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Methods of Psychological Research Online 2003, Vol.8, No.2, pp. 75-112 Department of Psychology

Internet: http://www.mpr-online.de © 2003 University of Koblenz-Landau

Structural Equations Modeling in Developmental Research

Concepts and Applications1

Alexander von Eye2

Michigan State University

Christiane Spiel and Petra Wagner

University of Wien

This article discusses structural models for developmental research. First, it gives a

brief overview of structural modeling using the well-known LISREL notation. Then,

characteristics of well-fitting models are reviewed. These characteristics are presented

with the goal in mind of presenting complete reports of structural models such that

readers are enabled to derive their own conclusions concerning a model. The features

of developmental models are discussed. Models are developmental in particular when

some variation of time or age is central to the hypotheses under study. Data examples

are presented from a study on the long-term effects of biological and social risks in in-

fancy and early childhood. These examples suggest that performance in academic sub-

jects in school at age 11 can be predicted based on information of early risks. The

models are discussed with reference to the characteristics of well-fitting models.

Keywords: developmental modeling, fit criteria, developmental hypotheses, structural

modeling

The purpose of this article is to discuss and illustrate structural equations modeling

(SEM) for research in disciplines concerned with development, for example, develop-

mental psychology, education, or developmental psychopathology. This article provides

an overview of SEM concepts and methods, and presents examples, using data from a

1 This research was in part supported by the Austrian “Fonds zur Förderung der wissenschaftlichen For-

schung” Grant No. P7630-SOZ and by the Johann Jacobs Foundation. 2 Correspondence concerning this article should be addressed to Alexander von Eye, Michigan State Uni-

versity, Department of Psychology, 119 Snyder Hall, East Lansing, MI 48824-1117, USA;

E-mail [email protected]

76 MPR-Online 2003, Vol. 8, No. 2

study on the prediction of performance in academic subjects in elementary school from

indicators of risk in early childhood.

We also discuss characteristics of well-fitting models. In this discussion, we posit that

models can be appreciated by readers only if they are described to the extent that read-

ers are enabled to derive their own conclusions.

Concepts of SEM for Developmental Research

Structural equations modeling (SEM) provides a methodology that can be viewed as

a rather general methodology in the contexts of regression analysis and factor analysis.

As Tomer (2003) notes in his “short history of SEM,” one root of SEM can be traced

back at least to Wright’s (1918) early specifications of path analysis. Another root is

factor analysis which is known to go back to Thurstone (1931). In the 1960s, Jöreskog

(1966) proposed methods of confirmatory factor analysis and published software, e.g.,

the well known program COFAM, to enable researchers to perform such analysis. The

general LISREL model (Jöreskog, 1977), combines path analysis and factor analysis in a

more general model.

The development of models for longitudinal data, of particular concern to develop-

mental researchers, goes parallel to the development of structural modeling. Bayley

(1956) identified the search for the best methods to perform longitudinal data analysis

as a major concern of much developmental research. In the meantime, a plethora of

SEM approaches to the analysis of longitudinal data have been proposed (see Little,

Schnabel, & Baumert, 2000; McArdle & Bell, 2000). The most popular among these

models are known as latent growth curve models which originated in work by Tucker

(1958) and Rao (1958), and have first been related to SEM by Meredith (see Meredith

& Tisak, 1990; Tisak & Meredith, 1990).

McArdle and Bell (2000) note that SEM methods provide mathematical and statisti-

cal tools for the specification and testing of theoretical propositions. In particular, the

authors list the following five options:

1. Organization of concepts about data analysis into scientific models;

2. Providing tools for the estimation of mathematical components of models;

3. Providing tools for the evaluation of statistical features of models;

von Eye et al: SEM in Developmental Research 77

4. Providing tools for the specification of models with unobserved or latent variables;

and

5. Providing flexible tools for the analysis of incomplete data.

Most of the commercially or freely available software packages allow researchers to

use all these options. In addition, most of these programs also provide graphical input

or output options that translate models into graphs. These graphs are created based on

a number of competing specifications (see McDonald & Ho, 2002). The general purpose

of depicting a model in graphical form is to provide an overview of all important fea-

tures in an unambiguous fashion. The probably most widely used SEM software package

is LISREL (Jöreskog & Sörbom, 1993), at the time of this writing available in release

8.52. Also among the more popular commercially distributed programs are Bentler’s

EQS (1995), and Arbuckle’s Amos (1997). For a comparison of these three packages see

von Eye and Fuller (2002). Free of charge is Neale’s Mx program (1997).

In the following paragraphs, we give a brief overview of structural modeling. This

overview is followed by a discussion of SEM for developmental research and data exam-

ples.

SEM –an Overview

For the following overview, we use the notation known as the LISREL notation

(Jöreskog & Sörbom, 1993). Alternatively, other, equivalent notations could be used,

e.g., the one introduced by Bentler and Weeks (1980).

The full LISREL model can be described by

η η ξ ζ= + +Β Γ

with

η η η' ( ,..., )= 1 m : m latent factors on the dependent variables side;

B: covariance matrix of the η factors

ξ ξ ξ' ( ,..., )= 1 n : n latent factors on the independent variables side;

Γ: matrix of the covariances of η and ξ factors; and

ζ ζ ζ' ( ,..., )= 1 m : vector of residuals.


The full model incorporates both components of factor analysis and path analysis.

The path analytic component of this model resides in the structural model part. This

part describes relations of dependency. These relations connect latent variables, mani-

fest variables (in path models), or both. Depending on the causality concept adopted by

researchers, these relations are often interpreted as causal. The factor analytic compo-

nent of the full model resides in the measurement models. These models specify the re-

lations of the p observed, that is, manifest variables to the m latent variables, or factors

on the dependent variables side, and the q manifest variables to their n latent variables

on the independent variables side. More specifically, the measurement model for the

dependent variables is

y y= +Λ η ε,

and the measurement for the independent variables is

x x= +Λ ξ δ ,

with

y’ = (y1, …, yp): observed dependent variables,

x’ = (x1, …, xq): observed independent variables,

ε’ = (ε1, …, εp): residuals on the dependent variable side;

δ’ = (δ1, …, δq): residuals on the independent variable side;

Λ: matrices of loadings of the x- or y-variables on their factors;

p: number of dependent variables; and

q: number of independent variables.

There exists a number of submodels of the full LISREL model. Most interestingly,

one of the submodels (Submodel 3B; Jöreskog & Sörbom, 1993) is more general than the

full model. Thus, all LISREL models can equivalently be expressed as special cases of

this submodel. This includes the full model.

Consider the situation in which there are no x-variables. In this situation, either all

variables are assumed to be on the y-side or there are indeed no manifest independent

variables. An example of the latter could be a second order factor model. A model with

no x-variables can be defined by the two equations yy = Λ η+ ε and Bη = η+ Γξ+ ς


from above. Solving the second of these two equations for η and inserting into the equa-

tion for y yields

y I By= − + +−Λ Γ( ) ( )1 ξ ζ ε .

This model is known as Submodel 3A. Going one step farther and assuming that there

are no ξ-variables either, the equation for η reduces to

η η ζ= +B ,

and the equation for y reduces to

y I By= − +−Λ ( ) .1ζ ε

The covariance matrix of y under this model is

Σ Λ Ψ Λ Θ= − − +− −y yI B I B( ) ( ' ) '1 1

ε

where

Ψ: Matrix of covariances of the η-factors; and

εΘ : Matrix of variances and covariances of the residuals ε.

This model is known as Submodel 3B (Jöreskog & Sörbom, 1993). There are two rea-

sons why researchers often prefer this model to the full model. The first reason is that

the number of matrices to consider is only 4, down from 8 of the full model. These are

the matrics Λy, Β, Ψ and εΘ . Thus, the submodel is simpler yet can be used to express

the same hypotheses as the full model. The price for simplicity of specification is that

each of the four remaining matrices is larger than the corresponding matrices in the full

model. When large numbers of variables are examined, the size of matrices can lead to

numerical accuracy problems. The second reason why this model may be advantageous

is that the covariances of residuals on the x-side and the y-side of the model appear in

just one matrix, the εΘ matrix. In the original version of LISREL, there was no provi-

sion for covariances of residuals from the x-side of the model with residuals on the y-

side of the model. In more recent versions (beginning with LISREL 8), the full model

was extended to allow for such covariances. They can be specified to appear in the rec-

tangular matrix δεΘ . The examples in the second part of this paper use both the full

model and Submodel 3A. In the first data example in the section titled “Model 1”, we

use Submodel 3b. In the second example, we use the Full LISREL model.


In the following section, we discuss characteristics of structural models for develop-

mental research. The focus is on latent growth curve models.

Characteristics of Structural Models

In this section, we discuss characteristics of structural models. We split the presenta-

tion of characteristics in two parts. The first part is general and contains characteristics

of models that can be considered well fitting. These characteristics apply to most struc-

tural models, and to path models and factor models as well. Many researchers use these

characteristics for the decision as to whether to retain or reject/further improve a mo-

del. The second part is specific to developmental models. This part has no evaluative

component to it. However, it lists features often found in models for developmental

data.

Characteristics of Structural Models –General

Researchers who examine their data and hypotheses using SEM often wonder what

the specifics of a “good” model are, and what to report about the models they retain.

There exists a number of articles and chapters providing guidelines. While being peri-

odically updated, these texts rarely reflect noteworthy changes (texts worth reading

were presented by Boomsma, 2000; McDonald & Ho, 2002; and Tomer, 2003). From a

user’s perspective problematic, these texts suggest that criteria that can be considered

commonly agreed-upon markers of the quality of a model do not exist. Therefore, we

adopt a perspective in this article that deviates slightly from the rather prescriptive per-

spective found in some articles. Rather than discussing criteria for the sole purpose of

model evaluation and rather than proposing rules of thumb, we take the perspective

that models can be evaluated and appreciated by readers only if they are described and

depicted to the extent that readers are put in a position in which they can reconstruct a

model or derive their own conclusions. The following paragraphs discuss some of the

more frequently proposed characteristics used to describe models (see Boomsma, 2000;

McDonald & Ho, 2002; Tomer, 2003).

Model specification. A first set of information to be conveyed concerns the specifica-

tion of a model. For both, the measurement and the structural models, one expects

theoretical reasons why particular relations were included and why the ones not consid-

ered were excluded from the model. Researchers often justify only relations included in

their models. In a review of papers in which SEM was used, McDonald and Ho (2002)


note that none of the screened articles presented justification for the presence or absence

of every relation in the model. The authors note that the “possibility of unspecified

omitted common causes is the Achilles heel of SEM” (2002, p. 67). In addition, for most

models there exist equivalent specifications. One would expect researchers to justify

their model choice in particular in the light of equivalent models (compare Hershberger,

1994; Raykov & Penev, 1999). We thus conclude that theoretical justification is needed

for every relation in a model, so readers are placed in a position in which they can use

their own substantive knowledge to evaluate the plausibility of a particular model. In

addition, the fit of alternative, equally plausible models should be reported for compari-

son. If these models are nested, they should be reported in an order such that the more

parsimonious ones appear first and the more complex ones follow (Boomsma, 2000).

Model identification. A second set of information to be reported concerns the identifi-

ability of a model. It is well known that identifiability of a model is a prerequisite for

proper estimation. Graphical conditions for the identifiability of path coefficients in a

path model have been provided (Pearl, 2000), and the corresponding algebraic condi-

tions have been formulated (McDonald, in press).

As far as the measurement models are concerned, a sufficient condition for identifi-

ability is pure identification of factors. This condition states that each observed variable

loads on only one factor. A weaker version of this condition is that there be an inde-

pendent cluster basis. This implies that each latent variable or factor have at least two

pure indicators, that is, indicators that load on no other factor. In addition to these

conditions, the number of indicators for the factors must be large enough. In most em-

pirical applications, all factors are purely identified, and mostly meet the desiderate of

unidimensionality. There exist, however, examples in which variables load on more than

one factor (e.g., Jöreskog & Sörbom, 1993). Some of these cases are easily justified, for

example, when a scale has both speed and power components.

There also exist conditions that must be met for a path model to be identified. A

strong condition is that the covariances of disturbances of variables involved in a causal

(or prediction) chain be zero. By way of inspection of the path diagram (or the corre-

sponding output), one can check this condition which was called the precedence rule

(McDonald, 1997). A stronger condition is placed by the orthogonality rule which im-

plies that all disturbances of endogenous variables be uncorrelated (McDonald, 1997).

The orthogonality rule implies the precedence rule. While it is not always easy to detect

violations by visual inspection, some errors are easily found. An example of such a


threat to identifiability is the existence of correlated disturbances between factors in-

cluded in a causal or predictive chain.

A third aspect related to identifiability is scaling. Researchers often resort to stan-

dardization either before or after estimation in order to avoid underidentification that is

due to arbitrariness of scale. Only some of the currently available software programs are

capable of providing standardized solutions with correct standard errors.

Estimation and data characteristics. The third issue to be discussed here concerns es-

timation. LISREL, just as any of the other programs for SEM, attempts to estimate

parameters such that some function of the discrepancies between the observed and the

model covariance matrices is minimized. The most frequently used estimation function,

the default option in some programs, is maximum likelihood (ML). Unweighted least

squares (ULS) and generalized least squares (GLS) are popular also. In addition, most

programs offer a number of weighted least squares options (WLS) with the weights pro-

vided by the users or estimated by the program. The first three of the options men-

tioned here require multivariate normality. WLS does not make this requirement. How-

ever, it may need very large samples. Boomsma (2000) notes that if the number of vari-

ables is 15 or greater, the sample should include several thousand cases.

The requirement of multivariate normality has been a problem for social science re-

search since the beginning of multivariate statistics, for two reasons. First, there are no

conclusive tests for multivariate normality. Researchers often check multivariate nor-

mality by inspecting the univariate distributions, or they use Mardia’s test which checks

multivariate normality by examining multivariate skewness and curtosis coefficients.

These tests, often criticized, are inconclusive because they allow one to make a clear

decision about multivariate normality only in the case in which the null hypothesis

must be rejected. If this is the case, it is very unlikely that the data come from a popu-

lation with multivariate normal distribution. In the affirmative case, that is, when the

null hypothesis can be retained, one cannot conclude that the data come for a multi-

variate-normal population. Therefore, the usefulness of such tests has been challenged.

The second reason why the assumption of multivariate normality is a problem is that

many data sets in the social and behavioral sciences fail to meet this assumption. Mic-

ceri (1989) examined the distributional characteristics of 440 large-sample achievement

and psychometric measures and found that all of them were significantly nonnormal at

the .01 level. There is a temptation to generalize and to assume that many more empiri-

cal data sets come from nonnormal populations.


In response to these problems, two developments are under way. The first involves

the derivation of asymptotically distribution-free estimators (Browne, 1984), or correc-

tions to the normal likelihood ratio statistic in the presence of elliptical distributions

(Satorra & Bentler, 1994). When nonnormality is caused by categorical variables,

Muthén’s (1984) continuous/categorical variable methodology allows one to analyze any

combination of dichotomous, ordered polytomous, and metric variables. The second de-

velopment involves the specification of robust estimators as they have become available

in a number of programs, for example, the recent versions of LISREL, beginning with

release 8.20.

Users have been reluctant to adopt these new methodologies, mainly for two reasons.

First, the standard ML estimators are assumed to be robust against “reasonable” viola-

tions of the normality assumption. Second, some of the robust or asymptotically distri-

bution-free estimators require extremely large sample sizes for reliable estimation. How-

ever, the degree of distortion that is due to non-normality is largely unknown and there

always remains some uneasy feeling when the normality assumption is untenable.

From the perspective of a researcher who reports results, there remains only one op-

tion. This option involves reporting data characteristics and the selected estimation

method in detail. The results of tests of normality can be reported, and the variance-

covariance matrices should be provided along with means, so that readers can recalcu-

late results and draw their own conclusions. When data are missing, it needs to be re-

ported what was done with the missing information. Options include estimating and

imputing missing data, and using the full information maximum likelihood method that

is available, for instance, in Amos. Missing data are of particular concern in repeated

measurement developmental studies because attrition is know to be non-random in most

cases. It is very important that researchers also report the version of the program that

they used for data analysis. One main reason for this is that the methods of estimating

standard errors have changed over the various releases.

Goodness-of-fit (GFI). Measures of GFI come in many forms. One way of classifying

these measures distinguishes between absolute and relative fit indices. Absolute indices

describe the discrepancies between model and empirical covariance matrices. Some of

these indices also take into account sample size and degrees of freedom. Relative indices

assess the discrepancy between the fitted model and some null model, for instance the

independence model.


It is well known that these indices are problematic. McDonald and Ho (2002) list four

problems. First, there is no established empirical or mathematical basis for their use and

interpretation. Second, there exists no foundation for using an absolute index over a

relative one or vice versa. Third, not always do the various indices suggest the same

decision as to whether to retain a model. Thus, there is the suspicion that users select

the index that makes their preferred model look most favorably. Fourth, the same GFI

can indicate that a few large discrepancies exist that may be correctable, or that there

is a general scatter of discrepancies. An inspection of standardized discrepancies is

therefore always recommended.

Only recently has it been proposed recognizing that the structural model is nested

within the measurement model (e.g., Bentler, 2000). Therefore, one can ask what the

contribution of the structural model is above and beyond the measurement model. For

examples see McDonald and Ho (2002).

From the perspective of a user who reports the results of model fitting, it is hard to

make a decision as to which global indices to use. Practically every one of them has

come under critique. For example, the rather popular root mean square error of ap-

proximation (RMSEA) has recently been criticized as being logically incompatible with

the usual procedure of testing nested models (Hayduk & Glaser, 2000). In addition, the

authors noted that the frequently cited target value of RMSEA = .05 is not a stable

target. In a rebuttal, Steiger (2000, p. 149) dismissed these two criticisms and states

that they describe “no problem at all.” We conclude that it may be useful to report

many GFI indices, even if the individual index is not interpreted in detail, just to dis-

perse the suspicion that indices were left unreported that describe a model in a less than

positive light.

Parameters and standard errors. In a structural model, there are two groups of pa-

rameters. The first belongs to the measurement models. It contains the loadings, their

residual variances, and their covariances. The second group belongs to the structural

model. It contains the path coefficients, the disturbances, and the covariances. A com-

plete report of a structural model should contain all estimated parameters and their

standard errors. The graphs that are often used to report the results of a model test

may not be the best way to report all this information. Therefore, researchers may con-

sider reporting this information in the form of a table or equations (for an example of

the latter see Hussy & von Eye, 1988). Criteria for the inspection of this information

include that the unique variances should not be close to zero, and the standard error


should not be very large. The disturbance variances indicate the portion of the variance

of endogenous variables that the model accounts for.

Characteristics of Developmental Models

The preceding section listed criteria that can be used when presenting the results of a

modeling effort. These criteria are general in the sense that they can be employed for

any structural or measurement model. In the present section, we ask a different ques-

tion. We ask what makes a model developmental.

Developmental research focuses on constancy and change in behavior over time. This

includes research of ontogenetic development, but also phylogenetic development and

the development of larger bodies of individuals as studied in sociology or biology. As is

well known from the discussion of research methodology (Baltes, Reese, & Nesselroade,

1977; von Eye, & Spiel, 1995; von Eye & Bergman, in press), variation of time can be

achieved in several ways. This can be illustrated using the variable age.

Age variation can be achieved in two ways. The first involves cross-sectional designs,

that is, designs in which samples from different cohorts are observed at the same point

in time. If this is repeated at a later point in time using independent samples, one cre-

ates a cross-sequential design. There are many options to specify structural models for

cross-sectional and cross-sequential designs. The second way involves repeated observa-

tion of the same individuals. The following paragraphs present four examples of the

analysis of age variation.

The simplest case of a cross-sectional study involves two cohorts. One can ask

whether the measurement and structural models are the same for these two cohorts. To

establish identity (compare von Eye & Bergman’s, in press, concept of dimensional

identity), one can compare mean structures, covariance structures, or both. The

LISREL program (Jöreskog & Sörbom, 1993), to give an example, offers a variety of

options (other programs offer comparable options). The program allows users to

1. test whether correlation matrices or covariance matrices are the same across cohorts

or age groups;

2. test whether the measurement models are the same in all cohorts or age groups;

3. test whether the residual structures are the same across all cohorts or age groups;

and


4. test whether differences among groups can be represented by differences in the dis-

tributions of latent variables.

Any degree of similarity or differences can be tested. One extreme case is that re-

searchers propose that parameters are invariant across groups. The other extreme case

is that there are no constraints across groups. An example of a less extreme case is the

hypothesis that indicators load on the same factors in each group, but that their load-

ings can differ in magnitude.

Multigroup analyses have been popular in developmental research in spite of their

shortcoming that sample sizes have to be rather large. More specifically, the sample size

for each cohort or age group has to meet the same standards as the sample size for a

one-group analysis, because parameters are estimated for each group. To the best of our

knowledge, there exist no studies on the power of SEM methods for the case in which

parameters are constrained to be equal.

The second possibility to achieve variation in age is the repeated observation of the

same individuals. This is typically done with either one cohort in longitudinal studies,

with several, time-shifted cohorts in cohort-sequential studies, or with stacks of short-

term longitudinal studies as in McArdle’s convergence analysis in which series of meas-

ures from age-adjacent cohorts are pieced together (McArdle & Hamagami, 1991;

Ghisletta & McArdle, 2001).

Most popular in the analysis of repeated observations are latent growth models, in

which repeatedly observed measures are hypothesized to follow trends that can be de-

scribed by continuous functions, e.g., orthogonal polynomials. In the typical case (for an

example, see McArdle & Bell, 2000), latent growth models display the following charac-

teristics:

1. The covariation among the temporally ordered measures is hypothesized to be due

to two groups of underlying variables: the trend variables that are represented by,

e.g., polynomial coefficients (which can be interpreted in a fashion parallel to poly-

nomial regression coefficients; see Part III in von Eye, 2002), and the unique vari-

ables. The latter can be viewed parallel to the usual residuals. The residuals contain

those portions of the variance of a variable that cannot be accounted for by the

growth model;

2. The level score, for example the parameter for the zero order polynomial, is typi-

cally constant and applies to each of the observed scores. The scores for the first


second, and higher order polynomials are chosen to represent the hypothesized shape

of the growth curve. The values of the parameters depend on the hypothesized

shape. Typically, none of these scores is estimated. Rather, researchers determine

them based on theory or earlier results. These scores are also called basis parame-

ters;

3. The basis parameters and the latent means are used to describe group characteris-

tics. Recent extensions allow one to employ latent growth curve modeling from a

person perspective so that group-specific parameters are estimated (e.g., Li, Duncan,

Duncan, & Acock, 2001; Muthén, & Muthén, 2000).

4. Individual differences within each group can be captured.

In addition to the already mentioned extension to latent growth modeling for the

person perspective, these models have been developed so that two-stage growth models

and multilevel growth models can be estimated. In addition, and for the purposes of

developmental research most important is that these models can be used to combine

longitudinal and cross-sectional information (see McArdle & Bell, 2000).

In many contexts, researchers process manifest variables rather than latent variables.

A typical manifest variable model connects measures observed over time including (a)

autoregression relations among repeated measures of the same behavior, and (b) rela-

tions among measures of different behaviors, observed cross-sectionally or at different

points in time. In manifest variable models, variables predict each other over time. The

person perspective can be considered by specifying stacked models, that is, multigroup

models. To test hypotheses about particular shapes of developmental change it has been

proposed specifying SEMs using the parameters of the polynomials estimated to describe

intra-individual development (Fuller, von Eye, Wood, & Keeland, 2003; compare Li et

al., 2000; Raykov, 2000).

Another most interesting and important yet underused option is available in the dy-

namic latent factor models proposed by Molenaar (1985, 1987, 1994). These models ac-

commodate the normal linear factor model to multivariate time-series data. One of the

most important aspects of these new developments is that the assumption of time-

invariant parameters does not need to be made any more. Rather, models can be time-

heterogeneous and parameters can be time varying.

These four examples, multigroup models for cross-sectional data, latent growth mod-

els, manifest variable path models for repeated observations, and dynamic factor models


indicate that the nature of developmental models reflects the nature of developmental

research in general. Constancy and change are central parts of this research, and the

models exemplified in these examples suggest that developmental research typically ap-

proaches constancy and change via variation of the age variable.

The following sections present two empirical data examples of covariance structure

models in developmental research.

Data Examples:

Predicting Academic Performance in Elementary School from

Early Childhood Risks and Early Childhood Behavior

In this section, we present two examples of models that are typical in developmental

research. In both models, data are used from the Vienna Developmental Study (VDS;

Spiel, 1995, 1996a, 1996b). The VDS is a combined prospective and retrospective longi-

tudinal study that focuses on social competence, achievement behavior, and academic

performance in infancy, childhood, and adolescence. Based on such conceptualizations as

Horowitz’s (1987, 1992) interactional models and Sameroff’s transactional model (Sam-

eroff & Chandler, 1975), the VDS assumed that development results from the continu-

ous interplay of organismic and environmental factors. It is assumed that, on one side,

an individual’s competencies change over time, and, on the other side, environmental

characteristics change as well. On both sides, at least part of these changes is the result

of the interplay of the two sides. The VDS focuses on the effects of biological and social

factors on development. One major focus of the study is school achievement.

Method. In 1981, a random sample of children was drawn from 13 daycare centers,

representatively located in Vienna, Austria. Two waves of data collection were con-

ducted at kindergarten age, a third wave was conducted at school age. Data were ob-

tained from the children, their parents, and their kindergarten teachers. One part of the

data was collected retrospectively using questionnaires; the other part was element of

the prospective component of the study. Here, standardized observations, question-

naires, and tests were used (for more detail see Spiel, 1995)

Sample. The examples presented here use data from the first and the third waves. At

the first wave, the sample consisted of 94 participant children, 44 of which were girls.

Mean age was 2.56 (range 2.06-3.06 y). The third wave sample consisted of 90 partici-

pating children, including 47 girls. Mean age was 12 years (range 12.22-13.06 y). As is


obvious from the gender distribution, the samples overlap only partly. To obtain a

large-enough sample, a total of 27 children were included for the third wave, mostly to

counter attrition.

Measures. The following measures were collected:

1. Developmental parameters

1.1 Developmental parameters for the first year of life (collected retrospectively):

Fine motor behavior, autonomy, concentration, and exploration. The last three

of these variables are hypothesized to be prerequisites of information processing

competence (McCall & Carriger, 1993)

1.2 Developmental parameters at school age: Crystallized and Fluid Intelligence

scores, measured using Kubinger’s and Wurst’s AID-test (Adaptive Intelligence

Diagnostic; 1985; see Spiel, 1999); Task Commitment (4 items; observational);

Social Behavior, both variables observed while taking the AID (3 items; observa-

tional; see Spiel 1996c); number of friends nominated by the respondent chil-

dren; emotional stability of children (parent-provided ratings)

2. Factors affecting development

2.1 Early influences (collected prospectively for part of the sample, retrospectively

for the rest of the sample): pre-, peri-, and postnatal biological risks (e.g., infec-

tions, vacuum extraction, low birth weight); socio-economic status (SES); hospi-

talizations in the first year of life (number of days in hospital); begin of unas-

sisted walking.

2.2 Later influences: Hospitalizations at kindergarten age and at school age (num-

bers of days); life events (parent-provided data; degree of negative-positive im-

pact was rated by independent raters on 7-point Likert scale); number of school

changes (only non-normative changes included).

2.3 Academic Achievement: Grades in German and in Mathematics (transformed to

ten-point Likert scales to take different performance tracks into account).

We selected these data for the following reasons. First, the characteristics of these

data are typical of many data sets collected in developmental research. They are longi-

tudinal, they include variables that vary in characteristics from observational to test

information, and the distributional characteristics of many of the variables are un-

known. Thus, in the following analyses we have to rely on the robustness of the estima-

tors. This is no different than in many other studies in developmental research. Second,


the sample is admittedly small. As is occasionally the case in developmental research,

the financial means of researchers are limited, and it is hard to find participants first

committing to a many-year study and then being steady. As was indicated above, there

was attrition of over 20%. Again, this is typical of developmental research.

To counter attrition, 27 children were added to the sample. For these children, all

the information of the years before school had to be created retrospectively. This proce-

dure has three important implications. First, the original sample and the school-age

sample overlap only in 71 children (Only 64 children provided information at all three

observation points). Second, the effects of the inclusion of these 27 children on compa-

rability are unknown at this point. Studies will be undertaken to determine whether

estimation and imputation of unavailable information would lead to a different render-

ing of variable relationships than reported here. The results reported here use the sam-

ple created for the school-age wave of data collection.

A third implication of the rather modest sample size is that the power of the model-

ing efforts is rather low. Using the often-cited rule of thumb that about 10 respondents

are needed for each variable in a structural model, we can estimate models with no

more than 10 variables. Using the power tables published by MacCallum, Browne, and

Sugawara (1996), we need a sample of N = 132 when testing a model with df = 100

(close fit; that is, ε0 = .05 and εa = .08 with α= .05, where ε0 is the RMSEA score under

the null hypothesis; for not-close fit, we would have α = .01; and ε0 = .05 and εa = .01,

respectively).

Again, we conclude that the models we can fit are rather limited in scope. Still, we

employ SEM, for two reasons. First, there is no alternative that would be recommended

given the small samples. Second, it is well known that for repeated observation data,

the required sample sizes can – all other things being equal – be smaller than in cross-

sectional studies (Muthén & Curran, 1997).

Still, the results presented here should be interpreted with caution. First, replications

are necessary. Second, it needs to be determined whether the true models have been

approximated.

The following sections present two examples of latent variable developmental models.

The first example uses the grades in German and Mathematics (these data have been

published before; see Spiel, 1998; however, the model presented here represents an im-

provement over the earlier publication). The second model predicts academic perform-

ance from factors comprising early risk and early behavioral patterns. Both models use


longitudinal data. The first model examines the same variable – grades in school -, ob-

served over four years. The second model examines the longitudinal effects of different

variables over time.

Both models were estimated using LISREL 5.2. In both cases, the final models re-

sulted from modifications of an original model. The modification were done using (1)

the modification indices provided by LISREL, and (2) theory-based decisions as to the

meaningfulness, interpretability, and defensibility of the added parameters.

Model 1: A Simplex Model of Academic Performance

The model presented here examines the autoregression structure of the end-of-year

grades in German and Mathematics in the first four year of elementary school of the 89

students for which these data were available at the third observation point. For each of

these students the transformed grades were related to each other under the following

assumptions:

1. The grades in the two subjects are indicators of a grade-specific performance factor;

2. The time-sequential factors predict each other as in a first-order Markov chain; thus,

there are no predictions with lag greater than one;

3. There is one exception to Assumption 2: The performance factor at the first year in

school predicts the performance factor at the fourth year in school. The reason for

this prediction is that in both years, students make a transition-related experience.

In the first year, this experience concerns the transition from pre-school to grade

school. In the fourth year, this experience concerns the decision as to having a tran-

sition to Gymnasium or Hauptschule. The final model appears in Figure 1.


grade11.00

grade20.80

grade30.46

grade40.23

g1 0.27

g2 0.29

g3 0.27

g4 0.42

m1 0.27

m2 0.30

m3 0.08

m4 0.00

Chi-Square=13.37, df=11, P-value=0.27012, RMSEA=0.049

0.85

0.85

0.84

0.84

0.85

0.96

0.76

1.00

0.45

0.73

0.22

0.78

0.26

0.13

0.25

0.20

0.28

Figure 1: Final simplex of academic performance in the first four grades of elementary

school (completely standardized parameters are given; all paths significant)

We now discuss the model with reference to the five characteristics discussed in

“Characteristics of structural models –general”, above.

Model Specification

The issue of model specification concerns the decisions as to which covariances to in-

clude and which covariances not to include. In the present model, it is hypothesized

that grades in school can be sufficiently predicted using information from the previous

year. The first year in school serves as external variable. This does not imply that pre-

dictions that skip one or two years cannot be made. However, the model proposes that

lags greater than one are not needed for a good description of the data. Therefore, none

of the paths for lags greater than one are included in the model. The only exception is

the path from the first to the fourth year, because these years share the transition ele-

ment in common.

Keeping the simplex model pure, that is, omitting the path from Year 1 to Year 4,

yields a model of lesser fit that is significantly worse than the model in Figure 1. More

specifically, for the model without this path, we obtain a X2 = 21.07 (df = 12; p = .049)


and for the difference between these two nested models we calculate ∆X2 = 7.7 (df = 1;

p = .006). We thus reject the alternative model and retain the one depicted in Figure 1.

Model Identification

The model in Figure 1 fulfils the sufficient condition of pure identification of factors.

Each indicator loads on only one latent variable. In addition, the weaker criterion of an

independent cluster basis is fulfilled. Each factor has two pure indicators. None of the

indicators in this model loads on more than one factor. Furthermore, considering that

the first loading of each latent variable in the above model is fixed, we note that the

number of indicators is large enough.

As far as the precedence rule is concerned, we realize that none of the covariances

among the factors involved in the prediction chain is unequal to zero. In addition, the

disturbances of the endogenous variables are uncorrelated. Thus, the orthogonality rule

is also met.

Finally, the parameter estimates reported in Figure 1 are completely standardized in

the LISREL sense. This implies that (a) the latent variables are scaled to have standard

deviations equal to unity and (b) the observed variables are also scaled to have stan-

dard deviations equal to unity.

Estimation and Data Characteristics

The parameter estimation for the model in Figure 1 used maximum likelihood. This

implies that the data stem from a population in which the variables show a multivari-

ate-normal distribution. This assumption was not tested for the present data. The

rather small sample size and problems with the conclusiveness of the tests of multivari-

ate normal distributions prevented us from going though the steps. We thus have to

rely on the robustness of the employed estimation.

We do consider it useful, however, to compare the robust and the normal theory-

based estimates of the overall goodness-of-fit X2–values. LISREL reports the Minimum

Fit Function Chi-Square = 14.80 (p = 0.19; df = 11) and the Normal Theory Weighted

Least Squares Chi-Square = 13.37 (p = 0.27; df = 11). These values are very close to

each other, and both suggest retaining the model. We thus conclude that the bias that

results from using normal theory may be small.


Goodness-of-Fit

It is recommended that many goodness-of-fit results be reported, mainly to (a) indi-

cate that discrepant appraisals of model fit are minimal, and (b) to disperse with the

suspicion that authors select the index that presents their final model in the most fa-

vorable light. For the present example, we decided to report the entire goodness-of-fit

information that LISREL makes available to the user. Table 1 contains this informa-

tion.

Table 1

Goodness-of-Fit Information for the Model in Figure 1 (From the LISREL Output)

Goodness of Fit Statistics Degrees of Freedom = 11 Minimum Fit Function Chi-Square = 14.80 (P = 0.19) Normal Theory Weighted Least Squares Chi-Square = 13.37 (P = 0.27) Estimated Non-centrality Parameter (NCP) = 2.37 90 Percent Confidence Interval for NCP = (0.0 ; 15.84) Minimum Fit Function Value = 0.17 Population Discrepancy Function Value (F0) = 0.027 90 Percent Confidence Interval for F0 = (0.0 ; 0.18) Root Mean Square Error of Approximation (RMSEA) = 0.049 90 Percent Confidence Interval for RMSEA = (0.0 ; 0.13) P-Value for Test of Close Fit (RMSEA < 0.05) = 0.45 Expected Cross-Validation Index (ECVI) = 0.72 90 Percent Confidence Interval for ECVI = (0.69 ; 0.87) ECVI for Saturated Model = 0.82 ECVI for Independence Model = 9.39 Chi-Square for Independence Model with 28 Degrees of Freedom = 810.16 Independence AIC = 826.16 Model AIC = 63.37 Saturated AIC = 72.00 Independence CAIC = 854.07 Model CAIC = 150.58 Saturated CAIC = 197.59 Normed Fit Index (NFI) = 0.98 Non-Normed Fit Index (NNFI) = 0.99 Parsimony Normed Fit Index (PNFI) = 0.39 Comparative Fit Index (CFI) = 1.00 Incremental Fit Index (IFI) = 1.00 Relative Fit Index (RFI) = 0.95 Critical N (CN) = 148.04 Root Mean Square Residual (RMR) = 0.23 Standardized RMR = 0.060 Goodness of Fit Index (GFI) = 0.96 Adjusted Goodness of Fit Index (AGFI) = 0.88 Parsimony Goodness of Fit Index (PGFI) = 0.29


Of the many indices presented in this table, we discuss only a few briefly. First, we

note again that both X2–values suggest that the discrepancies between the model-

created variance-covariance matrix and the empirical variance-covariance matrix are

non-significant. The inspection of the residuals follows below. Second, we find an

RMSEA smaller than the cut-off of 0.05 that is typically used to indicate close fit.

Third, the model AIC scores are smaller than the saturated and the independence AIC

scores. We conclude that the model enabled us to reduce uncertainty. The GFI is a

large 0.96. All other goodness-of-fit indices are very large too. We thus conclude that

none of the goodness-of-fit indicators for the present model would lead us to rejection.

In addition, the residual plot is very favorable to the model. This plot appears in

Figure 2. Qplot of Standardized Residuals 3.5.......................................................................... . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x . . . . . . x . . . x . N . . x . o . . x . r . . * . m . . * . a . . xx . l . . * . . . xx . Q . . xx . u . . x . a . . * . n . . xx . t . . x x . i . . x x . l . .x x . e . x . . s . x. . . x . . . x . . . . . . x . . . . . . . . . . . . . . . . . . . . . . . . . . . . . -3.5.......................................................................... -3.5 3.5 Standardized Residuals

Figure 2: Q-plot of the residuals for the model in Figure 1 (from LISREL output)


Figure 2 shows two of the important characteristics of residual plots that represent a

well-fitting model. First, there is a monotonic increase of residuals that is almost paral-

lel to the optimal line (dotted in the figure). This characteristic suggests that the distri-

bution of the residuals is very near to a normal distribution. Discrepancies are most

likely random. Second, there are no extreme residuals. None of the residual scores comes

even close to the value of 3.5, that is, the extreme ends of the graph. Note that there is

no agreement among methodologists concerning the percentage of extreme residuals that

can be accepted. One group of experts proposes that α-% of the residuals can be ex-

treme. Another group of experts proposes that the existence of extreme residuals sug-

gests that a model shows room for improvement.

Parameters and Standard Errors

It is recommended that all parameters and their standard errors be reported. Table 2

displays the parameters, their standard errors and the corresponding t-values from the

LISREL output.

Table 2 Parameters, Standard Errors, and t-Values for the Model in Figure 1 (From LISREL Output)

LISREL Estimates (Maximum Likelihood) LAMBDA-Y grade1 grade2 grade3 grade4 -------- -------- -------- -------- d1 1.80 - - - - - - m1 1.79 - - - - - - (0.30) 5.96 d2 - - 2.27 - - - - m2 - - 2.05 - - - - (0.27) 7.70 d3 - - - - 1.69 - - m3 - - - - 1.59 - - (0.12) 12.86 d4 - - - - - - 1.27 m4 - - - - - - 1.39 (0.14) 10.27 BETA grade1 grade2 grade3 grade4 -------- -------- -------- -------- grade1 - - - - - - - - grade2 0.45 - - - - - - (0.12) 3.60 grade3 - - 0.73 - - - - (0.10) 7.00


Table 2 (continued)

Parameters, Standard Errors, and t-Values for the Model in Figure 1 (From LISREL Output)

grade4 0.22 - - 0.78 - - (0.07) (0.08) 3.25 10.24 Covariance Matrix of ETA grade1 grade2 grade3 grade4 -------- -------- -------- -------- grade1 1.00 grade2 0.45 1.00 grade3 0.33 0.73 1.00 grade4 0.47 0.67 0.85 1.00 PSI Note: This matrix is diagonal. grade1 grade2 grade3 grade4 -------- -------- -------- -------- 1.00 0.80 0.46 0.23 (0.23) (0.19) (0.09) (0.05) 4.28 4.27 5.02 4.56 Squared Multiple Correlations for Structural Equations grade1 grade2 grade3 grade4 -------- -------- -------- -------- - - 0.20 0.54 0.77 THETA-EPS d1 m1 d2 m2 d3 m3 -------- -------- -------- -------- -------- -------- d1 1.22 (0.51) 2.37 m1 - - 1.21 (0.47) 2.59 d2 - - - - 2.09 (0.59) 3.52 m2 - - - - - - 1.76 (0.45) 3.94 d3 1.09 - - 0.67 - - 1.07 (0.20) (0.18) (0.19) 5.57 3.71 5.63 m3 - - - - - - - - - - 0.23 (0.10) 2.45 d4 0.89 - - 0.90 - - 0.92 - - (0.17) (0.20) (0.15) 5.08 4.62 6.14 m4 - - - - - - - - - - - - THETA-EPS d4 m4 -------- -------- d4 1.18 (0.18) 6.55 m4 - - 0.00 (0.09) 0.01


Table 2 shows the estimates for the unstandardized solution. LISREL does not pro-

vide standard errors for the standardized solutions (note, however, that the estimates

for the structural part of the model are the same in the non-standardized and the com-

pletely standardized solutions). We therefore inspect the above solution. None of the

standard errors in this solution is extremely large. This applies to both the parameters

for the measurement and the path models. In addition, the variances of the latent vari-

ables are significantly greater than zero. Their covariances are unequal to zero, too.

However, they are not needed for a satisfactory description of the data at hand.

Finally, we include the variance covariance matrix for the present model, so that

readers can recalculate this and alternative models and come to their own conclusions

about the structure of these data. This matrix appears in Table 3.

Table 3

Variance-CovarianceMmatrix of theDdata Used for the Model in Figure 1

Covariance Matrix d1 m1 d2 m2 d3 m3 -------- -------- -------- -------- -------- -------- d1 4.40 m1 3.16 4.40 d2 2.04 2.15 7.28 m2 1.18 1.19 4.75 5.95 d3 2.57 1.43 3.41 2.59 4.03 m3 1.41 1.36 2.54 2.41 2.75 2.77 d4 2.27 1.32 2.59 1.60 2.87 1.84 m4 1.49 1.42 1.84 1.81 2.09 1.97 Covariance Matrix d4 m4 -------- -------- d4 2.85 m4 1.82 2.01

Model 2: Predicting Academic Performance in Elementary School

from Early Childhood Risk and Early Childhood Behavior

For the example in this section, we use data from the same study as for the example

in section “Model 1”. We propose a model in which academic achievement in elemen-

tary school is predicted from early childhood risks and early childhood and school-age

competencies. More specifically, we propose the following four latent variables, listed in

the temporal order in which information was observed:


1. Risks. Indicators are biological risks (labeled as biorisk in Figure 3); SES (labeled as

SES with high scores indicating high SES); and number of days spent hospitalized

in the first year of life (hospital).

2. Competencies in the first year of life (labeled as Comp Y1). Indicators are fine mo-

tor skills (motor); Autonomy; and exploratory behavior (exploration).

3. Competencies at school age (labeled as comp S). Indicators are social behavior rat-

ings (social) and task commitment (on task).

4. Academic achievement (labeled as Achiev). The sole indicator of this latent con-

struct is the average grade in school in the German language and mathematics at

the end of the 4th year in elementary school.

At the structural level, the following paths were proposed.

1. Increased early risks predict lower levels of competencies;

2. Increased risks predict lower levels of competencies at school age;

3. Higher levels of competencies in the first year of life predict higher levels of school-

age competencies.

4. Higher levels of competencies at school age predict better academic performance in

the fourth year of elementary school.

This model appears in Figure 3.

biorisks0.83

SES0.49

hospital0.91

risks

Comp Y1

Comp S

Achiev GPA 0.31

social 0.79

on task 0.44

motor 0.59

autonomy 0.58

explorat 0.77

Chi-Square=22.66, df=24, P-value=0.53992, RMSEA=0.000

0.64

0.65

0.48

0.75

0.45

0.83

0.41

-0.71

0.300.65

1.00

-0.49

-0.48

Figure 3: Model of the prediction of school-age competencies from early childhood risk,

early competencies, and achievement-oriented behavior (completely standardized pa-

rameters are given; all paths significant)


In a fashion parallel to the first data example, we now discuss this model with refer-

ence to the five model characteristics presented in the section titled “Characteristics of

structural models –general”.

Model Specification

Obviously, the model in Figure 3 is an indirect effects model. That is, risk is pro-

posed to predict achievement via early competencies and achievement-oriented behav-

ior. To establish an indirect effects model, one needs to run at least the following three

models: (1) the complete model including all indirect and all direct effects; (2) the

model of only indirect effects; and (3) the model of only direct effects. The latter two

models are hierarchically related to the first, but not to each other. One can thus de-

termine whether the latter two model account for significantly less variance than the

complete model. However, there is no way that we are aware of to statistically compare

the two partial effect models. For lack of space, we only report the indirect effect model

here3. This applies accordingly to the direct versus mediated relationship between the

latent variables Comp Y1 and Achiev.

Model Identification

The model in Figure 3 is identified. It fulfills the criteria of pure identification and

independent cluster basis. The number of indicators per latent variable is relatively

small. However, both the path from the achievement factor and the residual of the

achievement indicator were fixed, and so was the variance of the achievement factor.

Thus, the explainable portion of achievement variance resides in the path from the fac-

tor of achievement-oriented behavior to achievement. Fixing these parameters made the

model identified. The discussion of the precedence rule carries over from the first exam-

ple.

Also as in the first example, the parameter estimates presented in Figure 3 are com-

pletely standardized in the LISREL sense.

3 Readers are invited to confirm that the complete model represents only a modest, non-significant im-

provement over the indirect effects model in Figure 3 (X2 = 21.28; df = 23; p = .56; ∆X2 = 1.38; ∆df =

1; n.s.); and that the direct effects-only model represents the data significantly worse than the complete

model (X2 = 28.13; df = 24; p = .25; ∆X2 = 6.85; ∆df = 1; p = .009). We thus retain the indirect effects

model. The command files for the models discussed in the two examples of this article can be obtained

from C. Spiel at [email protected] .


Estimation and Data Characteristics

The parameter estimation for the model in Figure 3 used maximum likelihood. As in

the first example, we are not certain that the variables used in the present model stem

from a population in which they follow a multivariate-normal distribution. So again, we

have to rely on the robustness characteristics of the estimators used by LISREL. As in

the first example, we note that the differences between the normal theory-based and the

robust X2–values are minimal. Specifically, we obtain a Minimum Fit Function Chi-

Square = 25.19 (p = .40) and a Normal Theory Weighted Least Squares Chi-Square =

22.66 (p = .54; df = 14 for both). Clearly, both of these values suggest that the model

can be retained.

Goodness-of-Fit

Following the recommendation to report a larger number of goodness-of-fit estimates

rather than a small number, we present the entire table of goodness-of-fit statistics pro-

vided by LISREL in Table 4.

As was noted above, both X2–values suggest good model-data fit. The discrepancies

between the model covariance matrix and the empirical covariance matrix are non sig-

nificant. The RMSEA is basically zero, indicating exact fit. The AIC scores indicate

that the model reduced uncertainty in comparison to both the independence and the

saturated models. The goodness-of-fit indices are all close to or at the maximum value

of 1.0. We conclude that none of these indicators suggests that the model misrepresents

the data. This applies accordingly to the Q-plot of the residuals, which appears in Fig-

ure 4.


Table 4:

Goodness-of-Fit Indicators for the Model in Figure 3 (From the LISREL Output)

Goodness of Fit Statistics Degrees of Freedom = 24 Minimum Fit Function Chi-Square = 25.19 (P = 0.40) Normal Theory Weighted Least Squares Chi-Square = 22.66 (P = 0.54) Estimated Non-centrality Parameter (NCP) = 0.0 90 Percent Confidence Interval for NCP = (0.0 ; 13.81) Minimum Fit Function Value = 0.28 Population Discrepancy Function Value (F0) = 0.0 90 Percent Confidence Interval for F0 = (0.0 ; 0.16) Root Mean Square Error of Approximation (RMSEA) = 0.0 90 Percent Confidence Interval for RMSEA = (0.0 ; 0.080) P-Value for Test of Close Fit (RMSEA < 0.05) = 0.78 Expected Cross-Validation Index (ECVI) = 0.74 90 Percent Confidence Interval for ECVI = (0.74 ; 0.90) ECVI for Saturated Model = 1.01 ECVI for Independence Model = 3.66 Chi-Square for Independence Model with 36 Degrees of Freedom = 308.14 Independence AIC = 326.14 Model AIC = 64.66 Saturated AIC = 90.00 Independence CAIC = 357.64 Model CAIC = 138.16 Saturated CAIC = 247.49 Normed Fit Index (NFI) = 0.92 Non-Normed Fit Index (NNFI) = 0.99 Parsimony Normed Fit Index (PNFI) = 0.61 Comparative Fit Index (CFI) = 1.00 Incremental Fit Index (IFI) = 1.00 Relative Fit Index (RFI) = 0.88 Critical N (CN) = 152.86 Root Mean Square Residual (RMR) = 0.12 Standardized RMR = 0.050 Goodness of Fit Index (GFI) = 0.94 Adjusted Goodness of Fit Index (AGFI) = 0.90 Parsimony Goodness of Fit Index (PGFI) = 0.50

The Q-plot of the residuals of the model in Figure 3 is close to perfect. The residuals

are almost perfectly normally distributed, and they are far from extreme. Each residual

in Figure 4 is indicated by an x, multiple residuals with the same values are indicated

by an asterisk. If the distribution of residuals in a Q-plot is monotonic and close to the

ideal, dotted, line, the residuals can be considered normally distributed. This is the case

in this example.


Qplot of Standardized Residuals

3.5.......................................................................... . .. . . . . . . . . . . . . . . . . . . . . . . . . . . x . . . . . . x . . . . . . x . N . . xx . o . . x . r . . * . m . xx . a . xxx . l . x x . . xx . Q . xxx . u . *. . a . * . n . xx . t . *x . i . x. x . l . x.x . e . x . . s . xx . . . x . . . . . . x . . . . . . x . . . . . . . . . . . . . . . . . . . . . . . . . -3.5.......................................................................... -3.5 3.5 Standardized Residuals Figure 4: Q-plot of the residuals of the model in Figure 3 (from LISREL output)

Parameters and Standard Errors

Table 5 presents the parameter estimates, their standard errors, and their t-scores for

the parameters of the model in Figure 3. As was indicated before, LISREL does not

provide standard errors and t-scores for the completely standardized model. Therefore,

Table 5 reports these estimates for the non-standardized solution.


Table 5


LISREL Estimates (Maximum Likelihood) LAMBDA-Y Comp Y1 achiev b achiev -------- -------- -------- MOTOR 0.86 - - - - AUTONOMY 0.74 - - - - (0.16) 4.59 EXPLOR 0.55 - - - - (0.15) 3.65 TASK - - 1.50 - - SOCIAL - - 0.85 - - (0.21) 4.01 GPA - - - - 1.50 LAMBDA-X risks -------- BIORISK 0.26 (0.08) 3.37 SES -1.27 (0.24) -5.27 HOSPITAL 0.46 (0.19) 2.39 BETA Comp Y1 achiev b achiev -------- -------- -------- Comp Y1 - - - - - - achiev b 0.65 - - - - (0.20) 3.33 achiev - - 1.00 - - (0.13) 7.64 GAMMA risks -------- Comp Y1 -0.49 (0.18) -2.65 achiev b -0.48 (0.17) -2.84 achiev - - Covariance Matrix of ETA and KSI Comp Y1 achiev b achiev risks -------- -------- -------- -------- Comp Y1 1.00 achiev b 0.88 1.00 achiev 0.88 1.00 1.00 risks -0.49 -0.79 -0.79 1.00 PHI risks -------- 1.00 PSI Note: This matrix is diagonal.


Table 5 (continued)


Comp Y1 achiev b achiev -------- -------- -------- 0.76 0.05 - - (0.30) (0.12) 2.51 0.36 Squared Multiple Correlations for Structural Equations Comp Y1 achiev b achiev -------- -------- -------- 0.24 0.95 1.00 Squared Multiple Correlations for Reduced Form Comp Y1 achiev b achiev -------- -------- -------- 0.24 0.63 0.63 Reduced Form risks -------- Comp Y1 -0.49 (0.18) -2.65 achiev b -0.79 (0.17) -4.71 achiev -0.79 (0.16) -4.96 THETA-EPS MOTOR AUTONOMY EXPLOR TASK SOCIAL GPA -------- -------- -------- -------- -------- -------- 1.05 0.74 0.99 1.74 2.77 1.00 (0.21) (0.15) (0.16) (0.34) (0.44) 5.14 5.03 6.05 5.16 6.36 Squared Multiple Correlations for Y - Variables MOTOR AUTONOMY EXPLOR TASK SOCIAL GPA -------- -------- -------- -------- -------- -------- 0.41 0.42 0.23 0.56 0.21 0.69 THETA-DELTA-EPS MOTOR AUTONOMY EXPLOR TASK SOCIAL GPA -------- -------- -------- -------- -------- -------- BIORISK - - - - - - - - 0.33 - - (0.11) 2.88 SES - - - - - - - - - - - - HOSPITAL - - - - - - - - - - - - THETA-DELTA BIORISK SES HOSPITAL -------- -------- -------- 0.34 1.56 2.18 (0.06) (0.50) (0.34) 6.06 3.09 6.41 Squared Multiple Correlations for X - Variables BIORISK SES HOSPITAL -------- -------- -------- 0.17 0.51 0.09

The values in Table 5 suggest that none of the standard errors is extremely large. In

all, there is nothing suspicious in this table that would lead us to doubt that this model


provides an excellent rendering of the data. The covariance matrix of the analyzed data

appears in Table 6. Readers should be able to reproduce the above results based on this

matrix.

Table 6

Variance-Covariance Matrix of the Data Used for the Model in Figure 3

Covariance Matrix MOTOR AUTONOMY EXPLOR TASK SOCIAL GPA -------- -------- -------- -------- -------- -------- MOTOR 1.79 AUTONOMY 0.56 1.28 EXPLOR 0.66 0.35 1.29 TASK 1.07 1.18 0.69 3.98 SOCIAL 0.66 0.41 0.50 1.25 3.51 GPA 1.09 1.03 0.59 2.37 1.31 3.12 BIORISK -0.10 -0.12 -0.04 -0.22 0.15 -0.30 SES 0.30 0.67 0.28 1.24 1.02 1.65 HOSPITAL -0.24 -0.14 -0.21 -0.28 -0.42 -0.81 Covariance Matrix BIORISK SES HOSPITAL -------- -------- -------- BIORISK 0.41 SES -0.34 3.16 HOSPITAL 0.18 -0.41 2.39

Summary and Discussion

This article discusses structural equations modeling (SEM) from a developmental

perspective. It is first noted that SEM in developmental research has the same charac-

teristics as SEM in any other discipline of empirical research. Specific to developmental

research is that researchers attempt to vary the age of the subjects under study. If age

is varied by way of a cross-sectional design, SEM in developmental research is no differ-

ent than standard multi-group modeling. As soon, however, as repeated observations are

concerned, modeling has characteristics that are typical of developmental research. We

now discuss three of these characteristics.

The first characteristic comes in the form of an implication. If the same variables are

observed t times on the same individuals, statistical power is not the same as the power

for a model with t independent groups. Rather, to obtain the same power one needs a

sample of size less than t×n, but larger than n (for details, see Muthén & Curran,

1996). Still, samples in developmental research need to be larger than in routine cross-

sectional research in which age is a covariate at best.


The second characteristic is that attrition is typical. In repeated observation studies,

attrition is almost impossible to prevent. The authors know of now long-term study in

which there was no attrition. There are many ways to deal with attrition. First, one can

employ the full information maximum likelihood method. Second, one can estimate and

impute missing information. A third option, which was realized in the examples given in

this article, involves collecting a random sample of data from a parallel cohort that is

then imputed. The characteristics and implications of this procedure are largely un-

known. Future studies will have to determine the degree to which this procedure creates

bias.

The third characteristic concerns the model specifications used to represent the data.

Growth curve models and path models of repeated observations differ from models of

curvilinear relations in cross-sectional studies. For example, growth curve models repre-

sent dependent data, because the same individuals are repeatedly observed. Curvilinear

models of functions such as the one that describes the Weber-Fechner law can be speci-

fied using cross-sectional data. McArdle’s convergence models reside between these ex-

tremes. The main difference is that covariances between time-adjacent observations or,

in more general terms, covariances between observations from different scale points of

some predictor variable can be estimated only if the same individuals are involved. In

cross-sectional designs, these covariances are not defined. The examples presented in

this article used longitudinal data and the models included parameters for the cross-

time covariances.

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