satisfying a simplex structure is simpler than it should be: a latent curve analysis revisit

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Satisfying a Simplex Structureis Simpler Than it Should Be: ALatent Curve Analysis RevisitTenko RaykovPublished online: 10 Jun 2010.

To cite this article: Tenko Raykov (1998) Satisfying a Simplex Structure is SimplerThan it Should Be: A Latent Curve Analysis Revisit, Multivariate Behavioral Research,33:3, 343-363, DOI: 10.1207/s15327906mbr3303_2

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Multivariate Behavioral Research, 33 (3), 343-363 Copyright O 1998, Lawrence Erlbaum Associates, Tnc.

Satisfying a Simplex Structure is Simpler Than it Should Be: A Latent Curve Analysis Revisit

Tenko Raykov Fordham University

This article is concerned with the utility of the structural equation modeling (SEM) methodology for studying change. Two broad classes of models are considered, individual and group change models. The relationship between the constant rate of change and simplex models, as popular representatives of either class, is examined. Both models are shown to be special cases of the comprehensive latent curve analysis (Meredith & Tisak, 1990'1. Sensitivity of its models to differences in individual growth curves is demonstrated on data from Rogosa and Willett (1 985a). Benefits of studying longitudinal change using SEMand related issues of model choice are d~scussed.

The past 35 years since the first conference on measuring change (Harris, 1963) have witnessed a nearly exponential increase in the number of longitudinal studies conducted in a wide variety of substantive domains across the behavioral, social, and educational sciences. This trend has been paralleled by a similar growth in the development and applications of models of change in these and related disciplines. A number of earlier models drew directly upon the seminal work in growth curve analysis by Potthof and Roy (1 964), Rao (1 965), Khatri (1 966) and Grizzle and Allen 19691, and were initially used mainly in biostatistics, biometrics, and applied statistics. Their far reaching ideas and approaches have precipitated and i~nfluenced subsequent development of models for studying change in the blehavioral ,sciences. As a consequence, the last 15 years or so have seen a siubstantial rise in the interest in growth curve analysis in these disciplines as well. During this time, the number of models of behavioral change has also increased markedly. Guidance through a number of them, along with insightful

This research has been partly supported by a grant from the Australian Research Council. 1 am grateful to J. J. McArdle and B. Muthen for valuable discussions on individual change structural equation models. R. E. Millsap and two anonymous Referees provided critical Comments on earlier versions of the article, which have contr~buted considerably to its improvement. I am indebted to R. E. Millsap for a number of editorial conlments and Suggestions that have swbstant~ally improved the text.

Correspondence relating to this article can be directed to Tenko Raykov, Department of psychology, Fordham University, Bronx, NY 10458.

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discussions of their bases and empirical utilizations, is provided in the instructive volume edited by Collins and Horn (1991) that focuses on strategies and methods in these fields for analyzing change. In particular, applications of the structural equation modeling (SEM) methodology to the study of growth or decline have become especially popular here, some of which exhibit close connections to earlier factor analytic approaches by Rao (1958) and Tucker (1958).

The present article is concerned with the utility of SEM for measuring change. From this perspective two main types of approaches appear to have been frequently utilized in behavioral research. They can be viewed as individual and group change models. The relationship between popular representatives of either class, the constant rate of change and simplex models, will be examined. Both will be obtained as special cases of the comprehensive latent curve analysis (Meredith & Tisak, 1990). Its models will be shown to be sensitive to interindividual differences in intraindividual developmental profiles. Caution will be raised against overinterpretation of earlier findings by Rogosa and Willett (1985a), and the utility of SEM applications for purposes of modeling change will be demonstrated on their numerical example. Benefits of studying longitudinal change and its correlates and predictors using SEM, as well as related issues of model choice, will be discussed.

Individual Change Models and Group Change Models

Over the past couple of decades, a number of methodological developments took place in the behavioral, social, educational and life sciences, which contributed to the wider use of models of change over time. In the early and mid 1980s, Rogosa and his colleagues (e.g., Rogosa, Brandt, & Zimowski, 1982; Rogosa & Willett, 1985a, 1985b; Rogosa, 1987a, 1987b) presented insightful descriptions of regression-based methods for studying individual and group change in behavioral and educational research, as well as illuminating discussions of psychometric and related issues pertaining to the measurement of growth or decline. At that time and later, pioneering contributions by McArdle and his colleagues - for example, McArdle (1986, 1988, 1989), McArdle and Epstein (1987), McArdle and Anderson (l990), McArdle and Aber (1 990) -dealt with SEM-based approaches to modeling change that focused on interindividual differences in intraindividual growth. Their influential work also demonstrated the wide applicability and flexibility of the SEM methodology to the examination of interindividual differences in growth or decline in a number of developmental settings across the behavioral and social sciences. Subsequently presented models of change, such as

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Browne and Du Toit's (199 l), MuthCn's (1 991,1993), Browne's ((1 993) and Willett and Sayer's (1994), demonstrated the potential of this methodology for examining individual differences in growth.

A unifying characteristic of these structural equation models is their being fitted to matrices (of variable covariance indices and means by n~inimizing discrepancy measures reflecting the distance between model implied and empirical moment structures, such as the maximum likelihood fit jhnction or others resulting from a generalized weighted least squares discrepancy measure (e.g., Browne & Arminger, 1995, pp. 188-193). A reason why they have quickly gained in popularity in the behavioral sciences and related disciplines is their allowing for fallible correlates or explanatory variables to be included into the analysis. Thereby explicit account of the associated measurement error is taken, which is nearly ubiquitous in these fields, with the possibility of fitting general error covariance structures (for identified models) and no need of additional assumptions such as sphericity and/or covariance matrix or regression homogeneity. Anumber of these models can be obtained as special cases of the comprehensive latent curve analysis (LCA) approach by Meredith and Tisak (1990). LCA represents a very general framework within which diverse models of change can be devised. A later section will show that the constant rate of change and simplex models -popular individual change and group change models - are special cases of LCA too. The sectioin will also demonstrate the utility of LCA for studying individual differences in intraindividual change as well as their correlates and predictors.

These structural equation approaches to change measurement complement a related stream of earlier progress in modeling change, which was initiated mostly in the applied statistics, biostatistical, and biometrical literature. Those developments have lead to alternative growth curve and repeated measurement models that found wider applications in those fields. A characteristic of many of them is their formulation within the comprehensive framework of the general linear model (e.g., Crowder & Hand, 1990), and the extension of its classical version with uncorrelated homogenous errors to more complex error covariance structures accounting for serial correlation typically found in longitudinal data (e.g., Kshirsagar & Smith, 1994). Such models can postulate conditionally uneorrelated, equicorrelated, autoregressive, or exponentially correlated errors (e.g., Diggle, Liang, & Z'eger, 1994), or impose a Markov structure on the latter, that of antedependence (e.g . , Kenward, 1987), or alternative covariance structures (Diggle, 1988). A number of these modeling approachels focus on Yhe individual developmental profiles, and their defining equations are formulated in terms of individual-specific parameters and variables, for example, in widely used random effects models (Laird & Ware, 1982). The


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models can be fitted directly to raw data by employing generalized least squares and/or maximum likelihood (e.g., Hand & Crowder, 1996), often after assuming a corresponding structuring of the associated error covariance matrix and (restricted) maximum likelihood estimation of its underlying parameters (e.g., Harville, 1977). Nonlinear extensions of them and generalizations to nonnormal error distributions have also been developed, and they exhibit close relations to change models for binary and categorical data (e.g., Hand & Crowder, 1996). All these approaches to modeling change seem to be still receiving comparatively less attention in the behavioral sciences. This may arguably be partly due to associated difficulties with incorporating fallible covariates andlor predictors, inclusion of which can be directly achieved in the above mentioned SEM models of change. On the other hand, in difference to the latter models the former approaches generalize easily to situations where different subjects have different number and times of assessment as well as covariates (e.g., Diggle et al., 1994), which is often the case with subject attrition and missing values that are common in longitudinal research. Therefore, the SEM-based models and these alternative growth curve approaches build an integral entity of methodological advances over the past 40 years or so that are available and bring important theoretical and empirical benefits to behavioral, social, educational, and life scientists in the study of change processes.

The Constant Rate of Change Model

A well-known member of the class of individual change models, which plays an important role in accomplishing the goals of this article, is the constant rate of change model (e.g., Rogosa & Willett, 1985a). It postulates the following latent relationships of q > 1 consecutive assessments of N individuals at times t,, t,, ..., tq:

whereby the relations between individual true, observed and error scores, q , , Yl,, and ciP respectively, are given by:

In Equation 1, p = 1, 2, ..., N is the person index, q is true initial status of ' 4

thept"subject, and Op is his/her rate of change over time. This rate remains constant across all repeated assessments, and as a person-specific variable may be related to initial ability status. For simplicity of the following


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considerations, it may be assumed that t , = i are the repeated assessment times, i = I, ..., q (but this need not be the case in general; see below). To complete the definition of the CRCmodel, initial ability status q , and random lope 0 can be considered zero-mean normal variables with variances o;l and

a: and general covariance o,, and the measurement errors E, such normal variables with variances (i = 1, ..., q), respectively. Equations 1 and 2 show that under the CRC model interindividual latent differences at the it" measurement occasion stem from such differences in 0, q, , and their possible interrelationship. Thereby, the individual slope 0,? is the maiin source of (latent) change over time in the model @ = I, ..., N). Because of this feature, for the purposes of the present article the CRC model will be referred to as "individual change model". The existence of such a random coefficient like 0,, reflecting a major aspect of temporal development within subjects is a main characteristic of i~~dividual change models. The CRC model is relatively parsimonious. In general, it has altogether q + 3 parameters: u.,12,, a:, oqH, and o:, (i = 1 , ..., q); with the assumption of a common error variance, o:, it has only 4 parameters. A special case of this CRC model, with uncorrelated initial status and rate of change, that is, a?, = 0, has only 3 parameters and will be of interest below.

The CRC model represents an instructive example of a model that may be seen as shared between the SEM-tradition and the alternative stream of growth curve models discussed earlier. It can be fitted using (geineralized) least squares prin~ciples (e.g., Rogosa, 1996), or SEM after appropriate formulation outlined in a later section. For the latter purpose, the individual slopes 0,, 0, = 1, 2, ..., N) are not considered model parameters, only their variance is (and their mean when the model is fitted to the mean structure). Estimation ofthe random slopes is also possible via SEMafter the CRCmodel is fitted, using methods for estimation of factor scores (e.g., Joreskog & Sorbom, 1993). Folr q = 2 the CRCmodel is just-identified with the restriction o = 0 and a common error variance; unless these constraints are imposed,

BH the model is under-identified. For q = 3 it is just-identified without the restriction of a common error variance, and over-identified with it. For q > 3 the model is over-identified irrespective of whether or not the con~straint of common error variance is imposed.

' This model could as well be called "random-slope model" or "varyinglrandom coefficient plodel" (e.g., MuthCn, 199 1,1993). in difference to "group-slope models" or ''group1 coefficient Models" that could be used for naming the type of models considered next, for example, the $iinplex model (Equations 2 and 4). The present classification is motivated by the ]purposes of (his article, and to stress characteristics of instances of model usefulness when studying thange. The distinction between these two classes is not sharp and is primarily of pragmatic value here, as implied from the following discussion.

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The CRC model builds upon a specific conception of the process of change, which is examined by seeing how observed-true-score covariances structure in terms of model parameters. Using Equation 1 and correspondingly subindexed symbol u for denoting variance or covariance, one obtains:

It is thereby noted that observed-true covariances are structured not only in terms of the model parameters but also in terms of the times of repeated assessments (which in this case are 1, 2, ..., and q according to the earlier convention).

Group Change Models

A large part of applications of SEM for purposes of measuring change during the 1970s and 1980s dealt with models focusing on features of change of group(s) of subjects considered as a whole.

The Simplex Model

A prominent member of this class is the simplex model defined as follows (e.g., Rogosa & Willett, 1985a):

with relations between true scores, q,, observed variables, TI,, and measurement errors, E ~ ~ , given by the above Equation 2. Thereby, the assumption of uncorrelated latent residuals S,, with earlier true scores 9, (i = 1, ..., q - 1) is also made for identifications purposes (see below). In Equation 4, the remaining new notation relative to Equation 1 involves the group- and assessment-specific slopes Pi (i = 1, ..., q - I). They generally change from one measurement point to another but not across subjects, and are the major parameters reflecting temporal development in the simplex


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model. This is the reason for considering it a "group change model" in the remainder (see Falotnote 1). The existence of such major model ]parameters as (i = 1, ..., q - 1) that reflect pattern(s) of change in the whole group is a main characteristic of group change models. From Equation 4 is seen that according to the ~jimplex model interindividual latent differences at each measurement occasion stem only from such differences in (a) immediately preceding ability status q,, and (b) structural disturbances a,,, ,i = 1,2, ..., q - 1 ; no other individual sources of variance are provided in the model. For completeness of this discussion, it can be assumed that q l , 8i and are zero- mean normal vairiables with variances a:,, a: and u:(i = 1, ..., q), respectively.

The simplex is in general considerably more complex than the CRC model. The former has 3q - 1 parameters: u:,, P1 (i = 1, ..., q - I), a:, (i = 2, ..., q), and u i (i = 1, ..., q); with the additional assumption of a colrrimon error variance, u:, the simplex has altogether 2q pakameten. For q = 2, it is just- identified with the constraint of a common error variance and fixecl structural regression slope (or, instead, a restriction placed on the relation between initial status variance and that of the structural residual); unless these constraints are imposed, the model is under-identified. For q = 3, it is just- identified with the assumption of equal measurement error variances, and under-unidentified without it. For q > 3 the simplex is over-identified with the restriction of equal measurement error variances (e.g., Joreskog 8c Sorbom, 1993). A comparison of the number of its parameters then with that of the corresponding CRC model with common error variance shows that this simplex has considlerably less degrees of freedom than the CRCmotlel - their difference is (4q - $)I2 and increases linearly with the number q (of repeated assessments.

To highlight the different conception of change according to the simplex model, it is instructive to examine the way observed-true-score co~variances structure in terms (of model parameters. To this end, from Equation 4 one obtains:

yq,, = Pq-I Py-2 ... PI TI/, + Pq-, Py-2 P 2 8Zp + ... + Pq-, 8q-l.p + isqp+ Eqp - From (5) follows:

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Equation 6 exhibits a clearly different parameterization in the simplex relative to this in the CRCmodel (see Equation 3 ). The presence of (products of) squared group-specific slopes f3: (i = 1, ..., q- I), as well as the lack of any model term covariances, is notable. In addition, time of assessment is not consequential for this parameterization in the simplex, whereas it is influential in the CRC model (see Equation 3). Apart from the covariance structure differences across Equation 3 and 6, this lack of time dependence is a characteristic of the simplex model, which contributes to a different conception of the modeled process of change. It is also interesting to note from Equation 6 that due to the complex multiplicative structure oftheirright- hand sides any change of a structural slope(s) in the simplex proliferates through a number of implied covariances and affects in a complicated manner observed-true-score interrelationships.

Relationship Between the Constant Rate of Change and Simplex Models

The CRC and simplex models, as popular representatives of two main classes of models that have been frequently used to study behavioral change, exhibit a specific interrelationship of relevance for this article.

A Model That is Both a Simplex and a Constant Rate of Change Model

With appropriate constraints on the simplex and on the CRC model, one obtains a model that is both a simplex and a CRC model. To see this, first rewrite the definition equations of the latter as follows (cf. Equations 1 and 2; since 2 is not formally changed, its number is preserved next):

(7) q , = q l , + Op(tr - tl) = q l p + (i - l)Op (i = 1, ..., q ; p = 1, ..., N,

(2) Yip = qlll + E,,, (i = 1, ..., q; p = 1, ..., N) .

A CRC model with o,, = 0 is simultaneously a simplex model with:

(8) Zi2 = ... = Ziq (= 8 say), and


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(see Equations 2 and 4). Indeed, under 8 and 9,

(10) T,+ = q,/, + 8,, and T,, = all, (i = 1, ..., q - 1; P = 1, ..., N)

hold true. Consequently,

which is identical to the right-hand side of 7 with a changed notation of 6 to 0. Thus, the CRC model with uncorrelated initial status and slope is also a simplex model wi1.h identical residuals and unitary regression sl~opes. It is emphasized, however, that this is not the case for the general CR67 model in Equations 1 and 2 that CRC model and the simplex with 8 and 9 are only overlapping modells that have as common most of their free parameters, but are not identical because that CRCmodel has a parameter unmatchled by such of the simplex - the covariance uq, between initial status and slope.

It is stressed that this discussion ofthe relationship between the restricted CRC and simplex models was carried out at the model definition level regardless of the method used to fit either. This is owing to the fact that the focus so far was on the model definition equations in terms of observed and unobserved variables (Equations 2, 7, and lo), in order to demonstrate the identity relationship between these special cases of the CRC and of the simplex models. To fit the latter, constrained simplex using SEM, however, only an implication of Equation 8 can be introduced, namely:

This is because in SEM a model is fitted to the sample matrix of variable interrelationships (and possibly means) by minimizing a discrepantly function reflecting a generalized distance between that empirical matrix and the corresponding one implied by the model. In the latter reproduced matrix, a random variable constraint like Equation 8 can be represented only by its consequence(s) for the implied moment structure, not at the level of model definition equations. Hence Equation 8 cannot be imposed/tested as such iwhen fitting the restricted simplex under consideration, but only in terms of its parameter restriction consequence(s) for the analyzed moment matrix, bamely as Equation 12.

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Near Cross-Reproducibility of Implied Covariance Matrices

The fact that the CRCmodel is "nearly" a special case of the simplex, in the sense of a restrictive version of the former being identical to a correspondingly constrained variant of the latter, leaves no room for a surprise in an instructive result by Rogosa and Willett (1 985a). They found that a simplex model yielded a nearly perfect fit to data generated by a CRC model (for q = 5 repeated measurements), which are presented in their Table I.

An explanation of their result can be given by the observation that since the CRCmodel is in general "nearly" a special case of the simplex model (in the above sense), data simulated by a version of the former might be explained satisfactorily by an appropriate variant of the simplex model as well. In fact, the data in Rogosa and Willett's (1985a) Table I were generated by a CRC model with a weak correlation between initial status 9, and individual slope 0: using the numerical values employed in the data simulation (Rogosa & Willett, 1985a, p. 1051, one obtains p, = -.26 that explains less than 7% of the interrelation between initial ability status and slope. That is, the data in their Table I was in practical terms generated by a fairly close to a CRCmodel with no correlation between initial status and individual slope, which as shown above is a special case of the simplex model -namely that with Equations 8 and 9. The good fit of the simplex to the covariance matrix in Rogosa and Willett's Table I cannot be unexpected also because in general more than one model may fit well a given set of observed data, as is well documented in the literature (e.g., Bollen, 1989; Breckler, 1990; Luijben, 1991 ; MacCallum, Wegener, Uchino, & Fabrigar, 1993).

Fitting a CRC Model to Data Generated by a Simplex Model

The fact that a CRC model is a special case of the simplex conversely suggests that data generated by a simplex model without the critical constraints in Equations 8 and 9 will generally not be reproduced well by a CRC model. To exemplify, the data in Table 1 was simulated for N = 500 individuals according to the simplex model in Equations 2 and 4; all structural slopes were thereby chosen equal to 1.5, the four consecutive structural disturbances were then introduced with variances of -1, .2, .2, and . l , respectively, and measurement errors were simulated with a common variance of .5.

When fitted to the covariance matrix in Table 1 (see next section), the CRC model in Equations 1 and 2 yielded a poor fit of x2 = 1324.50, for df = 12,p = 0.0. (Even relaxing the constraint ofuncorrelated initial status and individual slopes did not render this model any close to acceptable - the


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Table 1 Observed Score Covariance Matrix for Data Generated by a Simplex Model

Variable y 1 y, y3 y4 - y5

y , 1.24 5 1.50 2.50 y3 2.26 3.32 5.28 y4 3.30 4.96 7.45 11.29

y5 5.01 7.49 1 1.25 16.66 25.49

Note: Y,, Y,, Y,, Y,, and Y, represent indicators of a latent construct following the simplex model in Equations 2 and 4, which has been assessed at five consecutive times. 'The data was simulated under the simplex model with conditions P, = ... = P,= 1.5; Var(q,) = 1; Var(8,) = Var(8,) = . I ; Var(8,) - Var(8,) = .2; Var(e,) = ... = Var(e,) = .5 .

resulting fit indices were as follows: x2 = 807.26, for df = 1 1 ,p = 0.0; in this and preceding cases LISREL reported not being able to compute confidence limits of fit indices due to too smallp-value.) These findings can be explained by the fact that in the data generation process the requirement was violated of Equations 8 and 9 to hold in order for the so-constrained simplex model to represent a restricted CRC model (with (rg, = 0).

This numerical example is only one of many that could be constructed showing the general inability of a more specialized model (e.g., the restricted CRC model) to reproduceiemulate well data simulated by a more general model (e.g., the simplex where Equations 8 or 9 are violated) particularly after introduction of additional parameters reflecting such violations. In general, more parsimonious models like the CRC model cannot be expected to fit well data generated by more complex mechanisms like that reflected in the simplex.

Latent Curve Analysis, Constant Rate of Change, and Simplex Models

The CRC and the simplex models are special cases of yet a more general model, that of latenl. curve analysis (LCA; Meredith & Tisak, 1990), which is especially useful in the study of change and its correlates and predictors. These relations are examined and exemplified in the present sectioln.

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Latent Curve Analysis

For a group of subjects and a repeatedly measured variable at times ti, ..., t (q > I ) , the general LCA model postulates the existence of a set of curves that reflect basic patterns of change in the group. (For its descriptive purposes, this subsection draws upon Meredith & Tisak, 1990.) If Yp(t) denotes the observed time-path of thepth individual at time t, andg,(t) is the kthbasic curve (k = 1, ..., r; r 2 I), the general LCA model assumes that:

holds true ( p = 1, ..., N). In Equation 13, wp, are individual weights (saliences) that are attached to the p'" subject and kth basis curve in approximating hidher developmental curve, and are considered realizations of random variables W, (k = 1, ..., r). They reflect model-specific, individual aspects of change over time, such as initial status or shape (McArdle & Anderson, 1990), overall change or pretest-to-posttest change (Raykov, 1995), or linear and/or quadratic growth/decline pattern coefficients (MuthCn, 1991, 1993; Raykov, 1996). The term Ep(tl) represents a realization of random error at time t,, E(tl), which is assumed uncorrelated with any weight variable. To be useful in an empirical context, an LCA model should be based on a small number of substantively interpretable curves. In particular, LCA models with much less basis curves than repeated measurements achieve a considerable data reduction that is typically desirable in btehavioral research. For example, the Level and Shape model by McArdle and Anderson (1990) is based on only 2 basis curves: a constant curve representing the initial ability status, and a variable curve reflecting the shape of group change over time. The LCA model in Equation 13 is shown by Meredith and Tisak (1 990) to lead to a confirmatory factor analysis model. It is identified with standard restrictions (Meredith & Tisak, 1990, p. 110) and can be fitted using, for example, LISREL (Joreskog & Sorbom, 1993) or EQS (Bentler, 1995). Thereby, the generally unknown discrete values of the basis curves at measurement points, g,(tl), can be free model parameters (i =

1, ..., q, k = 1, ..., r; provided model identification is insured, for example, Browne & Arminger, 1995). When the curves are appropriately chosen, these values reflect substantively interesting aspects of group change (e.g., McArdle & Aber, 1990). Alternatively, special restrictions can be imposed upon them when the aim is to test - for example, via differences in the associated chi-square values - various meaningful hypothesis, such as linear growth/decline (e.g., McArdle & Anderson, 1990) or specific group identity in multiple-population contexts.


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The general LCA model in Equation 13 gives rise to numerous special cases that have been shown useful in the study of change. For example, the arlier mentioned models by Mc Ardle and Epstein (1 9871, McArdle (1 988, 989), McArdle and Anderson (1990), McArdle and Aber (1 990), MuthCn 199 1,19931, and Willett and Sayer (1 994) can be derived from Equation 13 ia special restrictions imposed upon the discrete-time values gk(tl) of the asis curves (k = 1, ..., r; i = 1, .. ., q) and for specific choices of these curves

and weight variables. On the other hand, the general LCA model can be onsidered a special case of Browne and Du Toit's (1991) stochastic arameter growth model (e.g., Browne, 1993).

The potential of LCA for modeling change processes can be appreciated by stressing the fact that it permits (a) the study ofboth individual and group patterns of change, and (b) explicit description and modeling of individual

evelopmental profiles. The group change patterns are reflected in the discrete-time values of the basis curves, gk(tl), k = 1, ..., r, i == 1, ..., q. Individual change patterns are studied by focusing on the properties of the weight variables R; (k = 1, ..., r), which reflect features of the developmental profile of each individual. By relating W, (k = 1, ..., r) to external variables via corresponding model extensions, correlates and predictors of change can be studied. The latter can be fallible or perfectly measured personal characteristics, latent or observed variables, once or repeatedly assessed dimensions.

The CRC and Simplex Models as Latent Curve Analysis Models

To see if a structural equation model is an LCA model, an attempt should be made to represent its definition equations as a special case oEEquation 13 that provides the general LCA model; this involves showing that the former equations result from 13 for a particular choice of basis curves and individual weights.

For the CRC model, Equation 7 - that follows from Equation 1 - and Equation 2 reveal that the CRC model is an LCA model with only d - 2 curves. These are: g,(t) = 1 and g2(t) = t - t ,, with discrete-time values g (1') = 1 and

1 . I g2(tl) = tL - t, (= i - 1 for simplicity in this article), i = 1, ..., q; the ilndividual weights are wp, = ylp, and wp2 = ep 0, = 1, ..., N). Thus, having only 2 basis burves the CRCmotlel accomplishes a substantial data reduction with regard $0 repeated assessment occasions, which becomes particularly pronounced with a large number of successive measurements.

The simplex model is also a special case of the general LCA m~odel, but pith as many basis curves as there are repeated assessments. Irtdeed, its Equation 5 is obtained from 13 for q basis curves with the following discrete- time values and corresponding weight variables:


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(14) g,(t,) = 1, g,(tJ = PL-, .. .PI, 2, 3, ..., q, and w,,, = q,,,; g2(tl) = 0, g2(t2) = 1, g2(t3) = 82, e e . 7 g2(fq) = Pq-, ,.. P 2 9 and w2,, = S2,,; g3(t,) = g,(t,) = 0, g3(t3) = 1, g3(t4) = b3, ..-, g3(tq) = Pq-, P,,

and w,,, = ti3,,;

Thus, as an LCA model with q basis curves, the simplex model does not achieve any data reduction with regard to repeated measurement occasions. In this sense, it is an "exceptional"LCA model, rather than an "ordinaryVLCA model that typically has (much) less basis curves than repeated assessments.

Fitting a Latent Cuwe Analysis Model to Data Generated by a Constant Rate of Change Model

To exemplify sensitivity of LCA models to interindividual differences in intraindividual time-paths, and thus illustrate their utility for measuring change, the Rogosa and Willett (1 985a) data in their Table I is reanalyzed next. To this end, the model in Equations 2 and 7 can be represented as follows (e.g., Joreskog & Sorbom, 1993; below, priming is used to denote transposition, underlining - column vector, and brackets to enclose matrix):

where I, is the identity matrix of size 5; in Equation 15, A,, is the fixed factor loadings matrix and OC is the diagonal error covariance matrix with a common diagonal element > 0 (error variance). The path diagram of this model is presented on Figure 1 following graphical conventions for displaying structural equation models (e.g., Joreskog & Sorbom, 1993). Fixed parameters are denoted by their constants attached to the paths symbolizing them (see matrix Ay).

Even though Rogosa and Willett (1 985a) did not provide observed means for the variables in their Table I, it is instructive to fit the model in Figure 1


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el €2 €3 €4 €5 Figure 1 A latent curve analysis model corresponding to the constant rate O F change model (see Equationsl, 2.7, and 15).

to their data.2 As could be expected, the LCA model in Equation 15 yields an essentially perfect fit: x2 = .015, df = 11, associated probabilityp = 1 .O, and covariance residuals being uniformly less than .0 1. On the one hand, these fit indices are not surprising, given its structural identity to the CRCrnodel that underlies the generation process of the covariance matrix in Rogosa and Willett's Table I. Similarly, this LCA model recovers nearly perfectly all parameters used in the data simulation and the associated measure reliabilities: their estimates are Identical (at least up to the second digit after the decimal point) to the values ofthe parameters actually used for this purpose by Rogosa and Willett (1985a~, pp. 105-106). Thus, true initial status variance was 'estimated (standard errors given in parentheses) at .469200 (.035536), individual slope variance at .007867 (.00 1545), covariance between initial position and individual slope at -.015733 (.00561), common error v,ariance at

As indicated in the earlier discussion of LCA, modeling the observed variable means kimu~taneousl~ with theur variances and covariances represents an integral part of the very idea of LCA (e.g., Meredith & Tisak, 1990). Also, the LCA approach affords alnuch more flexible inodeling of individual growth by parameterizing aspects of it in the basis curve values, rather than stipulating a fixed pattern of change over time as in the above constant factlor loadings inatsix A,,. (For substantively useful alternatives, see for example McArdle & Anderson, 1990; WcArdle & Aber, 1990; Meredith & Tisak, 1990.)


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.I49533 (.005466), and successive measure reliabilities at (rounded-off to third digit after decimal point) .758, .748, .745, .749, and ,758, respectively; thereby, the relatively small standard errors indicate high stability of the obtained estimates.

At least as important, however, is the finding that in the fitted LCA model of Equation 15 all parameters are significant. Hence, the variances of its latent variables indicate marked interindividual differences in intraindividual growth curves, both in initial ability status q, and individual change slopes 0 (upon which differences the data in Rogosa & Willett's, 1985a, Table I was actually based). Thus, the current application of SEM for purposes of studying longitudinal change via an LCA model exemplifies the sensitivity of SEM to interindividual differences in intraindividual growth curves. It is thereby emphasized that the presently considered model in Figure 1 explicitly describes and models individual time-paths; in addition, as indicated earlier with the general LCA approach, by relating initial status q, and slope 0 to external variables it becomes possible with this model to study correlates and predictors of longitudinal change.

Discussion and Conclusion

This article deals with the potential of the structural equation modeling methodology for studying behavioral change processes. Two broad categories of models were initially considered: (a) individual change models, as represented by the constant rate of change model; and (b) group change models, as exemplified by the simplex model. Although the differences between these classes are not sharp (see Footnote I), typically their major parameters of change are individual-specific or group-specific, respectively. It was subsequently shown that there are close connections between their popular members, the CRCand simplex models, and that either of them is a special case of the comprehensive latent curve analysis by Meredith and Tisak (1 990).

The above sections raise the question of model choice in empirical behavioral research. As discussed in a number of methodological sources (e.g., Bollen, 1989; Breckler, 1990; Lee & Hershberger, 1990; MacCallum et al., 1993; McArdle, 1988; McArdle & Aber, 1990; McArdle & Hamagami, 1992), and based on the preceding discussion, model choice depends decidedly on all available information about the data and mechanism having likely generated it, as well as the particular research question(s). As suggested by earlier subsections, individual change models - such as the CRC model -are of particular interest when the research question(s) relate to individual time-paths. When such a model provides satisfactory fit along with group-specific models, the parsimony of the former may become an


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rtant criterioin of model choice. Explicit account of the times of repeated sments, which as demonstrated earlier is a characteristic of the CRC and

ther correspondingly constructed individual change models but not of the implex (see Equations 3 and 6), may become a feature of special relevance hen modeling change in certain contexts, and thus contribute to the

ttraction of the former models rather than the latter. Also, recent nvestigations of the utility of the simplex model in studies of change orrelates (Stoolmiller & Bank, 1995) indicate that detection of important redictors of growth or decline may be difficult with it.

The above criterion of parsimony is closely related to statistical power hat may also contribute to consideration of less complex change models in ome empirical settings. Being more parsimonious, these models can be xpected to be associated with higher power relative to more coxriplex ones.

power differences may become more relevant with not very large le sizes, when the analyst may also be interested in using a model and/

parameter significance tests that are associated with higher power. To emplify, reference to the data in Rogosa and Willett's (1985a) 'Table I and

he CRC and simplex models is again instructive. Earlier resealrch on this ower issue by h4cArdle and Hamagami (1992, pp. 160-162) using the ethod of Satorra and Saris (1985) showed lower power with ithe simplex

han with a latent growth curve model. It is interesting to address this issue Iso using an alternative method that affords overall power estimation without recise parameter specification under the alternative. This is the recent ethod by MacCallum, Browne, and Sugawara (1996). With {.he data in ogosa and Willett" Table I, at the used sample size of 500, it reveals that the ower of the test oE'close fit' (i.e., y 5.05, where y is the root mean squared ror of approximation reflecting badness of fit per degree of freedom; rowne & Cudeck, 1993) with the simplex model is again clonsiderably lower

han that with the (7RCmodel. Indeed, this power with the simplex is found o be .40, while with the CRCmodel it is .65. Similarly, power of'the test of exact fit' (i.e., y = 0) with the simplexmodel is also lower, namely .45, while ith the CRC model it is found to be .69. However, the earlier discussed

onsiderations in model choice, 'namely of the substantive nature of the echanism having likely generated the data and the research qluestion(s),

annot be typically replaced by such power considerations. Less parsimonious models, such as the simplex and other group change

odels, may provide an attractive means of modeling change if lsiubstantive terest focuses on group patterns of change and previous knowledge in the

omain suggests multiple sources of variability in the analyzed data. As lified in this article, parsimonious models have in general deficiencies in ring data generated by more complex mechanisms. This suggests that


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the simplex and other group change models may be attractive when group pattens of development are of special concern and under circumstances characterized by complexity of sources of observed variance. In such cases, consideration of group change models may also be attractive given the possibility that the group means can follow across assessment occasions, or can be well approximated by, a curve belonging to a different class than the one used to model change at the individual level (e.g., Thissen & Bock, 1990).

Capitalizing on the LCA approach, this article also exemplified the fact that for a given covariance matrix there are in general more than one models - for example, an individual and a group change model - that fit it statistically acceptably well, regardless ofthe true mechanism underlying the process of data generation. This is not unexpected as it is well known that different models can be (nearly) equally consistent with the same data (e.g., Breckler, 1990). It is therefore of critical importance that the behavioral analyst incorporates all prior information about (a) the studied phenomenon, (b) research question(s), and (c) data specificities, into the process of choosing models to be fitted to the analyzed data, as well as subsequently in differentiating in the light of obtained results between these models as means of data description and explanation (e.g., Lee & Hershberger, 1990; MacCallum et al., 1993).

This article is not inconsistent with the message of Rogosa and Willett (1985a, pp. 99, 105) that "markedly different types of growth (learning) curves may generate indistinguishable covariance structures". In fact, in the context of the preceding discussion their statement is completely plausible. At the same time, however, the article demonstrates that their message does not imply that SEM applications to the study of change are generally insensitive to individual differences in growthldecline patterns. As discussed in an earlier section and shown on their data, a class of structural equation models - namely LCA models - focus on aspects of individual developmental profiles. These methods are capable of explicitly describing and modeling individual change profiles as a basis of longitudinal data analysis, and at the same time offer a very general and comprehensive devise of studying various aspects of change, including aspects of group development, as well as correlates and predictors of growth or decline. The general LCA model yields as special cases numerous models of change that can be of interest to the practicing behavioral analyst. Such models are those cited in the first section of the article, as well as other models that could be obtained given particular research concerns, design features, and substantive considerations on the nature of the studied phenomenon. In addition, these models offer all benefits of the SEM methodology. They are: (a) explicit consideration of measurement error in studied covariates and explanatory


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variables; (b) routine estimability and testability ofmore general than diagonal error structures (as long as the overall model is identified) and of indirect effects in complex multivariable models of change processes; as well as (c) lack of sensitivity to violations of covariance matrix homogeneity, sphericity, or regression homogeneity assumptions in (M)AN(C)OVA approaches to repeated measure analysis. These SEM-based models, together with the earlier mentioned alternative models of change employing corresponding extensions of the general linear model, represent an integral body of models for growth or decline that are available and bring important theoretical and empirical benefits to the behavioral researcher.

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Accepted November, 2997.


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satisfying a simplex structure is simpler than it should be: a latent curve analysis revisit

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