M. Eid - DGPs · A mo dels for p olytomous resp onse v ariables. Then, the prob-abilistic...

Methods of Psychological Research Online 1996, Vol.1, No.4Internet: http://www.pabst-publishers.de/mpr/

c 1996 Pabst Science Publishers

Longitudinal Con�rmatory Factor

Analysis for Polytomous Item Responses:

Model De�nition and Model Selection on

the Basis of Stochastic Measurement

Theory

Michael Eid

Abstract

Based on a distinction between four di�erent models of longitudinal con�r-matory factor analysis (LCFA) originally explained by Marsh and Grayson(1994) an analogous class of LCFA models for polytomous variables is de-scribed. Then, the probabilistic foundations of LCFA models for polytomousvariables are explained and it is shown that only two models of the initiallyconsidered four LCFA models can be de�ned as stochastic measurement mod-els on the basis of an explicated random experiment. For these two models therepresentation, uniqueness, and meaningfulness theorems are proven and it isshown how some implications of these models can be tested. The two stochas-tic measurement LCFA models are illustrated by a short empirical applica-tion. Finally, the results are discussed with respect to the role of stochasticmeasurement theory for the de�nition and selection of di�erent LCFA models.

Keywords: measurement of change, measurement theory, item response the-

ory, structural equation modeling

1 Introduction

Models of longitudinal con�rmatory factor analysis (LCFA) have been success-fully applied to the analysis of longitudinal data in various research areas (e. g.,Deinzer et al., 1995; Eid, Notz, Steyer, & Schwenkmezger, 1994; Kirschbaum et al.,1990; Marsh & Grayson, 1994; Ra�alovich & Bohrnstedt, 1987; Steyer, Ferring, &Schmitt, 1992). These models have been developed to deal with three general prob-lems: the problems of (a) occasion speci�city, (b) item- resp. test-speci�city, and (c)measurement error. In LCFA models a manifest variable is additively decomposedin various (temporary) occasion-speci�c and (lasting) occasion-unspeci�c as well asitem-speci�c resp. general latent variables.

LCFA models di�er from autoregressive models, another class of longitudinalmodels, in the way intraindividual change is explained (Eid & Ho�mann, 1995;Marsh, 1993). In autoregressive models on the one hand it is assumed that thetest or item score of an individual at a given point in time depends on his or herformer score and a residual indicating change (e. g., J�oreskog, 1979). Thus, inautoregressive models change is considered as a result of a time-related process.The size of the correlations between repeatedly measured variables will thereforedecline in a systematic manner as the time interval between di�erent occasionsbecomes longer. These models are suitable for analysing change processes like, e. g.,

M. Eid: Con�rmatory factor analysis 66

developmental processes. LCFA models on the other hand are based on the conceptof variability: Change is regarded as the occasion-speci�c deviation of a test oritem score from a stable person-speci�c level. In contrast to autoregressive modelsthe correlations between two latent variables of adjacent occasions are typically nothigher than those between two variables of nonadjacent occasions. LCFA modelsare typically used for analysing uctuating psychic states like moods or emotionswhich cannot adequately be analysed with autoregressive models (e. g., Hertzog &Nesselroade, 1987).

Like the models of generalizability theory (Cronbach, Gleser, Nanda, & Rajarat-nam, 1972; Shavelson, Webb, & Rowley, 1989) models of LCFA aim at identifyingdi�erent sources of interindividual di�erences in longitudinal research. Whereas themodels of generalizability theory (GT) are variance components models based on theideas of the analysis of variance methodology, models of longitudinal con�rmatoryfactor analysis were developed in the framework of structural equation modeling.In contrast to models of GT models of LCFA have the major advantages that (a)the error variables can be separated from occasion-speci�c e�ects, (b) some impli-cations of the models can be tested with computer programs for structural equationmodeling, (c) the �t of these models can be compared with competing longitudinalmodels, and (d) the associations of the LCFA variables with other latent variablescan be analysed in the framework of structural equation modeling.

In the LCFA models presented by Marsh and Grayson (1994) and other au-thors (e. g., Steyer, et al., 1992) there is a linear relationship between the manifestvariables and the latent variables. Hence, these models are suitable for continuousresponse variables only and this methodology is { strictly speaking { not applica-ble to categorical (dichotomous, polytomous) response variables. Therefore, thesemodels cannot be used for item analysis and test construction. To overcome thisproblem a class of LCFA models for polytomous response variables will be de�nedin this paper. Especially, it will be shown, how LCFA models for polytomous vari-ables can be de�ned as stochastic measurement models, i.e., (a) how the randomexperiment considered is explicated, (b) how the variables of the model are de�nedon the set of possible outcomes of this random experiment, and (c) how the con-ditions leading to a testable model are formulated (s. Steyer, 1989, and Steyer &Eid, 1993, for an extended discussion of psychometric models as stochastic mea-surement models). Furthermore, de�ning LCFA models for polytomous variablesas stochastic measurement models gives answers to the three important questionsof measurement theory (Coombs, Dawes & Tversky, 1970; Suppes & Zinnes, 1963):

1. How are the latent variables of the model de�ned (representation) and whatdo they measure?

2. How unique are the latent variables of the model de�ned?

3. Which theoretical terms are meaningful?

Based on these investigations it will be demonstrated that only some models of aclass of LCFA models for polytomous variables (explained in the next section) canbe de�ned as stochastic measurement models.

In summary, the two major aims of this paper are (1) to de�ne LCFA mod-els for polytomous variables as stochastic measurement models and to analyse therepresentation, uniqueness, and meaningfulness problems in this class of models;(2) to show how stochastic measurement theory can guide the selection of a LCFAmodel for polytomous variables. Therefore, the class of LCFA models developed byMarsh and Grayson (1994) is described in the next section and is transferred to ananalogous class of LCFA models for polytomous response variables. Then, the prob-abilistic foundations of LCFA models for polytomous variables are explained. After

MPR{online 1996, Vol.1, No.4 c 1996 Pabst Science Publishers


that, LCFA models for polytomous variables are de�ned as stochastic measurementmodels and the representation, uniqueness, and meaningfulness problems are anal-ysed. Furthermore, the meaning of di�erent LCFA models will be explained on thebasis of the measurement theoretical investigations and a small empirical example.Finally, the consequences of these investigations will be discussed.

2 Models of Longitudinal Factor Analysis

Marsh and Grayson (1994) distinguish between four types of LCFA models for aset of m items or tests (i = 1, 2, ...,m) administered on n occasions of measurement(k = 1, 2, ..., n). In each model an item (or test) variable Xik is determined by anoccasion-unspeci�c and item-speci�c factor �i, an occasion-speci�c common factor�i and an error variable �ik:

Xik = �ik�i + �ik�k + �ik; (1)

where �ik is the factor loading of the manifest variable on the item-speci�c factor �i,�ik is the factor loading on the occasion-speci�c common factor �k, and all variablesare assumed to be deviation variables (i. e., variables with zero means).

In all LCFA models presented by Marsh and Grayson (1994) it is assumed that(1) the error variables �ik are uncorrelated with the latent variables �i and �k aswell as with each other and that (2) the item-speci�c factors are uncorrelated withthe occasion-speci�c factors. In each model the variance of the variable Xik can bedecomposed into three sources:

V ar(Xik) = �2ikV ar(�i) + �2ikV ar(�k) + V ar(�ik); (2)

that is (1) a component due to lasting interindividual di�erences [�2ikV ar(�i)], (2)a component due to occasion-speci�c interindividual di�erences [�2ikV ar(�k)], and(3) an error component [V ar(�ik)]. Whereas this structure holds for all modelsdiscussed by Marsh and Grayson (1994) the models di�er in further assumptions(s. Fig 1).

Model I is characterized by uncorrelated item-speci�c factors �i and correlatedcommon factors �k. In this model temporal stability in interindividual di�erencesis explained by two sources: (a) lasting item-speci�c di�erences and (b) stability ofthe common factors �k. This model is discussed in detail by several authors (e. g.,Marsh & Grayson, 1994; Ra�alovich & Bohrnstedt, 1987; Steyer et al., 1992) andis historically the earliest LCFA model. A signi�cant limitation of this model stemsfrom the fact, that it is not possible to get separate estimates of variance componentsdue to stable resp. non-stable interindividual di�erences, when the common factorsare correlated.

This problem is solved in Model II by the introduction of a common (occasion-and item-unspeci�c) factor � accounting for the covariances of the common factors�k (see Marsh & Grayson, 1994; Steyer et al., 1992):

�k = �k� + �Rk : (3)

The residuals �Rk re ect the in uences of occasion-speci�c e�ects that are com-mon to all items/tests administered. In this model it is assumed that all latentvariables are uncorrelated with each other. Hence, the variance of the occasion-speci�c variable �k can be decomposed into two sources: (1) one component dueto common lasting interindividual di�erences [�2kV ar(�)] and (2) one componentdue to common occasion-speci�c interindividual di�erences and not due to stableinterindividual di�erences [V ar(�Rk)]. In the two other Models III and IV stabilityand change is explained in a di�erent way.



X11 X21 X31 X12 X22 X32 X13 X23 X33

.3.2.1

>3>2>1

X33

X11 X21 X31 X12 X22 X32 X13 X23 X33

.3.2.1

>3>2>1

X33

X11 X21 X31 X12 X22 X32 X13 X23 X33

.3.2.1

>3>2>1

X33

>

X11 X21 X31 X12 X22 X32 X13 X23 X33

.R3

.2.1

>3>2>1

X33

.Model I Model II

Model III Model IV

.3

>R3

Figure 1: Four models of con�rmatory longitudinal factor analysis (from Marsh &Grayson, 1994). The models are explained in the text. The factor loadings are not pointedout.

Model III is characterized by uncorrelated occasion-speci�c common factors �kand correlated latent item-speci�c variables �i (s. Eid, 1995; Marsh & Grayson,1994, for a more comprehensive discussion of this kind of models). In this modeltemporal stability is totally explained by the existence of stable item-speci�c factors.The common occasion-speci�c factors �k re ect occasion-speci�c in uences beingunique for each occasion of measurement. Hence, in this model it is possible to getseparate estimates of variance components due to stable (the item-speci�c factors �i)resp. unstable (the common occasion-speci�c factors �k) interindividual di�erences.The correlations between the item-speci�c variables re ect the occasion-unspeci�cinterrelationships between the variables. As there is no common general factor inthis model, the item-speci�c variance cannot be separated from the variance due tocommon general di�erences. This is possible in the next model.

In Model IV a second order common factor accounts for the correlations amongthe item-speci�c factors:

�i = �i� + �Ri: (4)

As it is assumed that all latent variables are uncorrelated with each other, thevariance of an item-speci�c factor �i can be decomposed into (1) a component dueto common lasting interindividual di�erences [�2i V ar(�)] and (2) one componentdue to lasting item-speci�c interindividual di�erences [V ar(�Ri)].

Because of the linear dependences of the variables Xik on the latent variables inthese models, they are only suitable for continuous manifest variablesXik . However,all four models presented by Marsh and Grayson (1994) can be transformed to LCFAmodels for polytomous variables, if it is assumed that the variables Xik in Equation(1) are not directly observed, but are latent response variables and are related to



a manifest polytomous response variable Yik with cik categories in the followingmanner:

Yik = s; if �isk < Xik � �i;s+1;k : (5)

s 2 f0; : : : ; cik � 1g, where �isk are threshold parameters with �i0k = �1; �i1k <�i2k < : : : < �i;cik�1;k, and �i;cik;k =1 (e. g., J�oreskog, 1990; Muth�en, 1984). Thus,with the Equations (1) to (5) four LCFA models for polytomous response variablescan be de�ned. For an empirical application of LCFA models the crucial questionrefers to the selection of one of these models. Usually, the choice is guided by twocriteria: (1) The model should �t the data, and (2) it should imply the most detaileddecomposition of variance. There is a third possibility, however. In this paper athird answer will be given on the basis of stochastic measurement theory. It will beshown that only two of the four models can be de�ned as stochastic measurementmodels (Model I and III) and that there are some theoretical advantages of applyinga model that can be de�ned as a stochastic measurement model. This will bedemonstrated in the next sections by presenting the probabilistic and measurementtheoretical foundations of LCFA models for polytomous variables.

3 Probabilistic Foundations of LCFA Models for

Polytomous Variables

As a measurement theoretical foundation of LCFA models Steyer and some col-leagues have developed the so-called latent state-trait theory (LST theory; e. g.,Steyer et al., 1992; Steyer & Schmitt, 1990). Latent state-trait theory is a gen-eralization of Classical Test Theory (Gulliksen, 1950; Lord & Novick, 1968) tothe measurement of persons in situations. Like Zimmerman's (1976) 'classical testtheory with minimal assumptions,' LST theory is formulated on the theory of con-ditional expectations. In this section LST theory is extended, so that polytomousresponse variables can be analysed as well within this theoretical framework. Inthis section, LCFA models for polytomous variables are not longer presented in theform of the 'underlying variable approach' (Bartholomew, 1987) like in Equation(5), but in the equivalent form as multidimensional graded response models (for acomprehensive discussion of the equivalence of both models, s. Takane & de Leeuw,1987; s. Samejima, 1969, for the �rst de�nition of a graded response model).

3.1 The Probability Space

Formulating a psychometric model as a stochastic measurement model requires thatthe variables of the model are de�ned on a probability space (;A; P ) (s. Eid, 1995;Steyer, 1989; Steyer & Eid, 1993; Zimmerman, 1975). A probability space consistsof a set of possible outcomes, a �-algebra A of subsets of , and a non-negative,countable additive set function P on A with P () = 1. The kind of randomexperiment considered in LST theory is de�ned by the following set of possibleoutcomes (Eid, 1995; Steyer et al., 1992):

= U � S1 � S2 � : : :� Sn � O1 �O2 � : : :�On: (6)

The set of possible outcomes is the cartesian product of three di�erent types ofsets:(1) U is the set of persons from which a subject is drawn, (2) Sk; k 2 K :=f1; : : : ; ng, is a set of (usually unknown) situations that might occur on occasion kof measurement, and (3) Ok is a set of possible outcomes of the items administeredon occasion k of measurement.



Each set Ok is a cartesian set product Ok = O1k� : : :�Omk, where the elementsof a set Oik are the categories of the item i; i 2 I := f1; : : : ;mg administered onthe kth occasion of measurement.

An example with two occasions of measurement (k = 2) and two items peroccasion of measurement (m = 2) may illustrate the set of possible outcomes. Inthis case can be written as:

= U � S1 � S2 �O1 �O2 = U � S1 � S2 �O11 � O21 �O21 �O22: (7)

An element ! = (u; s1; s2; o11; o21; o12; o22) of consists of

� a person u from the set of persons U ,

� a situation s1 2 S1, in which the person u may be on the �rst occasion ofmeasurement,

� a situation s2 2 S2, in which this person may be on the second occasion ofmeasurement,

� a possible outcome o11 of the �rst item on the �rst occasion of measurement,

� a possible outcome o21 of the second item on the �rst occasion of measurement,

� a possible outcome o12 of the �rst item on the second occasion of measurement,

� a possible outcome o22 of the second item on the second occasion of measure-ment.

The situations do not have to be known. In LST theory a situation can be de�nedas all inner and outer conditions under which a response to an item is assessed (s.Pawlik & Buse, 1992, for a similar concept of situations).

3.2 The Random Variables

The random variable Yik : ! N0, denotes a response variable indicating the�rst category of item i on occasion k with cik categories by taking the value 0, thesecond category by taking the value 1 and the rth category by taking the values := r � 1. For de�ning a graded response model on the basis of LST theorythe starting points are the conditional probabilities P (Yik � sjp0; pk), where thevalues of the mapping p0 : ! U are the persons and the values of the mappingpk : ! Sk; k = 1; : : : ; n, are the situations which might occur for a sampledperson on the kth occasion. Hence, a value P (Yik � sjp0 = u; pk = sk) of theprobability function P (Yik � sjp0; pk) is the (conditional) probability of a responseto item i on occasion k in a category higher than or equal to category s for a person uin a situation sk on the kth occasion of measurement. For each probability functionP (Yik � sjp0; pk); s 2 Iik := f1; : : : ; cik � 1g a probit variable is de�ned in thefollowing way:

�isk := ��1[P (Yik � sjp0; pk)]; (8)

where � is the (cumulative) distribution function of the standard normal distri-bution. The variable �isk characterizes the state of a person being in a speci�csituation on occasion k of measurement and is therefore called latent state variable.Without further assumptions the variable �isk can be additively decomposed intotwo latent variables: (1) the conditional expectation E(�isk jp0), i. e., the expecta-tion of �isk given p0, and (2) the residual �isk �E(�isk jp0):

�isk = �isk �E(�isk jp0) +E(�isk jp0): (9)



De�ning �isk := E(�isk jp0) and �isk := �isk��isk results in the basic decompositionof latent state-trait models for polytomous variables:

�isk = �isk + �isk : (10)

Whereas the latent state variable �isk characterizes a person-in-the-situation on anoccasion k the variable �isk characterizes the person itself. A value of the variable�isk is the value of the latent state variable �isk we would expect for a person u, if wedo not consider the speci�c situation of this person on an occasion of measurement.A value of this variable is the person-speci�c integral over all latent state valueswhich depend on the person and the situations to which the person sampled mightbe exposed on the occasion k of measurement. The variable �isk is called latenttrait variable. The residual �isk represents e�ects of the situations and/or person-situation-interactions, i. e., that part of a latent state variable �isk that is notdetermined by the person alone and is called latent state residual.

3.3 The Variance Components

As a consequence of Equation (10) the variances of the latent state variables �iskare additively decomposed into the variances of the latent trait variables �isk andthe latent state residuals �isk (Eid, 1995; Steyer, 1988):

V ar(�isk) = V ar(�isk) + V ar(�isk): (11)

Based on this decomposition two item coe�cients can be de�ned as variance com-ponents of the latent state variable:

Con(�isk) = V ar(�isk) + V ar(�isk) (12)

and

Spe(�isk) = V ar(�isk)=V ar(�isk): (13)

The consistency coe�cient Con(�isk) is the proportion of variance of a latent statevariable �isk due to true interindividual di�erences and not due to the di�erentsituations realized on occasion k. The occasion speci�city coe�cient Spe(�isk) onthe other hand is the proportion of variance of a latent state variable �isk due to thesituations and/or person-situation interactions on occasion k. The sum of both co-e�cients is one. If the consistency coe�cient is larger than 0.5 (and therefore largerthan the speci�city coe�cient), this means that true interindividual di�erences onone occasion of measurement depend to a larger degree on true interindividualdi�erences (not due to situational and/or interactional e�ects). If the speci�citycoe�cient is larger than 0.5 (and therefore larger than the consistency coe�cient),this means that true interindividual di�erences on one occasion of measurement aredetermined to a larger degree by situational and/or interactional e�ects. Items of aquestionnaire for the assessment of enduring traits should have high consistency andlow speci�city coe�cients. Items of a state mood questionnaire on the other handshould have high speci�city and low consistency coe�cients. Thus, this approachcan be used for the selection of state-like as well as trait-like items for psychometricquestionnaires (Eid, 1995, in press).

The de�nitions of the consistency and speci�city coe�cients in Equations (12)and (13) di�er from those in LST models for continuous manifest variables, wherethey are de�ned as variance components of manifest variables (s., e. g., Steyer et al.1992). With polytomous response variables the consistency and speci�city coe�-cients cannot be de�ned in this way, as the manifest variables are not linear-additivefunctions of the latent trait variables, the state residuals and the error variables. To



compare both coe�cients between polytomous and continuous response variables,they can be computed for continuous variables in the same way as for polytomousvariables, i. e., as variance proportions of the latent state variables.

The decomposition in Equation (9) as well as the other equations based on thisdecomposition are always true, as they do not depend on any restrictive assump-tions. However, without further assumptions it is impossible to identify the vari-ances and covariances of the latent variables. In order to estimate the consistencyand speci�city coe�cients of a probit variable and to analyse the interrelations ofdi�erent state and trait variables speci�c assumptions have to be made that de�nespeci�c measurement models. In the following section only those two models froma set of possible models will be de�ned that correspond to the Models I and IIIin Figure 1. Furthermore, it will be shown that the Models II and IV cannot bede�ned within the random experiment considered up to now.

4 LCFAModels for Polytomous Variables as Stochas-

tic Measurement Models

In this section three important questions of measurement theory will be analysed forLCFA models for polytomous item responses: Representation (existence), unique-ness, and meaningfulness.

4.1 Representation (existence)

In the LCFA models described in the �rst section it is assumed that (1) all itemsadministered on the same occasion k measure the same occasion-speci�c latent vari-ables �k and (2) that all variables with the same index i measure the same occasion-unspeci�c and item-speci�c latent variable �i. Both assumptions can formally bede�ned in the following way:

De�nition 1 ((�i; �k)-congeneric variables). The random variables Y11; : : : ; Yik;: : : ; Ymn; i 2 I := f1; : : : ;mg; k 2 K := f1; : : : ; ng, on a probability space(;A; P ) are called (�i; �k)-congeneric if and only if the following conditions hold:

(a) (;A; P ) is a probability space such that = U � S1 � S2 � : : :� Sn �O1 �O2 � : : :�On:

(b) The projections p0 : ! U; p1 : ! S1; : : : ; pn : ! Sn are randomvariables on (;A; P ).

(c) Yik : ! N0 are random variables on (;A; P ).

(d) The variables�isk := ��1[P (Yik � sjp0; pk)]; s 2 Iik := f1; : : : ; cik � 1g;

�isk := E(�isk jp0), and

�isk := �isk � �isk

are random variables on (;A; P ), where � is the probability distributionof the standard normal distribution, P (Yik � sjp0; pk) denotes the (p0; pk)-conditional probability that Yik takes a value equal to or larger than s, andE(�isk jp0) denotes the p0-conditional expectation of �isk.

(e) For each quintupel (s; t; i; k; l); s 2 Iik ; t 2 Iil; i 2 I; k; l;2 K, there is a�istkl 2 R and for each tripel (i; k; l); i 2 I; k; l;2 K, there is a �ikl 2 R+,such that

�isk = �istkl + �ikl�itl; (14)



where R+ is the set of positive real numbers.

(f) For all s 2 Iik ; t 2 Ijk there is for each tripel (i; j; k); i; j 2 I; k 2 K, a�ijk 2 R+, such that

�isk = �ijk�jtk : (15)

According to condition (e) (�i-congenerity), all latent trait variables with thesame index i are positive linear transformations from each other. This condition isthe formalization of the assumption that the repeatedly administered items measurethe same occasion-unspeci�c latent trait variable. In condition (f) (�k-congenerity)it is assumed that all items aministered on the same occasion of measurement aresimilarity transformations from each other. This condition is the formalization ofthe assumption that all items measure a common latent occasion-speci�c deviationvariable within each occasion of measurement. Both assumptions become clearerby considering the following equivalent formulation of the conditions (e) and (f).

Theorem 1 (Existence). The random variables Y11; : : : ; Yik; : : : ; Ymn; i 2 I; k 2K, are called (�i; �k)-congeneric if and only if the conditions (a) to (d) in De�nition1 hold as well as:

(e') For each i 2 I there is

� a real random variable �i,

� a vector �1i := (�i1; : : : ; �ik ; : : : ; �in) 2 Rn+, and

� a vector

�2i := (�i1l; : : : ; �i;ci1�1;1; : : : ; �i1k; : : : �i;cik�1;k; : : : ; �i1n; : : : ; �i;cin�1;n) 2 Ra;

a =P

k(cik � 1); k 2 K; such that for all s 2 Iik ; i 2 I and k 2 K:

�isk = �ik(�i � �isk): (16)

(f ') For each k 2 K there is

� a real random variable �k and

� a vector �k := (�1k; : : : ; �ik; : : : ; �mk) 2 Rm+ , such that for all s 2 Iik ; i 2

I and k 2 K:

�isk = �ik�k; (17)

(s. Proof A1 in the appendix).In this theorem it is shown that the assumptions of (�i; �k)-congeneric variables

imply the existence of (1) a common (item-speci�c and occasion-unspeci�c) latenttrait variable �i for all variables belonging to the same repeatedly administered itemi and (2) a common (occasion-speci�c and item-unspeci�c) latent state residual �kfor all items aministered on the same occasion k. As a consequence of this theoremeach latent state variable is a linear function of the item-speci�c common latenttrait variable �i and the occasion-speci�c deviation variable �k:

�isk = �ik(�i � �isk) + �ik�k: (18)

By de�ning �ik := �ik�i + �ik�k and �isk := �ik�isk it can be shown that all latentstate variables belonging to the same variable Yik are translations of one commonlatent state variable �ik:

�isk = �ik � �isk : (19)



Furthermore, as a consequence of the Equations (8) and (18) the following equationholds for the conditional probability functions P (Yik � sjp0; pk):

P (Yik � sjp0; pk) = P (Yik � sj�i; �k) = �[�ik(�i � �isk) + �ik�k]:1 (20)

Models ful�lling the conditions (a) to (d) in De�nition 1 as well as (e') and (f')in Theorem 1 will be called multistate-multitrait models, as in this class of modelsit is allowed that di�erent items measure di�erent latent state variables �isk anddi�erent latent trait variables �i.

De�nition 2 (Multistate-multitrait model).

M := h(;A; P );�; �; �;�1;�2; �i

is called multistate-multitrait model (msmt model) if and only if the conditions (a)to (d) in De�nition 1, the conditions (e') and (f ') in Theorem 1, and the followingde�nitions of the vectors �; �; �;�1;�2 and � hold:

� := (�111; : : : ; �1;c11�1;1; : : : ; �isk ; : : : ; �i;cik�1;k; : : : ; �m1n; : : : ; �m;cmn�1;n);(21)

� := (�1; : : : ; �i; : : : ; �m); (22)

� := (�1; : : : ; �k; : : : ; �n); (23)

�1 := (�11; : : : ; �m1; : : : ; �ik; : : : ; �1n; : : : ; �mn); (24)

�2 := (�111; : : : ; �1;c11�1;1; : : : ; �isk ; : : : ; �i;cik�1;k; : : : ; �m1n; : : : ; �m;cmn�1;n);(25)

� := (�11; : : : ; �m1; : : : ; �ik; : : : ; �1n; : : : ; �mn): (26)

4.2 Uniqueness

Neither the common latent trait variables �i and �k nor the �-parameters areuniquely de�ned in msmt models (s. Theorem A3 in the appendix). In msmtmodels positive linear transformations of the common latent trait variables �i areadmissible, if the item parameters �isk are transformed by the same positive lineartransformations and the item parameters �ik by corresponding similarity transfor-mations. Thus, the variables �i as well as the parameters �isk are measured on aninterval scale and the parameters �ik are measured on a ratio scale. Furthermore,similarity transformations of the common latent state residuals �k as well as theitem parameters �ik are admissible. Accordingly, the latent variables �k and theparameters �ik are measured on a ratio scale.

As the common latent variables �i and �k as well as the item parameters �ik ; �isk ,and �ik are not uniquely de�ned, some standardization have to be made in order toselect speci�c representatives. One possibility is to set one of the item parameters�ik for each i 2 I , one of each item parameter �ik for each k 2 K to any real valuelarger than 0, and one �isk for each (i; k); i 2 I; k 2 K, to any real value. Anotherpossibility { among other things { is to set the expectation E(�i) for each i 2 i toany real value and the variances V ar(�i) for each i 2 i and V ar(�k) for each k 2 K,to any real value larger than 0.

4.3 Meaningfulness

As the common latent trait variables �i and the common latent state residuals�k are not uniquely de�ned in the msmt model, it is important to �nd out whichstatements are invariant with respect to the admissible transformations (meaningful

1s. Proof A2 in the appendix



statements). In the msmt models the following statements { among other statements{ are meaningful (for n 2 R) (s. Corollary A4 in the appendix):

(1) The di�erence of the values of two individuals u1 and u2 on the trait variable�i is n-times the di�erence of the values of two other individuals u3 and u4 onthis trait variable.

(2) The occasion-speci�c deviation value �k(!1) of a person u1 is n-times (largeror smaller than) the value �k(!1) of a person u2.

(3) The ratio of the occasion-speci�c deviation values of two individuals u1 andu2 on an occasion l of measurement is equal to the corresponding ratio of thesame individuals on another occasion k plus a constant n.

(4) The di�erence between two item parameters �isk and �itl is n-times the dif-ference between two item parameters �ivo and �iwp.

(5) The item-parameter �ik of the item i on occasion k is n-times the correspond-ing parameter of the same item on occasion l.

(6) The item parameter �ik of an item i on occasion k is n-times the parameter�jk of another item j on the same occasion of measurement k.

(7) The ratio of the item parameters �ik and �jk of two items i and j on anoccasion k of measurement is equal to the corresponding ratio of the sameitems on another occasion l plus a constant n.

Furthermore, statements about the intercorrelations of the latent trait variables �iand the intercorrelations of the latent state residuals �k are meaningful. Invariantwith respect to the admissible transformations are also statements about the prod-ucts �2ikV ar(�i) resp. �2ikV ar(�k) and therefore statements about the consistencyand speci�city coe�cients:

Con(�isk) :=V ar(�isk)

V ar(�isk)=

�2ikV ar(�i)

�2ikV ar(�i) + �2ikV ar(�k); (27)

Spe(�isk) :=V ar(�isk)

V ar(�isk)=

�2ikV ar(�k)

�2ikV ar(�i) + �2ikV ar(�k): (28)

5 MSMT Models With Conditional Independence

In order to determine the parameters as well as to test the �t of an msmt model,further independence and uncorrelatedness assumptions have to hold. In the �rstindependence assumption it is further assumed that the manifest variables Yik arestochastically independent given all latent variables �i and �k . According to thesecond assumption the probability of a response to a speci�c category of item i onoccasion k does neither depend on the other trait variables �j ; j 6= i, nor on theother state residuals �l; l 6= k, when the trait variable �i and the state residual �k aregiven. If both assumptions hold, the model is called msmt model with conditionalindependence.

De�nition 3 (Multistate-multitrait model with conditional independence).M := h(;A; P );�; �; �;�1;�2; �i is called multistate-multitrait model with condi-tional independence, if for the realizations yik of the random variables Yik for all



i 2 I and k 2 K:

P (Y11 = y11; : : : ; Ymn = ymnj�1; : : : ; �m; �1; : : : ; �n)

=nY

k=1

mYi=1

P (Yik = yikj�1; : : : ; �m; �1; : : : ; �n); (29)

P (Yik = yikj�1; : : : ; �m; �1; : : : ; �n) = P (Yik = yikj�i; �k): (30)

The msmt model with conditional independence implies that the marginal prob-ability of a response vector is determined by (a) the conditional probability P (Yik =yikj�i; �k) and (b) the common distribution of the latent variables � and � only:

Corollary 1. If M := h(;A; P );�; �; �;�1;�2; �i is a multistate-multitrait modelwith conditional independence and P �1;::: ;�m;�1;::: ;�n is the common distribution ofthe latent variables �i and �k, then

P (Y11 = y11; : : : ; Ymn = ymn) =

Z "mYi=1

nYk=1

P (Yik = yikj�i; �k)

#dP �1;::: ;�m;�1;::: ;�n

(31)

holds for all values P (Y11 = y11; : : : ; Ymn = ymn) of the common distribution ofY11; : : : ; Ymn (s. Proof A5 in the appendix).

This marginal probability of an msmt model in the form of a multidimensionalgraded response model is formally equivalent to a factor analytic model of orderedvariables in the form of the underlying variable approach in the Equations (1)and (5) with independent error variables �ik (s. Takane & de Leeuw, 1987, for aformal proof of the equivalence of factor analytic models of ordered variables andmultidimensional graded response models). The item parameters �ik and �ik ofboth formalizations (underlying variable approach resp. graded response model)are equal and �isk = �isk�ik , when the variances of the error variables �ik are setequal to one.2 In factor analysis of ordered variables not the variances of the errorvariables are usually �xed, but other standardizations are set, e. g., the variancesof the variables Xik are set equal to 1. Using asterisks for the parameters of theunderlying variable model with these other standardizations, then the parameters ofthe multidimensional graded response model are related to the parameters ��ik ; �

�ik

and ��isk of the underlying variable model by the equations

��ik = �ik�ik ; ��ik = �ik�ik and ��isk = �ik�ik�isk ; (32)

where �ik is the standard deviation of the error variable �ik (Eid, 1995; Muraki& Carlson, 1995; Takane & de Leeuw, 1987). Hence, the parameters of the msmtmodel as well as the consistency and speci�city coe�cients (s. Eq. 27 and 28) canbe estimated and some consequences of the model can be tested with methods de-veloped in the framework of factor analysis for ordered items (for di�erent testableimplications of LCFA models for polytomous variables s. Eid, 1995). Further-more, the identi�cation rules for models of con�rmatory factor analysis describedby J�oreskog and S�orbom (1993) among others can be applied to LCFA models. Inthe estimation and testing methods for structural equation models of ordered vari-ables the latent variables and the error variables �ik are taken to be multivariatenormal, yielding multivariate normal Xik (for relaxations of these distributional as-sumptions, s. Mislevy, 1986). In the 'full-information' approach (Bock, Gibbons, &Muraki, 1988; Muraki & Carlson, 1995) the whole information of the contingencytable is utilized, whereas in the 'limited information' approach (e. g., Muth�en, 1984,1988) only the information in lower order (bivariate) margins are used.

2The variance of the error variables �ik in the underlying variable representation equals 1,because � in Equation (8) is the probability function of the standard normal distribution (s. Eid,1995; Eid & Ho�mann, 1995, for further explanations).



6 Meaning and Applications of LCFA Models

The Models I and III depicted in Figure 1 are multistate-multitrait models as theyful�ll the de�ning conditions of this class of models. In these models three kinds oflatent variables are considered: (1) latent state variables, (2) latent trait variables,and (3) latent state residuals (occasion-speci�c deviation variables). Both mod-els are msmt models with conditional independence but with di�erent covariancestructures of the latent variables �i and �k resulting from further uncorrelatednessassumptions. In Model I, it is assumed that the latent trait variables �i are uncor-related, but the correlations of the latent state residuals are not restricted in anyway. Thus, the de�ning assumption of Model I is:

Cov(�i; �j) = 0; for all i; j 2 I; i 6= j: (33)

In Model III it is assumed that the state residuals are uncorrelated between occa-sions of measurement, but that the correlations of the latent trait variables are notrestricted. The de�ning assumption of this model is

Cov(�k ; �l) = 0; for all k; l 2 K; k 6= l: (34)

In both models the latent trait variables �i and the state residuals �k are uncorre-lated as a consequence of the model de�nitions (s. A6 in the appendix):

Cov(�i; �k) = 0; for all i 2 I; k 2 K: (35)

What do both models mean on the basis of the random experiment considered inthis paper? This will be explained by a small empirical example. In this examplethe two items happy and unhappy of a mood adjective checklist for the assessment ofthe momentary mood are analysed with an msmt model (for a detailed descriptionof the study,3 s. Eid, 1995; Eid, Steyer, & Schwenkmezger, in press). For simplicityreasons only two items (with a �ve category intensity rating, 0 = not at all, 5 = verymuch; the categories of the item unhappy were recoded, N = 490) on two occasionsof measurement are considered and it is assumed that the parameters of the sameitems are equal between occasions of measurement. These items were analysed withthe limited information approach implemented in the computer program LISCOMP(Muth�en, 1988). The parameters of the multidimensional graded response modelgiven in Table 1 were computed on the basis of Equation (32) from the parameters ofthe underlying variable model. In this special application (two items, two occasions)both models �t equally well as they are data-equivalent (�211 = 6.55, p = .83). Theydi�er, however, in their psychological meaning. The uncorrelatedness of the latenttrait variables in Model I means (on the basis of De�nition 1) that the occasion-unspeci�c (trait)-happiness-values of the sampled individuals are unrelated to theiroccasion-unspeci�c (trait)-unhappiness-values. On the other hand this model im-plies that the occasion-speci�c deviation variables are correlated. This means thatthe occasion-speci�c e�ects on di�erent occasions of measurement are similar to acertain degree. In this model stability depends to a certain degree on the fact thatpersons are in similar situations and/or that there is a similar person-situation-interaction on di�erent occasions of measurement. According to this model theonly reason for the association of di�erent items is that they are in uenced by thesame situational and/or interactional e�ects and the cross-occasional stability of sit-uational and/or interactional e�ects. Furthermore, the consistency and speci�citycoe�cients are not interpretable as coe�cients of stability vs. variability (change),because the �-variables do not only measure the in uence of idiosyncratic e�ects

3This study was supported by a grant from the German Research Foundation (DeutscheForschungsgemeinschaft, Ste 411/3-1).



Table 1: Item parameters, variances and covariances of the latent variables as well asconsistency and speci�city coe�cients for two di�erent LCFA models

Model Item �i1k �i2k �i3k �i4k �ik �ik V ar(�i) Con(�isk)happy -3.47 -1.76 0.41 2.87 2.31 2.31 0.16 0.20

I unhappy -4.41 -3.02 -1.58 -0.30 2.31 2.31 0.16 0.20Corr(�1; �2) = :35happy -3.47 -1.76 0.41 2.87 2.31 2.31 0.38 0.47

III unhappy -4.41 -3.02 -1.58 -0.30 2.31 2.31 0.38 0.47Corr(�1; �2) = :59

The parameters �ik are set equal within and between occasions of measurement.The parameters�ik are set equal for each item i. The variances of the latent state residuals are set equal betweenoccasions of measurement and the variances of the latent trait variables are set equal between thetwo items.

unique to the situations resp. interactions on an occasion of measurement (s. Tab.1).

In Model III on the other hand the stable common trait variables completely ex-plain the stability in interindividual di�erences between occasions of measurement;there is no systematic covariation on the level of the occasion-speci�c deviation vari-ables. In this model the correlation of the latent trait variables are not restrictedin any way. These correlations mark the occasion-unspeci�c relationships betweenthe items administered. According to the application there is a signi�cant corre-lation of the latent trait variables indicating that both items are not independentwith respect to their latent trait variables. On the other hand, the state residualsre ect situation- and/or interaction-speci�c e�ects being unique for each occasionof measurement (s. Tab. 1). Hence, the consistency resp. speci�city coe�cientsindicate the in uence of stable resp. non-stable factors on the latent state variables.The speci�city coe�cients of both items are very high and di�er signi�cantly from0. Therefore, both items are suitable for the assessment of uctuating emotionalstates.

This small empirical example elucidates two problems of Model I solved in ModelIII: (1) The assumption of uncorrelated trait variables and (2) the fact that theconsistency coe�cient does not re ect all stable interindividual di�erences. Whyshould we assume that the happiness and the unhappiness traits are unrelated?Ignoring the correlations of the latent trait variables in Model I can result in spuriouscorrelations of the latent state residuals. In addition to this, only in Model III it ispossible to estimate separate variance components re ecting stable resp. occasion-speci�c in uences and thereby important information for item selection and testconstruction is provided by this model. Thus, for theoretical reasons Model III ispreferred to Model I in the application considered.

The two other models (Models II and IV) cannot be de�ned in the framework ofthe random experiment considered, because the general factors cannot be de�ned asrandom variables on the set of possible outcomes. Therefore, the general factors inthese models cannot be expressed as functions of stochastically well de�ned variableslike the conditional probability functions. From the scope of stochastic measurementtheory we do not know exactly what these general factors really measure. Thisargumentation shows that stochastic measurement theory is a strong basis for theinterpretation and selection of di�erent LCFA models. This role of measurementtheory will �nally be discussed in the next section.



7 Discussion

In this paper models of LCFA for polytomous response variables are de�ned asstochastic measurement models. The major advantage of de�ning LCFA models asstochastic measurement models results from the possibility to solve the problemsof measurement theory (representation, uniqueness, meaningfulness) for this classof models. This clari�es the meaning of the latent variables, their scale type, andthe statements that are invariant with respect to the admissable transformations.Furthermore, assumptions made in di�erent LCFA models can adequately be val-ued with respect to their usefulness for empirical applications. It was shown that(1) the Models II and IV cannot be de�ned as stochastic measurement models, andthat (2) in Models I and III stability and change is explained in a di�erent way.Both aspects will be discussed now.Ad (1) General factor models. The existence of a general factor in the ModelsII and IV could not be derived on the basis of the random experiment considered.Thus, from a strict measurement theoretical position only three kinds of variables(latent state variables, latent trait variables, and latent state residuals) should beconsidered in LCFA models. This is in accordance with other models of item re-sponse theory for analysing change, in which a latent response variable for an occa-sion of measurement (latent state variable) is decomposed into two latent variablesmeasuring (a) stable resp. (b) occasion-speci�c interindividual di�erences (s., e. g.,the system of test models in Rost & Spada, 1983). Though it is of course no problemto analyse a general factor from a more technical point of view, this form of dataanalysis goes along with a loss of theoretical strength from a measurement theoret-ical point of view. Only in the Models I and III we can test hypotheses concerningthe homogeneity of di�erent constructs and we can answer important questions ofmeasurement theory (like the uniqueness and meaningfulness questions).

Taking stochastic measurement theory as a basis for the de�nition of psycho-metric models, the assumption of a general latent trait variable is justi�ed, e. g.,when all latent trait variables �isk are linear functions from each other. In thiscase there is no item-speci�city, but perfect homogeneity on the level of the latenttrait variables and of the latent state residuals, and consequently on the level ofthe latent state variables as well. In this model all items measure the same com-mon latent trait variable, an assumption that is well known from the model of� -congeneric variables, another stochastic measurement model (s., e. g., Steyer &Eid, 1993). But, in contrast to the model of � -congeneric variables the items do notonly measure a common latent trait variable, but also a common occasion-speci�cstate residual.

If, e. g., we have to reject the hypothesis that all items measure the same latenttrait variables in Model IV, why should we then assume that there is a factor behindthe covarying traits? According to the well de�ned Models I and III personalityis characterized by a system of di�erent states and di�erent traits and there is no'super-factor' of personality. Assuming the existence of a general factor in the Mod-els II and IV, however, results in some interpretative problems. In Model IV, e. g.,one has to explain 'what' is lying behind a stable trait and this is really not aneasy task. As long as general factors in Models II and IV are not de�ned in a strictmeasurement-theoretical sense their interpretation has to be built on intuition. In-tuition can be a good adviser, but it can be misleading, too. This does not meanthat is not useful to analyse general factors even in the case of di�erent latent traitvariables. There are a lot of applications conceivable where one is interested in the'common part' of all items considered. But, from the scope of measurement theoryit would be better to de�ne a general latent variable as a function of well de�nedlatent variables instead of 'only assuming' a general factor. This has the advan-tage that also the general latent variables are well de�ned and the uniqueness and



meaningfulness theorems can be investigated for these variables too. Furthermore,it puts the interpretation of models with general and method factors on a soundtheoretical basis. In order to de�ne a general factor on the basis of the random ex-periment considered one has to think about its meaning from a substantive point ofview �rst. If their de�nition is clari�ed one can de�ne an appropriate measurementmodel. One possibility { among others { could be to de�ne a general factor as alinear combination of di�erent latent trait variables, but this would imply anothermodel than Models II and IV. Thus, developing LCFA models with general factors(and method factors) de�ned on the random experiment considered will be one im-portant topic of further methodological research in this �eld.Ad (2) Model I vs. Model III. Both models can be de�ned as msmt modelson the random experiment considered. For selecting one of both models for an em-pirical application one has to think carefully about whether or not the assumptionsmade in both models are appropriate for the application considered. In Model I,e. g., the uncorrelatedness assumption of the latent item-speci�c variables has tobe legitimated from theoretical considerations. If Model I is de�ned as an msmtmodel this uncorrelatedness assumption means that the latent trait variables areunrelated. The emprical example in the last section has shown that ignoring thecorrelations of latent trait variables can result in spurious correlations of the latentoccasion-speci�c residuals. As there might be no reasons to assume that latenttrait variables are unrelated in many applications, there are some theoretical rea-sons to prefer Model III to Model I. Furthermore, only in Model III it is possible toestimate separate variance components re ecting stable resp. occasion-speci�c in u-ences which provide important information for item selection and test construction.Nevertheless, the assumption of uncorrelated state residuals in Model III could betoo restrictive in some applications. But, this assumption could be relaxed if thereare enough occasions of measurement. Correlated state residuals, however, shouldonly be admitted, if there are substantive reasons for it (e. g., similar situationalcharacteristics on two di�erent occasions of measurement) and if there is no othermodel being more adequate (e. g., an LCFA model with latent trait change, s. Eid& Ho�mann, 1995).

The LCFA models considered in the last sections are based on the concept ofvariablity and they will { with the exception of Model I { not �t to longitudinal data,e. g., if there is an autoregressive change process. For this data structure the �rstorder autoregressive model with uncorrelated method factors is usually applied (s.,e. g., J�oreskog, 1979). Even this model, originally developed for continuous man-ifest variables, can be de�ned as a stochastic measurement model for polytomousvariables on the random experiment considered in this paper. Based on De�nition I,the uncorrelated 'method' factors in this model can be interpreted as uncorrelatedlatent trait variables and the autoregressive process is assumed on the level of thelatent state residuals. Thus, when there are few reasons for the uncorrelatednessof the latent trait variables in an empirical application, too, one should start withcorrelated latent trait variables and should assume an autoregressive process on thelatent state residuals only in the case, when an LCFA model of Type III does not�t (for a discussion of LCFA models and autoregressive model, s. Kenny & Zautra,1995; Steyer & Schmitt, 1994). Hence, Model III with an autoregressive structureon the latent state residuals is an alternative to the 'classical' autoregressive model(Model I with an autoregressive structure explaining the autocovariance structureof the latent state residuals).

The argumentation and discussion presented in this contribution do not onlyapply to polytomous, but also to continuous manifest variables and shed some lighton former applications of LCFA models which can be reconsidered on the basisof the considerations presented. This paper focusses on LCFA models originallypresented by Marsh and Grayson (1994) and Steyer et al. (1992), which represent



only one class of strutural models for longitudinal research. Their implications forother change models (s., e. g., Raykov, 1996) are beyond the scope of this article.This is left to further methodological research.

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[38] Steyer, R., & Schmitt, T. (1994). The theory of confounding and its application incausal modeling with latent variables. In A. v. Eye & C. Clogg (Eds.), Latent variableanalysis (pp. 36-67). Thousands Oaks: Sage.



[39] Suppes, P., & Zinnes, J. L. (1963). Basic measurement theory. In R. D. Luce, R.Bush, & E. Galanter (Eds.), Handbook of mathematical psychology. Vol. 1 (p. 1-76).New York: Wiley.

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[42] Zimmerman, D. W. (1976). Test theory with minimal assumptions. Educational andPsychological Measurement, 36, 85-96.

Appendix

A1 (Proof of Theorem 1). In order to derive Equation (16) from Equation(14) de�ne for each i 2 I; t = 1 and l = 1:

�i := �i11; �ik := �ik1; �isk := ��is1k1�ik1

(36)

Inserting these new de�ned parameters and variables in Equation (14) results inEquation (16). Equation (14) can be derived from Equation (16) because Equation(16) holds for two variables �isk and �itl and consequently

1

�ik�isk + �isk =

1

�il�itl + �itl (37)

holds. De�ning

�ikl =�ik�il

; �istkl = (�itl � �isk)�ik ; (38)

results in Equation (14). The equivalence of conditions (f) in De�nition 1 and (f')in Theorem 1 can be shown in the same way and is left to the reader.

A2 (Proof of Equation 20). De�ne Iiks : ! f0; 1g with Iiks(!) := 1, if! 2 f! 2 : (Yik)(!) � sg, and Iiks(!) := 0, else, then P (Yik � sjp0; pk) =E(Iiks jp0; pk) and E(Iiks jp0; pk) = �[�ik(�i � �isk) + �ik�k]. Taking the (�i; �k)-conditional expectation of both sides of the last equation results in

E[E(Iiksjp0; pk)j�i; �k] = E[�[�ik(�i � �isk) + �ik�k]j�i; �k] (39)

and consequently

E(Iiksj�i; �k) = �[�ik(�i � �isk) + �ik�k]; (40)

because (1) (�i; �k) is a (p0; pk)-measurable variable and thereforeE[E(Iiksjp0; pk)j�i; �k] =E(Iiks j�i; �k) (s. Steyer, 1988, Rule 4) and (2) �[�ik(�i��isk)+ �ik�k] is a functionof �i and �k and therefore E[�[�ik(�i��isk)+�ik�k]j�i; �k] = �[�ik(�i��isk)+�ik�k](s. Steyer, 1988, Rule 5).

A3 (Theorem: Uniqueness). (1) If M := h(;A; P );�; �; �;�1;�2; �i is amultistate-multitrait model and if for all s 2 Iik; i 2 I; k 2 K:

�0i := �i�i + �i; (41)

� 0k := k�k ; (42)

�0ik := �ik=�i; (43)

�0isk := �i�isk + �i; (44)

�0ik := (1= k)�ik; (45)



where �i; �i; k 2 R and �i; k > 0, then M0 := h(;A; P );�; �0; �0;�01;�02; �

0i is amultistate-multitrait model, too, with:

�0 := (�01; : : : ; �0i; : : : ; �

0m); (46)

�0 := (� 01; : : : ; �0k; : : : ; �

0n); (47)

�01 := (�011; : : : ; �0m1; : : : ; �

0ik; : : : ; �

01n; : : : ; �

0mn); (48)

�02 := (�0111; : : : ; �01;c11�1;1; : : : ; �

0isk ; : : : ; �

0i;cik�1;k; : : : ; �

0m1n; : : : ; �m;cmn�1;n);

(49)

�0 := (�011; : : : ; �0m1; : : : ; �

0ik; : : : ; �

01n; : : : ; �

0mn): (50)

(2) If bothM := h(;A; P );�; �; �;�1;�2; �i andM0 := h(;A; P );�; �0; �0;�01;�

02; �

0iare multistate-multitrait models, then there are for each i 2 I an �i 2 R and a�i 2 R+ and for each k 2 K a k 2 R+ such that Equations (41) - (50) hold.

Proof. Ad (1). If �0i := �i�i + �i; �0ik := �ik=�i; and �0isk := �i�isk + �i, then

�i = (1=�i)(�0i � �i); �ik = �0ik�i; �isk = (1=�i)(�

0isk � �i): Inserting the last three

equations in Equation (16) results in �isk = �0ik(�0i � �0isk): In the same way it can

be shown that �isk = �0ik�0k holds for � 0k := k�k and �0ik := (1= k)�ik : Ad (2). If

both M := h(;A; P );�; �; �;�1;�2; �i and M0 := h(;A; P );�; �0; �0;�01;�

02; �

0iare msmt models, then �ik(�i � �isk) = �0ik(�

0i � �0isk): Consequently, for all s 2

Iik ; i 2 I; k 2 K:

�0i =�ik�0ik

�i +

��0isk � �ik

�isk�0ik

�: (51)

As the ratio of the parameters �ik and �0ik as well as the term in the brackets haveto be the same real value for each s 2 Iik and each k 2 K, two real constants canbe de�ned for each i 2 I :

�i := �0isk � �ik�isk�0ik

and �i :=�ik�0ik

: (52)

The proof for the variables �k can be derived in the same way and is left to thereader.

A4 (Corollary: Meaningfulness). If M := h(;A; P );�; �; �;�1;�2; �i andM0 := h(;A; P );�; �0; �0;�01;�

02; �

0i are msmt models, then for !1; !2; !3; !4 2; s 2 Iik ; t 2 Iil; v 2 Iio; w 2 Iip; i; j 2 I ; k; l; o; p 2 K :

�i(!1)� �i(!2)

�i(!3)� �i(!4)=

�0i(!1)� �0i(!2)

�0i(!3)� �0i(!4)(53)

for [�i(!3)� �i(!4)] 6= 0,

�k(!1)

�k(!2)=

� 0k(!1)

� 0k(!2)(54)

for � 0k(!2) 6= 0,

�k(!1)

�k(!2)�

�l(!1)

�l(!2)=

� 0k(!1)

� 0k(!2)�

� 0l(!1)

� 0l(!2)(55)

for �k(!2) 6= 0; �l(!2) 6= 0,

�isk � �itl�ivo � �iwp

=�0isk � �0itl�0ivo � �0iwp

(56)



for (�ivo � �iwp) 6= 0;

�ik�il

=�0ik�0il

; (57)

�ik�jk

=�jk�0jk

; (58)

�ik�jk

��il�jl

=�0ik�0jk

��0il�0jl

(59)

�2ikV ar(�i) = �02ikV ar(�

0i); (60)

Corr(�i ; �j) = Corr(�0i ; �0j); (61)

�2ikV ar(�k) = �02ikV ar(�

0k): (62)

Corr(�k ; �l) = Corr(� 0k ; �0l); (63)

Proof. The statements of this Corollary are direct consequences of Theorem A3.The proof is simple and left to the reader.

A5 (Proof of Corollary 1). De�ne IA : ! f0; 1g with IA(!) := 1; if ! 2 f! 2 : (Y11; : : : ; Ymn)(!) = (y11; : : : ; ymn)g; and IA(!) := 0; else, then

P (Y11 = y11; : : : ; Ymn = ymn) = P (IA = 1) = E(IA) = E[E(IAj�1; : : : ; �m; �1; : : : ; �n)]

and

E[E(IAj�1; : : : ; �m; �1; : : : ; �n)] =

ZE(IAj�1; : : : ; �m; �1; : : : ; �n) dP

�1;::: ;�m;�1;::: ;�n

(64)

(s. Bauer, 1991, De�nition 3.3 and Equation 3.7; or Billingsley, 1986, p. 280). Equa-tion (31) results from the last equation, the equation E(IAj�1; : : : ; �m; �1; : : : ; �n) =P (Y11 = y11; : : : ; Ymn = ymnj�1; : : : ; �m; �1; : : : ; �n) and the Equations (29) and(30).

A6 (Uncorrelatedness of �i and �k). The variable �k is a similarity transfor-mation of the residual �isk (s. Equation (17). As �isk := �isk � E(�isk jp0), �iis p0-measurable and a residual is regressively independent from any measurablefunction of its regressor (Steyer, 1988, Equation 9), Equation (35) holds.

Submitted June 1, 1996Revised August 14, 1996

Accepted August 18, 1996


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