ONE-FACTOR LATENT CHANGE ANALYSIS MODEL

To introduce the reader to latent change modeling, the discussion of LCA begins with a simple one-factor model. In this model, to prepare the ground for the following empirical example it is assumed that a set ofk= 4 repeated measurements of cognitive ability give rise to a covariance matrix and means that can be explained in terms of a single latent variable (such a model is provided in Fig. 12). However, we stress that the following devel- opments are immediately generalizable to any larger number,k≥5, of lon- gitudinal assessments, and are equally well applicable even with 3 measurement points. This one-factor LCA model has been discussed in the literature on several earlier occasions, and McArdle (1988) has termed it a curve model because the latent variable can be interpreted as a time factor that governs the intraindividual latent change processes or curves (see also McArdle & Epstein, 1987). Alternatively, the time factor can be thought of as initial true status of the underlying ability that is being repeatedly mea- sured. As indicated later, the loadings of the repeated measures on this fac- tor can be interpreted as rates of mean change in the studied ability. Meredith and Tisak (1990) chose to term this model a monotonic stability model because, although significant changes in mean levels may occur in

the studied ability, the rank order of the observations tends to stay the same over repeated assessments (Duncan et al., 1999).

Analysis of Mean Structures

One important aspect in which LCA models differ from all preceding ones discussed in this book, is that the variable means and their development over time are taken into account in addition to variable variances and covariances. This is achieved by what is referred to as mean structure analy- sis (MSA). A MSA includes in the analysis not only the covariance matrix of the repeated measures, but also the manifest variable means. In order to ac- complish this, a model must be fit to the so-called covariance/mean matrix, rather than only to the covariance matrix that has been used for modeling purposes up to this point (the slash in ‘covariance/mean matrix’ is meant to indicate extension rather than division). The covariance/mean matrix re- sults after the observed variable means are added as a last row and column to the covariance matrix. That is, the covariance/mean matrix is the covariance matrix augmented by the manifest variable means as a last added row and column.

Why Is It Necessary to Include Variable Means into an Analysis of Change?

The inclusion of observed variable means and their development over time into a change analysis complies with the conceptual basis of the classical ap- proaches to studying change. Accordingly, temporal development in the means of observed variables under investigation is of special importance. Indeed, one cannot imagine a repeated measures ANOVA, for example, which would exclude the information about change over time that is con- tained in the means. Because the use of LCA has the same goal of studying development, it is only natural to include variable means in the analysis. If the model were fitted only to the covariance matrix, however, one would omit the observed variable means and their dynamic from the analysis. The reason is that any covariance coefficient, and similarly any correlation coef- ficient, is based on the sum of cross-products around the means (e.g., Hays, 1994). That is, any covariance (as well as correlation) disregards the means of the two involved variables, and as a result the observed means and their development over time are inconsequential for the covariance matrix. Hence, different patterns of change over time (e.g., increase over time; de- cline; or growth followed by decline, or vice versa) can be equally consis- tent with a given covariance matrix resulting in a repeated measure context. Therefore, to fit a model only to the covariance matrix is tantamount to be- ing wasteful of information about the development of the studied phenom-

enon over time, which information is contained in the observed variable means and their dynamic. Hence, in an analysis of change the observed means are to be included since without them one cannot achieve the goal of examining growth or decline over time.

As a result of this inclusion of variable means, there are more data points to which the model is fit. That is, in addition to the elements of the covariance matrix, there are also as many means to count as data points as there are observed variables. For example, withk= 4 repeated measure- ments assessments, there arek(k+ 1)/2 = 4(5)/2 = 10 nonredundant ele- ments of the covariance matrix plusk= 4 observed means that the model must also ‘emulate’. As a result, in a mean structure analysis ofk= 4 longi- tudinally administered measures there are altogetherq=k(k+ 1)/2 +k

= 4(5)/2 + 4 = 14 pieces of empirical information to which the model is fitted. From this numberqone needs to subtract the number of model pa- rameters in order to obtain the model degrees of freedom. As shown later, conducting a MSA makes necessary the introduction of additional model parameters concerning the structure of the variable means and in fact may be unique to this structure; that is, these mean structure specific parame- ters may not have implications for the variable variances and covariances. This issue will be discussed further after details concerning LCA models are presented.

How Is a Model Fit to the Covariance/Mean Matrix?

SEM accomplishes the inclusion of observed variable means in the analy- sis—that is, achieves fitting a model to the sample covariance/mean matrix—by extending the fit function with a special term (see discussion of fit function in Chap. 1 and its Appendix). Hence, when fitting models to the covariance/mean matrix, one uses an extended fit function relative to the case when the model is fitted only to the covariance matrix (covariance structure) of a given model. This special term that extends the original fit function represents a weighted sum of the squared differences between the observed means and those reproduced by the model, with the weights be- ing the corresponding elements of the inverse of the model-reproduced covariance matrixS(for the maximum likelihood method used throughout this book). Just as the model has certain implications for the variances and covariances of the analyzed variables, it also has consequences for the vari- able means. These consequences can easily be worked out by noting that the mean of any linear combination of variables, whether observed or latent or of both types, is simply the same linear combination of their means. In or- der to understand and conceptualize the LCA model implications and the reproduced covariance/mean matrix, as relevant for this introductory text, a fifth law is now added to those presented earlier in Chap. 1.