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Interval Estimation of LatentCorrelations in StructuralEquation Models With A PrioriRestricted ParametersTenko Raykov & Spiridon PenevPublished online: 15 Jun 2010.

To cite this article: Tenko Raykov & Spiridon Penev (2002) Interval Estimation ofLatent Correlations in Structural Equation Models With A Priori Restricted Parameters,Understanding Statistics, 1:3, 139-155, DOI: 10.1207/S15328031US0103_01

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Interval Estimation of LatentCorrelations in Structural EquationModels With A Priori Restricted

Parameters

Tenko RaykovDepartment of Psychology

Fordham University

Spiridon PenevDepartment of Statistics

University of New South Wales

We propose a procedure for estimating standard errors and confidence intervals (CIs)for particular latent variable correlations. Our approach is based on a utilization of thegenerally applicable delta method and can be used to evaluate precision of estimation offactor correlations in structural equation models with factor loadings a priori restrictedto given constants. This procedure allows comparison of strengths of interrelationacross pairs of latent constructs. Similarly, the approach can be used to evaluate stan-dard errors and CIs or parametric expressions of substantive interest in covariancestructure models, which are not directly accessible as parameters of the latter. The out-lined interval evaluation method is illustrated with a numerical example.

Keywords: confidence interval, delta method, latent correlation,parameter constraint, standard error, structural equation modeling

Applications of structural equation modeling (SEM) in behavioral, social, and edu-cational research have been steadily increasing over the past three decades. The

UNDERSTANDING STATISTICS, 1(3), 139–155Copyright © 2002, Lawrence Erlbaum Associates, Inc.

Requests for reprints should be sent to Tenko Raykov, Department of Psychology, Fordham Univer-sity, Bronx, NY 10458.

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methodology offers attractive means of studying complex multivariable phenom-ena at the construct level of typical interest in those sciences, in which measure-ment error in all fallible indicators is explicitly accounted for. In addition, specialparameterizations are possible within structural models, which allows evaluationand tests of hypotheses about substantively interesting aspects of the phenomena.Such opportunities are frequently provided by models incorporating a priori re-strictions on some of its parameters, for example, factor loadings.

The popular latent curve analysis (Meredith & Tisak, 1990) has offered a com-prehensive approach to the study of growth or decline along latent dimensions un-der investigation. Within this approach, the level-and-shape model (McArdle &Anderson, 1990) permits examination of true baseline position (level factor) andamount of change across measurement points (shape factor) of a repeated assess-ment study as well as their interrelationship. Similarly, the intercept-and-slopemodel (Rogosa & Willett, 1985; see also Raykov, 1998) and its higher order exten-sions in terms of random coefficient models (e.g., Muthen, 1991) permit the re-searcher to study the latent change curves of all examined individuals. This isaccomplished by focusing on its intercept and slope factors (and higher order fac-tors if applicable) and their interrelationships. These and related models also repre-sent useful means of studying correlates and predictors of change. This goal isachieved by interrelating their latent variables to background characteristics suchas initial ability, personality dimensions, age, gender, and socioeconomic status(e.g., Raykov, 1995, 1996).

CONSTRAINTS TO ACHIEVE DESIREDPARAMETERIZATION

The type of models discussed have in common the feature that some of their factorloadings are fixed a priori to particular constants. In the level-and-shape model, forexample, all variable loadings on the level factor and one on the shape factor arefixed at unity before the model is fitted to data (e.g., McArdle & Anderson, 1990).A version of this model is displayed in Figure 1 following a widely adopted nota-tional convention for representing covariance structure models (e.g., Jöreskog &Sörbom, 1993). In this figure, η1 and η2 represent the level and shape factors, re-spectively, and are placed within circles. Their correlation ρ12 is frequently of inter-est in applications of the model, as it reflects the degree of (linear) relationship be-tween true initial status and latent change over time, and the present note is alsoconcerned with its interval estimation. The observed variables Y1 through Y4 are po-sitioned within squares. One-way arrows indicate that the variables at their begin-ning are presumed to play an explanatory role for the variables at their ends, andtwo-way arrows symbolize assumed relationships between the variables they areconnecting, whose direction is not specified further.

140 RAYKOV AND PENEV

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Similarly, in the intercept-and-slope model, a version of which is displayed inFigure 2, all variable loadings on the intercept factor (η1) are set at unity beforehand,whereas those on the slope factor (η2) are restricted to be equal to the time points ofsuccessive assessments (Muthen, 1991; Rogosa & Willett, 1985). The latent corre-lation ρ12 indicates the extent of (linear) relationship between the intercept and slopefactors, and is oftentimes of concern when studying patterns of change. The exten-sion of this model to capture the curvature of individual change curves contains anadded third factor, whereas the repeated measure loadings on it equal the squares ofassessment time points (e.g., Raykov, 1996).

Figures 1 and 2 represent examples of restricted covariance structure modelshaving some of their factor loadings fixed at particular constants to begin with,which remain unchanged throughout the parameter estimation process. A numberof other models that can be obtained, for example as special cases of the general la-tent curve analysis model (Meredith & Tisak, 1990), share this distinctive featureas well having parameters fixed at appropriate constants by model definition.These a priori restrictions enable the models to achieve the specificparameterizations that reflect theoretically and empirically relevant concerns oftheir developers when addressing substantive research questions.

POINT AND INTERVAL ESTIMATION OF LATENTCORRELATIONS IN RESTRICTED STRUCTRUAL

MODELS

A natural question that arises with such models concerns evaluation and testing ofhypotheses about interrelationship indexes of their latent variables. This query isfrequently of special relevance for behavioral researchers applying the models.Similarly, other substantive concerns can be phrased in terms of appropriate ex-

LATENT CORRELATION STANDARD ERROR 141

FIGURE 1 The level-and- shapemodel for four repeated assessments(McArdle & Anderson, 1990). Fixingof the four loadings on the level factorand the last on the shape factor is indi-cated by unities, placed within smallrectangles, attached to their paths. η1 =level factor, η2 = shape factor.

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pressions of parameters of a model under consideration, such as ratios of latentmeans or of other parameters in multiple-group mean structure models. As specificexamples in terms of latent interrelationships, questions such as “How stable is theestimate of the correlation between cognitive decline and neuroticism in late life?”or “Are individual intercept and slope of change more saliently related than rate ofdecline and covariate depression score?” can be addressed via estimation and ex-amination of particular assertions about latent correlations as well as their compari-sons in studied population(s).

This and similar types of queries require that the researcher be in a position notonly to estimate latent correlations but also to obtain measures of their stabil-ity/precision of estimation, such as standard errors. Once the latter are made avail-able, one can construct confidence intervals (CI) for the latent correlations ofinterest. These intervals provide ranges of plausible values for the populationcounterpart parameters of actual concern, and allow one to assess precision of theirestimation with the available sample. Standard errors and CIs therefore represent anecessary addendum to latent correlation point estimates that still seem to be pre-occupying much of the attention in behavioral and social research. In fact, intervalestimates are valuable and needed elements of the process of drawing inferencesabout construct interrelationships from the sample to the population of concern forthe scientist (Hays, 1994).

In typical empirical applications of the previous and similar a priori restrictedcovariance structure models it is not standard to parameterize latent factor corre-lations, and thus the possibility of obtaining directly standard errors associatedwith them is lacking (see also next paragraph and footnote 1). The reason is thatwith the involved factor loading constraints imposed beforehand, one actuallyfixes the scales of the pertinent latent variables to a metric that is meaningful forparticular substantive questions to be addressed with the models. However, toparameterize a latent correlation and obtain a standard error of it a change of the

142 RAYKOV AND PENEV

FIGURE 2 The intercept-and-slope model for four repeated assess-ments (Rogosa & Willett, 1985).Fixing of the four loadings on the in-tercept factor and those on the slopefactor are indicated by the constantsattached to their paths, placed withinsmall rectangles. For the purposes ofthis article and without limitation ofgenerality, the latter factor’s loadingsare fixed at the successive numbers 0,1, 2, and 3 (e.g., Raykov, 1998). η1 =intercept factor, η2 = slope factor.

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underlying metric would be needed, which is not typically carried out in practicewhen imposing those constraints. More specifically, in the originallevel-and-shape model the correlation between the level and shape factors is notestimated directly (i.e., as a single parameter of the model; see Figure 1); toachieve this via their covariance that is a model parameter would require theirvariances to be set at 1 before the model is fitted to data. This is because by vir-tue of the first indicator loading restriction on the level factor, as well as on thelast of the shape factor, the metric of each factor is correspondingly locked atthat of the fixed indicator. Under these circumstances, to additionally constrainthe factor variances to 1 would mean to impose in general misspecifying restric-tions. Similarly, in the original intercept-and-slope model (see Figure 2), themetric of its latent variables is fixed beforehand as a result of the a priori im-posed restrictions on its factor loadings, to address the substantive questions ofinterest that the model has been developed for (e.g., Rogosa & Willett, 1985).

These limitations do not have any consequential relevance for purposes of pointestimation of the corresponding latent correlations in such models. Indeed, consis-tent estimates of construct correlations obviously can be obtained by dividing theircovariance by the product of their standard deviations, all three of which are consis-tently estimated when fitting the model (e.g., Bollen, 1989; Rao, 1973). The previ-ously discussed limitations become relevant, however, when one is interested inobtaining routinely standard errors of latent correlations (as opposed to theircovariances whose standard errors are typically furnished by fitting the model be-cause they are model parameters). If a rescaling of the latent variable metric could becarried out so that their variances became one—as readily accomplished in otherstructural models without such a priori constraints—the large-sample standard er-rors of their covariances would furnish large-sample standard errors of the corre-sponding correlations that are then equal to their covariances. Yet such a rescaling isnot routinely carried out in applications of the currently considered a priori con-strained models, for the aforementioned reasons, but can be accomplished via alter-native methods that aim to obtain standard errors of these latent correlations.1

The purpose of this article is to discuss a relatively simple analytic method for es-timation of a standard error for each latent correlation in a structual equation modelwith factor loading(s) fixed beforehand to specific constants. The following ap-proach represents a utilization of the so-called delta method that is frequently used instatistical applications to obtain standard errors of expressions involving model pa-rameters (e.g., Rao, 1973). This is a general method that furnishes an approximate

LATENT CORRELATION STANDARD ERROR 143

1Reparameterizations leading to equivalent models are possible, for each of the models depicted inFigures 1 and 2, with which the discussed correlations become accessible for point and interval estima-tion. These reparameterizations are not pursued here, however, due to the instructional aims of this noteand their more advanced nature. (Further details can be obtained from the authors on request. We aregrateful to an anonymous referee for pointing out such reparameterizations).

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standard error of any parameter estimator expressible as an appropriate (i.e., contin-uously differentiable) function of other parameters for which estimators with theirvariances and covariances are available. Beyond its specific focus, this article pro-vides a didactic discussion of how the delta method can be employed for purposes ofevaluation of approximate standard error of any substantively meaningful functionof model parameters (e.g., ratio, product, or a more complicated expression) by us-ing their estimates, standard errors, and related information obtained when a struc-tural equation model of interest is fitted to data.

APPLICATION OF THE DELTA METHODFOR OBTAINING STANDARD ERRORS

AND CONFIDENCE INTERVALS OF LATENTCORRELATIONS

In this section, a general approach is discussed that obtains an approximate stan-dard error and CI for any function of covariance structure model parameters, suchas product, ratio, or a nonlinear function of model parameters such as a correlation.Although the following discussion will be concerned with latent correlations, ap-plications of its underlying method to other parametric expressions evolves directlyalong the following lines (for an example of reliability interval estimation, see e.g.,Raykov, in press).

Because the correlation ρ between two random variables is defined as the ratioof their covariance to the product of their standard deviations, this correlation canbe considered a function of these three quantities. That is, symbolically

(1)

where σ12 denotes the variables’ covariance, σ12 and σ 2

2 their variances, and f(., ., .)stands for a function of the arguments listed within parentheses. (Throughout thisarticle, the square root symbol will denote positive square root of the following ex-pression.) For the purposes of this article, we will often use the empirical counter-part of Equation 1 in terms of sample correlation, variances, and covariance:

(2)

where a carat signals sample estimator.The delta method of fundamental relevance to this article is based on a linear

approximation of a continuously differentiable function, which is implied by the

144 RAYKOV AND PENEV

12 2 212 1 22 2

1 2

( , , ),fσρ σ σ σσ σ

� �

12 2 212 1 22 2

1 2

ˆˆ ˆ ˆ ˆ( , , ),ˆ ˆ

fσρ σ σ σσ σ

� �

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Taylor expansion of the function in the vicinity of a particular point from its defini-tion domain (e.g., Stewart, 1991). For the function �ρ in Equation 2, which is seen tobe continuously differentiable in its definition domain, that linear approximationin the vicinity of the population variances and covariance, σ σ σ12 1

222, , and (ex-

cluding zero variances), has the following form:

(3)

where “≈” stands for approximately equal while ∂ σ σ σ ∂σf ( , , ) /12 12

22

12 ,∂ σ σ σ ∂σf ( , , ) /12 1

222

12 and ∂ σ σ σ ∂σf ( , , ) /12 1

222

22 denote partial derivatives of

f(., ., .) with respect to the variable covariance and variances, at the point of pop-ulation covariance and variances. Denoting for simplicity these partial deriva-tives by D1, D2, and D3, respectively, Equation 3 is rewritten into a form that iseasier to manipulate for our purposes and represents the basis of the estimationapproach of this article:

(4)

Equation 4 furnishes an approximation of the latent variable correlation esti-mator of interest, �ρ, as a linear combination of three quantities, apart from theconstant f ( , , )σ σ σ12 1

222 —that is inconsequential for the application of the

method in this article as seen later. These quantities are the distances betweensample estimates and pertinent true values of the latent covariance and vari-ances. The weights of this combination are the corresponding partial derivatives,evaluated at the point of these population parameters. Using well-known rulesfor function differentiation (e.g., Stewart, 1991), these derivatives are obtainedas follows (see Equation 1):

D1 = 1/(σ1 σ2), (5)

(6)

(7)

Because the standard error of any estimator is defined as the standard deviationof its random sampling distribution (e.g., Hays, 1994), the square of the standard

LATENT CORRELATION STANDARD ERROR 145

2 2 2 2 2 212 12 12 12 12 121 2 1 2 1 2

2 2 2 2 2 2 2 2 2 212 121 1 1 2 1 2 2 1 2 2

ˆ ˆ ˆ ˆ ˆ( , , ) ( , , ) ( ) ( , , ) /ˆ ˆ( ) ( , , ) / ( ) ( , , ) / ,

f f ff f

ρ σ σ σ σ σ σ σ σ σ σ σ σσ σ σ σ σ σ σ σ σ σ σ σ

� � � � � �

� � � � � � � �

2 2 2 2 2 212 12 12 1 2 31 2 1 1 2 2ˆ ˆ ˆ ˆ( , , ) ( ) ( ) ( ) .f D D Dρ σ σ σ σ σ σ σ σ σ� � � � � � �

32 12 21/(2 ), andD σ σ σ��

33 12 1 2/(2 ).D σ σ σ��

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error of a latent correlation �ρ under scrutiny is obtained by taking variance of bothsides of Equation 4. Given that f ( , , )σ σ σ12 1

222 ; the population parameters

σ σ σ12 12

22, , and ; and the three derivatives in Equations 5 to 7 are all constants,

this yields:

(8)

where Var(.) denotes variance of the random variable within parentheses and Cov(.,.) covariance of the two involved variables. Equation 8 yields now an approximatestandard error of a considered latent correlation:

(9)

where SE stands for standard error.To use Equation 9 in practice, we first note that when a considered model is fit-

ted to data one obtains consistent estimates of the parameters appearing in theright-hand sides of Equations 5 to 7 (use of any fit function currently availablewithin the framework of SEM yields consistent parameter estimates; Bollen,1989). Hence, consistent estimates of the derivatives D1 to D3 are obtained by sub-stituting those estimates of the latent variances and covariance into Equations 5 to7 (e.g., Rao, 1973):

(10)

(11)

(12)

This yields the estimator of the standard error of the latent correlation estima-tor �ρ as:

(13)

146 RAYKOV AND PENEV

2 2 2 2 2121 2 1 3 2

2 2 2 21 2 12 1 3 12 2 31 2 1 2

ˆ ˆ ˆ ˆ( ) ( ) ( ) ( )ˆ ˆ ˆ ˆ ˆ ˆ2 ( , ) 2 ( , ) 2 ( , ).

Var D Var D Var D VarD D Cov D D Cov D D Cov

ρ σ σ σσ σ σ σ σ σ

� � �

� �

2 2 2 2 2121 2 1 3 2

2 2 2 2 ½1 2 12 1 3 12 2 31 2 1 2

ˆ ˆ ˆ ˆ( ) [ ( ) ( ) ( )ˆ ˆ ˆ ˆ ˆ ˆ2 ( , ) 2 ( , ) 2 ( , )] ,

SE D Var D Var D VarD D Cov D D Cov D D Cov

ρ σ σ σσ σ σ σ σ σ

� � �

� � �

1 1 2ˆ ˆ ˆ1/( , ),D σ σ�

32 12 21

ˆ ˆ ˆ ˆ/(2 ), andD σ σ σ��

33 12 1 2

ˆ ˆ ˆ ˆ/(2 ).D σ σ σ��

2 21 12 2 31 2

2 2 2 2 ½1 2 12 1 3 12 2 31 2 1 2

ˆ ˆ ˆˆ ˆ ˆ ˆ( ) [ ( ) ( ) ( )ˆ ˆ ˆ ˆ ˆ ˆˆ ˆ ˆ ˆ ˆ ˆ2 ( , ) 2 ( , ) 2 ( , )] .

SE D Var D Var D Var

D D Cov D D Cov D D Cov

ρ σ σ σσ σ σ σ σ σ

� � �

� � �

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We stress that all quantities appearing in the right-hand side of Equation 13 areobtained once the model is fitted to data. Indeed, all involved variance andcovariance estimates are such of model parameters and hence contained in thesoftware output used to fit the covariance structure model of concern. As men-tioned earlier, the estimates of the partial derivatives D1 to D3 are obtained viaEquations 10 to 12 by substituting them into corresponding model parameter esti-mates. The estimates of the covariances of the variance and covariance estimators,which appear as last three elements of the right-hand side of Equation 13, are ob-tained as pertinent entries of the parameter estimates’ covariance matrix. Spe-cifically, when LISREL (Jöreskog & Sörbom, 1996) is used for model fitting andestimation purposes, and one requires all possible output (by including the key-word ALL in the last line of the submitted input), this matrix is obtained and titledCovariance Matrix of Parameter Estimates. An SPSS file implementing the calcu-lation of the standard error according to Equation 13 is shown in the Appendix (aswell as a subsequent 90% CI construction).

An approximate 100(1 – γ)% CI for the latent correlation ρ of interest can beobtained by capitalizing on the asymptotic normality of its estimator as ex-pressed in Equation 4 (0 < γ < 1). This results from the asymptotic normality ofthe estimators of latent variances and covariance appearing in the right-handside of Equation 4 (e.g., Bollen, 1989). Thus, adding and subtracting zγ/2 times

the standard error in Equation 13 to/from its estimate � � / ( � � )ρ σ σ σ= 12 12

22 , where

zγ/2 is the γ/2th fractile of the normal distribution (e.g., Hays, 1994), yields thisapproximate CI as follows:

(14)

By comparing CIs of latent correlations, one can explore differences in strengthof (linear) interrelationships between the involved latent variables; this questionmay arise both in single-group as well as multiple-group analyses. Specifically, ifthese intervals do not overlap, the correlation whose interval is to the right may beconsidered more salient (at the used confidence level; e.g., Raykov, 1995). Other-wise, the amount by which they overlap indicates the degree to which, at the usedsample size and confidence level, the population latent correlations cannot be dif-ferentiated between (with this procedure, based on the analyzed data).2

LATENT CORRELATION STANDARD ERROR 147

2As indicated, this is an explorative rather than statistical hypothesis testing method of comparingstrength of (linear) latent interrelationships, offered within the specific analytic context followed in thisarticle.

/ 2 / 2ˆ ˆ ˆ ˆ[ ( ); ( )].z SE z SEα αρ ρ ρ ρ� �

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ILLUSTRATION ON DATA

In this section, the method is demonstrated on simulated data. Using LISREL8.30(Jöreskog, Sörbom, Du Toit, & Du Toit, 2000), multinormal data were simulatedon k = 6 variables.3 The data on the first 4 of these 6 variables are employed in Fig-ure 3 as four successive assessments of the same latent dimension, and the last twovariables are used as two indicators of a latent covariate/predictor of the underlyingchange process along that dimension. The data for N = 500 cases were simulatedfollowing a direct extension of the level-and-shape model for purposes of studyingcorrelates of change (McArdle & Anderson, 1990), which for completeness of thisdiscussion is presented in Figure 3 (cf. Figure 1).

In terms of underlying parameters, the model used for data generation had thefollowing defining equations:

(15)

where the level factor (true initial status) η1, the shape factor (true overall growth)η2, and the latent correlate η3 were set to have means of 100, and all error varianceswere set equal to 3. The correlations of η3 with η1 and η2, as well as that of the lattertwo factors, are frequently of special interest in behavioral research studying corre-lates and predictors of ability growth or decline (e.g., McArdle & Anderson, 1990;Muthen, 1991; Rogosa & Willett, 1985; see also Raykov, 1995, 1996, 1998). Thepopulation values of these latent correlations were set as follows: ρ12 = Corr(η1,η2)= .40, ρ13 = Corr(η1,η3) = .60 and ρ23 = Corr(η2, η3) = .80, where Corr(.,.) denotescorrelation of the variables in parentheses (this was achieved by setting the three la-tent variances at 10, and the corresponding latent covariances at 4, 6, and 8).

The covariance matrix and means of the 6 variables are given in Table 1. Fittingthe extended level-and-shape model (Figure 3) to this data yields χ2(12, N = 500) =16.34,p=.18.Therootmeansquareerrorofapproximation is .027,witha90%CI(0;.056) (Jöreskog & Sörbom, 1996). These indexes suggest an acceptable fit of themodel. Using the output covariance and variance estimates, the latent correlationsare estimated as follows (rounded off to second digit after decimal place):� . , � . , � .ρ ρ ρ12 13 2340 57 82= = =and , which are either identical or fairly close to theabove true correlations. This method furnishes .07, .04, and .04 (rounded off to sec-

148 RAYKOV AND PENEV

1 1 1

2 1 2 2

3 1 2 3

4 1 2 4

5 3 5

6 3 6

,.70 ,.85 ,

,.90 ,

,

YYYYYY

η εη η εη η εη η ε

η εη ε

� �

� � �

� � �

� � �

� �

� �

3Details on the simulation procedure can be obtained from the authors.

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TABLE 1Covariance Matrix and Means of the Six Simulated Variables

Variable Y1 Y2 Y3 Y4 Y5 Y6

Y1 12.15Y2 11.28 21.55Y3 12.37 20.24 25.59Y4 12.56 21.35 23.96 28.38Y5 5.13 10.08 11.24 11.94 11.06Y6 5.06 10.10 11.91 12.99 8.80 12.49Means 100.07 169.97 185.07 199.96 90.10 100.06

Note. N = 500. Y1 to Y4 = four successive assessments (of same latent dimension); Y5 and Y6 = twoindicators of a latent change covariate/predictor (η3 in Equation 15).

FIGURE 3 The extended level-and-shape model (McArdle & Anderson, 1990), including alatent correlate/predictor of change construct, η3 (point and especially interval estimation of itscorrelations with true initial status/level and overall change/shape is frequently of main interestin its applications in behavioral research).

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ond digit after decimal place) as their respective standard errors (for the softwarecode for the calculations, see Equation 13 and the Appendix). This approach alsoyields the 90% CIs of the latent correlations as follows: (.28, .52) for ρ12, (.50, .64)for ρ13, and (.76, .88) for ρ23 (see Equation 14). The three true correlations are posi-tioned well within their corresponding CIs. Comparing the three CIs to one anothersuggests that the (linear) interrelationship of the correlate construct with starting po-sition (ρ13) was (a) markedly higher than that between starting position and growth(ρ12) across the four assessment points, (b) considerably smaller than that relation-ship between latent correlate and growth (ρ23) across the repeated measurements,and (c) that relationship between initial status and growth (ρ12) was substantiallysmaller than the one between latent correlate and growth (ρ23) across all assess-ments. These conclusions could be obtained because use was made of the methodoutlined in this article (particularly Equations 13 and 14); these conclusions cannotbe obtained with conventional/routine applications of the consideredlevel-and-shape model (Figure 3). This is due to the fact that conventional analysisdoes not yield a standard error for the three latent correlations in the model—as indi-cated earlier, this limitation is due to a priori imposed factor loading restrictions in itthatprecludestandardparameterizationof thesecorrelations(seealsofootnote1).

CONCLUSION

In this article we proposed an approximate method of obtaining standard errors andCIs for latent correlations in restricted structural models with factor loadings fixeda priori at specific constants to achieve particular parameterizations of substan-tively interesting aspects of studied phenomena. These restrictions imply scaling ofthe latent metric that cannot be routinely used in practice for direct point and inter-val estimation of latent correlations (in terms of model parameters) between factorswith such restricted loadings. The same general method is also used when one is in-terested in obtaining approximate standard error and CI for any substantively inter-esting expression (function) of parameters of covariance structure or other models,such as ratios of latent means in multiple-group models (e.g., with at least threegroups), products or ratios of other parameters, or complex nonlinear functions ofmodel parameters (e.g., Raykov, in press).

This approach is analytic in nature rather than computer intensive. This methodis based on such a linear approximation of a latent correlation of interest as a func-tion of latent variances and covariance, which results from the first-order terms ofits Taylor expansion (e.g., Stewart, 1991). The procedure used in this article is eas-ily applied once the model is fitted to data, and does not require potentially compli-cated reparameterizations into equivalent models (see footnote 1). All constituentsof the corresponding formulas for standard error (Equation 13) and CI (Equation14) are obtained from the resulting output of a SEM application (for their computa-tion, see Appendix).

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The outlined approach is related to the extended delta method provided byJamshidian & Bentler (2000), and this discussion is similar to theirs in general aimand spirit. The differences primarily lie in the fact that Jamshidian and Bentler fo-cused on standardized variance-covariance matrices and studied asymptotic rela-tive biases, whereas we concentrate on asymptotic CIs for individual correlations.Furthermore, the extended delta method is based on first- as well as second-orderterms from the Taylor expansion of the parametric function of concern. Althoughthe second-order terms contribute to improved approximation, practical applica-tions of this method require computation of second-order derivatives that compli-cates routine use of it. By comparison, the linear approximation method used inthis article is applied in a straightforward method, requesting only first-order de-rivatives. Given this simplicity, until automated utilization of the extended deltamethod is offered in popular SEM software, our procedure seems to be morewidely applicable in empirical behavioral research. We stress however that incases where the linear approximation underlying our procedure is unsatisfactory,the extended delta method is likely to bring desired improvement in accuracy ofestimation (Jamshidian & Bentler, 2000).

An alternative approach to the estimation of approximate standard errors andCIs of latent correlations is obtained via application of the popular bootstrap meth-odology (e.g., Efron & Tibshiriani, 1993; Jamshidian & Bentler, 2000). Within it,the model of concern is repeatedly fitted to resamples (with replacement) from theoriginal raw data set, and an estimated standard error obtained from the resultingempirical distribution of latent correlations across these resamples (e.g., Raykov,2002). In contrast with this method that involves fitting of a single model, an appli-cation of the bootstrap would be computer and time intensive. Moreover, althoughgeneral theoretical statements of satisfactory behavior of bootstrap estimates areavailable in the asymptotic case (Efron & Tibshiriani, 1993), it is not knownwhether this will always be the case within the framework of SEM and if some pa-rameter constellations may be associated with incorrect bootstrap interval esti-mates. Further theoretical and practical work remains to be carried out in themethodological area of covariance structure modeling before more trustworthystatements about large sample behavior of bootstrap estimates can be made (e.g.,Bollen & Stine, 1993).

Limitations of the outlined method stem from its large sample requirement andlinear approximation nature. The former follows from the fact that the approachcapitalizes on estimates obtained via an utilization of the covariance structuremodeling methodology that itself is based on a large-sample theory of estimationand testing (e.g., Bollen, 1989). Hence, the larger the sample size, the more trust-worthy the results of applications of our approach. This recommendation is addi-tionally reinforced by the fact that standard errors of correlation coefficients tendto be relatively large with small and even moderate samples (e.g., Hays, 1994).Thus, with larger samples the CIs of considered latent correlations would tend to

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be shorter, hence allowing more precise statements about construct interrelation-ships of interest. Furthermore, results with this method will be more trustworthy inempirical cases in which the linear approximation of focused correlations is moretenable in the vicinity of the true variances and covariances.

Our approach offers a valuable and relatively inexpensive analytic tool thatprovides necessary elements of inference about latent correlations or other para-metric expressions from samples to studied populations, namely, standard errorsand CIs. Although current behavioral and social research still seems to be some-what biased toward obtaining and interpreting point estimates of model parame-ters (at times, at face value), particularly construct correlations, the outlinedmethod provides a useful addition to the armory of behavioral scientists. It al-lows a more complete use of analyzed data sets, as opposed to focusing mainlyon point estimates of unobserved variable correlations and related parameters.This approach achieves this aim by yielding (a) measures of parameter estimateprecision/stability (standard errors); (b) ranges of plausible values for latent cor-relations (CIs), for a priori constrained structural models such as the ones men-tioned in the introduction; and (c) interval estimates for any other parametricfunctions of substantive interest.

In conclusion, the general delta method underlying this article can be appliedalong the same lines when the purpose is to obtain approximate standard error andCI of substantively interesting functions of parameters of covariance structure orother models, which functions cannot be obtained as single-model parametersgiven the concerns when using the particular model. The quality of these obtainedinterval and related estimates will depend on the validity of the approximation ofthat parametric expression in terms of first-order elements of its Taylor expansion(e.g., Jamshidian & Bentler, 2000; Stewart, 1991).

ACKNOWLEDGMENTS

This research was supported by a grant from the School of Mathematics of the Uni-versity of New South Wales, Sydney. We thank A. Satorra, R. E. Millsap, B.Everitt, and an anonymous referee for valuable discussions and helpful suggestionson an earlier version of this article.

REFERENCES

Bollen, K. A. (1989). Structural equations with latent variables. New York: Wiley.Bollen, K. A., & Stine, R. A. (1993). Bootstrapping goodness-of-fit measures in structural equation

models. In K. A. Bollen & J. S. Long (Eds.), Testing structural equation models (pp. 111–135).Newbury Park, CA: Sage.

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Efron, B. J., & Tibshiriani, R. J. (1993). An introduction to the bootstrap. London: Chapman & Hall.Hays, W. L. (1994). Statistics. Orlando, FL: Harcourt Brace Jovanovich.Jamshidian, M., & Bentler, P. M. (2000). Improved standard errors of standardized parameters in

covariance structure models: Implications for construct explication. In R. D. Goffin & E. Helmes(Eds.), Problems and solutions in human assessment (pp. 73–94). Dordrecht, The Netherlands:Kluwer Academic.

Jöreskog, K. G., & Sörbom, D. (1996). LISREL8 user’s guide. Chicago: Scientific Software.Jöreskog, K. G., Sörbom, D., Du Toit, S. H. C., & Du Toit, M. (2000). LISREL8: New statistical fea-

tures. Chicago: Scientific Software.McArdle, J. J., & Anderson, E. (1990). Latent variable growth models for research on aging. In J. E.

Birren & K. W. Schaie (Eds.), The handbook of the psychology of aging (Vol. 2, pp. 21–43). NewYork: Plenum.

Meredith, W., & Tisak, J. (1990). Latent curve analysis. Psychometrika, 55, 431–462.Muthen, B. (1991). Analysis of longitudinal data using latent variable models with varying parameters.

In L. Collins & J. Horn (Eds.), Best methods for the analysis of change. Recent advances, unan-swered questions, future directions. Washington, DC: American Psychological Association.

Rao, C. R. (1973). Linear statistical inference and its applications. New York: Wiley.Raykov, T. (1995). Multivariate structural modeling of plasticity in fluid intelligence of aged adults.

Multivariate Behavioral Research, 30, 255–287.Raykov, T. (1996). Plasticity in fluid intelligence of older adults: An individual latent growth modeling

application. Structural Equation Modeling, 3, 286–307.Raykov, T. (1998). Satisfying a simplex structure is simpler than it should be: A latent curve analysis re-

visit. Multivariate Behavioral Research, 33, 276–299.Raykov, T. (2002). Automated procedure for interval estimation of composite reliability. Understand-

ing Statistics, 1, 75–84.Raykov, T. (in press). Analytic estimation of standard error and confidence interval of scale reliability.

Multivariate Behavioral Research.Rogosa, D. R., & Willett, J. (1985). Understanding correlates of change by modeling individual differ-

ences in growth. Psychometrika, 50, 203–228.Stewart, J. (1991). Calculus. Monterey, CA: Brooks/Cole.

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APPENDIX

SPSS Command File for Computing Approximate StandardError of Latent Correlations in Constrained StructuralEquation Models, and LISREL Input Filefor the Extended Level-and-Shape Model

TITLE‘POINTANDINTERVALESTIMATIONOFLATENTCORRELATIONS’.COMP S1=SQRT(9.33).COMP S2=SQRT(9.75).COMP S12=7.82.COMP VAR_S12=.46.COMP VAR_S11=.92.COMP VAR_S22=.53.COMP COV_S121=.46.COMP COV_S122=.36.COMP COV_S112=.25.COMP D1=1/(S1*S2).COMP D2=-S12/(2*S1**3*S2).COMP D3=-S12/(2*S1*S2**3).COMP VAR_RHO=D1**2*VAR_S12+D2**2*VAR_S11+D3**2*VAR_S22.COMPVAR_RHO=VAR_RHO+2*D1*D2*COV_S121+2*D1*D3*COV_S122+2*D3*D3*COV_S112COMP SE_RHO=SQRT(VAR_RHO).COMP RHO=S12/(S1*S2).COMP CI90_LOW=RHO-1.645*SE_RHO.COMP CI90_UPP=RHO+1.645*SE_RHO.EXECUTE.

This file is utilized separately for each latent correlation considered. The researcherneeds to supply the numbers appearing in the right-hand sides of the first 9 rows (af-ter the title line), which quantities are obtained as appropriate entries in the LISRELoutput (see main text). (The particular numbers entered in this Appendix pertain tothe calculation of the standard error and confidence interval of ρ23 = Corr(η2,η3),the correlation between latent correlate and amount of change over the four re-peated assessments, for the data used in the illustration section; see Figure 3.)

S1, S2, and S12 = (estimated) standard deviations and covariance of the latentvariables whose correlation is under consideration. (Note that the indexes need notbe identical to those of the corresponding elements in the LISREL matrix nota-tion.) VAR_SIJ (I, J = 1,2) = variances of the estimates of latent variances andcovariance; COV_SIJK (I, J, K = 1, 2) = covariances of the estimates of latent

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covariance and variances; RHO = latent correlation (ρ23 in the present example);VAR_RHO, SE_RHO = variance and standard error of the latent correlation (ρ23);CI90_LOW and CI90_UPP = lower and upper endpoints of the 90% confidenceinterval for the latent correlation (ρ23).

LISREL Input for the Extended Level-and-Shape ModelWith a Latent Covariate (Assessed by a Pair of Indicators)Related to the Level and Shape Factors(see McArdle & Anderson, 1990)

DA NI=6 NO=500CM=<see Table 1>ME=<see Table 1>LAMEASMT1 MEASMT2 MEASMT3 MEASMT4 COVIND1 COVIND2MO NY=6 NE=3 AL=FR PS=SY,FRLELEVEL SHAPE LAT_COVVA 1 LY 1 1 LY 2 1 LY 3 1 LY 4 1VA 1 LY 4 2FR LY 2 2 LY 3 2VA 1 LY 6 3FR LY 5 3EQ TE(1)-TE(4)OU ALL

MEASMTi = ith repeated assessment (i = 1, 2); COVINDi = ith indicator of latentcovariate/predictor of change (i = 1, 2); LAT_COV = latent covariate/predictor ofchange (related to the level and shape factors).

This LISREL input file is not original, and is presented here only for purposesof completeness of the current discussion (e.g., McArdle & Anderson, 1990; therestriction of equal error variance is not essential for applications of the model andmay be appropriate in some repeated assessment contexts, e.g., Raykov, 1995).

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