Definition of the multitrait-multimethod LST model

Although the mathematical framework presented above is sufficient to define the observed and latent variables, it is not sufficient to estimate the variances and covariances of the observed variables (Σ) as well as the variance components of the latent variables. For that, some further assumptions about the latent variables already defined must be introduced. There are several dif-ferent sets of possible assumptions and restrictions which can be used to pro-perly determine a model. The model of (T_ij, T M_ijk, O_ijl, OM_jkl)-congeneric variables (Steyer & Eid, 1993; Steyer, 1989a; J¨oreskog, 1969) will be defined.

In this model, the loadings and the intercepts may differ. The goal of this section is to show how this model can be defined. It is clear that this trans-formation should not change the meaning of the latent variable but only its scale. The goal of this section is to define latent variables and to determine the necessary conditions for their existence.

Figure 5.2 (page 96) provides a visual description of the model defined below. To facilitate understanding, Latin letters corresponding to points e through i of the following definition and theorem are introduced in the figure to indicate examples of the relations.

Definition 1. The random variables{Y1111, . . . , Ymnop}on a probability space (Ω,A, P) are called (T_ij, T M_ijk, O_ijl, OM_jkl)-congeneric variablesif and only if the following conditions hold :

a (Ω,A, P) is a probability space such that with finite and positive variances and covariances.

d The variables

are real-valued random variables on (Ω,A, P) with finite and positive variances.

e Definition of common indicator-specific trait variables

For each indicator i and trait j measured by the standard method (k= 1) and for each pair (l, l⁰) ∈ L×L (l 6= l⁰), there are real numbers α_Tij1ll0 and λ_Tij1ll0 such that

T_ij1l =α_Tij1ll0 +λ_Tij1ll0T_ij1l⁰. (5.4.6.1) f Assumption that the regression of the indicator-specific variables mea-sured by method k on the common indicator-specific trait variables is linear

For each indicator i, trait j, occasion l, and method k ∈ K(k 6= 1), there are real numbers α_Tijkl and λ_Tijkl such that

E(T_ijkl|T_ij1l) = α_Tij1l+λ_Tij1lT_ij1l. (5.4.6.2) g Definition of common trait-specific method variables

For each indicatori and traitj measured by methodk and for each pair (l, l⁰)∈L×L, there is a real number λ_{T Mijkll}0 such that²

T M_ijkl =λ_{T Mijkll}0T M_ijkl⁰. (5.4.6.3)

h Assumption that the regression of the indicator specific variables mea-sured by method k on the common indicator-specific occasion-specific variables is linear ³

For each indicator i, trait j, occasion l, and method k ∈ K(k 6= 1), there is a real number λ_Tijkl such that ⁴

E(O_ijkl|O_ij1l) =λ_Oij1klO_ij1l. (5.4.6.4)

i Definition of common occasion-specific method variables

For each trait j, methodk, occasion l and pair (i, i⁰)∈(I×I), there is a real number λ_OMii0jkl such that

OM_ijkl =λ_OMii0jklOM_i⁰_jkl. (5.4.6.5)

2There is no intercept αT M_ijkll0 because the latent trait-specific method variables are

residualsT Mijkl and have, by definition, an expected value of 0.

3The definition of common occasion-specific variables is not necessary because they are

uniquely defined by their corresponding observed variable

4There is no interceptα_Oij1kl because the latent occasion-specific method variables are

residuals and have, by definition, an expected value of 0.

These conditions assume that the latent trait and trait-specific method variables with the same index l are linear transformations from each other.

Moreover, they suppose a linear regression of the indicator-specific trait (res-pectively, occasion) variables measured by the standard method and the same variables measured by the non-standard methods. All assumptions become clearer by considering the following equivalent formulation of the conditions (e, f, g, h, i) as presented in the theorem below.

Theorem 1. (Existence)

The random variables {Y₁₁₁₁, . . . , Y_mnop} are (T_ij, T M_ijk, O_ijl, OM_jkl )-congeneric variables, if and only if Conditions a through d of Definition 1 hold, inserting these parameters in Equation 5.4.6.1 in (e) of Definition 1 yields :

T_ij1l =α_Tij1l +λ_Tij1lT_ij.

In the same way, inserting these parameters in Equation 5.4.6.2 in (f) of Definition 1 yields :

E(T_ijkl|T_ij1l) = α_ijkl+λ_TijklT_ij,

with α_ijkl =α_Tij1ll0 +λ_Tij1ll0 ·α_Tij1l and λ_Tijkl =λ_Tij1ll0 ·λ_Tij1l.

(2) For two different trait variablesT_ij1landT_ij1l⁰, it follows from Equa-tion 5.4.6.6 that :

Therefore, T_ij1l =α_Tij1l +_λ^λTij1l these parameters in equation 5.4.6.3 in (g) of Definition 1 yields :

T Mijkl =λT MijklT Mijk.

(2) For two different trait-specific method-specific variablesT M_ijkl and T M_ijkl⁰, it follows from Equation 5.4.6.8 that T M_ijk = _λ^{T Mijkl} these parameters in equation 5.4.6.5 in (i) of Definition 1 yields :

OMijkl =λOMijklOMjkl.

(2) For two different occasion-specific method-specific variablesOM_ijkl and OM_i⁰_jkl, it follows from Equation 5.4.6.10 that OM_jkl = _λ^OMijkl

This equation is equal to Equation 5.4.6.5 with λ_OMii0jkl := _λ^λOMijkl

OMi0jkl

.

Fig. 5.2: Multitrait-multimethod LST model with indicator-specific traits and occasions (i= 2 indicators,j = 2 traits,k = 2 methods,l = 2 occasions).

Correlations between occasion-specific and occasion-specific method-specific factors are admissible only when the factors bear an identical letter a, b, c or d. The letters e, f, g, h, i show the effect of the definition points e, f, g, h, i on the first trait, trait-specific method and occasion-specific factors. Factor loadings are not depicted. An error variable is only depicted for Y₁₁₁₁.

c

Explanations. In this theorem it is shown that the assumptions of (T_ij, T M_ijk,O_ijl,OM_jkl)-congeneric variables imply the existence of (1) acommon (occasion-unspecific) latent trait variable T_ij for all variables belonging to the same repeatedly administered itemi (2) acommon trait-specific method variable T M_ijk for all variables belonging to the same item i and method k (3) a common (occasion-specific and item-specific) latent state residual O_ijl for all items belonging to itemiadministered on the same occasionl and 4) a common occasion-specific method variableOM_jkl for all variables belonging to the same item i and method k administered on the same occasion l. As a consequence of this theorem, each observed variable is a linear function of the item-specific common latent trait variable T_ij, the trait-specific method variable T_ijk, the item-specific common latent occasion-specific variable O_ijl and the occasion-specific method variable OM_jkl.

First, without loss of generality, the first method was chosen as the stan-dard (Eid, 2000; Eid et al., 2003). Then, the model of (Tij, T Mijk, Oijl, OMjkl )-congeneric variables assumes that :

1. All observed variables with the same index i and j are influenced by the same occasion-unspecific and item-specific latent trait variables Tij

with the standard method. These variables represent common latent trait variables which can also be called trait factors.

2. All observed variables with the same index i, j and k (k 6= 1) are in-fluenced by the same occasion-unspecific, trait-specific and item-specific latent trait-specific method variablesT M_ijk. These variables represent common latent trait-specific method variables which can also be called trait-specific method factors.

3. All variables with the same indexiandj observed on the same occasion l are influenced by the same occasion-specific, trait-specific and item-specific latent variablesO_ijl with the standard method. These variables represent common latent occasion-specific variables which can be called occasion-specific factors.

4. Finally, all variables with the same index j and k (k 6= 1) observed on the same occasion l are influenced by the same item-unspecific, occasion-specific trait-specific latent occasion-specific method variables OM_jkl. These variables represent common latent occasion-specific method-specific variables which can also be called occasion-method-specific method fac-tors.

The MM-LST model can now be defined by the set of its parameters.

Definition 2. LetM:=h(Ω,A, P),S,T,TM,O,OM,αijkl,λTij1l,λT Mijkl,

λ_Oij1l, λ_OMijkli be a MM-LST model with :

T := (T₁₁· · ·T_ij· · ·T_mn)^t, (5.4.6.11) TM := (T M₁₁₁· · ·T M_ijk· · ·T M_mno)^t, (5.4.6.12) O := (O₁₁₁· · ·O_ikl· · ·O_mop)^t (5.4.6.13) OM := (OM₁₁₁· · ·OM_jkl· · ·OM_nop)^t, (5.4.6.14) α_ijkl := (α₁₁₁₁· · ·α_ijkl· · ·α_mnop)^t, (5.4.6.15) λT := (λT1111· · ·λTij1l· · ·λTmn1p)^t, (5.4.6.16) λT M := (λT M1111· · ·λT Mijkl· · ·λT Mmnop)^t, (5.4.6.17) λ_O := (λ_O1111· · ·λ_Oij1l· · ·λ_Omn1p)^t, (5.4.6.18) λ_OM := (λ_OM1111· · ·λ_OMijkl· · ·λ_OMmnop)^t. (5.4.6.19) The complete linear regression determining the observed variables Yijkl

is then :

Y_ijkl = α_ijkl+λ_Tij1lT_ij +λ_{T Mijkl}T M_ijk

+λ_Oij1lO_ijl+λ_OMijklOM_jkl+E_ijkl, (5.4.6.20) if the latent variables T Mijk and OMjkl are defined as equal 0 when k = 1.