## 1. INTRODUCTION

For expository purposes, we consider first the two-class latent class model (with, say, latent classes 1 and 2) applied to the observed data in a four-way 2 × 2 × 2 × 2 cross-classification of four dichotomous variables (say, variables *A*, *B*, *C*, and *D*). There are 16 possible response patterns in the four-way cross-classification table; and we let *f _{ijkl}* denote the observed number (i.e., the observed frequency) of individuals whose response pattern is (

*i*,

*j*,

*k*,

*l*), with response

*i*(for

*i*= 1, 2) on variable

*A*, response

*j*(for

*j*= 1, 2) on variable

*B*, response

*k*(for

*k*= 1, 2) on variable

*C*, and response

*l*(for

*l*= 1, 2) on variable

*D*. When the latent class model is applied to the cross-classification table, the observed data in the table are used to estimate the corresponding expected frequency

*F*under the latent class model; and, for an individual whose response pattern is (

_{ijkl}*i*,

*j*,

*k*,

*l*), we can also estimate the conditional probabilities, and , of this individual being in latent class 1 and latent class 2, respectively. When the latent class model fits well the observed data in the cross-classification table (i.e., when the estimated

*F*are sufficiently close to the corresponding observed

_{ijkl}*f*, for

_{ijkl}*i*= 1, 2,

*j*= 1, 2,

*k*= 1, 2, and

*l*= 1, 2), the following question can arise: For an individual whose response pattern is (

*i*,

*j*,

*k*,

*l*), how can we use the corresponding estimated conditional probabilities, and , to assign this individual to one of the two latent classes?

The above question was considered very briefly in a short two-page subsection of a long (81-page) article (Goodman 1974a) that was focused on many other matters pertaining to latent class analysis. More needs to be said on the assignment of individuals to latent classes.

In the short subsection of the long article cited above, an assignment procedure was described, but no criterion was introduced there to help assess when the assignment procedure is satisfactory and when it is not. In the present article, an additional assignment procedure is introduced, and two different criteria are also introduced that can be applied to each of the two assignment procedures in order to help assess each of them.

When the two-class latent class model is applied to the observed data in a four-way 2 × 2 × 2 × 2 cross-classification of four dichotomous variables, the 16 possible response patterns correspond to the 16 cells in the cross-classification table. We can think of these cells as the 16 possible observed classes into which each of the individual respondents can be classified in accordance with his or her response pattern. The two-class latent class analysis seeks to determine whether the observed classification of the individual respondents into the 16 observed classes can be described in terms of just two latent classes (say, latent classes 1 and 2), where the proportion of respondents in latent class 1 is π^{X}_{1} and the proportion in latent class 2 is π^{X}_{2} (with π^{X}_{1}+π^{X}_{2}= 1); and where these two proportions in the latent class model, and some other characteristics of the respondents in each of the two latent classes in the model, are estimated using the observed data. The estimates are then used to estimate the corresponding expected frequency *F _{ijkl}* under the latent class model. When the estimated

*F*are sufficiently close to the corresponding observed

_{ijkl}*f*(for

_{ijkl}*i*= 1, 2;

*j*= 1, 2;

*k*= 1, 2;

*l*= 1, 2), we can view the latent class model with its two latent classes as possibly an underlying description, and a basic explanation, of the subject under study—that is, the observed classification of the respondents in the 16 observed classes. In this case, the latent class model with its two latent classes can be viewed as possibly more fundamental than the observed classification of the respondents in the 16 observed classes. This article considers two possible procedures for assigning the respondents in the 16 observed classes to the two latent classes.

In these introductory comments we have, for expository purposes, considered the two-class latent class model applied to the four-way cross-classification table of four dichotomous variables. These introductory comments can be directly generalized to the case where the *T*-class latent class model (for *T*= 2, 3, …) is applied to the *m*-way cross-classification table of *m* dichotomous variables (for *m*= 3, 4, …).

The two assignment procedures that will be considered here for assigning individuals to latent classes, and the two criteria that will be introduced here for assessing the assignment procedures will be viewed as a contribution to the methodology of the subject of “classification.” In the present context, a classification study would determine whether a set of individuals, who have responded to a set of questions, can validly be described in terms of a small number of classes (or clusters) of similar individuals, whose responses to the questions are similar in some sense. (For views of the subject of classification applied in many other contexts, see, e.g., Gordon 1999.)

The classification issues considered in the present article are viewed here in the situation in which the *T*-class latent-class model (for *T*= 2, 3, …) is applied to the *m*-way cross-classification of *m* dichotomous variables (for *m*= 3, 4, …). The same kinds of classification issues could also be considered in the situation in which other finite mixture models are applied in the analysis of various kinds of data; but this would be beyond the scope of the present article. (For views of other mixture models, and some extensions of latent class models, see, e.g., MacLachlan and Peel 2000; Magidson and Vermunt 2001.)