Meaningfulness

As the latent trait variables, trait-specific method variables, and the occasion-specific variables and occasion-specific method variables are not uni-quely defined in the MM-LST models, it must be determined which state-ments are invariant with respect to the admissible transformations. These statements will be called meaningful statements. Many statements are inva-riant. Only a selection of those that are of practical importance are presented below.

Corollary 2. (meaningfulness)

If bothM:=h(Ω,A, P),S,T,TM,O,OM,αijkl,λTij1l,λT Mijkl,λOij1l, λ_OMijkli and M⁰ := h(Ω,A, P),S,T⁰,TM⁰,O⁰,OM⁰,α⁰_ijkl, λ⁰_Tij1l, λ⁰_{T Mijkl}, λ⁰_Oij1l, λ⁰_OMijkli are MM-LST models, then for ω1, ω2, ω3, ω4 ∈ Ω; i, i⁰ ∈ I, j ∈J, k ∈K, and l, l⁰, l⁰⁰, l⁰⁰⁰ ∈L :

T_ij(ω₁)−T_ij(ω₂)

T_ij(ω₃)−T_ij(ω₄) = T_ij⁰ (ω₁)−T_ij⁰ (ω₂)

T_ij⁰ (ω₃)−T_ij⁰ (ω₄), (5.4.9.1) for [T_ij(ω₃)−T_ij(ω₄)] and [T_ij⁰ (ω₃)−T_ij⁰ (ω₄)]6= 0,

T M_ijk(ω₁)

T M_ijk(ω₂) = T M_ijk⁰ (ω₁)

T M_ijk⁰ (ω₂), (5.4.9.2) for T M_ijk(ω₂) and T M_ijk⁰ (ω₂)6= 0,

T M_ijk(ω₁)−T M_ijk(ω₂)

T M_ijk⁰(ω₁)−T M_ijk⁰(ω₂) = T M_ijk⁰ (ω₁)−T M_ijk⁰ (ω₂)

T M_ijk⁰ 0(ω₁)−T M_ijk⁰ 0(ω₂), (5.4.9.3) for [T M_ijk⁰(ω₁)−T M_ijk⁰(ω₂)] and [T M_ijk⁰ 0(ω₁)−T M_ijk⁰ 0(ω₂)]6= 0,

O_ijl(ω₁)

Oijl(ω2) = O_ijl⁰ (ω₁)

O_ijl⁰ (ω2), (5.4.9.4) for O_ijl(ω₂) and O⁰_ijl(ω₂)6= 0,

O_ijl(ω₁)−O_ijl⁰(ω₁)

Oijl(ω2)−Oijl⁰(ω2) = O_ijl⁰ (ω₁)−O⁰_ijl0(ω₁)

O_ijl⁰ (ω2)−O⁰_ijl0(ω2), (5.4.9.5) for [O_ijl(ω₂)−O_ijl⁰(ω₂)], and [O⁰_ijl(ω₂)−O⁰_ijl0(ω₂)]6= 0,

OM_jkl(ω₁) D´emonstration. The proofs for equations 5.4.9.1, 5.4.9.4, 5.4.9.8, 5.4.9.13 and 5.4.9.14 are presented as examples. The proofs for the other statements follow the same principle and are, therefore, straightforward.

5.4.9.1 Replacing T_ij⁰ in equation 5.4.9.1 by γ_Tij+β_Tij·T_ij verifies the equality 5.4.9.4 Replacing O⁰_ij1l in equation 5.4.9.4 by βOijl ·Oijl verifies the equality

Oijl(ω1)

Explanations. The explanations will be illustrated by an example based on data already presented by Cole and Martin (2005). Children were evalua-ted on depression and anxiety (j = 2) either by self-report, peer report or parent report (k = 3). The depression and anxiety questionnaires were then each separated in two test-halves (i = 2). The traits were measured four times (l= 4) on a six months interval.

Eq. 5.4.9.1 The difference of the value of two individuals u₁ and u₂ on the trait variable T_ij is n-times the difference of the values of two other indivi-duals u₃ and u₄ on this trait variable. In other words, the ratio of the difference of the value of two individualsu₁ andu₂ on the trait variable T_ij on the difference of the value of two other individualsu₃ and u₄ on this trait variable is equal to the same ratio for the same individuals on a transformed trait variable.

For example, the difference in the depression scores of two subjects would be n-times the difference in the depression scores of two other subjects.

Eq. 5.4.9.2 The trait-specific method deviation value T Mijk(ω1) of a person u1 is n-times (larger or smaller than) the value T M_ijk(ω₂) of a personu₂. For example, the deviation of the depression score as measured by the peer rater from the value expected given the self-rating isn-times larger for one subject than the same deviation for another subject.

Eq. 5.4.9.3 The difference between the trait-specific method deviation valueT Mijk(ω1) of a person u₁ and the same deviation of a person u₂ is n-times (lar-ger or smaller than) the corresponding difference between the same individuals for another trait-specific method deviation valueT Mijk⁰

For example, the difference of the deviation of the depression score as measured by the peer rater from the value expected given the self-rating between two individuals is n-times larger the corresponding difference between the same individuals evaluated by parent rating and not by peer rating.

Eq. 5.4.9.4 The occasion-specific deviation valueO_ijl(ω₁) of a person u₁ isn-times (larger or smaller than) the value O_ijl(ω₂) of a person u₂.

For example, the occasion-specific deviation of the depression score from the trait score on the first occasion of measurement is n-times larger for one subject than the same deviation score for another subject.

Eq. 5.4.9.5 The difference of two occasion-specific deviation values O_ijl and O_ijl⁰ for a person u₁ is n-times (larger or smaller than) the corresponding

difference for another person u₂.

For example, the difference between the occasion-specific deviation of the depression score from the trait score on the first occasion of measu-rement and the same deviation on the second occasion of measumeasu-rement is n-times larger than the same difference for another individual.

Eq. 5.4.9.6 The occasion-specific method deviation value OM_ijkl(ω₁) of a person u₁ isn-times (larger or smaller than) the valueOM_ijkl(ω₂) of a person u₂.

For example, the occasion- and rater-specific deviation score of peer rater on the first occasion of measurement is n-times larger for one subject than the same deviation score for another subject.

Eq. 5.4.9.7 The difference between the parameters α_ijkl and α_ijkl⁰ is n-times the difference between two item parameters αijkl⁰⁰ and αijkl⁰⁰⁰.

For example the difference of mean change between the first and the second occasion is n-times the difference of mean change of the third and fourth occasions.

Eq. 5.4.9.8 The item parameter λTij1l of the item i on occasion l is n-times the corresponding parameter of the same item on occasion l⁰.

For example, the loading of depression on the first occasion is n-times the same loading on the second occasion.

Eq. 5.4.9.9 The item parameter λT Mijkl on occasion l is n-times the parameter λ_{T Mi}0jk on another occasion of measurement l⁰.

For example, the loading of the peer reported depression on the first occasion isn-times the same loading on the second occasion.

Eq. 5.4.9.10 The item parameter λOMijkl on occasion l is n-times the parameter λ_OMi0jkl on another occasion of measurementl⁰.

For example, the loading of the occasion- and rater-specific deviation score of peer rater on the first test-half is n-times the same loading on the second test-half.

Eq. 5.4.9.11 The ratio of the item parameters λ_{T Mijkl} and λ_{T Mi}0jk0l of two items i and i⁰ measured by a method k is equal to the corresponding ratio of the same items measured by another method k⁰ plus a constantn.

For example, the ratio of the loading of peer reported depression of the first and second test-half is equal to the corresponding ratio of the same items measured by parent report plus a constant n.

Eq. 5.4.9.12 The ratio of the item parameters λ_OMijkl and λ_OMi0jk0l of two items i and i⁰ measured by a method k is equal to the corresponding ratio of the same items measured by another method k⁰ plus a constantn.

For example, the ratio of the loading of the occasion- and rater-specific deviation score of peer rater on the first occasion of the first and second test-half is equal to the corresponding ratio of the same items measured by parent report plus a constant n.

Equations 5.4.9.13, 5.4.9.15, 5.4.9.17 and 5.4.9.19 indicate that, although variances of latent variables by themselves are not meaningful, variances of latent variables multiplied by their squared loadings are. This is particularly important because these products are used for consistency and specificity coefficients. Equations 5.4.9.14, 5.4.9.16, 5.4.9.18 and 5.4.9.20 indicate that the correlations between latent variables are meaningful and can therefore be interpreted.