probabili-ties. If there weren’t individual differences in the two classes considered in our example, the response probabilities of the two classes reported in Figure 8.2 would indicate the general response probabilities of a two-class latent class model. In this case, each individual belonging to the same class would have the same response probabilities. Eid and Diener (2001) as well as Eid, Langheine, and Diener (2003) have applied latent class analysis to analyze intercultural differences in norms for emotions and in life satisfaction judgments.
Other Models of Item Response Theory
The basic ideas of IRT have been introduced with respect to binary response variables. Analogous approaches exist for items with more than two categories.
Baker, Rounds, and Zevon (2000) applied models for polytomous items to the measurement of subjective well-being. There are also IRT models in which items can differ in the form of their item characteristic curve. The handbook of van der Linden and Hambleton (1997) provides an overview of many IRT mod-els. Recent extensions of IRT models to multidimensional models are described by Rost and Walter (2006). Von Davier and Carstensen (2007) give an overview of binary and polytomous mixed Rasch models (also called mixture distribution Rasch models) and their usefulness in different areas of research. Overviews of latent class models are provided by Hagenaars and McCutcheon (2002).
Measuring the Different Temporal Facets
utilized by each individual to rate situational influences. As Eid, Schneider, and Schwenkmezger (1999) have shown, floor and ceiling effects can be found in the state ratings of subjective being because people with a high subjective well-being trait usually use the high categories of subjective well-well-being state scales.
Ratings of subjective well-being deviations, however, might not be affected by these ceiling and floor effects because people with generally high and generally low subjective well-being have the same room to rate situational deviations because the middle of the scale is the individual subjective well-being trait. A problem of the direct assessment is that the direct rating of one’s subjective well-being trait might be affected particularly by recall biases, such as influences of current mood and other characteristics of the subjective well-being process (Kahneman, 1999; Schwarz & Strack, 1999; Stone & Litcher-Kelly, 2006).
Aggregation Approach
To avoid recall biases, all three temporal facets can be measured on the basis of repeatedly assessed subjective well-being states using the aggregation approach. A subjective well-being trait could be measured as the mean of repeatedly measured subjective well-being states. The momentary deviation is then the current state minus the trait (aggregated state) value. The aggregation approach is a reasonable and rather simple approach. Moreover, it is not limited to self-report data but can also be applied to other types of measures, such as physiological data. It also has, however, several shortcomings. One of these is found in the fact that the aggre-gation approach does not inform us about what we are aggregating out. If we want to measure a stable subjective well-being trait by the aggregation of several states, we have to be sure that there is, for example, no trait change. The aggre-gation approach does not imply a measurement model of the different temporal facets of subjective well-being that can be used to test hypotheses about the structure of subjective well-being. This can be done by models of latent state–
trait theory described in more detail in the next section. Moreover, applications of this approach are presented to provide further insights into the reliability and validity of subjective well-being measures.
Separating Stable from Variable Determinants
of Subjective Well-Being by Latent State–Trait Modeling
Models of latent state–trait (LST) theory (Steyer, Schmitt, & Eid, 1999) analyze repeatedly measured variables. These models are based on the idea that the cur-rent state of an individual can be decomposed in a value characterizing the gen-eral subjective well-being level (trait, setpoint) and a value representing occasion-specific influences. In addition, it as assumed that measurement error affects each
measurement. Thus, to separate the different temporal facets and measurement error from each other, at least two occasions of measurement and two indicators measuring subjective well-being are needed. If the index i indicates the measure and k the occasion of measurement, an observed subjective well-being variable Yik is decomposed in the following way:
Yik= Sik + Eik
observed subjective well-being variable latent state variable error variable Moreover, the latent state variable Sikis decomposed into a latent trait variable Tikand a residual Oik, which represents occasion-specific influences:
Sik= Tik+ Oik
latent state variable latent trait variable occasion-specific residual Occasion-specific residuals represent all influences that do not depend on the trait. These occasion-specific deviations can be caused by the fact that individuals can be in different (“inner”) situations and that individuals interact in different ways in various situations. Taking both equations together—that is, by replacing the latent state variable in the first equation by the components of the latent state variable in the second equation, the basic decomposition of an observed variable in LST theory results:
Yik = Tik + Oik+ Eik
It is important to note that each observed subjective well-being measure on each occasion of measurement is decomposed into a trait variable, an occasion-specific variable, and an error variable without making any assumption about the struc-ture of the different subjective well-being measures.
Several models have been defined to test different hypothesis about the structure of subjective well-being. A model that has often been applied is depicted in Figure 8.6 for four subjective well-being measures (i = 1, . . . , 4) and two occasions of measurement k (k = 1, 2). In this model it is assumed that the four measures are unidimensional on the level of the occasion-specific variables but multidimensional on the level of the trait variables. The fact that measures are not unidimensional on the trait level is often found in longitudinal studies because the repeated measurement of the same items makes it possible to identify stable item-specific effects. Usually, however, the intertrait correlations are very high. The model depicted in Figure 8.6 is a model of confirmatory factor
analy-sis: The parameters λTik are the loadings of the observed subjective well-being variables on the trait factors, and the parameters λOik are the loadings of the observed variables on the occasion-specific factors. The occasion-specific factors are uncorrelated because it is assumed that the whole stability is explained by the trait factors, and occasion-specific influences of different occasions are indepen-dent. The fit of an LST model such as the one presented in Figure 8.6 can be analyzed by comparing the covariances of the observed variables with the covariances of the observed variables predicted by the model using the different fit criteria of structural equation modeling (Schermelleh-Engel, Moosbrugger, &
Müller, 2003).
In the model depicted in Figure 8.6 (and also in other models of LST the-ory), the variance of the observed variable can be decomposed into one part that is determined by the trait variable (consistency), one part that is determined by the occasion-specific variable (occasion-specificity), and one part that is due to measurement error (unreliability). The consistency coefficient
CO Y Var T
Var Y
ik
ik i
ik
( ) ( )
( )
= λT 2
FIGURE 8.6. Latent state–trait model for four observed variables. Yik, i, indicator; k, occasion of measurement; Ti, indicator-specific trait variables; Ok, common occasion-spe-cific variables; Eik, error variables;λTik, trait factor loadings;λOik, occasion-specific factor loadings. For each factor one loading was fixed to 1 for identification reason. For simplic-ity reasons not all error variables are marked.
indicates the degree of the variance of an observed variable [Var(Yik)] that is due to interindividual trait differences [λ2TikVar(Ti)]. A high value for this coefficient indicates that interindividual differences in subjective well-being on an occasion of measurement are mainly determined by stable interindividual (trait) differ-ences. Trait measures of subjective well-being should have high consistency val-ues.
The specificity coefficient
SPE Y Var O
Var Y
ik
ik k
ik
( ) ( )
( )
= λO 2
represents the proportion of variance of an observed variable that is due to occasion-specific influences and not due to trait and error influences. State mea-sures of subjective well-being that should be sensitive to change (occasion-specific influences) ought to have rather high (occasion-specificity coefficients.
The consistency and occasion-specificity coefficient add up to the reliability coefficient, which is 1 minus the proportion of interindividual differences that is due to measurement error:
Rel( ) ( )
( )
( )
( ) –
Y Var T
Var Y
Var O Var Y
ik
ik i
ik
ik k
ik
= λT +λO =
2 2
1 Var E Var Y
ik ik
( )
( ) All observed variables should have high reliability values.
The model presented so far is a model for continuous subjective well-being measures. It is usually applied to subjective well-being scales consisting of multi-ple items. In order to obtain two (or more) indicators i, a scale can be split into two test halves (e.g., each test half is the mean of the responses to one half of the items) or multiple parcels.
Eid (1996) has shown how LST models can be defined for binary and ordinal variables. These models allow the estimation of the consistency and occasion-specificity for single items. This information can be used to select items in such a way as to maximize the consistency or occasion-specificity of a scale measuring subjective well-being. Eid and Langeheine (1999) defined LST latent class models to analyze typological subjective well-being differences on the state and trait levels. Cole, Martin, and Steiger (2005) have shown how LST models can be extended to consider an additional autoregressive process on the level of the occasion-specific variables. An autoregressive process is needed when there is a change process on the level of the occasion-specific variables, which simply means that the occasion-specific deviation on one occasion of measurement depends on an occasion-specific deviation of another occasion of measurement.
Applications of Latent State–Trait Models
in the Realm of Subjective Well-Being Measurement
Models of LST theory have been applied to analyze different questions concern-ing the reliability, validity, and structure of subjective well-beconcern-ing measures. The main results of these studies illustrate the different ways in which models of LST theory can be applied.
The Polarity and Dimensionality of the Affective Component of Subjective Well-Being
One of the most hotly debated issues concerning the structure of the affective component of subjective well-being is the question of whether positive and neg-ative affects are independent or not (see Schimmack, Chapter 6, this volume).
According to a bipolarity model of positive and negative affect, the two types of affect are opposite poles of one continuum. In contrast, the monopolarity model supposes that positive and negative affects cannot be ordered on a single contin-uum. Eid (1995) analyzed this research question in detail on the basis of LST modeling. He applied LST models for ordinal variables to analyze the structure of two pairs of semantically opposite items measuring momentary positive and negative affect (happy, unhappy, satisfied, dissatisfied). LST models for ordinal vari-ables correct for differences in the distribution of positive and negative affect items. Whereas positively keyed state items usually show a symmetric distribu-tion, negatively keyed state items are usually skewed. If items differing in their distributions are analyzed with factor models for continuous variables (as it is often done), artificially monopolar factors can be produced. Eid (1995) analyzed the structure of affect ratings by comparing the fit of several LST models. The model presented in Figure 8.6 was the only model that fit the data (Eid [1995]
analyzed three occasions of measurement; the model structure was the same as the model in Figure 8.6 but contained one more occasion of measurement).
According to this model, the different items measuring positive and negative affect are bipolar on the level of occasion-specific influences but monopolar on the trait level. Situational influences that make people happier are exactly those that make people less unhappy. The unidimensional structure on the occasion-specific level means that one can perfectly predict the latent deviation score of unhappiness (feeling more or less unhappy in this situation) by the latent happi-ness deviation score. On the trait level there were item-specific differences.
However, the different trait variables are highly correlated (between r = .67 and r = .92), with the correlations between the items of the same valence being higher (between unhappy and dissatisfied: r = .92; between happy and satisfied: r = .83) than between the semantically opposite items (happy and unhappy: r = .67;
satisfied and dissatisfied: r = .82). The nonperfect correlations on the trait level
show that each item has a stable component (unique meaning) that is not shared with the other items.
Mood Influences on Life Satisfaction Judgments
Whereas a state–trait distinction has generally been accepted for the affective component of well-being, life satisfaction judgments are considered as a stabler subjective well-being component. Life satisfaction judgments should be sensitive to changes in the conditions in which people live, but they should not be sensi-tive to immediate mood fluctuations (Campbell, Converse, & Rodgers, 1976).
The assumption that life satisfaction items are immune to short-time fluctuations such as mood has been questioned strongly (Schwarz & Strack, 1999). Eid and Diener (2004) analyzed mood influences on subjective well-being judgments in a longitudinal study with three occasions of measurement using LST models. They found that life satisfaction judgments are much stabler than mood judgments.
Whereas the occasion-specificity coefficients were between .40 and .52 for the mood ratings, they were much lower for the life satisfaction judgments (between .12 and .16). The correlations between the occasion-specific variables of mood and life satisfaction were very low and not significant for the first two occasions (r
= .13 and r = .23) and higher and significant for the third occasion of measure-ment (r = .55). These low correlations indicate that the occasion-specific vari-ability of mood can only explain between 1.7 and 30% of the varivari-ability of life satisfaction judgments. Eid and Diener (2004) assumed that the higher correlation on the third occasion might be due to the fact that the last occasion of measure-ment was at the end of the semester (participants were college students) and that typical events at this time (e.g., exams) might have had an influence on mood and life satisfaction. In general, the much higher stability of life satisfaction judg-ments and the rather small correlations between the variability of mood and life satisfaction demonstrate that mood effects on life satisfaction judgments seem to be rather small in nonexperimental survey research.
Validity of Direct Assessment Methods
Eid, Notz, Steyer, and Schwenkmezger (1994) analyzed the validity of the Mood Level subscale of Underwood and Froming’s (1980) Mood Survey in a longitudi-nal study with four occasions of measurement. An example of an item from this scale is “I usually feel quite cheerful.” The consistency coefficients from the four occasions of measurement are very high (between .78 and .85) and are close to the reliability coefficients (between .89 and .92). Hence, occasion-specific influ-ences are very small (occasion specificities: between .06 and .11), proving that this subjective well-being scale assesses very stable aspects. Moreover, the latent
trait variable of this scale is highly correlated with the latent trait variable of the mood state ratings (r = .78), verifying high convergent validity.
Eid et al. (1999) scrutinized the validity of the direct assessment of occasion-specific deviations (“Do you feel better or worse?”) and of trait assessments (“How do you feel in general?”) using the same adjectives of a pleasantness–
unpleasantness mood scale. Applying LST models, they found the following results: (1) The deviations ratings are uncorrelated with the trait ratings, indicat-ing that they represent—as intended—pure situational and/or interactional effects. (2) The occasion-specific variables of the state ratings are highly corre-lated with those of the deviation ratings (between .70 and .90), proving their validity. (3) The consistency coefficients of the deviation ratings (between .18 and .32) are smaller then those of the state ratings (between .27 and .43), show-ing that deviation ratshow-ings represent mainly situational influences. However, they differ significantly from 0. This unexpected small proportion of stable differences of the deviation ratings can be explained by the fact that all participants (all were students) had been repeatedly assessed after the same lecture. The stability of deviation ratings indicates the stability of (objective) situations. The trait value of repeatedly measured deviations represents, therefore, how the students generally felt after the lecture compared to how they felt in general. Eid et al. (1999) showed that the mean of repeatedly measured deviations can be used to suppress that part of the repeatedly measured states that are due to stable situational aspects. As a consequence, the corrected correlation between the indirectly measured subjec-tive well-being traits (on the basis of repeatedly measured states) with the directly assessed subjective well-being trait (trait self-rating) is higher (r = .79) than the uncorrected correlation (r = .71). The effect is even stronger for the Mood Level subscale of the Mood Survey (Underwood & Froming, 1980), where the cor-rected and uncorcor-rected correlations are r = .76 and r = .62, respectively. These results substantiate a rather high convergent validity of direct and indirect assess-ment procedures. Moreover, they also demonstrate that it is useful to suppleassess-ment state ratings of subjective well-being by deviations ratings in order to find out whether the subjective well-being people experience during a specific period of time is representative of their lives in general. Thus, even the indirect assessment of a subjective well-being trait by repeatedly measured states can profit from the additional direct assessment of the different temporal facets. As in other domains of the social and behavioral sciences, multimethod approaches offer deeper insights into the questions at hand than single-method designs (Eid & Diener, 2006).
Interindividual Differences in Intraindividual Variability
Subjective being states characterize random fluctuations of subjective well-being around the subjective well-well-being trait level. The degree of these
fluctua-tions is not unsystematic but highly stable because there are strong interindividual differences in intraindividual variability. Using daily assessment data of the affec-tive component of subjecaffec-tive well-being, measured each evening over a period of 52 days, Eid and Diener (1999) applied LST modeling to examine whether intraindividual variability is stable over time. They calculated the intraindividual standard deviation over all the days of 1 week for two test halves of seven scales, measuring three positive and four negative affects. The intraindividual standard deviations were the observed variables in a LST model, such as the one depicted in Figure 8.6. The estimated occasion-specificity coefficients (between .00 and .18 for the positive affects and between .07 and .57 for the negative affects) showed that the intraindividual variability of subjective well-being during 1 week is highly stable and has, particularly for positive emotions, the character of a trait.
Separating Stable from Variable and Resilient from Nonresilient Individuals with Mixture Distribution Latent State–Trait Models
Models of LST theory assume that the estimated parameters of the model charac-terize all individuals of the population appropriately. This means, for example, that all individuals stem from the same population in which the degree of expected variability is described by the variance of the occasion-specific factors.
Interindividual differences in intraindividual variability are allowed but only to a certain degree. For example, if the variances of the occasion-specific variables are small, the expected variability is also small, and if the variances are large, there could be strong differences in the variability. The population, however, might not be homogeneous with respect to the expected variability. Given the strong interindividual differences in intraindividual variability, there is good reason to assume that some subpopulations differ in the degree of intraindividual variability they allow. Eid and Langeheine (2003, 2007) have extended LST models for cat-egorical variables to mixture distribution models, and Courvoisier, Eid, and Nussbeck (2007) defined mixture distribution LST models for continuous vari-ables. These models assume that the population consists of subpopulations differ-ing in the variances of the occasion-specific variables. Courvoisier et al. (2007), for example, found that there were two latent classes. The larger class (76%) was characterized by higher intraindividual variability of subjective well-being mea-sures and generally lower subjective well-being mean values. The smaller class (24%) was stabler and showed a high degree of subjective well-being. Most inter-estingly, the two latent classes differed in the influences of daily hassles and uplifts on the occasion-specific mood difference. Whereas in the first, more variable class, daily hassles had a negative and daily uplifts had a positive influence on sub-jective well-being, in the second class, only daily uplifts had a significant positive influence, whereas the influence of daily hassles was not significant. People in the
second class seem to profit from positive daily events and to be rather resilient to the influences of negative life events.
Separating Trait Change from Occasions-Specific Variability
In the LST models presented here, the latent trait variables do not change within the time period considered. However, LST models can be extended easily to allow trait change as well. Intervention methods for enhancing subjective well-being (see Emmons, Chapter 23, this volume; Fredrickson, Chapter 22, this vol-ume; King, Chapter 21, this volume) aim not only at changing momentary sub-jective well-being states but also the habitual subsub-jective well-being level. If one measures subjective well-being states several times before an intervention and several times after an intervention, the general mood level (i.e., the mean of the different states) should be higher after the treatment than before. LST models with trait change allow distinguishing between the short-term state variability caused by situational influences and more enduring trait changes (Eid &
Hoffmann, 1998). Trait change represents the degree of change on the trait level free of influences due to measurement error and occasional influences in the period before and after the treatment. Steyer (2005) provides a detailed discussion of the conclusions that can be drawn about the effect of a treatment when it is evaluated by LST trait change models.
Psychometric Modeling of Single-Case Data
Models of LST theory require a sample of individuals and a sample of occasions of measurement. The analysis of different individuals allows an examination of interindividual differences in subjective well-being, and the consideration of dif-ferent occasions of measurement enables the analysis of intraindividual differences over time. To apply these models, however, it is necessary that the models fit for the total population or at least for subpopulations (mixed models).
If the research interest is in a single individual, factor models for single-case data can be applied. These models focus on the structure of the occasion-specific deviation variables of a single individual. If one is, for example, interested in the structure of the four positive and negative affect items, a single-case factor model, with a factor for positive affect and a factor for negative affect, would have the form presented in Figure 8.7. This model has no trait factor because the trait is a constant for an individual, and there is only one positive and one negative affect factor because the values of the observed variable are now the different individual values at different occasions of measurement. The values of the factors are the true occasion-specific values for a single individual on the different occasions of measurement. This single-case factor model is a model of p-technique factor
analysis (Nesselroade, 2001). Ong, Horn, and Walsh (2007), for example, used p-technique factor models to analyze the structure of hedonic and eudomanic well-being of nine individuals who recorded their well-well-being 10 times a day over a period of 61 days. They found that the two constructs showed discriminant validity for all individuals analyzed. However, the two-factor model did not fit for all individuals in the same way. Feldman (1995) explored whether the struc-ture of affect differs between individuals. She analyzed the response to 16 affect markers that were repeatedly measured over 62–91 days. She found a two-factor structure for nine participants, a three-factor structure for 12 participants, and a three-factor structure for two participants. Moreover, the participants differed in the degree to which valence and arousal components were present in their rat-ings. Larsen and Cutler (1996) analyzed 21 mood adjectives with p-technique factor analysis. They found strong individual differences in emotional complex-ity, as indicated by the fact that the individuals differed in the number of factors needed to explain the structure of affect (between two to five factors). Emotional complexity was negatively correlated with subjective well-being for men but positively correlated for women.
Models of p-technique factor analysis are appropriate only if the repeated measurements are independent. If there is a change process, the dependency between the occasions of measurement has to be considered. For example, there can be an autoregressive process indicating that a subjective well-being state depends on the subjective well-being states measured before. In this case, single-case dynamic factor analysis that models the temporal structure can be applied (e.g., Nesselroade, 2001). Ferrer and Nesselroade (2003), for example, showed how dynamic factor analysis can be applied to examine the structure of positive and negative affect. Moreover, they showed how this approach can be used to compare the structure of affective subjective well-being between a husband and FIGURE 8.7. Single-case factor model for repeatedly measured indicators of positive and negative affect. PAi, indicators of positive affect; NAi, indicators of negative affect;
PA, NA, occasion-specific factors representing true occasion-specific values for a single individual; EPAi, error variables. For simplicity reasons not all error variables are marked.
wife and how the mutual influences of affect can be tested. They found that the husband’s negative affect particularly influenced the wife’s positive and negative affect.
If there are nonlinear changes such as daily or weekly cycles, frequency domain analysis (spectral analysis) can be applied (Larsen, 1987). Chow, Grimm, Ram, and Fujita (2007) found, for instance, strong individual differences in the extent to which individuals’ emotions follow a weekly cycle. They also show how modeling strategies of IRT and spectral analysis can be combined.
Whereas mixed models such as the mixed Rasch models and the mixed LST models follow a top-down strategy (searching for subgroup differences if the model does not hold for the population), single-case factor analysis can be applied for bottom-up strategies—that means, the search for a common structure of subjective well-being based on the comparison of single cases. In order to combine idiographic methods such as single-case analysis with nomothetic approaches for explaining individual differences, the parameters of single-case analysis can be related to other (personality) variables (Larsen, 2007).
Conclusion
This chapter focused on the psychometric models (latent variable models) that can be applied to analyze the reliability and validity of subjective well-being measures. These models offer many ways to analyze the psychometric quality of subjective well-being both on the individual level as well as on the level of indi-vidual differences. Besides the latent variable models, all other statistical methods can be applied to analyze subjective well-being data. The Handbook of Methods in Positive Psychology, edited by Ong and van Dulmen (2007), shows how advanced statistical approaches such as structural equation models, hierarchical linear mod-els, IRT modmod-els, single-case methods, experimental methods, and many others can be applied to learn more about the structure, condition, and consequences of subjective well-being.
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