The cognitive facet of subjective well-being is typically assessed by self-report items. These items refer to a person’s life as a whole and/or the different life domains. Figure 8.2 shows the items of the Life Satisfaction subscale of the Freiburg Personality Inventory (FPI; Fahrenberg, Hampel, & Selg, 1984), one of the most widely applied personality questionnaires in German-speaking coun-tries. All items are answered on a binary response scale with the categories yes and no. When analyzing the responses to life satisfaction items, one finds typically two results. (1) There are strong individual differences in life satisfaction—in other words, there are people who are more or less satisfied. (2) Items differ in their distributions: There are items that many people affirm, such as the last item in Figure 8.2, which 74% of the sample agreed with (“I am usually very confident about the future”). There are also items that only a small number of individuals agree with: Only 33% concurred with the second item in Figure 8.2 (“I am at peace with myself and have no inner conflicts”). Therefore, to enable us to ana-lyze the interindividual as well as the interitem differences and evaluate the psy-chometric property of this scale, models of item response theory (IRT) were applied. In IRT models, the probability of an individual item response is a func-tion of an individual’s standing on a latent variable representing true (“error-free”) individual differences in life satisfaction as well as the standing of the item on this latent variable, representing the degree of life satisfaction necessary to affirm an item with a certain probability (“difficulty” of an item).
Measuring Quantitative Differences: The Rasch Model
The simplest model of IRT is the Rasch model (also called the one-parameter logistic model; Rasch, 1960). In this model the probability of an item response depends on only two parameters: a person parameter indicating the latent (life satisfaction) value of an individual, and an item parameter indicating the difficulty of an item.
The curves that describe the dependency of the response probability from the latent (life satisfaction) variable are depicted for three items in Figure 8.3.
Because all items are binary, it is sufficient to present the probability curve for only one category (the probability of the first category is 1 minus the probability
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FIGURE8.2.SelecteditemsoftheLifeSatisfactionsubscaleoftheFreiburgPersonalityInventoryanditemdistributionsinthetotalsample =480)andthetwolatentclasses(relativefrequenciesofthecategoriesindicatingsatisfaction).
of the second category). This model assumes that all items are unidimensional, meaning that they measure the same latent trait (life satisfaction). Individuals dif-fer in their latent life satisfaction values. The higher the latent life satisfaction value of an individual, the higher is his or her probability to agree with a life sat-isfaction item, which simply means saying yes to a positively keyed item (such as the first one in Figure 8.2) and saying no to a negatively keyed item (such as the fourth one). All item responses are coded in such a way that a “1” indicates a response representing life satisfaction (“positive response”) and a “0” represents a nonsatisfaction answer (“negative response”). The response probabilities of the categories “1” are depicted in Figure 8.3 meaning that a high value of the latent trait indicates life satisfaction. There are differences between the items: Some of the items are considered to be easy; even individuals with a comparatively low life satisfaction value have a high probability of responding positively to the item (e.g., item at the left in Figure 8.3), and some items practically require a high life satisfaction value for a positive response (difficult items, e.g., item at the right).
The ordering of the difficulties of the items does not differ between individuals.
All individuals have higher response probabilities for the easier items than for the more difficult items. The Rasch model is defined by the following equation
P Y e
i e
i
i
( | )
–
= = –
1 +
θ 1 θ θσσ
where a value of the conditional probability function P(Yi= 1|θ) is the probabil-ity of positive satisfaction judgment for a value of the latent (life satisfaction) vari-FIGURE 8.3. Item characteristic curves of three items in the Rasch model. Item parameters:σ1= –1, σ2= 0, σ3= 1.
ableθ;σiis the item difficulty; and e is the exponential function. The probability function that describes the dependency of the response probability from the latent variable is called the item characteristic curve. As can be seen in the item char-acteristic curves shown in Figure 8.3, the shape of the item charchar-acteristic curves is always the same, but the items differ by their location on the latent variable.
This location is marked by the item parameterσi. The value of an item parame-ter is the value of the latent (life satisfaction) variable, where the response proba-bility for a category of this item equals 0.50. An item becomes easier with the decreasing value of the item parameter. Easiness means, in this context, that even individuals with a comparatively low life satisfaction value answer this item with a relatively high probability in the category that indicates life satisfaction. In addi-tion, the assumption of local independence is made. Local independence means that the latent variable explains all associations between the observed variables. The items are associated because they are measuring the same trait (life satisfaction).
But within a group of individuals with the same latent life satisfaction value, the different items are not related. Differences between item responses that are not determined by the latent satisfaction variable reflect unsystematic random influ-ences that are not due to life satisfaction or another common trait.
The Rasch model has many important statistical properties. For example, it is the only psychometric model for binary items for which the sum score (the number of all affirmed items) contains all information about interindividual dif-ferences with respect to life satisfaction. However, for analyzing life satisfaction items the model has one disadvantage in its assumption that the item difficulties do not differ between individuals. This means, for example, that the difficulties of the items “I have a good relationship/marriage” and “I have (had) a job that is (was) very satisfying” have to be the same for all individuals. However, it seems to be more likely that there are qualitative (structural) differences between indi-viduals as well. There might be indiindi-viduals for whom it would be easier to say that they are satisfied with their jobs than to say that they are satisfied with their marriage, and vice versa, thus suggesting that the Rasch model is an appropriate model for analyzing individual (quantitative) differences but only in the case when there are no structural differences between individuals. When all items measure, for example, general life satisfaction, and no subgroup differences with respect to the item differences can be expected, the Rasch model would be an appropriate model to separate measurement error and random influences from true life satisfaction differences. In the current case, however, where different life domains are assessed as well, structural (qualitative) differences can be expected.
Measuring Quantitative and Qualitative Differences in Life Satisfaction Judgments: The Mixed Rasch Model
To provide researchers with the tools to consider both structural (qualitative) dif-ferences as well as individual (quantitative) difdif-ferences, the Rasch model has been
extended to the mixed Rasch model. This extended model assumes that the pop-ulation consists of different subpoppop-ulations. For each subpoppop-ulation a Rasch model fits the data, however, the item parameters can differ between sub-populations. Thus, this model allows quantitative and qualitative individual dif-ferences. Qualitative differences are represented by different latent classes that differ in the item parameters. Quantitative differences are allowed within latent classes because individuals can have different latent life satisfaction values. If g denotes a subpopulation (latent class), the mixed Rasch model is defined by
P Y e
i g e
g ig
g ig
( | )
–
= = –
1 + θ 1
θ θ
σ σ
(Rost, 1990). The difficulty of an item i (σig) and the value of an individual v (θvg) can differ between classes g.
The model assumes that an individual can (and must) belong to only one latent class. However, it cannot be perfectly determined to which latent class an individual belongs. Therefore, a latent life satisfaction value will be estimated for each individual and each class. The probabilities of belonging to the different latent classes (assignment probabilities) can be estimated for each individual on the basis of his or her observed responses to the different items. Individuals can then be assigned to the latent class for which their assignment probability is maxi-mal, and the latent satisfaction value in the latent class chosen can be taken as the best estimate of the individual’s latent satisfaction. The mean of the assignment probabilities of all individuals assigned to the same class indicates the reliability of class assignments and the separability of classes.
In order to find out how many classes are necessary, a Rasch model with several latent classes must be specified and tested. The fit of the different models can be compared by information criteria such as Akaike’s information criterion (AIC) or the Bayesian information criterion (BIC). These criteria weigh the fit of the model with the complexity of the model (the mixture distribution model is a more complex model). The best fitting model—that means, the most parsimoni-ous model that explains the observed data appropriately—is the model with the smallest values of both these criteria. The BIC is superior to the AIC when the number of possible response patterns (2K, where k is the number of items) is much larger than the sample size (Rost, 2004). The model with the lowest value of the information criteria is the best fitting model. A second possibility is to compare the observed frequencies of the different response patterns with the expected frequencies of these response patterns, given the parameters of the model. A model fits the data well when the expected frequencies of the different response patterns are close to the observed frequencies. That means that the model is able to predict reality. There are several statistical tests that can be used to determine whether the differences between the observed and the expected
frequencies are significant; for example, the Pearson χ2test, the likelihood ratio test, or the Cressie–Read test. These statistics are asymptotically χ2 distributed (when all expected frequencies are at least larger than 1); in cases of large item patterns and smaller samples, the distribution of the statistics can be estimated by bootstrapping analysis. Von Davier (1997) has shown that the bootstrap works well for the Pearsonχ2test and the Cressie–Read test but not for the likelihood ratio test.
An Application of the (Mixed) Rasch Model
The Rasch model and the mixed Rasch model with different latent classes have been applied to the items presented in Figure 8.2. Because the sample size is small (n = 480) in relation to the number of possible response pattern (512), the selec-tion of the model was based on the BIC. The BIC shows that the model with two latent classes fits the data best (1 class: 5170.96; 2 classes: 5131.10; 3 classes:
5172.46). Also the bootstrap fit indices indicate that the two-class model shows a good fit to the data (p values of the bootstrapped distributions: Pearson: p = .16, Cressie–Read: p = .04). The mean assignment probabilities are .96 for the first class and .91 for the second class. These assignment probabilities indicate a high reliability of class assignments and high separability of classes. The item difficulty parameters are given in Figures 8.4 and 8.5. Figure 8.4 reveals some interesting differences between the two classes. First of all, there is a strong difference with respect to item 7 (“All in all, I am very satisfied with my life”). This item is easi-est in class 1 and rather difficult in class 2. Moreover, there is another strong dif-ference with respect to item 2 (“I am at peace with myself and have no inner conflicts”). This item is the most difficult item in both classes but is much more difficult in class 2. Figure 8.5 contains the same information as Figure 8.4 but presents it in a somewhat different way. In Figure 8.5, the different ordering of the items in the two different classes is more visible than in Figure 8.4, whereas in Figure 8.4, the differences between the two classes with respect to the single items are more obvious. One very interesting difference between the two classes concerns the item “All in all, I am very satisfied with my life.” This item is the easiest in class 1. People in class 1 have a generally positive view of their life (item 7) and their futures (item 9), and in order to have this generally positive view, they do not have to be satisfied with all domains of their life.
In class 2 the item “All in all, I am very satisfied with my life” is a rather dif-ficult one. A respondent needs a rather high value on the latent life satisfaction variable to affirm this item with a comparably high probability. For people belonging to this class, it is easier for them to admit that they are satisfied with their relationship or their job than with life in general. This might also explain why it is very difficult for people in this class to be at peace with themselves without experiencing inner conflicts. Practically no one in this class is at peace
with him- or herself (see the distribution of the item categories of the two classes in Figure 8.2).
The differences between the two classes might be explained partially by the distinction between top-down and bottom-up processes found in the theories of subjective well-being (Diener, 1984). Top-down process theories assume that there are temperamental differences in subjective well-being that exert an influ-ence on the evaluation of different life domains: People who are generally happy evaluate their life domains in a more positive way. According to bottom-up pro-cess theories, general life satisfaction is the result of the life satisfaction in different domains. Individuals of the first class might be more top-down driven, because the general life satisfaction comes first on the latent life satisfaction dimension.
People of the second class might be more bottom-up driven, which means that FIGURE 8.4. Item parameters for the two classes in the mixed Rasch model. C1, Class 1; C2, Class 2. The items indicating dissatisfaction were recoded for the analysis so that a high value indicates satisfaction. To facilitate the understanding of the results, the nega-tively keyed items were reworded.
the satisfaction with one’s life might depend on the fulfillment of different expec-tations and aspirations, and these people might only be satisfied if all their aspira-tions are fulfilled. An alternative explanation would be that people belonging to the first class respond so positively to the items representing general life satisfac-tion because of self-decepsatisfac-tion mechanisms, whereas they have a more appropri-ate view of their lives when they are asked more concretely about the specific life domains.
The estimated distributions of the latent life satisfaction variables and the sum scores revealed that the first class consists primarily of individuals with high life satisfaction values (mean sum score: 6.26, SD: 1.66), whereas the second class is a class of relatively dissatisfied individuals (mean sum score: 2.02, SD: 1.26).
Because of the differences between the latent classes in the distribution of the latent variables as well as the item parameters, the two latent classes also differ in the expected frequencies of the item responses in the two classes (see Figure 8.2).
The frequencies of the satisfaction categories are very high in the first class and rather low in the second class. The differences are particularly strong for the item
“All in all, I am very satisfied with my life”: Whereas 93% of all individuals of the first class agree with this statement, only 13% of the individuals in the second class do so.
FIGURE 8.5. Comparison of the item parameters in the two latent classes.
The two-class structure has some interesting consequences for the individual assessment of subjective well-being as well as for research on the determinants and consequences of individual differences in life satisfaction judgments. With respect to the individual assessment of life satisfaction, a two-step assessment pro-cedure has to be applied. First, individuals must be assigned to latent classes based on their assignment probabilities. The latent class to which an individual is assigned characterizes his or her life satisfaction type. Then the individual life sat-isfaction score can be estimated. This latent value describes his or her degree of life satisfaction.
To analyze the conditions and consequences of life satisfaction differences, the two types of information (qualitative and quantitative) have to be taken into consideration. One interesting question concerns the differences between the two classes and whether class membership can be explained or predicted by other variables. In order to explain interclass differences, the point–biserial correlation between the latent class variable and the extraversion, neuroticism, and openness (social desirability) scales of the FPI were calculated. These correlations show that people belonging to class 2 respond in a less socially desirable manner (r = –.13, p < .01) and are more neurotic (r = .44, p < .01) and less extraverted (r = –.20, p < .01) than people belonging to class 1.
Within the two classes individual differences in life satisfaction are signifi-cantly negatively correlated with neuroticism (class 1: r = –.47, class 2: r = –.32).
Social desirability is significantly positively correlated with life satisfaction only in class 1 (r = .27) but not in class 2 (r = –.02). Extraversion and life satisfaction are unrelated in the two classes (class 1: r = .02, class 2: r = .11). These correlation analyses revealed that personality variables are able to predict class membership (qualitative differences), and individual (quantitative) differences but the predic-tion structures differ between the two levels: Whereas extraversion is related to class membership, it is unrelated to individual differences within classes. Al-though these results are preliminary, they do show that mixture distribution Rasch models offer quite interesting possibilities for representing and predicting qualitative and quantitative differences simultaneously.
Analyzing Qualitative Differences: Latent Class Analysis
The mixed Rasch model assumes that there are quantitative and qualitative indi-vidual differences. It is a general model that not only comprises the Rasch model as a special case (when there are no qualitative differences) but also the latent class model (when there are no individual differences within classes). Latent class anal-ysis assumes that the population consists of different subpopulations. Each subpopulation is characterized by the class-specific response probabilities of the items. In contrast to the mixed Rasch model, latent class analysis assumes that all individuals belonging to the same class do not differ in their response
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).