growth curve analysis of ability means and variances in measures of fluid intelligence of older...
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Growth curve analysis ofability means and variances inmeasures of fluid intelligenceof older adultsTenko Raykov aa Department of Psychology , University ofMelbourne , Parkville, Victoria, 3052, Australia E-mail:Published online: 03 Nov 2009.
To cite this article: Tenko Raykov (1997) Growth curve analysis of ability meansand variances in measures of fluid intelligence of older adults, Structural EquationModeling: A Multidisciplinary Journal, 4:4, 283-319
To link to this article: http://dx.doi.org/10.1080/10705519709540078
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STRUCTURAL EQUATION MODELING, 4(4), 283-319Copyright © 1997, Lawrence Erlbaum Associates, Inc.
Growth Curve Analysis of AbilityMeans and Variances in Measures of
Fluid Intelligence of Older Adults
Tenko RaykovDepartment of Psychology
University of Melbourne
This article is concerned with an application of growth curve analysis to modelingtest performance changes demonstrated by older adults following cognitive trainingor based on experience with repeatedly presented fluid intelligence measures. Groupdifferences and invariance in change over time in ability means and indices ofinterindividual differences at the ability level in a 2-group cognitive intervention study(Baltes, Dittmann-Kohli, & Kliegl, 1986) are examined using 3 latent curve analysis(Meredith & Tisak, 1990) models. The results indicate (a) group equivalence in thepattern of temporal development of ability variances and (b) training effects in theexperimental group that are stronger than the practice/experience effects in the controlgroup, whereby both types of effects are maintained over the 6-month testing period.The findings provide further suggestive evidence for plasticity in fluid intelligenceof elderly adults (cf. Baltes et al., 1986; Raykov, 1995, 1996).
Over the past 20 years, a considerable part of cognitive aging research has focusedon the query if cognitive performance of older adults can be improved by meansof training interventions or by practice with criterion-relevant materials withouttutor guidance into the nature of problem solving. Much of that research, partlyconcerned with plasticity in intellectual functioning, has been carried out as areaction to the earlier dominant focus in gerontology, which considered agingprimarily as a process of universal and gradual decline (cf. Baltes & Lindenberger,1988). The concept of plasticity in fluid intelligence (Cattell, 1971; Horn, 1970,1982), advanced as an alternative to such decline, presumes modifiability of this
Requests for reprints should be sent to Tenko Raykov, Department of Psychology, University ofMelbourne, Parkville, Victoria 3052, Australia. E-mail: [email protected]
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intellectual ability cluster that has been found to exhibit pronounced aging loss (e.g.,Baltes, 1987). During the past couple of decades, in operational terms, plasticity influid intelligence has become synonymous to enhancement of cognitive test per-formance after practice or tutor-guided training in test-relevant cognitive skills(e.g., Baltes, Dittmann-Kohli, & Kliegl, 1986; see also Hayslip, 1989a, 1989b;Kooken & Hayslip, 1984; Labouvie-Vief & Gonda, 1976; Raykov, 1995, 1996;Schaie, Willis, Schulenberg, & Hertzog, 1987; Verhaegen, 1993).
Following this research tradition, this article is concerned with a growth curveanalysis of aspects of reserve capacity of the aged in the fluid intelligence clusterof intellectual abilities. The following discussion focuses on the temporal develop-ment not only in mean test performance (e.g., Baltes et al., 1986; Baltes & Willis,1982), but also in variability indices. In contrast to Raykov (1996), this article dealswith group, rather than individual, change. In addition to Raykov (1995), theconcern here is with explicit modeling of temporal change in indices of interindi-vidual differences at the ability level and in ability means.
METHOD
Data
This article is based on a data set from a two-group cognitive intervention study byBaltes et al. (1986; see also Baltes & Willis, 1982; Blieszener, Willis, & Baltes,1981). For the purposes of this article, a brief description of the study design seemsnecessitated (cf. Raykov, 1995,1996). At its four measurement occasions, a pretestand three posttests after a cognitive intervention in the experimental group, anintelligence test battery was administered to all participants in the study. After thefirst assessment, participants were randomly assigned to an experimental group anda no-contact control group. Tutor-guided instructions into rules and conceptsunderlying well-established tests of Figurai Relations and Induction were given tothe members of the experimental group immediately after pretest. The threeposttests with all participants in the study were conducted 1 week, 1 month, and 6months after the training was completed in the experimental group. Further detailsconcerning the training program and the particular tests administered can be foundin Baltes et al. (1986) as well as Baltes and Willis (1982) and Blieszener et al.(1981). Recently, Raykov (1995, 1996) reported findings suggesting (a) groupequivalence in the indices of interrelationship among initial ability status andmaximal attained change in tests of Figurai Relations and Induction; (b) groupidentity in the correlative pattern among the intercepts, slopes, and quadratic bendparameters of fitted individual latent growth curves; and (c) group differences inthe means of these parameters.
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GROWTH CURVE ANALYSIS 2 8 5
In the remainder of this article, data on three repeatedly presented fluidintelligence tests are focused on. These are measures of the fluid intelligencesubabilities Figurai Relations and Induction trained in the initial phase of the study.They are the so-called ADEPT Induction and ADEPT Figurai Relations tests(Baltes et al., 1986), as well as Thurstone's Standard Induction test (Ekström,French, Harman, & Derman, 1976; Thurstone, 1962). The analyses reported laterare based on data from all 248 elderly participants with no missing value on a testand assessment. In difference to a considerable part of previous adult cognitiveintervention research, this article is concerned not only with exploration of thetraining effects on these tests, but also with the nature of practice effects. Specifi-cally, the focus of the following discussion is on such aspects of longitudinalchanges in indices of interindividual differences and means at the latent abilitylevel, which were not addressed previously by Baltes et al. (1986) or Raykov(1995, 1996). In particular, the interest here will be in studying the developmentover time of group differences and similarity in indices of interindividual variabil-ity at the latent dimension.
Models
This article employs three growth curve models that focus on change over time inability means and indices of latent interindividual variability (Model 1), on thetemporal development of group differences in ability means (Model 2), and onspecific group differences in ability change (Model 3) for the three repeatedlypresented fluid measures. Additionally, group identity in the pattern of temporalchange in ability variances is tested using the first model.
Each of the models described next is a special case of the comprehensive latentcurve analysis (LCA) by Meredith and Tisak ( 1990) and can be obtained via specificconstant restrictions of appropriate factor loadings, as demonstrated later. TheAppendix provides the LISREL 8 source code for each model. As elaborated in thefollowing paragraphs, the characteristic features of these models suggest their wideapplicability in behavioral, educational, and social research. First, they focus on thetemporal change pattern in interindividual differences at the latent ability dimensionby simultaneously modeling longitudinal means, variances, and covariances. Sec-ond, they can be considered useful alternatives to multivariate longitudinal analysesof variance when fundamental assumptions of the latter are violated, such ascovariance matrix homogeneity across the levels of between-subject factors orsphericity (e.g., Timm, 1975). Third, these models permit imposing/testing lessrestrictive assumptions about the covariance structure of measurement errors thanroutine applications of multivariate analysis of variance in these sciences. This isdue to the fact that the present models can accommodate any error structure thatleaves them identified (e.g., Cole, Maxwell, Arvey, & Salas, 1993; see also Willett
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& Sayer, 1994). Fourth, any of the models described next permits inclusion ofcovariates or extraneous (independent) variables, such as gender, education level,age, and so forth, that can be (a) once or repeatedly assessed over the study periodand (b) fallible or perfectly measured. In the provision of incorporating such falliblevariables, the models to follow are unmatched by routinely utilized, conventionalstatistical models within the general linear modeling context in the behavioral,educational, and social sciences.
Model 1
This model can be considered a relaxation of a three-dimensional version of theone common factor model of McArdle and Epstein (1987), as well as viewed as ageneralization of a model later proposed by Rudinger (1988) that is equivalent toMcArdle and Epstein's (1987; see Footnote 1). Model 1 assumes that two factorsunderlie the interrelationship structure of the repeated assessments with the threefluid intelligence tests. For each repeated measure, the first factor encompasses thesimilarities in test performance across the four consecutive assessments, whereasthe second factor does so with the deviations from ability means. The path diagramof the model is presented in Figure 1. Within the framework of classical test theory,a lower dimensional version of a special case of this model, based on a commonsense argument, was discussed inRaykov (1992; see following).
In this figure largely following a widely used graphical standard for displayingstructural equation models (e.g., Jöreskog & Sörbom, 1993), the notation used byMeredith and Tisak (1990) is adopted with respect to observed scores Xy andcorresponding variable specificities E¡¡, i = 1, 2, 3; y = 1, 2, 3, 4. (Here and in thefollowing discussion, the index i denotes fluid measure in the order ADEPTInduction, ADEPT Figurai Relations, and Thurstone Standard Induction test,whereas j designates measurement occasion.) For each fluid measure, the factorencompassing similarities in mean test performance across the four assessmentssubsumes their means (McArdle & Epstein, 1987). It is therefore referred to asability means factor hereafter. In the figure this dummy factor, with no variance,is denoted by the suffix "_M" following the abbreviation used with the correspond-ing repeated test. The second factor encompassing the similarities, across repeatedmeasurements, in latent deviations from means is called ability variances factorhereafter. It has the suffix "_V" for each repeated test in Figure 1. To ensure modelidentifiability, for each fluid measure the loading of initial measurement on its twofactors is fixed to unity. The diamond in the middle of the figure is used only tosignal that the mean structure of the data is analyzed. The paths leading from it intothe first of the couple of latent variables associated with each repeated measuredesignate its initial ability mean (McArdle & Epstein, 1987; because the mean of
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E31 E32 E33 E34
1X31
1X32
1X33
1X34
E24
E23
E21
X11
1 ,
X12
< 1
r
X13 X14
E11
î ÎE12 E13 E14
FIGURE 1 Modell.
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the latent deviation from mean is zero, the paths from the unit diamond to the abilityvariances factors are missing).1
Fixing to unity in this model, the pretest loading per factor is particularly usefulin the context of this empirical study because the training in its experimental grouptook place immediately after the first assessment and was completed before thesecond measurement occasion. Hence, Model 1 permits examination of measure-and group-specific patterns of performance improvement relative to the study'sstarting point, its pretest. These patterns are reflected in the ratios of ability meansand of deviations from means at each assessment, to these respective quantities atinitial measurement. The ratios are parameterized in the loadings on the correspond-ing factors of the consecutive assessments with the three fluid measures (seefollowing) and will be of main interest in the model applications reported in thenext section. For ease of reference, for each fluid measure the loadings of its fourassessments on the ability means factor will be referred to as ability means ratios.Because they equal the ratios of ability mean at each posttest to initial ability mean(cf. McArdle & Epstein, 1987), they reflect the rate of growth in ability means inthe metric of initial latent mean.
This discussion and Figure 1 show that Model 1 is saturated in its mean structure:It has as many parameters for the mean structure of the observed variables as thereare analyzed means. Hence, when fitted to data from a single group it will reproduceexactly the analyzed means, and thus no data reduction will be achieved with respectto them (cf. Bentler, 1989). However, if (a) this model is fitted to data from morethan one group, and then (b) cross-group restrictions are imposed that have animpact on the mean structure (as will be the case later in this article), Model 1 willnot be saturated. Thereby, the unique feature of the model is that it allows thesimultaneous estimation of (group) ability means and variances, as well as a test ofthe introduced cross-group restrictions. In particular, in the next section this modelwill be used also to address the issue of resemblance between training effects in the
1 Model 1 is a three-dimensional generalization of a model discussed by Rudinger (1988) becausefor each repeated measure the present model does not impose the (within-group) constraints of equalityof each posttest loading on the first factor to this assessment loading on the second factor. (That is, therestrictions b12 = c12, b13 = c13, b14 = c14, b22 = c22, b23 = c23, b24 = c24, b32 = c32, b33 = c33, b34 = C34, in thenotation used later in this section, are not introduced in Model 1.) This generalization of Rudinger'smodel which is identical to the same multivariate extension of the latent growth curve model ofMcArdle and Epstein (1987), is nested in Model 1 because it results from Model 1 by imposing theaforementioned nine pairwise constraints. McArdle and Epstein's (1987) model is more restrictive thanModel 1 as the former parameterizes both the covariance and mean structure of the analyzed variablesin the same set of parameters (apart from the factor means that are not used in the former structure). Atthe same time, as indicated in the main text, Model 1 has one set of parameters for the mean structure,and another set of parameters for the covariance structure. On the other hand, if tenable for a given dataset, McArdle and Epstein's (1987) model will provide a more parsimonious means of data descriptionand explanation than Model 1 because the former has fewer parameters.
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GROWTH CURVE ANALYSIS 2 8 9
experimental group and practice effects in the control group. This will be achievedby a capitalization on the mentioned characteristic of the model, namely theprovision for a test of the plausibility of cross-group constraints on the pattern ofchange in ability variances. This test is of substantive interest in this empiricalcontext because the pattern of change in ability variances reflects the (group-spe-cific) extent of interindividual variability in the ability scores at each assessmentas well as its stability/change over time (e.g., Schaie & Hertzog, 1985). Accord-ingly, the loadings on the second factor per repeated measure in Figure 1, the abilityvariance factor, will be of particular interest. These loadings will be referred to asability variances ratios. They equal the ratios of posttest ability deviations frommean to this deviation at pretest and hence reflect the rate of change in abilityvariances in the metric of initial ability mean deviation (see following). Thus, thegroup restriction of invariant pattern of change in the variances, which will beimposed and tested with this model in the next section, addresses the theoreticallyand empirically important issue of identity of training and practice effects withrespect to the pattern of temporal change in the indices of interindividual differencesat the ability level.
Relation to LCA. As mentioned earlier, Model 1 can be obtained as a specialcase of the comprehensive LCA approach by Meredith and Tisak (1990). To showthis and clarify practically useful features of the model, it is helpful to highlighttheir relation within the classical test theory framework (e.g., Lord & Novick,1968), in particular by capitalizing on the observed score decomposition into trueand error score. Thereby, no restriction of the discussion in the preceding subsectionwill be introduced because, at each parameter vector, the preceding and followingconceptualizations of the model yield identical implied covariance matrices andobserved means.
For each of the three repeatedly presented fluid tests, Model 1 builds on thefollowing identity for the measurement of the/rth participant atjth assessment (Ndenotes sample size):
XpJ = Tpj + Epj=[E(.TJ)] + [TpJ-E(.Tj)] + EpJ,p = l,...,N,j = l 4 (1)
representing the observed score Xpj as a sum of (a) ability mean in the group, E(Tj);(b) individual ability deviation from the latter, Tpj - E(Tj); and (c) measurementerror Epj (cf. Raykov, 1992). (In the remainder of this article, the italicized symbolE(.) without any subindex will denote expectation with respect to individuals, andshould be kept apart from the unitalicized one of measurement error, E, to whichat least one subindex will be attached.) The model then assumes that each of thefirst two terms on the right-hand side of Equation 1 is linearly related to its analog
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at first assessment, E(Ti) and Tpi - E(Ti), respectively. This relation is representedby the equalities £(7}) = b¡ • E(7i) and Tp} - E(T}) = c, • [Tpi - E(Ti)], respectively,with unknown parameters b¡ and q,j= 1, ..., 4, the ability means and variancesratios discussed in the preceding subsection. It is noted that according to thepostulated model, within jth assessment occasion the same parameters bj and c,areassumed valid across all participants, j = 1,..., 4. This suggests reference to it as agroup change model rather than individual change model like those discussed, forexample, in Willett and Sayer (1994) and Raykov (1996).
To see the relation of Model 1 to LCA, first the general definition equation ofthe latter is restated:
(la)
In Equation la, Xp¡ is the value the pth individual receives at jth consecutiveassessment of a given repeatedly followed variable, gi(f), ..., gt(t) are the basiscurves representing basic patterns of change in a studied group, wp\, ..., wPk areindividual weights that thepth individual attaches to the Jtth basis curve in shapinghis or her own time path, and Epj is the model error term, j = 1,..., 4, p = 1,..., N.(Equation la is identical to Equation 1 in Meredith & Tisak, 1990.)
Equation 1 in this article, with its following model assumptions, is now restatedas
Xpj = E(Tj)»bj+lTpl-E(Tl)]*Cj+EpJ,p = l N,j = l 4 (lb)
From Equation (lb) it can be seen that Model 1 is a special case of the general LCAmodel defined in Equation 1 of Meredith and Tisak (1990; see Equation la). Thisis so because the former is obtained from the latter by taking (a) d=2 basis curves,and (b) as individual weights wpi = E(Ti) for the first basis curve and wpz = [TP{ -E(Ti)] for the second basis curve, p = 1,..., N, j = 1,..., 4. The basis curves, denotednext by giW(t) and g2(1)(0> have the following discrete-time approximations (i.e.,values at consecutive assessment points):
and
) = c2, g2(1)(i3) = c3, g2
(1)(f4) = c4 (Id)
(The superscript(1) is used here to signal "basis curve of Model 1"; correspondingsuperscripts will be used for basis curves with Model 2 later.)
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GROWTH CURVE ANALYSIS 2 9 1
Direct algebraic rearrangements to obtain the reproduced covariance matrix andmeans (e.g., Bollen, 1989) show that with the three longitudinally followed char-acteristics the latent means E(T¡i), i =1,2,3, are influential only in parameterizingthe manifest variable means implied by Model 1. In difference, the true meandeviations Tpn — E(Tn), i = 1, 2, 3, are important only in parameterizing thereproduced covariance matrix. The b and c parameters of Model 1 reflect usefulaspects of change along repeatedly assessed dimensions and are of main interest inapplications of the model. Should Model 1 represent a tenable means of datadescription and explanation of a studied phenomenon, taking expectation from bothsides of Equation 1 and straightforward algebra show that the parameters b¡, j = 1,.... 4, represent the rate of change over time, relative to pretest (initial assessment),in the means of true scores on a given longitudinally followed characteristic.Similarly the parameters c> j = 1,..., 4, are shown to represent the rate, relative topretest (initial assessment), with which true deviation from contemporary meanchanges over time. It is noted that by fixing (to unity) a different assessmentoccasion loading on the model's basic variables E(T{) and T\ - E{T\), the rate ofchange over time in true means and deviations from them can be studied in relationto another anchor point distinct from the pretest used as such so far. It is also worthemphasizing that the true mean deviation variable(s) Ti - E(Ti) have as scores theindividual deviations from mean at pretest. In conjunction with the c parameters,this variable(s) captures the latent profile of each individual, in terms of his or herdeviations from group mean at respective assessment occasion. This feature ofModel 1 may be of particular interest in studies of correlates and predictors ofchange, whereby one may extend the model by regressing/correlating this deviationlatent variable(s) on other variables presumed to explain change in the latentdimension(s) of a studied phenomenon, which is reflected in the deviation vari-ableís) T, - £(r,).
Beyond the ability means and variances ratios, other empirically interestingparameters of Model 1 are: (a) the initial true means of each of the three repeatedlyfollowed abilities, E(T¡\), / = 1,2,3; and (b) the variances and covariances of theirpretest true mean deviations, Var[7n- E(Tn)] and Cov[r a- E(Tn), Tki- E(Tn)], i,k= 1,2,3. Thereby, the parameters in (b) represent disattenuated statistics that havebeen obtained after accounting for measurement error in the observed variables (byintroducing the error variance parameters). The correlations corresponding to thecovariance parameters in (b) indicate the degree of similarity of individual rankingat retest across repeatedly followed latent dimensions. In multi-group studies, alsothe parameters in (a) and (b) can be tested for group differences by the differencein chi-square values test, based on Model 1 and its nested version assumingcorresponding cross-group equality restrictions (e.g., Jöreskog & Sörbom, 1993;see also next section). Alternatively (or if the difference in chi-square values test isnonsignificant), these parameters can be constrained for equality across groups ifthe design of the study is such that random group assignment took place after pretest,
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2 9 2 RAYKOV
as in the empirical study under consideration in this article (see next section). Inthe latter case, one will increase the model degrees of freedom and thus arrive atmore precise, on average, estimates and statistical tests.
Model 2
This model allows exploration of the way training effects in the experimentalgroup differ from practice effects in the control group, specifically the temporalpattern of group differences. Given that both in the experimental and control groupspractice/experience effects are at work over time, group differences at consecutiveassessment occasions reflect "pure" training gains, that is, the gains in the experi-mental group disentangled from the confounding retest effects there. Thus, Model2 allows modeling the temporal development of the "pure" training gains.2
To achieve this goal, beyond the two latent factors of Model 1, Model 2incorporates in the experimental group a third latent variable representing groupmean differences. At the same time, all features of Model 1 remain unchangedwithin, and set invariant across, the two groups (see following). That is, Model 2can be obtained from Model 1 following a sequence of three modification steps:(a) postulating all features of Model 1 in both the experimental and control groups,(b) constraining then for group equality all model parameters, and (c) introducingonly in the experimental group a third latent variable for each of the repeatedlyadministered tests. For each fluid measure, the third factor in the experimental groupcomprises the common features, across posttests, of the degree to which this groupoutperforms the control group across the three posttests. For simplicity of reference,this third factor is called the group-differences factor. It is denoted in the next figureby the suffix "_GD" following the abbreviation used with the pertinent fluidmeasure. Model 2 is graphically displayed for the experimental group in Figure 2.A similar model is used in Raykov (1995), which, however, does not address theissue of concern here, viz. the study of the temporal development of abilityvariances; its exploration becomes possible with Model 2.
To achieve model identifiability, for each fluid measure the scale of the groupdifferences factor is set identical to the one of the first posttest, that is, that factor'sloading of the first posttest observed variable is fixed to unity. The remaining two
2 For simplicity of reference, this article uses frequently the term training gains to denote the "pure"effects of the training in the experimental group, whereas the notion training effect is used to refer tothe compound effect of practice effects and training gains in the experimental group. Therefore,conceptually, in the rest of the article training gains is understood to denote the discrepancy betweentraining effects (in the experimental group) and practice/experience effects (in either group), that is, thesuperiority of training effects over practice/experience effects.
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E34 E33 E32 E31
IX34
1X33
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1X31
-Mi
E24
E23
E22
E21
E11 E12 E13 E14
FIGURE 2 Model 2 (experimental group only; see text).
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posttest loadings on this factor are postulated free model parameters, as are its mean,variance, and covariances with the ability variance variables and the two othermeasures' group differences factors. Because all remaining model parameters areheld invariant across the two groups, as mentioned earlier, for each fluid measuresignificance of the mean and of the second and third posttest loadings on this groupdifferences factor would imply significant differences between the training effectsin the experimental group and the practice/experience effects in the control group,that is, significant training gains. To achieve the desired parameterization in Model2, the variances of the three latent group difference factors, as well as their meanand covariances with the remaining latent variables, are introduced as free parame-ters only in the experimental group; at the same time, these parameters are set tozero in the control group (see following). The pattern of change over time in theposttest loadings on the group differences factors is also informative in empiricalapplications of the model, as it indicates the dynamic of group differences: If theyare increasing/decreasing/constant across posttests, the superiority of the experi-mental group is increasing/diminishing/constant (cf. Raykov, 1995).
Relation to LCA. For each of the three longitudinally administered tests, Model2 postulates the same practice curves in both groups as did Model 1, with the followingdiscrete-time approximations (the superscripted prefix (l) signals parameter identityacross groups; the superscript(2) indicates basis curve of Model 2):
and
(2a)
In the experimental group, in addition, Model 2 postulates for each repeat-edly assessed characteristic the training curve with following discrete-timeapproximation:
2Hh) = 0, gf\t2) = 1, g3™(t3) = W 2 ) ( f 4 ) = hA (2b)
This training curve is tailored to the context of the study whose data is analyzed inthe next section. Because random group assignment took place after pretest andwas followed by the training in the experimental group, there are no training effectsat pretest. Thus g3(2)(fi) = 0 is set. However, in general the definition of Model 2
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GROWTH CURVE ANALYSIS 2 9 5
can be easily changed to accommodate group differences, if any, already at firstassessment; this is accomplished by allowing g3(2)(fi) to be a free model parameter.(The issue of model identification will have to be dealt with alternatively.)
To see more transparently the relation of Model 2 to LCA, denote by x = Xu,Xn, Xu, Xu, Xii, X22, X23, X24, Xn, X32, X33, X34' the strung-out vector of observedvariables (the four repeated assessments with the three fluid tests), and by e = [Eu,En, En, En, £21, E22, £23, £24, £31, £32» £33, £34]' the vector of correspondingmeasurement errors ((" used to denote matrix transposition). If w stands for thevector of random variables whose realizations are the individual weights attachedto basis curves (cf. Equation la), Model 2 has the following definition equation inthe experimental group:
(2c)
where
1
¿12
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01
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0
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00
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1
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¿23
¿24
0
0
00
0
0
1
C22
c23
C24
0
0
00
0
0
0
1
¿23
¿24
0
0
00
0
0
0
0
0
01
bit
00
0
0
0
0
0
0
1
00
0
0
0
0
0
0
0
1
0 0 0 0 0 0 ¿>33 c3 3 ¿ 3 3
O O O O O O bu c3 4 d.;34.
is the pattern matrix relating observed to weight variables. Thereby, the addedloadings on the third, group differences latent variable, symbolized by the dparameters, reflect the extent to which training effects in the experimental groupoutperform the experience/practice effects in the control group in the parameteri-zation context of Model 1. Simultaneously, in the control group the followingdefinition equations are valid:
(2d)
whereby in
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296 RAYKOV
' 1 1 0 0
¿12 C12 0 0
0 0 0
0 0 0¿l3 q 3 0 0 0 0 0¿14 c14 0 0 0 0 0
0 0 1 1 0 0
0 0 ¿22 C22 ° °
0 0 ¿23 C23 0 0
0 0 ¿24 C24 0 0
0 0 0 0 0 1
0 0 0 0 0 ¿32
0 00 00 00 00 00 00 00 01 0
cv 00 0 0 0 0 ¿33 c33 00 0 0 0 0 ¿34 c34 0
the same b and c parameters are postulated, whereas all parameters pertaining tothe group-differences variables are fixed to zero (i.e., their loadings, variances, andcovariances among themselves and with any other variable as well as their meansare set equal to zero in this group). Equations 2c and 2d with the pattern matricesGz£ and G^c represent, in a two-group context, a special case of Equation 2 on p.108 of Meredith and Tisak (1990) defining alternatively the general LCA model.(Their Equation 2 is derived from their Equation 1.)
Due to the postulation of Model 1 in the control group and the imposed groupequality restrictions on all but the h parameters, means and covariances of the groupdifference variables (and possibly the error variances, if this does not significantlydegrade the fit), the empirical interpretation of the b and c parameters of Model 2is the same as that of the identical parameters in Model 1, with the provision that inModel 2 they relate to practice effects. That is, the interpretation of the b parametersof this model in the experimental group is as rates of change over time of true meansfor the practice effects in this group (relative to pretest). Similarly, the interpretationof the c parameters of this model is as rates of change in the true mean deviations(relative to pretest) of the practice effects in the experimental group.
In addition to the b and c parameters, the ones of main interest in empiricalapplications of Model 2 are (a) its h parameters, (b) the means of the individualweight variables associated with the added training effects curve per repeatedlyfollowed characteristic in the experimental group, and (c) the variances andcovariances of the weights' variables in (b) as well as their covariances with theinitial true mean deviation variables (cf. Model 1). Should Model 2 be found atenable means of description for a given data set (covariance/mean matrix), theseparameters reflect all group differences to which it is sensitive. As Model 2 has asingle training curve per longitudinally followed characteristic, fixed at unity atfirst posttest, straightforward algebra—like that mentioned with Model 1—showsthat the mean of its weight variable pertaining to the training curve equals the mean
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GROWTH CURVE ANALYSIS 2 9 7
difference at first posttest. Similarly, the consecutive h parameters represent theratios of mean difference at following posttests to that difference at first posttest.
In applications of Model 2, the researcher will be interested in testing forsignificance the h parameters and the means of the group difference latent variables.Should they be found significant for a given data set, there will be evidence for truegroup differences that cannot be explained by chance fluctuations only, that is, forpure training gains that are the only (not error-related, in general) aspects in whichthe two groups differ. Moreover, as with Model 1, any trend in the h parameterswill be indicative of a trend in group differences, that is, in the training gains, over time(see earlier). Such a finding may be of main interest in some empirical applications ofthe model when analyzing multivariate longitudinal data. Last but not least, in someempirical cases of interest may also be the degree of interrelationship between thegroup-difference variables across the repeatedly studied abilities. This degree is re-flected in the aforementioned covariances and the correlations of these latent variables.They are indicative of the extent to which rank ordering in group differences at firstposttest is consistent across longitudinally assessed characteristics. The relations ofthese differences with initial true mean deviations may also be of interest. Theseinterrelations are reflected in the model correlations between the training curve weightvariables and the true mean deviation variables. They are indicative of the degree towhich rank ordering in initial true mean deviations on one repeatedly studied ability isrelated to or predicting that on a group difference variable pertaining to this or anotherability. Finally, the variances of the group difference variables are indicators of theextent of interindividual variability in group differences at first posttest on the pertainingrepeatedly assessed characteristic.
Model 3
This model capitalizes on the parameterization idea underlying Model 2, but indifference to the latter implements it directly at the latent ability level. Although inModel 2 the group differences were parameterized with a reference to the manifestlevel, in the sense of being so in the mean of the three measure-specific, groupdifference factors as well as in its loadings on the observed posttests with each fluidtest, Model 3 furnishes this parameterization at the level of ability change over time.Another important difference between the two models is that in Model 3 the focusis in expressing posttest ability as a linear function of pretest-to-first-posttest abilitychange (cf. McArdle & Anderson, 1990), rather than decomposing it into the sumof a mean change element and one depending on latent pretest variance as in Model2. The path diagram of Model 3 is presented in Figure 3.
This figure reveals that Model 3 can be formally obtained as a three-dimensionalextension of the Level and Shape model by McArdle and Anderson (1990), to whicha latent group-difference-in-shape factor is added for each repeated measure. In thefigure, the first factor per fluid test is the level factor (cf. McArdle & Anderson,
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=11 = 12 "13 E-,
X 1 4
•4-E 21
"22
• E 2 3
' 33 '34
32 : 3 3
FIGURE 3 Model 3 (experimental group only; see text).
298
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GROWTH CURVE ANALYSIS 2 9 9
1990). Its scores represent the initial ability status of each participant. Hence, itsloadings on all repeated assessments with the corresponding measure are set equalto unity. This level factor is denoted by the suffix "_L" added to the symbol usedwith the pertinent repeated test.
The second factor per fluid measure is the shape factor. It is denoted in Figure1 by the suffix "_S" added to the symbol used with the corresponding fluid test.The scores of this factor represent the pretest-to-first-posttest change for eachparticipant. This construction of the shape factor is particularly useful for thisempirical study because the training in its experimental group was conductedbetween pretest and first posttest. To introduce the shape factor, for each repeatedmeasure its loading on the first assessment is fixed to zero, whereas that on firstposttest is set to unity but the remaining loadings are freed (e.g., McArdle &Anderson, 1990). It is noted that other constructions of a shape factor are alsopossible, for example, via fixing to unity of the third or last assessment loading onit (e.g., Raykov, 1995). The particular choice for the present or alternative fixingsis made following substantive considerations and the research questions asked.
The final, third factor per repeated measure is the latent-group-difference-in-shape factor. It is denoted in Figure 3 by the suffix "_LGSD" following the symbolused with the pertinent repeated test. This factor reflects group differences in abilitychange over time. Model 3 uses a different latent change parameterization idea fromthe earlier Model 2 as well as the first model used by Raykov (1995): In differenceto them, Model 3 parameterizes the group differences directly at the latent, ratherthan via the manifest level (as indicated earlier; compare Figures 2 and 3). Althoughin Model 2 the group difference factor loads on the manifest variables, thiscounterpart factor in Model 3 does not have loadings on the observed variables butin effect "adds" the group difference in latent change to the shape factor in theexperimental group only. By analogy to Model 2, this latent group difference factoras well as its mean and covariances with the remaining latent variables areintroduced as free parameters only in the experimental group; these parameters areset to zero in the control group.3
3 The particular nature of the factors underlying the three structural equation models described heremay warrant in certain empirical situations the consideration or introduction of correlated observedvariable specificities associated with different repeatedly administered measures at same assessmentoccasions (cf. Raykov, 1995). This is because the factors are so constructed as to explain onlymeasure-specific common variance that is shared between consecutive assessments with each of thefluid intelligence tests. Hence, at a given assessment occasion some observed variability, which is dueto existing relations between two fluid measures, may remain unexplained by these measure-specificfactors and their interrelationships. (This variability would stem from the relation between test-specific"pure" variable specificities that remain after the corresponding factor variances and "pure" measure-ment error variances are taken out of observed variable variances; e.g., Mulaik, 1972). As a result, insuch empirical settings the corresponding measurement error covariance parameters can be introduced(cf. the argument advanced in favor of interrelated measurement errors in longitudinal common factoranalysis; e.g., Jöreskog & Sörbom, 1993; see next section).
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3 0 0 RAYKOV
Relation to LCA. To show that Model 3 can be obtained as a special case ofLCA, it is first noted that its observed score definition equations are based onidentical parametric structure in the experimental and control groups. With theearlier introduced specific vector and matrix notation, in the experimental groupthese equations are:
(3a)
where
11
1
1
0
0
0
0
0
0
0
0
01
¿13
Sl4
0
0
0
0
0
0
0
0
0
0
0
0
1
1
1
1
0
0
0
0
00
0
0
1
1
£23
«24
0
0
0
0
00
0
0
0
0
0
0
1
1
1
1
00
0
0
0
0
0
0
1
1
£S3
is the pattern matrix here. In the control group, the observed score equations are asfollows:
(3b)
where
1 0 0 0 0 01 1 0 0 0 01 913 0 0 0 01 9,4 0 0 0 00 0 1 1 0 00 0 1 1 0 0
0 0 1 ?23 ° °0 0 1 924 ° °0 0 0 0 1 10 0 0 0 1 10 0 0 0 1 9330 0 0 0 1 934
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GROWTH CURVE ANALYSIS 3 0 1
is the pattern matrix here, with unconstrained across groups q loadings.The group differences are also captured by the latent variable definition equa-
tions. In the control group these are:
wc = I9wc (3c)
with Ig being the 9 x 9 identity matrix and wc being a 9 x 1 vector of model latentvariables (weights' variables). Its last three components have vanishing means, aswell as such variances and covariances among themselves and with any of theremaining six components.
In contrast, in the experimental group these latent variable equations are:
we =Bwe+z (3d)
In Equation 3d, we is a 9 x 1 vector of model latent variables (weights' variables)in this group, and B is a 9 x 9 matrix defined as
0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 1 0 0
0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 1 0
0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 1
0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0
O O O f O O O O O
(3e)
Equations 3a to 3e result as an appropriately constructed special case of Equation2 on p. 108 of Meredith and Tisak (1990) in a two-group context, and hence Model3 is a special LCA model.
In empirical applications, of particular interest with this model will be the means,variances, and covariances of its latent variables. Specifically, the means andvariances of the latent-group-difference-in-shape variables, their intercorrelations,and their correlations with the initial ability status variables (level variables) willbe representing—for each repeatedly followed characteristic—the average latentchange at second and third posttest, relative to first posttest, as well as its interin-dividual variability and interrelationships among those group differences and withthe initial ability levels.
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3 0 2 RAYKOV
RESULTS AND DISCUSSION
The results reported here are obtained in an objective, substantively meaningfulmetric. It is the percentage correct score introduced after dividing each individualassessment score by the maximal possible number of correct answers for a repeat-edly administered measure. Covanances, variances, means, standard deviations, andcorrelations of the 12 analyzed variables in both groups are presented in Table 1.
Initially, the uni- and multivariate variable distributions were examined. Noevidence was found of serious deviations from normality, and no case was identifiedas a potential uni- or multivariate outlier to be taken out of the ensuing analyses.The application of the maximum likelihood method was therefore consideredjustified (Jöreskog & Sörbom, 1993).
Model 1 Results
For this model, the results of a fitted series of nested models are presented in Table2 (e.g., Jöreskog & Sörbom, 1993; the LISREL 8 input file for the last, tenablemodel variant is provided in the Appendix.)
Model 1 was first fitted without imposing any within- or cross-group parameterconstraints. This starting version yielded a%2 value of 172.35 for df= 102,p < .001.In it, two modification indices were quite large (Jöreskog & Sörbom, 1993). Theypertained to the fixed measurement error covariances of the ADEPT Induction andThurstone's Induction tests at pretest and at final posttest in the experimental group.These error covariances could be explained as a consequence of the fact that the twotests were measures of the same fluid intelligence subability, Induction (see alsoFootnote 2). They could additionally be justified as model parameters resulting fromthe uniformity in response across measures that may have been induced by thetraining program in this group. Introducing these error covariances led to a significantdecrease in the chi-square value by Aj£2=46.01 for difference in degrees of freedombeing Adf= 2. The associated goodness-of-fit indices of the resulting nested versionof Model 1 pointed to a plausible model as a means of data description andexplanation: %2= 126.34, df = 100, p = .039, comparative goodness-of-fit index(CFI) = .99; noncentrality parameter (NCP) = 26.34 with a 90% confidence interval(1.61; 59.25), root mean square error of approximation (RMSEA) = .033 with a90% confidence interval (.008, .049), and/» = .96 for the test of "close" fit (i.e., thatRMSEA < .05, or in other words that the model is plausible as a means of datadescription means; Jöreskog & Sörbom, 1993; Browne & Cudeck, 1993). All theseindices, and notably the facts that (a) the RMSEA was less than the assumedthreshold for a plausible model fit of .05 and (b) even its upper endpoint was smallerthan this threshold (Browne & Cudeck, 1993), pointed to an acceptable model fit.
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GROWTH CURVE ANALYSIS 3 0 3
This version of Model 1 is denoted by Vo in Table 2. Because freeing theaforementioned two error covariances led to a significant improvement in fit, theirprevious restriction to zero in the experimental group was abandoned in thesubsequent analyses with further variants of Model 1. The latter were nested inversion Vo that, based on its fit indices, was judged tenable as a means of datadescription.
Because the two groups were randomly formed, it was not expected that theywould differ in initial ability means, variances, and covariances for the threerepeated measures. Indeed, constraining across groups the corresponding parame-ters led to a nonsignificant increase in the chi-square value up to 141.18, df = 109,p = .021 (A%2 = 14.84, Adf= 9, relative to model Voin which this variant Vi wasnested (see Table 3). The additional goodness-of-fit indices were as follows (hereand throughout the remaining discussion, 90% confidence intervals of these indices,where appropriate, are given in parentheses after their estimates): CFI = .99, NCP= 32.18 (5.50; 66.92), RMSEA = .035 (.014; .050), andp = .95 for the test of closefit. It was concluded that the two groups were identical with regard to these initialcharacteristics. Consequently, this cross-group restriction was retained in the restof the analyses with the following variants of Model 1, which were nested in thisbaseline version Vi.
To test if the rates of change over time in ability means across all posttests wereidentical to the corresponding rates of change in ability variances, within each groupthe pairwise equality constraints were imposed on the pertinent ratio parameters.The resulting three-dimensional model of McArdle and Epstein (1987) was asso-ciated with a significant increase in the chi-square value up to 201.17, for df= 127(cf. model version Vo in Table 1, in which this model V2 was nested; see Footnote1). It was concluded that within each group the dynamic of ability means was notthe same as the one of ability variances. Therefore, this within-group restrictionwas abandoned in the ensuing analyses with further versions of Model 1. Toexamine then if the ability means ratios for the ADEPT Induction test were groupinvariant, these parameters were next restricted for equality in both groups (seemodel version V3 in Table 2). This resulted in a significant increase in the chi-squarevalue up to 201.11, df= 112 (cf. version Vi in which V3 was nested). It wasconcluded that the rates of change over time in ability means for this Inductionmeasure were not equal across groups. Hence, this constraint was abandoned in theensuing analyses with the following variants of Model 1. The ability variances ratiosfor the same fluid measure were then tested for group equality: The correspondingfactor loadings were then restricted for group identity (see model version V4 inTable 2). The associated change in the chi-square value was nonsignificant, relativeto model version Vi in which V4 was nested. This result indicated group invariancein the rates of change in the ability mean deviations for this measure. The lastrestriction was adopted in the next fitted models of the series.
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TABLE 1Covariances (Below Main Diagonal), Variances (Along Main Diagonal), Correlations (Above Main Diagonal; Italics), Means(Second Last Row), and Variable Standard Deviations (Bottom Row) tor the Four Assessments With the ADEPT Induction,
ADEPT Figurai Relations, and Thurstone's Standard Induction Tests (from Baltes, Dittmann-Kohli, & Kliegl, 1986)
Group
Experimental"AI(1)AI(2)AI(3)AI(4)AF(1)AF(2)AF(3)AF(4)TI(1)Tl(2)TI(3)TI(4)
MSD
Al(l)
218.057219.379218.208225.193155.152136.153138.844155.559220.921228.180226.843223.08630.82614.767
AI(2)
.820328.112310.488305.800212.465200.050202.093219.358276.086324.969322.896315.55149.08318.114
AI(3)
.804
.932338.014312.179207.086200.811202.286220.427276.473330.119333.189325.06351.65718.385
Al(4)
.820
.908
.913345.960210.479197.065202.484224.301277.108321.715324.459326.81448.57718.600
AF(1)
.651
.726
.698
.701260.760159.402170.659180.436203.467229.249221.491229.92349.48316.148
AF(2)
.653
.782
.774
.751
.699199.235173.222187.923178.463220.124225.760220.47571.41514.115
AF(3)
.644
.764
.753
.745
.724
.840213.317197.847188.281224.539228.492223.49873.36714.605
AF(4)
.664
.763
.756
.760
.704
.839
.854251.710200.173244.277242.081243.73570.77915.865
Tl(l)
.853
.869
.858
.850
.719
.721
.735
.720307.461296.517295.020291.01937.48217.535
TI(2)
.796
.924
.925
.891
.731
.803
.792
.793
.871377.212365.100355.87753.29919.422
Tl(3)
.775
.900
.915
.881
.692
.807
.790
.770
.849
.949392.469358.24754.81719.811
TI(4)
.778
.897
.911
.905
.733
.805
.788
.791
.855
.944
.932376.83952.63319.412
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Control1"AI(1)AI(2)AI(3)AI(4)AF(1)AF(2)AF(3)AF(4)TI(1)TI(2)TI(3)TI(4)
MSD
217.993221.817222.480214.599149.861146.690128.545134.393219.873245.323240.450246.66030.09114.765
.849313.093294.480298.148167.855188.827178.426178.333275.127312.579316.587323.39341.50717.694
.837
.924324.279302.417174.157178.220179.935180.596282.240322.843322.794338.91444.67718.008
.801
.929
.926328.862171.889179.609177.530183.742282.863317.520325.081343.05243.79018.135
.626
.585
.597
.585262.786195.643164.783171.480171.006209.405215.960213.75951.97116.211
.625
.671
.622
.623
.759252.786203.173206.127173.837222.849228.101230.81362.04315.899
.577
.668
.662
.648
.673
.846228.009183.048175.164223.404218.256227.42466.14815.100
.629
.696
.693
.700
.731
.895
.837209.659172.447215.706221.780223.87865.76514.480
.833
.870
.877
.873
.590
.612
.649
.667319.222326.420318.796333.06238.61617.867
.820
.872
.885
.864
.638
.692
.730
.735
.902410.251378.983388.67944.95620.255
.805
.885
.886
.886
.659
.709
.715
.757
.882
.925409.067390.59347.88020.225
.803
.879
.905
.909
.634
.698
.724
.743
.896
.923
.928432.69748.01120.801
Note. AI(i) = ADEPT Induction Test at ith assessment, i = 1, 2, 3, 4; AF(i) = ADEPT Figurai Relations Test at ith assessment, i = 1, 2, 3, 4; TI(i) = Thurstone'sStandard Induction Test at ith assessment, i = 1, 2, 3, 4.
"n = 161. bn = 87.
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306 RAYKOV
TABLE 2Comparison of Nested Variants of Model 1
201.17201.11141.40215.78149.63216.35152.63
127112112115115118118
59.9959.93
.2274.388.23
66.723.00
18333333
Model Version %2 df A%2 Mf
Least restrictive (Vo) 126.34 100 — —Initial ability means and variances equal across groups (V¡) 141.18 109 14.84 9Ability means ratios identical to ability variances ratios
(pairwise) within groups (V2)Ability means ratios for AI equal across groups (V3)Ability variances ratios for AI equal across groups (V4)Ability means ratios for AF equal across groups (V5)Ability variances ratios for AF equal across groups (V6)Ability means ratios for TI equal across groups (V7)Ability variances ratios for TI equal across groups (Vs)
Note. AI = ADEPT Induction Test; AF = ADEPT Figurai Relations Test; TI = Thurstone'sStandard Induction Test; X2 = Chi-Square Goodness-of-Fit Index; A%2 = difference in chi-squarevalues; Adf = difference in degrees of freedom.
The described testing approach was followed then in turn with each of the othertwo repeatedly presented measures, the ADEPT Figurai Relations and Thurstone'sStandard Induction tests. The same pattern of results was found with them, too (seemodel versions V5 to V8 in Table 3; V5 and V6 were nested in V4, whereas V7 andV8 were so in V6). It was therefore concluded that for each fluid measure the twogroups differed in the rates of change over time in ability means but not in theserates for ability variances. The resulting most restrictive version of Model 1, Vs, inwhich the nine rates of change in ability variance parameters were restricted forequality across the two groups (as were the initial ability means, variances, andcovariances; cf. model version Vi underlying this series of nested models), wasassociated with the following goodness-of-fit indices: %2 = 152.63, df= 118,/» =.018, CFI = .99, NCP = 34.63 (6.8; 70.57), RMSEA = .035 (.015; .049), and/? =.96 for the test of close fit. These indices suggested that model V8 was tenable as ameans of data description and explanation. The estimates and standard errors of thesubstantively interesting parameters in it, the ability means and variances ratios(with the latter being identical in the two groups) are presented in Table 3.
To explore further the group-specific dynamic in performance improvement aswell as occasion-specific group differences with this tenable version of Model 1, itwas focused next on the 95% confidence intervals (e.g., Steiger, 1990; Raykov,1995, 1966) of the ability means ratio parameters in it, which reflected the groupdifferences found in the reported analyses. Bearing in mind the imposed cross-group equality restriction on the initial ability means in this model variant V8, allnine rates of change over time in ability means were found larger in the experimentalgroup for the three repeated fluid measures and three posttest assessments, becausethe left-endpoints of all intervals in the experimental group were higher than the
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TABLE 3Ability Means Ratios and Ability Variances Ratios for the
Most Restrictive Version of Model 1 (Version V8):Parameter Estimates and Standard Errors
Parameter
Experimental group
bub12
b13
b14
c,2
c n
C14
b22
b23
24
Cni
c 2 2
c22
b31
b32
b33
b34
C32
C33
<=34
Control group
bub12
b J 3
b,4
C12c13
Cl4
b21
b22
b2 3
b24
C21c22
c23
C-M
Estimate
11.611.691.59
11.381.411.40
.43
.47
.41
.07
.06
.11
11.421.461.40
11.191.201.19
11.341.451.42
11.381.411.40
11.221.301.29
1.071.061.11
Standard Error
—.03.03.03
.06
.06
.06
.02
.02
.02
.06
.06
.06
—.02.03.02
.04
.04
.04
—.03.03.03
.06
.06
.06
—.02.03.02
—.06.06.06
95% Confidence Interval
(1.55, 1.67)(1.63, 1.75)(1.53, 1.65)
(1.39, 1.47)(1.43, 1.51)(1.37, 1.45)
(1.38, 1.46)(1.40, 1.52)(1.36, 1.44)
(1.28, 1.40)(1.39, 1.51)(1.36, 1.48)
(1.18, 1.26)(1.24, 1.36)(1.25, 1.33)
(Continued)
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TABLE 3
(Continued)
Parameter
b31
b32
b33
b34
C31C 3 2
<=33
C34
Estimate
11.171.251.25
11.191.201.19
Standard Error
—
.03
.03
.03
—.04.04.04
95% Confidence Interval
(1.11, 1.23)(1.19, 1.31)(1.19, 1.31)
Note, by, = ability means ratio parameter at ith assessment for ADEPT Induction Test (i = 1,2, 3, 4; cf. Appendix 1); ¿>2; = ability means ratio parameter at ith assessment for ADEPT FiguraiRelations Test (/ = 1, 2, 3, 4); ¿>3; = ability means ratio parameter at ith assessment for Thurstone'sStandard Induction Test (i = 1, 2, 3, 4); c\¡ = ability variances ratio parameter at the ith assessmentfor ADEPT Induction Test (i = 1,2,3, 4); c2; = ability variances ratio parameter at the ith assessmentfor ADEPT Figurai Relations Test (1 = 1, 2, 3, 4); c3; = ability variances ratio parameter at the ithassessment for Thurstone's Standard Induction Test (1 = 1, 2, 3, 4).
Integer numbers stand for fixed model parameters; numbers with decimal point symbolizeestimated parameters. The estimates and standard errors of the parameters cnk, n = 1, 2, 3; k = 2, 3,4, are set identical in both groups (see text). Initial ability means, variances, and covariances arealso set identical in the two groups.
right-endpoints of the corresponding intervals in the control group (see right-mostcolumn in Table 3). Moreover, each of the 18 confidence intervals in Table 3 waslocated completely above unity that symbolizes in this model the initial ability mean(divided by itself) ratio for each fluid measure. Hence, at each posttest there wasconsiderable improvement in test performance in both the experimental and controlgroups, as compared to initial level. That this improvement was maintained for eachfluid measure in either group over the total 6-month testing period is seen from thefact that for each measure the left endpoint of the last interval in the test-specifictriple of ability means ratios, which was associated with final, delayed posttest, waslocated completely to the right of unity.
Model 2 Results
To examine further the differences between training effects in the experimentalgroup and practice/experience effects in the control/experimental group in a singlemodeling session, and specifically the dynamic of these differences, Model 2 wasnext fitted. (As specified in its LISREL 8 input file in the Appendix, the errorassociated with the Thurstone's Standard Induction test at second assessment wasfreed from the cross-group identity imposed on the remaining residual variances;
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see the discussion of Model 2 in the preceding section. This can be seen as aconsequence of a test- and assessment-specific lack of equality of error variancesin the trained and untrained groups. The resulting goodness-of-fit indices indicateda tenable model: %2 = 131.26, df= 114, p = .128, CFI = .996, NCP = 17.260 (0;49.798), RMSEA = .025 (0; .042), and p = .995 for the test of close fit. In it, foreach fluid measure the estimates of the posttest assessments' loadings on thegroup-differences factors, as well as their means, were as follows (in the experi-mental group; standard errors are presented in the parentheses following eachestimate, and the endpoints of the corresponding 95%-confidence intervals followin the brackets; see Figure 2):
1. For ADEPT Induction test: 1; 1.013 (.09) [.85; 1.31]; .68 (.10) [.48; .88];mean: 7.97 (1.06).
2. For ADEPT Figurai Relations test: 1; .76 (.09) [.58; .94]; .58 (.08) [.42;.74]; mean: 10.95 (1.20).
3. For Thurstone's Standard Induction test: 1; .95 (.09) [.77; 1.13]; .75 (.08)[.59; .91]; mean: 8.88 (1.16).
The fact that all six factor loadings on, as well as means of, the three group-dif-ferences factors were significant was interpreted as corroborative evidence for theclaim that at all posttests the training effects found in the experimental group weremore salient than the practice/experience effects in the control group. Moreover,comparison of confidence intervals across time reveals that the extent to which thetraining effects in the experimental group outperformed the practice effects in thecontrol group (with these practice effects being equalized in the model to thepractice effects in the experimental group) was decreasing across posttests. Indeed,using the earlier comparative procedure, for the Induction measures the last posttestfactor loading is found to be markedly smaller than the first posttest loading on thegroup differences factor. For the ADEPT Fluid Relations test this was the case withthe last two assessments relative to first posttest. These findings suggested that withtime the salience of the pure training gains was becoming smaller. At the same time,however, it is noted that because none of the above presented six confidenceintervals contains zero, for each repeated fluid test the corresponding training gainsdid not vanish across the 6-month testing period.
Model 3 Results
To examine additional aspects of group differences in the latent patterns of change,Model 3 was then fitted. It yielded a %2 of 139.47, for df= 119, p = .097; NCP =20.47 (0; 54.06), RMSEA = .026 (0; .043) and/? = .99 for the test of close fit. Allits remaining goodness-of-fit indices also demonstrated an acceptable fit. The
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means of the latent group-difference-in-shape variables were all significantlypositive and found to be 8.02(1.02), 10.63(1.18), and 9.27(1.08) for the threerepeated tests, respectively (in the experimental group). This suggested that therewas a significant group difference in the latent change from pretest to firstassessment after the cognitive training in the experimental group, with this changein the latter group outperforming that in the control group. Of the three intercorre-lations for these three latent change variables, only the one between those pertainingto the ADEPT Induction and Thurstone Induction tests was found to have asignificant covariance, and estimated at .70. (The remaining two correlations wereestimated at. 17 and .21, and were associated with nonsignificant covariances.) Thisfinding suggested a moderate tendency of experimental participants with aboveaverage (latent) training gains at first posttest, relative to pretest, on Induction asassessed by one of the tests, to have above average such gains on it as assessed bythe other test. No such tendency could be found, however, for the relations between(a) these latent group difference variables pertaining to the ADEPT Induction andFigurai Relation tests and (b) these factors for the ADEPT Figurai Relations andThurstone Induction tests (cf. Raykov, 1995).
CONCLUSION
The results presented in this article suggest considerable reserve capacity in olderadults' fluid intelligence. The following findings are indicative of marked plasticityin fluid intelligence of elderly adults: (a) the experimental and control groups wereequivalent in patterns of change in the amount of interindividual differences inability scores; (b) the practice/experience effects were maintained over the total6-month testing period; (c) the training gains were maintained over the 6-monthtesting period in the experimental group; and (d) the training effects in theexperimental group were more salient than the practice/experience effects in thecontrol group, with the superiority of the former, that is, of the training gains,decreasing over time.
The results of the present growth curve modeling concerning the nature of thepractice effects for control participants provide additional support for modifiabilityof intellectual performance of the elderly beyond findings presented in other sources(cf. Baltes et al., 1986; Raykov, 1995,1996), in particular with regard to the currentresults of group invariant pattern of temporal change in the indices of interindividualdifferences at the ability level. As discussed in those cited sources, the practiceeffects in this study have occurred as a function of a limited, no-feedback (no-con-tact) retest condition and hence can be considered an index of plasticity in fluidintelligence. That cited research and the results presented in this article suggest thathealthy elderly adults without the aid of training instructions into test-relevantcognitive skills may be capable of markedly increasing their performance in fluid
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intelligence tests (as a result of experience with repeatedly presented measures).This may be occurring in a manner that resembles, with regard to the pattern oftemporal change in interindividual ability differences, tutor-guided training gains(cf. Baltes, Sowarka, & Kliegl, 1989). Relating the present findings to thosereported in Raykov (1995, 1996), it may be further hypothesized that possiblycapitalizing on interdependencies between fluid intelligence subabilities to a degreesimilar to the one experimental participants may have relied on test-relevant trainedskills, control participants may have been able to produce on their own considerablegains in the studied fluid intelligence measures, although these effects have beenless salient than the training-related ones.
A few cautionary remarks are in order at this point (cf. Baltes et al., 1986;Raykov, 1995). The results reported in this article should not be taken to imply that:(a) aged adults may profit more than younger adults from a cognitive training/prac-tice of the format in the ADEPT studies, (b) the specific training program couldhave raised level of intelligence as a whole, (c) all aged individuals could beexpected to demonstrate the plasticity of fluid intelligence suggested by the presentand previous analyses, or (d) that there are no limits to plasticity in fluid intelligenceof the aged (Baltes & Lindenberger, 1988). A consequential limitation of the presentsample that warrants such caution with respect to the point brought up in (c) is thefact that the analyzed sample is biased toward healthy elderly with a relatively higheducational status (Baltes et al., 1986). Specifically with a view to the practice/ex-perience effects found in the control group of this study, it is worthwhile stressingthat their nature remains to be further explored in order to obtain more informationallowing disentanglement of pure retest effects from possible domain-relevantpractice gains in typical fluid intelligence subabilities such as Induction and FiguraiRelations, as well as permitting further study of aspects of their maintenance overtime. This caution is additionally reinforced by the observation that limitations ofthe interpretations of the findings reported in this article lie also in the facts that:(a) they are based on multiple statistical hypotheses testing procedures that possessunknown type I error rates, (b) some capitalization on chance may have been doneby inspecting the modifications indices of the first fitted version of Model 1 andsubsequently freeing a pair of error covariance parameters that have been kept freein all analyses with Model 1, and (c) the sample is not particularly large. Conse-quently, a replication study is required before more confidence could be placed inthe results and substantive interpretations of this article.
ACKNOWLEDGMENTS
This research was supported by the Max Planck Society for Advancement ofScience and the Research Group Psychological Gerontology of the Free UniversityBerlin, and conducted while I was a visiting fellow at these institutions.
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I am indebted to J. J. McArdle for introduction to latent growth curve modeling,and to R. Kliegl, J. J. McArdle, P. M. Bentler, and G. Rudinger for valuablediscussions concerning applications of the structural equation modeling methodol-ogy. I am grateful to Paul B. Baltes, Freya Dittmann-Kohli, and Reinhold Kliegl(Max Planck Institute for Human Development and Education, Berlin) for permis-sion to use data from their project "Aging and Plasticity in Fluid Intelligence"; toR. Kliegl and P. B. Baltes for valuable and helpful discussions on plasticity in fluidintelligence; and to B. Hayslip, Jr., anonymous reviewers, and the journal editorfor valuable comments and criticism of an earlier draft of this article that contributedto its improvement.
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Verhaegen, P. (1993). Teaching old dogs new memory tricks: Plasticity in episodic memory performancein old age. PhD thesis, Center for Developmental Psychology, Catholic University of Leuven,Belgium.
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APPENDIXLISREL 8 Input Files for Models 1,2, and 3
In the following input files, explanations are added at appropriate places, prefixed by a neededexclamation mark (e.g., Jöreskog & Sörbom, 1993).
LISREL8 INPUT FILE FOR MODEL 1 * EXPERIMENTAL GROUP *DANI=12NO=161NG=2CMFI=r«*feiMEFI=Taife/LAALI AI_2 AI_3 AI_4 AF_1 AF_2 AF_3 AF_4 TI_1 TI_2 TI_3 TI_4MO NX=12 NK=6 PH=SY,FI KA=FI TD=SY,FILEAI_LMEAN ALLVAR AF_LMEAN AF_LVAR TIJLMEAN TI_LVAR! THE NEXT 6 MODEL LINES FREE THE LOADINGS ON THE LATENT MEANS AND¡VARIANCE FACTORSFRLX2 1LX3 1LX41FRLX22LX32LX42FRLX63LX73LX83FRLX64LX74LX84FRLX105LX115LX125FRLX106LX116LX126! THE NEXT LINE IS NEEDED FOR MODEL IDENTIFICATIONV A 1 L X 1 1 L X 1 2 L X 5 3 L X 5 4 L X 9 5 L X 9 6! THE NEXT LINE INTRODUCES THE LATENT MEAN PARAMETERSFRKA1KA3KA5! THE NEXT LINE INTRODUCES THE LATENT VARIANCES AND COVARIANCESFRPH22PH44PH66FRPH42PH62PH64! THE NEXT 2 MODEL LINES ADD THE ERROR VARIANCESF R T D 1 1 T D 2 2 T D 3 3 T D 4 4 T D 5 5 T D 6 6FRTD7 7TD8 8TD9 9TD1010TD11 11TD1212ST 100 ALL! FOLLOWING ARE DATA-SPECIFIC START VALUESMALX1 1 0 0 0 01.21.1500001.5 1.300001.7 1.400000 0 1 1 0 000 1.21.10000 1.5 1.20000 1.61.30000001 100001.21.100001.4 1.200001.61.3ST 150 PH 2 2 PH 4 4 PH 6 6ST30KA1KA3KA5OUNS
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LISREL8 INPUT FILE FOR MODEL 1 * CONTROL GROUP *DA NO=87CMFI=ra*feiME FI=TflWe 1
LAAI_1 AI_2 AI_3 AI_4 AF_1 AF_2 AF_3 AF_4 TI_1 TI_2 TI_3 TI_4MO LX=PS PH=PS TD=PS KA=PSLEALLMEAN ALLVAR AF.LMEAN AF_LVAR TI_LMEAN TLLVAR! THE NEXT TWO MODEL LINES ADD DATA-SPECIFIC PARAMETERSFRTD191FRTD1 124!THE FOLLOWING 9 MODEL LINES IMPOSE THE CROSS-GROUP RESTRICTIONS !OF
INTEREST: FIRST THE INITIAL ASSESSMENT RESTRICTIONS ARE COMING ¡(SEE TEXT)EQ KA 1 1 KA 1EQ KA 1 3 KA 3EQ KA 1 5 KA 5EQ PH 1 2 2 PH 2 2E Q P H 1 4 4 P H 4 4E Q P H 1 6 6 P H 6 6E Q P H 1 4 2 P H 4 2E Q P H 1 6 2 P H 6 2E Q P H 1 6 4 P H 6 4! NOW THE ADDED/TESTED RESTRICTIONS ARE COMINGE Q L X 1 2 2 L X 2 2E Q L X 1 3 2 L X 3 2E Q L X 1 4 2 L X 4 2E Q L X 1 6 4 L X 6 4E Q L X 1 7 4 L X 7 4E Q L X 1 8 4 L X 8 4EQLX1 106LX106EQLX1 116LX116EQ LX 1 12 6 LX 12 6OU
Note, AI = ADEPT Induction test; AF = ADEPT Figurai Relations test; TI = Thurstone StandardInduction test. The labels AI_1, AI_2, ..., and TI_4 denote the first assessment with the ADEPTInduction test, the second assessment with this test,..., and the last assessment with the ThurstoneStandard Induction test, respectively. The labels AI_LMEAN, ALLVAR,..., and TI_LVAR denotethe latent means and variances factors per repeated measure, respectively. First, third, and fifth latentvariables symbolize the ability means factors, whereas second, fourth, and sixth latent variablesdenote the ability variances factors. Starting values are specific for the analyzed data set and may beinappropriate for other data. As mentioned in the input file, the lines "FR TD 1 9 1" and "FR TD 112 4" are also specific for this data and may well be unnecessary for other data sets.
LISREL8 INPUT FILE FOR MODEL 2 * CONTROL GROUP *DA NI=12 NO=87 NG=2CM FI=7Vzífe iME FI=Toífe /
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LAAI_1 AI_2 AI_3 AI_4 AF_1 AF_2 AF_3 AF_4 TI_1 TI_2 TI_3 TI_4MO NX=12 NK=9 PH=SY,FI KA=FI TD=SY,FILEALLMEAN AI_LVAR AF_LMEAN AF_LVAR TI_LMEAN TI_LVARAI_GD AF_GD TI_GD! THE NEXT 6 MODEL LINES FREE APPROPRIATE LOADINGS ON THE LATENT! MEANS AND VARIANCES FACTORSFRLX21LX31LX41FRLX22LX32LX42FRLX63LX73LX83FRLX64LX74LX84FRLX105LX115LX125FRLX106LX116LX126! THE NEXT LINE FIXES INITIAL FACTOR LOADINGS FOR IDENTIFIABILITY!VA1LX11LX12LX53LX54LX95LX96! THE NEXT MODEL LINES INTRODUCE THE FACTOR MEANS AND VARIANCES! AS WELL AS ERROR VARIANCES AS MODEL PARAMETERSFRKA1KA3KA5FRPH22PH44PH66FRPH42PH62PH64FRTD11TD22TD33TD44TD55TD66FRTD77TD88TD99TD1010TD1111TD1212EQLX22LX32LX42EQLX64LX74LX84EQLX106LX116LX12 6ST 100 ALL! FOLLOWING ARE DATA-SPECIFIC START VALUESMALX1100000001.1 1.1500000001.21.200000001.3 1.3000000000110000000 1.1 1.10000000 1.2 1.10000000 1.3 1.10000000001100000001.1 1.100000001.21.100000001.3 1.1000ST150PH22PH44PH66ST30KA1KA3KA5OU NS SCLISREL8 INPUT FILE FOR MODEL 2 * EXPERIMENTAL GROUP *DA NO=161CMFI=Toífe/ME FI=Ta6ie /
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LAAI_1 AI_2 AI_3 AI_4 AF_1 AF_2 AF_3 AF_4 TI_1 TI_2 TI_3 TI_4MO LX=IN PH=IN TD=IN KA=INLEALLMEAN AIJLVAR AF.LMEAN AF_LVAR TI_LMEAN TI.LVARAI_GDAF_GDTI_GDFRLX3 7 L X 4 7FR LX 7 8 LX 8 8FRLX119LX129F R P H 7 7 P H 8 8 P H 9 9! THE NEXT LINE IS NEEDED FOR MODEL IDENTIFICATIONV A 1 L X 2 7 L X 6 8 L X 1 0 9! THE NEXT LINE INTRODUCES THE THREE GD-FACTOR MEANSFRKA7KA8KA9! THE NEXT LINE ALLOWS THE GD-FACTORS TO BE CORRELATEDF R P H 9 7 P H 9 8 P H 8 7! THE FOLLOWING 2 LINES GIVE GROUP-SPECIFIC START VALUESVA10KA7KA8KA9V A 1 0 P H 9 7 P H 9 8 P H 8 7! THE FOLLOWING LINE INTRODUCES THE RELATIONSHIPS BETWEEN THE GD-! FACTORS AND THE REMAINING LATENT VARIABLESF R P H 7 2 P H 8 4 P H 9 6 P H 7 4 P H 7 6 P H 8 2 P H 8 6 P H 9 2FRPH9 4! GROUP-SPECIFIC START VALUES ARE GIVEN NEXT TO THE LATTER! PARAMETERS AND THE GD-FACTORS' LOADINGSVA 10 PH 7 2 PH 8 4 PH 9 6 PH 7 4 PH 7 6 PH 8 2 PH 8 6 PH 9 2VA10PH94V A 1 L X 2 7 L X 3 7 L X 4 7V A 1 L X 6 8 L X 7 8 L X 8 8V A 1 L X 1 0 9 L X 1 1 9 L X 1 2 9! THE NEXT 3 MODEL LINES ARE DATA-SPECIFIC RELAXATIONS OF THE CROSS-! GROUP EQUALITY RESTRICTION IMPOSED EARLIERFRTD191FR TD 1 12 4FRTD1010OU
Note. In addition to the symbolism used with Model 1, AI_GD, AF_GD, and TI_GD denote thegroup-difference (GD-) factors for the AI, AF, and TI measures, respectively. As with Model 1,starting values are specific for the analyzed data set and may be inappropriate for other data. Thelines "FR TD 1 9 1," "FR TD 1 12 4," and "FR TD 10 10" are also specific for this data and maywell be unnecessary for other data sets.
LISREL8 INPUT FELE FOR MODEL 3 * CONTROL GROUPDA NI=12 NO=87 NG=2CM ¥\=TabU 1ME FI=rnWe 1LAAI_1 AI_2 AI_3 AI_4 AF_1 AF_2 AF_3 AF_4 TI_1 TI_2 TI_3 TI_4
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APPENDIX(Continued)
MO NY=12 NE=9 PS=SY,FI BE=FU,FI AL=FRLEAI_L ALS AF_L AF_S TI_L TI_S AIJLGDS AFJLGDS TI_LGDS! THE NEXT 3 MODEL LINES INTRODUCE THE UNITY RESTRICTIONS ON! APPROPRIATE FACTOR LOADINGS FOR EACH OF THE THREE REPEATED! MEASURESVA 1 LY 1 1 LY2 1 LY3 1 LY4 1 LY2 2V A 1 L Y 5 3 L Y 6 3 L Y 7 3 L Y 8 3 L Y 6 4VA1LY9 5LY105LY115LY12 5LY12 6! THE NEXT LINE FREES THE APPROPRIATE FACTOR LOADINGS PER TESTFRLY32LY42LY74LY84LY116LY126! THE NEXT 3 LINES FREE ALL LATENT VARIANCES AND COVARIANCESFR PS 1 IPS 22 PS 3 3 PS 44 PS 5 5 PS 66FR PS 2 1 PS 3 1 PS 3 2 PS 4 1 PS 4 2 PS 4 3 PS 5 1 PS 5 2 PS 5 3 PS 5 4FR PS 6 IPS 6 2 PS 6 3 PS 6 4 PS 6 5! THE NEXT MODEL LINE FIXES THE MEANS ON THE LGDS FACTORS: THEY! ARE "NONEXISTING" IN THIS, CONTROL GROUPFI AL(7)-AL(9)¡THE NEXT 3 MODEL LINES EQUALIZE OVER TIME THE MEASUREMENT ERROR! VARIANCES PER TESTEQTE(1)-TE(4)EQTE(5)-TE(8)EQ TE(9)-TE(12)MALY1000000001100000001100000001 1 0 0 0 0 0 0 00 0 1 0 0 0 0 0 0001 1000000 0 1 1 0 0 0 0 00 0 1 1 0 0 0 0 00 0 0 0 1 0 0 0 00 0 0 0 1 1 0 0 000001 10000 0 0 0 1 1 0 0 0MAPS192.0528.59 53.24156.87 51.49 224.87-12.45 8.96-53.08 66.48221.63 52.82 204.55 -15.22 285.015.09 40.47 22.21 17.54 9.02 53.77000000000000000000000000MATE27.72 27.72 27.72 27.72 35.7 35.7 35.7 35.7 22.29 22.29 22.29 22.29
(Continued)
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APPENDIX(Continued)
MA AL30.82 18.26 49.46 22.01 37.48 15.74 0 00OU AD=OF IT=99 SCLISREL8 INPUT FILE FOR MODEL 3 * EXPERIMENTAL GROUPDA NO=161CM FI=07STable 1MEFI=07STablelLAAI_1 AI_2 AI_3 AI_4 AF_1 AF_2 AF_3 AF_4 TI_1 TI_2 TI_3 TI_4MO LY=IN BE=IN PS=IN TE=IN AL=INLEAI_L AI_S AF_L AF_S TI_L TI_S AIJLGDS AF_LGDS TI_LGDS! THE FOLLOWING LINE RELAXES A DATA-SPECIFIC PARAMETER RESTRICTIONFRTE(4)! THE FOLLOWING LINE "ADDS" THE LGSD FACTORS IN THIS, EXPERIMENTAL! GROUP: THESE FACTORS ARE IN EFFECT PRESENT ONLY IN THIS GROUPV A 1 B E 2 7 B E 4 8 B E 6 9! THE FOLLOWING MODEL LINE "ADDS" THE MEANS OF THE LGSD FACTORS! IN THIS GROUPFR AL(7)-AL(9)¡THE FOLLOWING 2 MODEL LINES INTRODUCE THE VARIANCES AND! COVARIANCES OF THE LGSD FACTORS IN THIS GROUPFR PS 7 7 PS 8 8 PS 9 9 PS 9 7 PS 9 8 PS 8 7FR PS 7 1 PS 8 3 PS 9 5 PS 7 3 PS 7 5 PS 8 1 PS 8 5 PS 9 1 PS 9 3! THE FOLLOWING MODEL LINE ALLOWS THIS GROUP TO HAVE DIFFERENT! MEAN-CHANGE-RELATED FACTOR LOADINGSFRLY3 2 L Y 4 2 L Y 7 4LY8 4LY116LY12 6! THE FOLLOWING SIX MODEL LINES ARE REQUIRED FOR CONSISTENCY OF! MODEL PARAMETERS: LATENT RESIDUALS SHOULD BE INDEPENDENT OF! LATENT PREDICTORS (E.G., JORESKOG & SORBOM, 1993)FI PS 2 1 PS 2 2 PS 3 2 PS 4 2 PS 5 2 PS 6 2FI PS 4 1 PS 4 2 PS 4 3 PS 4 4 PS 5 4 PS 6 4FI PS 6 1 PS 6 2 PS 6 3 PS 6 4 PS 6 5 PS 6 6VA 0 PS 2 1 PS 2 2 PS 3 2 PS 4 2 PS 5 2 PS 6 2VA 0 PS 4 1 PS 4 2 PS 4 3 PS 4 4 PS 5 4 PS 6 4VA 0 PS 6 1 PS 6 2 PS 6 3 PS 6 4 PS 6 5 PS 6 6! THE FOLLOWING 2 MODEL LINES GIVE STARTING VALUES FOR THE "ADDED"! PARAMETERS IN THIS, EXPERIMENTAL GROUPST 100 PS 7 7 PS 8 8 PS 9 9 TE(4)ST10AL(7)-AL(9)OU
Note. AI_L, AF_L, TI_L, AI_S, AF_S, TI_S = level and shape factors for AI, AF, and TI tests; LGDS= latent group difference in shape. Starting values are specific to this data set and may be inappropriatefor other data sets.
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