In Latent Trait and Latent Class Analysis (Langeheine & Rost, 1988), "local independence" plays a central role. The term was brought into the literature by Lazarsfeld and Henry (L&H, 1966, 1968) in their study of Latent Class Analysis (LCA). Even though it is central to latent trait analysis, it has created confusion. This note clarifies the way L&H used the term, and relates LCA to Rasch model analysis (RMA).
LCA usually involves a small number of dichotomously scored items similar to those found in educational and psychological testing. Its purpose is to partition the sample of persons into a minimum number of homogeneous classes and to explain the data in terms of how the persons in the different classes responded differently to the items. Homogeneity is defined as, first, all persons in a class are placed at an identical location value in terms of, say, ability or attitude; second, the relationship among the responses to the items is considered to be, effectively, the Rasch model for dichotomous responses.
In a typical data analysis, it is common to examine the fit between model and data by examining relationships between residuals and the values of parameters. This can be done in RMA by checking whether residuals are correlated with abilities or difficulties. When there is no correlation, then the data and the model accord with each other. When they do not accord, one can examine the data to see whether anything is wrong in the data collection, or consider the model inappropriate and search for another. (The case for measurement models is made independently of data, and, therefore, when one wants measurement, the data must accord with a Rasch model, and not the other way around.)
Neither of these approaches, reconsideration of the data as a whole, or the model as a whole, is used in LCA. Instead, persons are classified into a number of location points so that the Rasch model holds at each point. When all the persons in a class are at one point and the Rasch model holds, then the correlation among responses for all items taken in pairs is zero. L&H focussed on this correlation of zero for each pair of items and called it local independence. This, however, turns out to be no more and no less than requiring the Rasch model to hold at one point. The term local is used to stress that the zero correlation pertains to the people in a class who are all at the same point. This requirement of zero correlation drives the equations that identify classes, so that, within each class and by definition, there is no correlation between responses, and the data fit the Rasch model, not as a whole, but as subsets within each class.
When the model for each class is the Rasch model, how are classes distinguished? One way is that items in different classes may have different relative difficulties. In the case of only two items, since the people can only be located at a single finite ability (a score of 1), the classes are identified by showing different relative difficulties for the pair of items. When two or more classes are identified in this way, the difference between the classes is said to be qualitative.
The case of three or more items is more interesting. In LCA, persons within a class are again located at one point. In contrast, in RMA, it is possible for persons to be located at different finite points (corresponding to scores 1,2,3...). Thus RMA may show that the data accord with the model even when LCA implies that there are, say, two classes. However, when data accord with RMA, that means that the relative difficulties of the items are the same across different partitions of the persons, and that what distinguishes the classes must be a difference in the location of persons: a difference of this kind between two LCA classes is seen to be quantitative, though this situation was not originally considered in LCA.
When data do not fit RMA, it is possible through an analysis of fit to partition the data so that within each subset the data do fit the model. Within each class, the items will have different relative difficulties, and the differences between the classes would be described again as qualitative. Thus this is an alternative approach to LCA.
The point is that using RMA as a criterion for discovering classes is entirely consistent with LCA. However, RMA is more general in that, when there are more than two items, then, in addition to items having the same relative difficulties, the criterion is that the persons within a class must be on the same continuum, but not necessarily at the same point. By explicitly writing the Rasch model in terms of individual person-item responses, the local independence that L&H strove for prevails, but at the level of the person rather than at the grosser level of class or group. These points are elaborated in Andrich (1991).
Andrich D 1991. Review of Latent Trait and Latent Class Models, R Langeheine and J Rost (Eds). Psychometrika, 56,1 155-168.
Langeheine R & Rost J (Eds.) 1988. Latent trait and latent class models. New York: Plenum Press.
Lazarsfeld PF & Henry NW 1966. Readings in Mathematical Social Science. Chicago: Science Research Associates Inc.
Lazarsfeld PF & Henry NW 1968. Latent Structure Analysis. Boston: Houghton Mifflin Co.
Local Independence and Latent Class Analysis, D Andrich Rasch Measurement Transactions, 1991, 5:3 p. 160
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