Identifiability

The parameters of a model can only be estimated uniquely if they are identified. To verify this condition, each estimated parameter must be uni-quely identified by a set of observed variances or covariances. For each type

of parameter, a general set of observed variances and covariances will be presented which define its value. A general rule in structural equation mo-deling is that for each latent factor either the variance of the latent variable or a loading and in addition the intercept of an observed variable or the mean of the latent variable has to be fixed. In order to simplify the proofs of identification, I will, without loss of generality, apply the rule fixing the intercepts with lowest indices to 0 and the loadings with lowest indices to 1. For example, the loading λ₂₂₁₁ of the trait variable T₂₂ is fixed to 1. Fur-thermore, in order to reduce complexity, parameters that have already been identified in a previous step will be used for the identification of parame-ters presented later on without replacing the newly identified parameparame-ters by the parameters of the observed variables. In the following corollary, it will be shown that the parameters are identified for at least two indicators, two traits, two methods and three occasions of measurement. In the case of two indicators and two occasions (traits and methods equal to two or more), the model is not identified without further restrictions on the parameters. For example, a model in which all loading parameters of the latent variables are set equal to one is identified.

Corollary 4. (Identifiability : MM-LST model with conditional regressive independence)

IfM:=h(Ω,A, P),S,T,TM,O,OM,α_ijkl,λ_Tij1l,λ_{T Mijkl},λ_Oij1l,λ_OMijkli is a MM-LST model with conditional regressive independence, then for i, i⁰ ∈ I, j, j⁰ ∈J, k, k⁰ ∈K, and l, l⁰ ∈L and l6=l⁰, k, k⁰ 6= 1 and α_ij11 = 0, λ_Tij11 = 1, λ_{T Mijk1} = 1, λ_Oij1l = 1 and λ_OM1jkl = 1 :

Identification : Parameters of the trait variables

E(T_ij) =E(Y_ij11) (5.4.11.1)

α_ijkl=E(Y_ij11)−λ_TijklE(Y_ijkl) (5.4.11.2)

λ_Tijkl = Cov(Y_ij1l, Y_ij1l⁰)

Cov(Y_ij1l, Y_ij1l⁰) (5.4.11.3)

V ar(T_ij) = Cov(Y_ij11, Y_ij1l)Cov(Y_ij11, Y_ij1l⁰)

Cov(Y_ij1l, Y_ij1l⁰) (5.4.11.4)

Cov(T_ij, T_i⁰_j⁰) = Cov(Y_i⁰_j⁰₁₁, Y_i⁰_j⁰_1l⁰)Cov(Y_ij1l, Y_i⁰_j⁰_1l)

Cov(Y_i⁰_j⁰_1l, Y_i⁰_j⁰_1l⁰) (5.4.11.5) for (ij)6= (ij)⁰.

Identification : Parameters of the trait-specific method variables

λ_{T Mijkl} =Cov(Y_ijkl, Y_ijkl⁰)−λ_Tijk1λ_Tijkl0V ar(T_ij)

Cov(Y_ijkl, Y_ijkl⁰)−λ_Tijklλ_Tijkl0V ar(T_ij) (5.4.11.6)

V ar(T M_ijk) =Cov(Y_ijk1, Y_ijkl)−λ_Tijk1λ_TijklV ar(T_ij)

λ_Tijkl (5.4.11.7)

Cov(T_ij, T M_i⁰_j⁰_k) =Cov(Y_ij11, Y_i⁰_j⁰_kl)−λ_Ti0j0klCov(T_ij, T_i⁰_j⁰) λ_{T Mi}0j0kl

, (5.4.11.8)

for (ij)6= (ij)⁰.

Cov(T M_ijk, T M_i⁰_j⁰_k⁰) = Cov(Y_ijk1, Y_i⁰_j⁰_k⁰_l)−λ_Tijk1λ_Ti0j0k0lCov(T_ij, T_i⁰_j⁰) λ_Ti0j0k0l ,

(5.4.11.9) for (ijk)6= (ijk)⁰.

Identification : Parameters of the occasion-specific variables

Cov(O_ijl, O_i⁰_j⁰_l) =Cov(Y_ij1l, Y_i⁰_j⁰_1l)−λ_Tij1lλ_Ti0j01lCov(T_ij, T_i⁰_j⁰), (5.4.11.10) for (ij)6= (ij)⁰.

λ_Oijkl = Cov(Y_i⁰_j1l, Y_ijkl)−λ_Ti0j1lλ_TijklCov(T_i⁰_j, T_ij)

Cov(O_i⁰_jl, O_ijl) (5.4.11.11)

V ar(O_ijl) = Cov(Y_ij1l, Y_ijkl)−λ_Tij1lλ_TijklV ar(T_ij)

λ_Oijkl (5.4.11.12)

Identification : Parameters of the occasion-specific method variables

Cov(O_1jl, OM_j⁰_kl) =Cov(Y_1j1l, Y_ij⁰_kl)−λ_T1j1lλ_Tij0klCov(T_1j, T_ij⁰)

−λ_T1j1lλ_{T Mij0kl}Cov(T_1j, T M_ij⁰_k)

−λ_Oij0klCov(O_1jl, O_ij⁰_l), (5.4.11.13) for j6=j⁰.

λ_OMij1l= Cov(Y_i⁰_j⁰_1l, Y_ijkl)−λ_Ti0j01lλ_TijklCov(T_i⁰_j⁰, T_ij) Cov(O_ijl, OM_j⁰_kl)

+−λ_Ti0j01lλ_{T Mijkl}Cov(T_i⁰_j⁰, T M_ijk)−λ_OijklCov(O_i⁰_j⁰_l, O_ijl)

Cov(O_ijl, OM_j⁰_kl) (5.4.11.14) for i6=i⁰ and j6=j⁰.

V ar(OM_ijkl) = Cov(Y_1jkl, Y_ijkl)−λ_T1jklλ_TijklCov(T_1j, T_ij) λ_OMijkl

+−λ_T1jklλ_{T Mijkl}Cov(T_1j, T M_ijk)−λ_{T M1jkl}λ_TijklCov(T_ij, T M_1jk) λ_OMijkl

+−λ_{T M1jkl}λ_{T Mijkl}Cov(T M_1jk, T M_ijk) λ_OMijkl

+−λ_O1jklλ_OijklCov(O_1jl, O_ijl)

λ_OMijkl , (5.4.11.15)

for i6= 1.

Cov(OM_jkl, OM_j⁰_k⁰_l) = Cov(Y_ijkl, Y_ij⁰_k⁰_l)−λ_Tijklλ_Tij0k0lCov(T_ij, T_ij⁰) λ_OMijklλ_OMij0k0l

+−λ_Tijklλ_{T Mij0k0l}Cov(T_ij, T M_ij⁰_k⁰)−λ_{T Mijkl}λ_Tij0k0lCov(T_ij⁰, T M_ijk) λ_OMijklλ_OMij0k0l

+−λ_{T Mijkl}λ_{T Mij0k0l}Cov(T M_ijk, T M_ij⁰_k⁰)−λ_Oijklλ_Oij0k0lCov(O_ijl, O_ij⁰_l) λ_OMijklλ_OMij0k0l

+−λ_Oijklλ_OMij0k0lCov(O_ijl, OM_j⁰_k⁰_l) λ_OMijklλ_OMij0k0l

+−λ_OMijklλ_Oij0k0lCov(O_ij⁰_l, OM_jkl)

λ_OMijklλ_OMij0k0l (5.4.11.16) for i6= 1 and(jk)6= (jk)⁰.

Identification : Error variables

V ar(E_ijkl) =V ar(Y_ijkl−λ²_TijklV ar(T_ij)−λ²_{T Mijkl}V ar(T M_ijk)

−λ²_OijklV ar(O_ijl)−λ²_OMijklV ar(OM_jkl) (5.4.11.17)

V ar(E_ij1l) =V ar(Y_ij1l−λ²_Tij1lV ar(T_ij)−V ar(O_ijl) (5.4.11.18) D´emonstration. Identification

Identification of α_(ijkl)⁰ and E(T_ij).

Any observed variable Yijkl can be defined as follows : Y_ijkl=α_ijkl+λ_TijklT_ij +λ_{T Mijkl}T M_ijk

+λ_OijklO_ijl+λ_OMijklOM_jkl+E_ijkl. The expected value of Y_ijkl is

E(Y_ijkl) =E(α_ijkl) +E(λ_TijklT_ij) +E(λ_{T Mijkl}T M_ijk) E(λOijklOijl) +E(λOMijklOMjkl) +E(Eijkl).

The expected value of the residual variables T M_ijk, O_ijl, OM_jkl and E_ijkl are all equal to zero (see Definition 1) and E(α_ijkl) =α_ijkl because α_ijkl is a constant. Therefore, the equation simplifies to :

E(Y_ijkl) =α_ijkl+λ_TijklE(T_ij).

This shows that the expected value of the trait variable E(T_ij) can be esti-mated as E(Y_ij11) since α_ij11= 0 and λ_Tij11 = 1 :

E(T_ij) = E(Y_ij11).

E(Tij) can then be replaced by E(Yij11) in any of the other five equations (two per occasion minus one already used) containing this parameter, thereby defining the intercepts α_ijkl in terms of E(Y_ijkl)−λ_TijklE(Y_ij11).

Identification of λ_Tijkl.

For any observed variable measured by the standard method Y_ij1l, Cov(Y_ij11, Y_ij1l⁰) = λ_Tij11λ_Tij1l0V ar(T_ij)

Cov(Y_ij1l, Y_ij1l⁰) = λ_Tij1lλ_Tij1l0V ar(T_ij)

Replacing λ_Tij1l0V ar(T_ij) in the first equation by ^Cov(Y_λ^ij1l,Yij1l0)

Tij1l yields :

Cov(Y_ij11, Y_ij1l⁰) =λ_Tij11Cov(Y_ij1l, Y_ij1l⁰) λ_Tij1l Therefore,

λTij1l

λ_Tij11 = Cov(Y_ij1l, Y_ij1l⁰) Cov(Y_ij11, Y_ij1l⁰) λTij1l is uniquely defined, since λTij11 = 1.

Identification of V ar(T_ij) Since λ_Tij1l = _Cov(Y^Cov(Y_ij11^ij1l,Y_,Y^ij1l0)

ij1l0) and λ_Tij11 = 1, it follows that : Cov(Y_ij11, Y_ij1l) = λ_Tij11λ_Tij1lV ar(T_ij)

= Cov(Yij1l, Yij1l⁰)

Cov(Y_ij11, Y_ij1l⁰)V ar(Tij).

Therefore,

V ar(Tij) = Cov(Yij11, Yij1l)Cov(Yij11, Yij1l⁰) Cov(Y_ij1l, Y_ij1l⁰)

Identification of Cov(T_ij, T_i⁰_j⁰),(ij)6= (ij)⁰ For any observed variable Y_ijkl,

Cov(Y_ij11, Y_i⁰_j⁰_1l) =λ_Tij11λ_Ti0j01lCov(T_ij, T_i⁰_j⁰) Since λ_Tij1l = _Cov(Y^{Cov(Yij1l,Yij1l0)}

ij11,Y_ij1l0) and λ_Tij11 = 1, it follows that : Cov(T_ij, T_i⁰_j⁰) = Cov(Y_i⁰_j⁰₁₁, Y_i⁰_j⁰_1l⁰)Cov(Y_ij11, Y_i⁰_j⁰_1l)

Cov(Y_i⁰_j⁰_1l, Y_i⁰_j⁰_1l⁰) Identification of λ_{T Mijkl}

For any observed variables Y_ijkl,

Cov(Y_ijk1, Y_ijkl⁰) = λ_Tijk1λ_Tijkl0V ar(T_ij) +λ_{T Mijk1}λ_{T Mijkl}0V ar(T M_ijk) Cov(Y_ijkl, Y_ijkl⁰) = λ_Tijklλ_Tijkl0V ar(T_ij) +λ_{T Mijkl}λ_{T Mijkl}0V ar(T M_ijk)

The second equation can be reformulated to isolate λ_{T Mijkl}0V ar(T M_ijk) : λ_{T Mijkl0}V ar(T M_ijk) = Cov(Y_ijkl, Y_ijkl⁰)−λ_Tijklλ_Tijkl0V ar(T_ij)

λT Mijkl

Replacingλ_{T Mijkl}0V ar(T M_ijk) in the first equation by ^{Cov(Yijkl,Yijkl0)−λ}_λ ^{TijklλTijkl0V ar(Tij)}

T Mijkl

yields :

Cov(Y_ijk1, Y_ijkl⁰) =λ_Tijk1λ_Tijkl0V ar(T_ij)+λ_{T Mijk1}Cov(Y_ijkl, Y_ijkl⁰)−λ_Tijklλ_Tijkl0V ar(T_ij) λ_{T Mijkl}

Therefore,

λ_{T Mijkl}

λ_{T Mijk1} = Cov(Y_ijk1, Y_ijkl⁰)−λ_Tijk1λ_Tijkl0V ar(T_ij) Cov(Y_ijkl, Y_ijkl⁰)−λ_Tijklλ_Tijkl0V ar(T_ij) λT Mijkl is uniquely defined, since λT Mijk1 = 1.

Identification of V ar(T M_ijk) For any observed variables Y_ijkl,

Cov(Y_ijk1, Y_ijkl) =λ_Tijk1λ_TijklV ar(T_ij) +λ_{T Mijk1}λ_{T Mijkl}V ar(T M_ijk).

Therefore, sinceλT Mijk1 = 1,

V ar(T M_ijk) = Cov(Yijk1, Yijkl)−λTijk1λTijklV ar(Tij) λTijkl

Identification of Cov(T_ij, T M_i⁰_j⁰_k),(ij)6= (ij)⁰ For any observed variables Yijkl,

Cov(Y_ij11, Y_i⁰_j⁰_kl) =λ_Tij11λ_Ti0j0klCov(T_ij, T_i⁰_j⁰)+λ_Tij11λ_{T Mi}0j0klCov(T_ij, T M_i⁰_j⁰_k) Since λ_Tij11 = 1,

Cov(Tij, T Mi⁰j⁰k) = Cov(Yij11, Yi⁰j⁰kl)−λT_i0j0klCov(Tij, Ti⁰j⁰) λ_{T Mi}0j0kl

Identification of Cov(T M_ijk, T M_i⁰_j⁰_k⁰), (ijk)6= (ijk)⁰

For any observed variables Y_ijkl,

Cov(Y_ijk1, Y_i⁰_j⁰_k⁰_l) =λ_Tijk1λ_Ti0j0k0lCov(T_ij, T_i⁰_j⁰)+λ_{T Mijk1}λ_{T Mi}0j0k0lCov(T M_ijk, T M_i⁰_j⁰_k⁰).

Since λ_{T Mijk1} = 1,

Cov(T Mijk, T Mi⁰j⁰k⁰) = Cov(Y_ijk1, Y_i⁰_j⁰_k⁰_l)−λ_Tijk1λ_Ti0j0k0lCov(T_ij, T_i⁰_j⁰) λ_Ti0j0k0l

Identification of Cov(Oijl, Oi⁰j⁰l), (ij)6= (ij)⁰ For any observed variables Y_ijkl,

Cov(Y_ijkl, Y_i⁰_j⁰_kl) =λ_Tijklλ_Ti0j0klCov(T_ij, T_i⁰_j⁰) +λ_Oijklλ_Oi0j0klCov(O_ijl, O_i⁰_j⁰_l) Therefore,

Cov(Oijl, Oi⁰j⁰l) = Cov(Y_ijkl, Y_i⁰_j⁰_kl)−λ_Tijklλ_Ti0j0klCov(T_ij, T_i⁰_j⁰) λ_Oijklλ_Oi0j0kl

Cov(Oijl, Oi⁰j⁰l) is uniquely defined whenk= 1 since bothλOij1l andλO_i0j01l = 1.

Identification of λ_Oijkl

For any observed variables Y_ijkl,

Cov(Yi⁰j1l, Yijkl) = λT_i0j1lλTijklCov(Tij, Ti⁰j) +λO_i0j1lλOijklCov(Oijl, Oi⁰jl) Therefore, since λ_Oi0j1l = 1,

λ_Oijkl = Cov(Y_i⁰_j1l, Y_ijkl)−λ_Ti0j1lλ_TijklCov(T_ij, T_i⁰_j) Cov(O_ijl, O_i⁰_jl)

Identification of V ar(O_ijl) For any observed variables Yijkl,

Cov(Y_ij1l, Y_ijkl) =λ_Tij1lλ_TijklV ar(T_ij) +λ_Oij1lλ_OijklV ar(O_ijl) Therefore, sinceλOij1l = 1,

V ar(Oijl) = Cov(Y_ij1l, Y_ijkl)−λ_Tij1lλ_TijklV ar(T_ij) λ_Oijkl

Identification of Cov(O_ijl, OM_j⁰_kl), j 6=j⁰ For any observed variables Yijkl,

Cov(Y_ij1l, Y_ij⁰_kl) =λ_Tij1lλ_Tij0klCov(T_ij, T_ij⁰) +λ_Tij1lλ_{T Mij}0klCov(T_ij, T M_ij⁰_k) +λ_Oij1lλ_Oij0klCov(O_ijl, O_ij⁰_l) +λ_Oij1lλ_OMij0klCov(O_ijl, OM_j⁰_kl) Therefore, sinceλ_Oij1l = 1,

Cov(O_ijl, OM_j⁰_kl) = Cov(Y_ij1l, Y_ij⁰_kl)−λ_Tij1lλ_Tij0klCov(T_ij, T_ij⁰) λ_OMij0kl

+ −λ_Tij1lλ_{T Mij0kl}Cov(T_ij, T M_ij⁰_k)−λ_Oij0klCov(O_ijl, O_ij⁰_l) λOM_ij0kl

When i= 1, Cov(O_ijl, OM_j⁰_kl) is uniquely defined, since λ_OM1jkl = 1.

Identification of λ_OMijkl

For any observed variables Y_ijkl,

Cov(Yi⁰j⁰1l, Yijkl) =λT_i0j01lλTijklCov(Ti⁰j⁰, Tij) +λT_i0j01lλT MijklCov(Ti⁰j⁰, T Mijk) +λO_i0j01lλOijklCov(Oi⁰j⁰l, Oijl) +λO_i0j01lλOMijklCov(Oijl, OMj⁰kl) Therefore, sinceλ_Oi0j01l = 1,

λ_OMijkl = Cov(Y_i⁰_j⁰_1l, Y_ijkl)−λ_Ti0j01lλ_TijklCov(T_i⁰_j⁰, T_ij) Cov(O_ijl, OM_j⁰_kl)

+−λ_Ti0j01lλ_{T Mijkl}Cov(T_i⁰_j⁰, T M_ijk)−λ_OijklCov(O_i⁰_j⁰_l, O_ijl) Cov(O_ijl, OM_j⁰_kl)

Identification of V ar(OM_ijkl)

For any observed variables Y_ijkl,

Cov(Y_1jkl, Y_ijkl) = λ_T1jklλ_TijklCov(T_1j, T_ij) +λ_T1jklλ_{T Mijkl}Cov(T_1j, T M_ijk)

+λ_{T M1jkl}λ_TijklCov(T_ij, T M_1jk) +λ_{T M1jkl}λ_{T Mijkl}Cov(T M_1jk, T M_ijk) +λ_O1jklλ_OijklCov(O_1jl, O_ijl) +λ_OM1jklλ_OMijklV ar(OM_jkl)

Therefore, sinceλOM1jkl = 1,

V ar(OMjkl) = Cov(Y_1jkl, Y_ijkl)−λ_T1jklλ_TijklCov(T_1j, T_ij) λ_OMijkl

+−λT1jklλT MijklCov(T1j, T Mijk)−λT M1jklλTijklCov(Tij, T M1jk) λ_OMijkl

+−λ_{T M1jkl}λ_{T Mijkl}Cov(T M_1jk, T M_ijk)−λ_O1jklλ_OijklCov(O_1jl, O_ijl) λ_OMijkl

Identification of Cov(OM_ijk, OM_ij⁰_k⁰), (jk)6= (jk)⁰

For any observed variables measured by the first indicator Y1jkl,

Cov(Y_ijkl, Y_ij⁰_k⁰_l) = λ_Tijklλ_Tij0k0lCov(T_ij, T_ij⁰) +λ_Tijklλ_{T Mij}0k0lCov(T_ij, T M_ij⁰_k⁰)

+λ_{T Mijkl}λ_Tij0k0lCov(T_ij⁰, T M_ijk) +λ_{T Mijkl}λ_{T Mij}0k0lCov(T M_ijk, T M_ij⁰_k⁰) +λ_Oijklλ_Oij0k0lCov(O_ijl, O_ij⁰_l) +λ_Oijklλ_OMij0k0lCov(O_ijl, OM_j⁰_k⁰_l)

+λ_OMijklλ_Oij0k0lCov(O_ij⁰_l, OM_jkl) +λ_OMijklλ_OMij0k0lCov(OM_jkl, OM_j⁰_k⁰_l)

Therefore,

Cov(OM_jkl, OM_j⁰_k⁰_l) = Cov(Yijkl, Yij⁰k⁰l)−λTijklλT_ij0k0lCov(Tij, Tij⁰) λOMijklλOM_ij0k0l

+−λTijklλT M_ij0k0lCov(Tij, T Mij⁰k⁰)−λT MijklλT_ij0k0lCov(Tij⁰, T Mijk) λ_OMijklλ_OMij0k0l

+−λT MijklλT M_ij0k0lCov(T Mijk, T Mij⁰k⁰)−λOijklλO_ij0k0lCov(Oijl, Oij⁰l) λ_OMijklλ_OMij0k0l

+−λ_Oijklλ_OMij0k0lCov(O_ijl, OM_j⁰_k⁰_l)−λ_OMijklλ_Oij0k0lCov(O_ij⁰_l, OM_jkl) λ_OMijklλ_OMij0k0l

Identification of V ar(Eijkl) For any observed variables Y_ijkl,

V ar(Y_ijkl) = λ²_TijklV ar(T_ij) +λ²_{T Mijkl}V ar(T M_ijk) +λ²_OijklV ar(O_ijl) +λ²_OMijklV ar(OM_jkl) +V ar(E_ijkl)

Therefore,V ar(E_ijkl) is identified by :

V ar(E_ijkl) = V ar(Y_ijkl)−λ²_TijklV ar(T_ij)−λ²_{T Mijkl}V ar(T M_ijk)

−λ²_OijklV ar(Oijl)−λ²_OMijklV ar(OMjkl) For k = 1, V ar(T Mij1), λOij1l and V ar(OMj1l) do not exist. Moreover, λ²_Oij1l = 1. Therefore,V ar(E_ij1l) is identified by :

V ar(Eij1l) =V ar(Yij1l)−λ²_Tij1lV ar(Tij)−V ar(Oijl)

Chapitre 6

Application : Depression and anxiety in school children

The multitrait-multimethod LST model presented above will now be illustrated with data on depression and anxiety in children (Cole, Martin, Powers, & Truglio, 1996; Cole, Truglio, & Peeke, 1997). Moreover, the ap-plication of the MM-LST will be compared with an apap-plication of the mul-ticonstruct LST model (Schermelleh-Engel et al., 2004), an alternative LST model that can be applied to analyze multimethod data and has been pre-sented in chapter 5, section 5.3, page 80. The comparison of these two models will illustrate the peculiarities of the two approaches. The data will first be analyzed with the multiconstruct LST model to show all the information that it can provide. They will then be analyzed with the multimethod LST model and the results will again be interpreted with respect with all the information that this new model can provide. The comparison of the results of the two models will then provide the opportunity to put in perspective what can be gained by using the newly developed model presented in this thesis.

6.1 Multiconstruct LST models

In this approach, an LST model is defined for each trait-method unit, for example depression measured by self-report, depression measured by teacher report, anxiety measured by self-report, and anxiety measured by teacher re-port. In a second step, all models will be combined by allowing for correlations between the trait variables and for correlations between the occasion-specific variables belonging to the different trait-method unit but the same occasion of measurement. Figure 6.1 illustrates this model. This yields a multiple trait LST model. Correlations between the trait variables belonging to the same

construct (e.g., depression) but different method (e.g., self-report and teacher report) indicate convergent validity on the trait level. Correlations between the trait variables of different constructs (e.g., depression and anxiety) re-present discriminant validity. In a similar vein, the correlations between the occasion-specific variables of different methods but the same construct in-dicate convergent validity on the occasion-specific level. For example, the correlation between the occasion-specific variable of depression measured by self-report on Wave one can correlate with the occasion-specific variable of depression measured by teacher report on Wave 1 but not with this variable on Wave two. In this specific example, the correlation can be interpreted as the similarity of occasion-specific influences between self- and teacher report.

If the situation is mostly composed of external influences, like the weather for example, then the similarity can be high. For example, if a sunny weather lowers the scores of depression, then occasion-specific variables of self- and teacher report may correlate highly. If, however, the situation is mostly com-posed of momentary but unobservable internal influences, like for example, the quality of sleep, then the similarity might not be very high because these situation influences are not shared by the two raters. This model will now illustrated with an empirical data set.

Fig. 6.1: Example of a multiconstruct LST model with indicator-specific traits (depression and anxiety) and occasions (i= 2 indicators, j = 2 traits, k = 2 methods, l = 2 occasions). For simplicity reasons, only two occasions of measurement are presented. DS (resp.DT) : depression trait variables for self-report (resp. for teacher report) ; AS (resp. AT : anxiety trait variables for self-report (resp. for teacher report) ;ODS(resp.ODT) : occasion-specific variables for self-report (resp. for teacher report) of depression ; OAS (resp.

OAT) : occasion variables for self-report (resp. for teacher report) of depres-sion.

Dans le document Unfolding the constituents of psychological scores : development and application of mixture and multitrait-multimethod LST models (Page 121-136)