As the LST model, the MM-LST model will be defined as a stochastic measurement model on the basis of stochastic measurement theory (Steyer, 1989b). Other models defined that way include LST models (Steyer et al., 1992) and CTC(M-1) models (Eid, 2000; Eid et al., 2003). All trait, method and situation factors are defined as functions of the true-score variables. The-refore, the trait, method, and situation factors are well-defined and have a clear meaning. Moreover, this definition is particularly useful because the mo-del can be defined on the basis of only five assumptions. All other properties of the model, for example the uncorrelatedness of trait variables with method and occasion-specific factors and the uncorrelatedness of the error variables with all other latent variables, are logical consequences of the definition of the model. Therefore, this model provides a rationale for the uncorrelatedness of trait and method factors in MM-LST models.
To formally define a stochastic measurement model, it is necessary to determine the probability space on which it is defined. The observed variables that are defined on this probability space are decomposed into a true-score variable and an error variable. The true score can then be decomposed into
different latent variables representing the different sources of true variability.
Based on this decomposition, the variance of each component can be deter-mined. The issues of representation (existence), uniqueness, meaningfulness, identifiability and testability must then be addressed to obtain a testable model (Steyer, 1989b). The representation of indicators formalizes the as-sumptions of the model, that is, the fact that several latent variables are well-defined by their indicators. In other words, the representation theorem shows the existence of latent variables, if certain assumptions about the true-score variables are made. The uniqueness of the factors determine to which degree the factors and parameters are uniquely defined and which transfor-mations of the model parameters are admissible. Moreover and related to the question of uniqueness, the meaningfulness of the factors demonstrates which statements are invariant (i.e. remain true or false) under the admis-sible transformations. The question of identifiability must then be addressed to verify if the model parameters can be uniquely determined and if the model is estimable. A necessary but not sufficient condition is that the in-formation available must be greater than the number of parameters tested.
In other words, the number of equations must be greater than the number of parameters estimated. Finally, the testability of the model will show the consequences of the model definition with respect to the expected covariance and mean structure of the observed variables.
5.4.1 Probability space
Formulating a psychometric model as a stochastic measurement model requires that the variables of the model are defined on a probability space (Ω,A, P) (Eid, 1995, 1996; Steyer, 1989b; Steyer & Eid, 1993; Zimmerman, 1975, 1976). A probability space consists of a set Ω of possible outcomes, a σ-algebra A of subsets of Ω, and a non-negative, countable additive set functionP onAwith P(Ω) = 1. The kind of random experiment considered in MTMM-LST models is defined by the following set Ω of possible outcomes A (Eid, 1996; Steyer et al., 1992) :
Ω =U ×Sit1×. . .×Sitp×A...1 ×. . .×A...p. (5.4.1.1) The set Ω of possible outcomes is the Cartesian product of three different types of sets : 1) U is the set of persons from which a subject is drawn, 2) Sitl, l∈L:={1, . . . , p}, is a set of (usually unknown) situations that might occur on occasion l of measurement, 3) A...l is a set of possible outcomes of the items administered on occasion l of measurement.
Each set A...l is a Cartesian set product Aijkl = A111l × . . .×Amnol
where the elements of a set Aijkl are the possible values of a scale or an item i, i∈I :={1, . . . , m}, or another kind of measurement (e.g., decibels).
This indicator measures a trait j, j ∈ J := {1, . . . , n} with a method k, k ∈K :={1, . . . , o} on an occasionl, l∈L:={1, . . . , p}.
An example with two occasions of measurement (l = 2), two traits (n = 2), two methods (o = 2) and two indicators (m = 2) may illustrate the set of possible outcomes. Even though this example has the smallest possible size to differentiate the effects of traits, methods and occasion, still 2 indicators × 2 traits × 2 methods × 2 occasions = 16 observed variables are necessary. In this case, Ω can be written as :
Ω = U×Sit1×Sit2×A1×A2 elements. The complete enumeration of the components ofωare in Appendix D. Below is a cursory presentation of some representative units.
– A person u from the set of persons U,
– A situation sit1 ∈ Sit1, in which the person u may be on the first occasion of measurement
– A situation sit2 ∈Sit2, in which the person u may be on the second occasion of measurement
– A possible outcomea1111 of the first indicator on the first occasion of measurement on the first trait with the first method
– A possible outcomea2111 of the second indicator on the first occasion of measurement on the first trait with the first method
– etc
The situations do not have to be known. In LST theory a situation can be defined as all inner and outer conditions under which a response to an indicator is assessed (Steyer et al., 1999). This means that situations may differ for each subject even though they are assessed at the same time.
5.4.2 Observed random variables
Each random variable Yijkl : Ω→R,i={1, . . . , m}, j ={1, . . . , n}, k= {1, . . . , o} and l, l = {1, . . . , p} maps the possible outcomes into the set of real numbers R where this set includes −∞ and ∞. The values of one
random variable Yijkl are the scores on an item or scale i measuring a trait j by a method k on an occasion l. The variances of the variables Yijkl are assumed to be strictly positive and finite. Let E(Yijkl) be the expectation of each observed variables Yijkl and Σ be the covariance matrix of the m× n×o×p observed variables Yijkl. The starting point for defining the latent variables are the conditional expectationsE(Yijkl|pU, pSit) wherepU andpSit are defined below.
5.4.3 Latent true-score and residual variables
The values of the mappingpU : Ω→U are the observational unitsu∈U (e.g., individuals), that is,
pU(ω) =u, for each ω∈Ω. (5.4.3.1) The mapping pU is a random variable, even though its values are not numbers but qualitative elements (in this case, observational units). The values of the mappings pSitl : Ω → Sitl, l = {1, . . . , p} are the situations sitl∈Sitl, that is,
pSitl(ω) = sitl, for each ω ∈Ω. (5.4.3.2) Both the persons and the situations are supposed to be picked at ran-dom during the experiment. Moreover,pU(ω) and pSitl(ω) are assumed to be independent. As mentioned above, in multimethod latent state-trait theory, the indicators and methods are chosen by the experimenter for theoretical reasons. In contrast, the occasions are supposed to cause random fluctua-tions around the trait. This means that the situafluctua-tions are random and the occasion-specific variables are random variables. They could be compared to random effect in mixed effects models, while indicator and method could be compared to fixed effects.
To define a testable model, it is necessary to use several indicators for each measurement unit, that is, each combination of occasion, trait, and method. Within this conceptual framework, the conditional expecta-tion E(Yijkl|pU, pSitl) may now be considered. This conditional expectation is a random variable because persons and situations are sampled randomly, that is, according to some distribution. This distribution does not need to be known for the multitrait LST model to be defined.