally stable biases being sorted out.
Now, let us turn to dynamic bipolarity. Taking Y+ and Y– as distinct poles (e.g., joy
vs. sadness), dynamic bipolarity expresses the idea that change rated using Y+ is negatively related to change rated using Y–. Considering a given self-rater i, let δη+i and
δη–i denote his or her change rated using Y+ and Y–, respectively. Now, considering
randomly sampled self-raters, we get two random change variables δη+ and δη–. Perfect and imperfect dynamic bipolarity may be distinguished. Both change variables
are linearly (and negatively) related (they correlate –1). In this case, although Y+ and
Y– offer distinct empirical assessments of a psychological state, variables δη+ and δη–
contain exactly the same information; that is, one of them is redundant. This is perfect dynamic bipolarity. Imperfect dynamic bipolarity refers to the imperfect negative correlation between δη+ and δη–. This is what Vautier and Raufaste (2003) found
for PA and NA as measured with the PANAS scales.
Although the use of antonymous adjectives suggests that perfect (linear) dynamic bipolarity may work, such an intuition needs to be framed into a testable
model. What can be tested is a structural equation model, within which change assessed by positively formulated items is linearly related to change assessed by negatively formulated items.
FRAMING THE PROBLEM WITH
POLYCHORIC CORRELATIONS
Ratings can be gathered using either analogue or Likert-type response formats.
The former refer to continuous measurements (up to rounding error) and are compatible with the assumption of interval variables, whereas the latter refer to ordered
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categories, which are only compatible with the assumption of ordinal variables. As
stated by Jöreskog (2001a), “Means, variances, and covariances of ordinal variables have no meaning” (p. 2). At first glance, analogue rating scales are appealing,
but continuity is not enough. SEM requires the assumption of linearity of a set of
measurement variables. Broadly, if observed data reflect one common variable,
their observed histograms will be expected to exhibit homogeneous shapes, due to
the effect of the common variable. Strong heterogeneity of the observed histograms questions the validity of a linear model, because it makes it difficult to imagine how the observed data were produced. Analogue rating scales often suffer from
floor or ceiling effects, depending on the semantic properties of the items (e.g.,
scared is weaker than terrified, and terrified is likely to capture a stronger ceiling
effect than scared). Moreover, our own experience with analogue rating scales or
graduated continuous rating scales used in self-administered questionnaires suggests that it can be difficult to fit alternative probability density functions to the observed distributions, which would be compatible with robust estimation methods.
In addition, the observed distributions are difficult to normalize by usual transformations (e.g., Tabachnik & Fidell, 1996). Even the use of parceling does not suffice to get normal-like histograms.
Vautier and Raufaste (2003) used parcels based on analogue ratings. Parceling
seems to be a safe practice if one has strong reasons to believe that the summed
variables do measure the same attribute (Bandalos, 2002; Bandalos & Finney,
2001; Kim & Hagtvet, 2003; Little, Cunningham, Shahar, & Widaman, 2002). Although this article does not focus on parceling, a specific issue deserves attention
here. Suppose that the observed item-level histograms exhibit heterogeneous
shapes, and that a one-factor model holds nevertheless, few error covariances
being added if necessary. Would the model explain how measurements were produced? A model that fits the covariance structure from data with unknown distributions reproduces the covariance structure, but not necessarily the observed histograms. A measurement model corresponding to the theory of one-dimensional
attribute beyond the data can be expected to reproduce not only the covariance
structure, but also the histograms as they are observed. If it does not, the notion of
an attribute as an effect of our ability to model a one-factor model is questionable
(see Borsboom, Mellenbergh, & van Heerden, 2003).
An alternative to analogue scales or parcels is using Likert-type rating scales,
which yield ordinal data. For instance, using a 4-point Likert-type scale, a respondent is asked to choose among agree strongly, agree, disagree, and strongly disagree
(Jöreskog, 2001a). Parceling of ordinal data is meaningless merely because addition
of ordered responses is meaningless (see also Stine, 1989). A standard solution to
treating ordinal data consists of assuming underlying continuous dimensions, and
then scaling them given the conventional normal probability function (probit). One
dimension by ordinal variable is defined, such that the probability of choosing one
response category corresponds to an interval on the dimension (see Jöreskog, 1990;
DYNAMIC BIPOLARITY
395
Takane & de Leeuw, 1987). By contrast to the observed variables Y+ and Y–, these underlying variables will be labeled Y*+ and Y*–. Then, the target model may be fitted to
the polychoric correlation matrix, which is the correlation matrix of the underlying
variables. Readers interested in the issue of estimating polychorics are referred to
Jöreskog (2001a). In addition, bivariate normality can be tested.
Using an SEM approach to dynamic bipolarity, the intuition of inverse-related
changes can be translated into a workable problem, formulated as follows: Is it
possible to estimate the parameters of a model stipulating that δη+ and δη– correlate negatively, in such a way that this model fits the polychoric correlation structure based on the observed data?
FORMAL FEATURES FOR DYNAMIC BIPOLARITY
Before presenting the datasets and the analyses, the concept of a latent change
score is detailed first, and then the technical solutions to model negative correlation of latent change score variables are introduced. True-change scores are differences between true scores as defined in classical test theory (Steyer, 2001), pertaining to two occasions of measurement, or waves. For the sake of brevity, the
probabilistic foundations of true score variables are not detailed (see Steyer, 1989,
2001). In SEM, true-score variables are modeled as latent variables and latent
change variables can be modeled as shown in Figure 1, where two items yield underlying variables at Waves 1 (Y*11 and Y*21) and 2 (Y*12 and Y*22). Disregarding
the latent change variable η2 – η1, Figure 1 shows a special case of what Steyer et
al. (2000) called a multi-state model with invariant loadings. Because η2 = (η2 –
η1) + η1, the latent change variable η2 – η1 is easily added, permitting an estimate
of the variance of the latent change variable, without computing the latent difference scores. As shown later, such a latent change variable can easily be related to
FIGURE 1
A latent change model.
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FIGURE 2
Latent change modeling for imperfect and perfect bipolarity: A two-wave design.
other latent change variables pertaining to other items. Of course, these latent
change variables can also be related to other manifest and latent variables.
Now, how to model imperfect and perfect dynamic bipolarity? As shown in Figure 2a, the basic idea suggested by Vautier and Raufaste (2003) to model (and then
test) imperfect linear dynamic bipolarity consists of connecting a second
multistate model for variables Y to the first model based on variables Y+, by introducing a covariance between both change variables. Figure 2a also shows that five
additional covariances have to be allowed.1
1Vautier and Raufaste’s (2003) model specification can be improved using unstandardized latent
variables and relaxing undue nullity of some covariances between them. The reanalysis of their data
suggests an acceptable goodness of fit, χ2ML(14, N = 214) = 26.90, SRMR = 0.038, CFI = 0.983,
RMSEA = 0.061 (to interpret these indexes, see Hu & Bentler, 1999), and yielded comparable results:
Using bootstrapped 90% confidence intervals, the correlation between latent change scores ranges
from –.69 to –.40, and the correlation between latent scores before change ranges from –.40 to –.08.
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As shown in Figure 2b, perfect dynamic bipolarity can be modeled by regressing one change variable, say δη–, on the other one (δη+). Importantly, perfect correlation requires that δη– has no disturbance term. Hence, the structural model has six
latent variables, four latent state variables, and two latent change variables. However, the latent change variables are deterministic functions of their corresponding
latent state variables, and one latent change variable is also a deterministic linear
function of the other latent change variable. Because latent variables have units of
measurement without intrinsic meaning, there is no rationale for hypothesizing
about the absolute value of the regression coefficient β. Moreover, means have no
substantive meaning, and the hypothesis of perfect bipolarity is only stated in
terms of a perfect correlation of the latent change variables, which means that the
variables of interest are linearly dependent.
Appendix A presents the implied covariance structure of a two-wave model. It
is demonstrated that three variances, the regression coefficient β, and three covariances suffice to parameterize the structural model. Testing perfect dynamic bipolarity against imperfect bipolarity, the χ2 difference test will have 3 df in a
two-wave design.
METHOD
Data
This study used the four-wave data that were analyzed by Steyer and Riedl (2004). If
perfect dynamic bipolarity is assumed, the more waves, the more convincing is the
test of the corresponding model, as perfect bipolarity can be false for each couple of
successive waves. (To model dynamic bipolarity requires at least two waves.) Our
own analyses paralleled Steyer and Riedl’s (2004) analyses, using their datasets, but
replacing the single-trait–multistate model with latent state-change models for dynamic bipolarity. The data were collected from German adults, using the German
version of the Multidimensional Mood Questionnaire (Steyer, Schwenkmezger,
Notz, & Eid, 1997), which measures the following three bipolar mood constructs:
feeling well versus feeling bad, being awake versus being tired, and feeling calm versus feeling tense. (Schimmack & Reisenzein [2002] labeled these constructs valence, energy, and tension.) The waves were 3 weeks apart.
Each dataset refers to a tetrad of items. A tetrad comprises items Y+1 and Y+2,
which are almost synonymous, and items Y–1 and Y–2, which also are almost synonymous. In addition, items of type Y+ and items of type Y– are antonymous. Then, all
four items along with their associated 5-point rating scale were used as a measurement device in a wave. Thus, any observed variable needs a second subscript: For
instance, the label Y23 refers to the ratings from Item 2 used at Wave 3. Employing
English translations of the original German items, Tetrad 1 is defined by [Y+1 =
good, Y+2 = content, Y–1 = bad, Y–2 = discontented] (valence), Tetrad 2 is defined by
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[Y+1 = well, Y+2 = happy, Y–1 = unwell, Y–2 = unhappy] (valence), Tetrad 3 is defined by [Y+1 = calm, Y+2 = relaxed, Y–1 = uneasy, Y–2 = tense] (tension), Tetrad 4 is
defined by [Y+1 = awake, Y+2 = lively, Y–1 = tired, Y–2 = exhausted] (energy), and
Tetrad 5 is defined by [Y+1 = good, Y+2 = well, Y–1 = bad, Y–2 = unwell] (valence).
Tetrad 6 is deliberately ill-defined by [Y+1 = good, Y+2 = well, Y–1 = tense, Y–2 = uneasy]. Because in Tetrad 6 items of type Y+ and items of type Y– are not exactly antonymous, it serves as a counterexample, expecting that perfect bipolarity is not
tenable. The sample sizes for each dataset ranged from 486 to 501, without missing
data (see Figure 4, first row).
Analyses
A test for the bivariate normality of underlying variables scaled with equal thresholds—means and variances not having to be invariant—is available in PRELIS
(see Jöreskog, 2001b). Thus, it was of interest to assess this data feature before estimating the models to assess the suitability of polychorics estimated using the assumption of equal thresholds.
Two main versions (A and B) of dynamic bipolarity were compared. The restricted model for perfect dynamic bipolarity (MA) is partially depicted in Figure
3. Technically, these are type 3B LISREL models (Jöreskog & Sörbom, 1996,
FIGURE 3
Latent change model for perfect dynamic bipolarity: Four-wave design.
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Chapter 6). They comprise 16 eta variables, including two method factors for
which the covariance is not depicted, but is free. The use of one method factor for
each pole follows Eid’s (2000) approach to modeling of method factors (see also
Eid, Lischetzke, Nussbeck, & Trierweiler, 2003). Because the items corresponding
to one pole are not the same, their semantic specificity must be taken into account
in the responding process. Because the same items are used across several waves,
systematic effects can be expected. One item is used as a reference, and the method
factor is defined conditionally to the true-score variable of the reference item.
Some general restrictions are defined as follows:
1. The loadings of the method factors are fixed to unity. Only their variances
and covariance have to be estimated.
2. The loadings of the state factors η+1i and η–1i, i = 1, …, 4, are fixed to unity
to fix the scales of the latent state variables, and invariance of the free loadings is assumed, as shown in Figure 3: Variables Y*+2i have the same loadings, and variables Y*–2i have the same loadings as well.
3. The error variances of Y*+ or Y*– variables corresponding to a given item
are invariant across waves.
In model MA, apart from the method factors, the structural part is based on five independent variables, two first-wave state variables, and three change variables.
Ten covariances (dotted lines) have to be estimated between the η variables, and
the regression coefficients linking the change variables are freely estimated.
Model MA has 109 df. Its detailed LISREL specification is provided in Appendix
B. Model MA can be weakened by relaxing the invariance of error variances and
invariant loadings, yielding model MAE with 91 df.
Loadings invariance in model MA is used as a parsimonious starting point. We
see no a priori reason why some set of analogous loadings should be different, nor
how they should differ. From a substantive viewpoint, the use of unjustified degrees of freedom in an unrestricted model may be somewhat useful to fit the
polychoric correlation structure, but it stresses our ignorance about data. However,
rejection of model MA can be due to problems in fit other than perfect bipolarity.
This possibility can be investigated using the unrestricted model MAE. Rejecting
model MAE would suggest imperfect bipolarity.
The restricted model for imperfect dynamic bipolarity (MB) permits the change
variables to not be perfectly correlated. Apart from the method factors, its structural part is based on two latent state variables and six change variables. Model MB
has 91 df. The detailed LISREL specification of model MB is provided in Appendix B. The unrestricted model for imperfect dynamic bipolarity (MBE) has 73 df.
If MA does not hold, but not because of equality constraints (which can be
tested using the contrast with MAE), the need for imperfect bipolarity can be
tested using MB. In the case of unequal loadings or error variances, the need for
imperfect bipolarity can be assessed by comparing MAE and MBE. The strategy
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FIGURE 4 Synopsis of the SEM analyses. Note. MA = restricted model for perfect dynamic
bipolarity (DB); MAE = unrestricted model for perfect DB; MB = restricted model for imperfect DB; MBE = unrestricted model for imperfect DB. Values between parentheses are the
bounds of the 90% confidence interval of the root mean squared error of approximation. Estimated correlations (rs) correspond to standardized covariances of the relevant change variables
(see text for more details).
for comparing the models was to estimate model MA first, and then to test an alternative model MB or MAE, depending of the goodness of fit of MA. Then, depending on the χ2 difference test, model MB or MBE was tested and compared to the selected model of type A. Figure 4 provides the synopsis of the decision trees for all
datasets. If imperfect dynamic bipolarity was retained, we estimated the correlations between the relevant change variables: The estimated correlation between
η+(2–1) and η–(2–1) was labeled r1, and so on.
Each dataset provided one polychoric correlation matrix and one asymptotic
covariance matrix, which were used as inputs along with the weighted least
squares estimation method in LISREL (Jöreskog & Sörbom, 1999). The complete
dataset is available electronically.2 As Steyer and Riedl’s (2004) models exhibited
2See
http://www.univ-tlse2.fr/cerpp/annuaire/vautier/
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close fit, we expected that the latent change analyses would also provide close fit
indexes. For the sake of simplicity, we used only the χ2 statistic, the root mean
squared error of approximation (RMSEA) and its 90% confidence interval.
RESULTS
Only items happy, unwell, and tired suggested measurement invariance across
waves, with no χ2 difference test being significant at the 5% level. The rest of the
items exhibited significant departure from bivariate normality when equal thresholds were assumed; for each item, six tests were performed, and the percentage of
positive tests at the 5% level ranged from 17% to 83%, depending on the items.
More details can be obtained electronically. Thus, polychorics were estimated
without specifying equal thresholds. Each dataset was thus compatible with
bivariate normality of the underlying variables (no test exceeding the 5% significance level). Readers interested in the polychoric correlations may find the appropriate output files electronically.
Figure 4 provides the goodness-of-fit summaries and the associated decision
trees. Although the estimated parameters were inspected for each model, it is not
possible to provide detailed path diagrams or tables in this article, because 17 models were estimated. However, no anomalies were found (the output files are also
available electronically). Overall, perfect dynamic bipolarity could be retained for
the valence datasets only (Sets 1, 2, and 5 with one exception detailed later).
Valence
Regarding Set 5, model MB was statistically better than model MA. This is because the first correlation is not compatible with perfect dynamic bipolarity, r1.MB
= .79. Only in Set 2 did the models not fit perfectly the input data.
Energy and Tension
Set 4 was more compatible with imperfect dynamic bipolarity, although the correlations between the relevant change variables ranged from –.93 to –.97. Goodness-of-fit indexes of the RMSEA related to model MA suggested close although
not perfect fit, whereas model MB reached the level of a nonsignificant χ2.
Set 3 was also more compatible with imperfect dynamic bipolarity, and the relevant correlations were more marked toward imperfect bipolarity, ranging from
–.84 to –.88. Perfect fit could be rejected at the .05 level, nonetheless.
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Test Set
Finally, Set 6, which was used as a test set, clearly suggested imperfect bipolarity,
with correlations ranging from –.64 to –.74. Interestingly, close fit (i.e., RMSEA <
.05) of model MA could not be rejected at the .05 level.
DISCUSSION
The contribution of this article is twofold: We technically refined Vautier and
Raufaste’s (2003) SEM approach to dynamic bipolarity, providing a solution to
specify and to contrast imperfect and perfect dynamic bipolarity. From a substantive viewpoint, we complemented Steyer and Riedl’s (2004) approach to
static bipolarity of antonymous affective judgments by a model for perfect dynamic bipolarity. It seems that perfect dynamic bipolarity can be found as a tenable view for some empirical data re-expressed using polychoric correlation
structures. In addition, this study sheds light on some general issues that are detailed next.
Could static and dynamic bipolarity be connected? If one assumes that the latent state variables related to ratings Y+ of Waves 1 and 2 follow η+1 = ξ+ + ζ+1 and
η+2 = ξ+ + ζ+2 (with the latent trait variable ξ+ and the latent state residuals ζ+i),
then latent change variable related to ratings Y+ of Waves 1 and 2 expresses η+2 –
η+1 = ζ+2 – ζ+1. Using the same assumption for ratings Y–, one has η–2 – η–1 = ζ–2 –
ζ–1. Thus, the correlation of latent change scores pertaining to ratings Y+ and Y– of
Waves 1 and 2 would be the correlation of the latent state residual differences.
However, the latent trait variables ξ+ and ξ– must have constant loadings, which is
not the case in Steyer and Riedl (2004, Appendix). Without this peculiar specification constraint, the definition of change factors cannot be closely linked to the definition of state residuals. Static bipolarity rests on correlated trait factors that permit
isolation of stable biases from the target state residuals, whereas dynamic bipolarity rests on latent change variables, without need for correlated trait factors. Static
and dynamic bipolarity do not necessarily reduce to each other and provide complementary approaches to the data.
From a methodological viewpoint, a first question relates to the usefulness of
the complicated modeling of latent change based on Steyer, Eid, et al. (1997;
Steyer et al., 2000). One could object that (a) raw difference scores would permit
modeling of perfect dynamic bipolarity, merely using a congeneric model based
on the four relevant observed difference variables; and (b) perfect dynamic bipolarity would be more convincing if more items were introduced in the measurement of the relevant attributes. A raw difference score for a given respondent is
expressed as Yi2 – Yi1, i indexing the item. Remember that because responses are
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collected using a Likert response format, the numerical values are only of an ordinal nature. For instance, the raw difference 5 – 3 has no meaning. Because a
raw difference score has no meaning, a covariance structure based on several
raw difference score variables is meaningless as well. Considering analogue ratings, it could be asserted that with the assumption of interval scales, raw differences could be computed, and the error components could be sorted out statistically using latent variables. This possibility would be attractive without floor or
ceiling effects, which suggest scaling problems (e.g., censoring). (Analogue
scales may reveal some interesting phenomena related to the behavior of the histograms, but this is a different topic.)
The second question relates to the small number of items that are managed by
the latent change models for dynamic bipolarity. Indeed, using polychorics limits
the number of observed variables, and also the number of response categories implemented in the response format, because (a) correlations are estimated based on
contingency tables with empty cells, and (b) the asymptotic covariance matrix requires few variables and a lot of respondents (Browne & Cudeck, 1993). We recognize that these models were estimated in nonoptimal conditions for stability of the
results, because the samples had only moderate sizes (about 500 cases). Such limits are inherent to the present state of the art in SEM. There is hope, however, that
new estimation methods implemented in Mplus (Muthén & Muthén, 2002) improve this situation. The use of parcels cannot be excluded as an alternative to
model data related to some broader constructs. However, we urge readers to carefully consider the related conceptual issues of forming composite variables (i.e.,
sums or means) without a firm sense of what addition refers to substantively (see
also Vautier, 2004).
We showed that the correlations of the underlying variables based on tetrads
belonging to the linguistic valence category are generally compatible with perfect dynamic bipolarity, whereas the corresponding correlational pattern based
on tetrads related to the energy or tension categories are more compatible with
imperfect dynamic bipolarity. If the change variables that were modeled are
credible representations of individual differences on affective attributes, the results support the view of a strong to perfect linear synchronism of the judgmental processes that yield the ratings. Even the models based on Tetrad 6 suggest
strong synchronism. However, we do not know whether positive and negative
judgments are elaborated by two different evaluative centers, as suggested by
Cacioppo and Berntson (1994; Cacioppo et al., 1999). In addition, our results
suggest that different tetrads in the valence category are not equivalent regarding
the goodness-of-fit summaries: Some tetrads worked better than other ones. Further empirical research is required to gain better knowledge of what is conveyed
by self-ratings related to core affect, and how judgments work dynamically in
similar or different experimental conditions.
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Working with the concept of perfect bipolarity, the results are based on the
probit convention for scaling. Under the hypothesis of the normal density function
to specify response probabilities, our results suggest that it is possible to model dynamic bipolarity using a given set of adjectives. Because we lack a detailed quantitative theory linking neural activity and judgment making, we must be cautious
about extrapolating on more hidden properties of hypothetical neural systems. Basically, what is measured using self-ratings is soaked in the essence of words. Furthermore, although the modeling of dynamic bipolarity is a confirmatory technique, its main purpose here is rather exploratory and empirical. It is a broader
issue to derive the epistemological consequences of the need for scaling conventions in data modeling (but see Essex & Smythe, 1999).
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APPENDIX A
COVARIANCE MATRIX
OF THE STRUCTURAL MODEL
DEPICTED IN FIGURE 2B
The covariance matrix is obtained as shown in the following table. Parameters in
parentheses are derived from parameters to be estimated, as detailed next.
η+1
η+2
δη+
η–1
η–2
δη–
η+1
η+2
δη+
η–1
η–2
δη–
Var(η+1)
(1)
ψ2
ψ1
(2)
(3)
(4)
(5)
(6)
(7)
(8)
Var(δη+)
ψ3
(9)
(10)
Var(η–1)
(11)
(12)
(13)
(14)
(15)
(1) Cov(η+1,η+2) = Cov(η+1, η+1 + δη+) = Var(η+1) + ψ2.
(2) Cov(η+1,η–2) = Cov(η+1, η–1 + βδη+) = ψ1 + βψ2.
(3) Cov(η+1,δη –) = Cov(η+1, βδη+) = βψ2.
DYNAMIC BIPOLARITY
(4)
(5)
(6)
(7)
(8)
(9)
(10)
(11)
(12)
(13)
(14)
(15)
407
Var(η+2) = Cov(η+1 + δη+, η+1 + δη+) = Var(η+1) + Var(δη+) + 2ψ2.
Cov(η+2,δη+) = Cov(η+1 + δη+, δη+) = ψ2 + Var(δη+).
Cov(η+2,η–1) = Cov(η+1 + δη+, η–1) = ψ1 + ψ3.
Cov(η+2,η–2) = Cov(η+1 + δη+, η–1 + βδη+) = ψ1 + βψ2 + ψ3 + βVar(δη+).
Cov(η+2,δη–) = Cov(η+1 + δη+, βδη+) = βψ2 + βVar(δη+).
Cov(δη+,η–2) = Cov(δη+, η–1 + βδη+) = ψ3 + βVar(δη+).
Cov(δη+, δη –) = Cov(δη+, βδη+) = βVar(δη+).
Cov(η–1, η –2) = Cov(η–1, η–1 + βδη+) = Var(η–1) + βψ3.
Cov(η–1, δη–) = Cov(η–1, βδη+) = βψ3.
Var(η–2) = Cov(η–1 + βδη+, η–1 + βδη+) = Var(η–1) + β2Var(δη+) + 2βψ3.
Cov(η–2, δη–) = Cov(η–1 + βδη+, βδη+) = βψ3 + β2Var(δη+).
Var(δη–) = Cov(βδη+, βδη+) = β2Var(δη+).
APPENDIX B
LISREL COMMANDS
FOR MODELS MA AND MB
TI Model Ma
Perfect Dynamic Bipolarity
4 waves with method factors
DA NI = 16 NO = 499 MA=PM
! Y* in the text replaced by y
LA
y+11 y+21 y+12 y+22 y+13 y+23 y+14 y+24
y–11 y–21 y–12 y–22 y–13 y–23 y–14 y–24
PM FI = set6.pm
AC FI = set6.acm
MO NY=16 NE=16 LY=FU,FI PS=SY,FI TE=DI,FR BE=FU,FI
LE
ETA+1 ETA+2 ETA+3 ETA+4
ETA*1 ETA*2 ETA*3 ETA*4
MetF+1 MetF–1
Et+(2–1) Et+(3–2) Et+(4–3)
Et–(2–1) Et–(3–2) Et–(4-)3
!Fixing the scales of the latent state variables
VALUE 1.0 LY(1,1) LY(3,2) LY(5,3) LY(7,4)
VALUE 1.0 LY(9,5) LY(11,6) LY(13,7) LY(15,8)
!Freeing and equating the other loadings on the latent state variables
FREE LY(2,1) LY(4,2) LY(6,3) LY(8,4)
FREE LY(10,5) LY(12,6) LY(14,7) LY(16,8)
EQUAL LY(2,1) LY(4,2) LY(6,3) LY(8,4)
EQUAL LY(10,5) LY(12,6) LY(14,7) LY(16,8)
!Freeing the variances and the covariance of the initial states
FREE PS(1,1) PS(5,5) PS(5,1)
408
VAUTIER, STEYER, JMEL, RAUFASTE
!Freeing the variance of the method factors
FREE PS(9,9) PS(10,10)
START 0.05 PS(9,9) PS(10,10)
!Fixing and equating the loadings of the method factors
VALUE 1.0 LY(2,9) LY(4,9) LY(6,9) LY(8,9)
VALUE 1.0 LY(10,10) LY(12,10) LY(14,10) LY(16,10)
!Freeing the covariance between the method factors
FREE PS(9,10)
START 0.02 PS(9,10)
!Defining latent change variables
VA 1.0 BE(2,1) BE(2,11)
FREE PS(11,11)
ST 0.5 PS(11,11)
VA 1.0 BE(3,2) BE(3,12)
FREE PS(12,12)
ST 0.5 PS(12,12)
VA 1.0 BE(4,3) BE(4,13)
FREE PS(13,13)
ST 0.5 PS(13,13)
VA 1.0 BE(6,5) BE(6,14)
!PS(14;14) linearly dependent
VA 1.0 BE(7,6) BE(7,15)
!PS(15,15) linearly dependent
VA 1.0 BE(8,7) BE(8,16)
!PS(16,16) linearly dependent
FREE BE(14,11) BE(15,12) BE(16,13)
ST –1.0 BE(14,11) BE(15,12) BE(16,13)
FREE PS(11,12) PS(11,13)
FREE PS(12,13)
FREE PS(11,1) PS(12,1) PS(13,1) PS(11,5) PS(12,5) PS(13,5)
EQUAL TE(1,1) TE(3,3) TE(5,5) TE(7,7)
EQUAL TE(2,2) TE(4,4) TE(6,6) TE(8,8)
EQUAL TE(9,9) TE(11,11) TE(13,13) TE(15,15)
EQUAL TE(10,10) TE(12,12) TE(14,14) TE(16,16)
PD
OU ME=WL ND=2 WP SI=LISOUT.MAT AD=OFF MI SE SC
TI Model Mb
Imperfect Dynamic Bipolarity
4 waves with method factors
DA NI = 16 NO = 499 MA=PM
LA
! Y* in the text replaced by y
LA
y+11 y+21 y+12 y+22 y+13 y+23 y+14 y+24
y–11 y–21 y–12 y–22 y–13 y–23 y–14 y–24
PM FI = set1.pm
AC FI = set1.acm
MO NY=16 NE=16 LY=FU,FI PS=SY,FI TE=DI,FR BE=FU,FI
LE
DYNAMIC BIPOLARITY
409
ETA+1 ETA+2 ETA+3 ETA+4
ETA*1 ETA*2 ETA*3 ETA*4
MetF+1 MetF–1
Et+(2–1) Et+(3–2) Et+(4–3)
Et–(2–1) Et–(3–2) Et–(4–)3
!Fixing the scales of the latent state variables
VALUE 1.0 LY(1,1) LY(3,2) LY(5,3) LY(7,4)
VALUE 1.0 LY(9,5) LY(11,6) LY(13,7) LY(15,8)
!Freeing and equating the other loadings on the latent state variables
FREE LY(2,1) LY(4,2) LY(6,3) LY(8,4)
FREE LY(10,5) LY(12,6) LY(14,7) LY(16,8)
EQUAL LY(2,1) LY(4,2) LY(6,3) LY(8,4)
EQUAL LY(10,5) LY(12,6) LY(14,7) LY(16,8)
!SET FREE the variances and the covariance of the initial states
FREE PS(1,1) PS(5,5) PS(5,1)
FREE PS(9,9) PS(10,10)
START 0.05 PS(9,9) PS(10,10)
!Fixing and equating the scales of the method factors
VALUE 1.0 LY(2,9) LY(4,9) LY(6,9) LY(8,9)
VALUE 1.0 LY(10,10) LY(12,10) LY(14,10) LY(16,10)
!Freeing the covariance between the method factors
FREE PS(9,10)
START 0.02 PS(9,10)
!Defining latent change variables
VA 1.0 BE(2,1) BE(2,11)
FREE PS(11,11)
ST 0.5 PS(11,11)
VA 1.0 BE(3,2) BE(3,12)
FREE PS(12,12)
ST 0.5 PS(12,12)
VA 1.0 BE(4,3) BE(4,13)
FREE PS(13,13)
ST 0.5 PS(13,13)
VA 1.0 BE(6,5) BE(6,14)
FREE PS(14,14)
ST 0.5 PS(14,14)
VA 1.0 BE(7,6) BE(7,15)
FREE PS(15,15)
ST 0.5 PS(15,15)
VA 1.0 BE(8,7) BE(8,16)
FREE PS(16,16)
ST 0.5 PS(16,16)
FREE PS(11,12) PS(11,13) PS(11,14) PS(11,15) PS(11,16)
FREE PS(12,13) PS(12,14) PS(12,15) PS(12,16)
FREE PS(13,14) PS(13,15) PS(13,16)
FREE PS(14,15) PS(14,16)
FREE PS(15,16)
FREE PS(11,1) PS(12,1) PS(13,1) PS(11,5) PS(12,5) PS(13,5)
FREE PS(14,1) PS(15,1) PS(16,1) PS(14,5) PS(15,5) PS(16,5)
410
VAUTIER, STEYER, JMEL, RAUFASTE
EQUAL TE(1,1) TE(3,3) TE(5,5) TE(7,7)
EQUAL TE(2,2) TE(4,4) TE(6,6) TE(8,8)
EQUAL TE(9,9) TE(11,11) TE(13,13) TE(15,15)
EQUAL TE(10,10) TE(12,12) TE(14,14) TE(16,16)
PD
OU ME=WL ND=2 WP SI=LISOUT.MAT AD=OFF MI SE SC