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FIGURE 18.2. Item profiles of four items in three latent classes njc.

It is an important characteristic of LCA that subjects are not assigned to classes in a deterministic way, but rather are assigned with certain probabilities to any class. Therefore, incorrect assignments do not emerge, and it is not necessary to apply an error model to a classification made by LCA. But if subjects are assigned to classes according to their highest (modal) probability, a manifest classification of all subjects is obtained, and the probability of this assignment is a measure of the quality.

To illustrate, imagine that three latent classes were identified by LCA, and each subject is assigned to the class to which he/she has the highest probability of belonging, given his/her response pattern. Let the mean probability of all subjects assigned to the first manifest class be 0.91, for instance. The counterprobability for these subjects, that is, the probability of being assigned to Class 1 but actually belonging to latent Class 2 or 3, represents the measurement error. In this example the measurement error is 1 - 0.91 = 0.09.

Similar to the Rasch model, the parameters are estimated using maximum likelihood procedures. In contrast to the Rasch model, however, the likelihood function of LCA cannot be simplified to a function of the marginal sums because only the pattern frequencies provide the sufficient information for parameter estimation.

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