Decomposition of variance

As a consequence of the equation defining the observed variable as a sum of conditional expectations and method deviations (equation 5.4.4.3), the va-riances of the observed variablesY_ijkl are additively decomposed into the va-riances of the latent trait variablesTij1l, latent trait-specific method variables T M_ijkl, latent occasion-specific variables Oij1l, latent occasion-specific me-thod variables OM_ijkl, and latent error variables (Eid, 1995; Steyer, 1988).

The decomposition contains no covariance because, for each observed va-riable, the latent variables are independent :

V ar(Yijkl) = V ar(E(Tijkl|Tij1l)) +V ar(T Mijkl) +V ar(E(O_ijkl|O_ij1l)) +V ar(OM_ijkl)

+V ar(E_ijkl). (5.4.5.1)

The proof of this decomposition relies on the latent variables having been defined on the basis of conditional expectations (Steyer, 1988; Eid, 2000; Eid et al., 2003). It has already been shown (Steyer, 1988; Tack, 1980) that

V ar(Yijkl) =V ar(Tijkl) +V ar(Oijkl) +V ar(Eijkl). (5.4.5.2) Because the trait-specific method variables are residuals with respect to the latent trait variables, they are uncorrelated and the covariances are also zero. Therefore, the variance of the person-specific variable can be decompo-sed into two parts : one part that is determined by the standard method and one part that is specific to the method considered :

V ar(T_ijkl) = V ar(T_ij1l) +V ar(T M_ijkl). (5.4.5.3) Based on this decomposition, several coefficients can be defined as va-riance components of the observed variable. The equations and interpreta-tions of nine particularly interesting coefficients are presented below : Trait coefficients

T Con(Y_ijkl) = V ar(E(T_ijkl|T_ij1l)) +V ar(T M_ijkl)

V ar(Y_ijkl) . (5.4.5.4) The trait consistency coefficient TCon(Y_ijkl) is the proportion of variance of an observed variable explained by the stable internal dispositions represented by the latent trait variable T_ij1l and by the trait-specific method variables T M_ijkl. This means that this proportion is due to true interindividual dif-ferences and not due to the different situations realized on occasion l. This coefficient can be decomposed into two more specific coefficients :

T MCon(Y_ijkl) = V ar(E(T_ijkl|T_ij1l))

V ar(Y_ijkl) , (5.4.5.5) T MSpe(Y_ijkl) = V ar(T M_ijkl)

V ar(Y_ijkl) . (5.4.5.6) The trait specificity coefficient TSpe(Yijkl) is the proportion of variance of an observed variable belonging to a non-standard method explained by the latent trait variableT_ij1l. This means that this proportion is due to true inter-individual differences as measured by the standard method and not due to the different situations realized on occasionl. Thetrait-specific method coefficient TMSpe(Y_ijkl) is the proportion of variance of an observed variable explained

by the latent trait-specific method variableT M_ijkl. This means that this pro-portion is due to differences in observed interindividual differences caused by method k of measurement or by the interaction between the person and the method. But this proportion is not due to the trait measured by the standard method or to the different situations realized on occasion l.

It is also possible to define a trait convergency coefficient T Conv es-timating the convergent validity on the trait level, that is the proportion of consistency explained by the trait and not by the trait-specific method deviation :

T Conv = V ar((E(T_ijkl|T_ij1l))

V ar((E(T_ijkl|T_ij1l)) +V ar(T M_ijkl). (5.4.5.7) Occasion-specific coefficients

In the same way, four occasion coefficients can be defined.

OSpe(Yijkl) = V ar(E(O_ijkl|O_ij1l)) +V ar(OM_ijkl)

V ar(Y_ijkl) . (5.4.5.8) Theoccasion consistency coefficient OCon(Y_ijkl) is the proportion of variance of an observed variable explained by momentary influences (situations) re-presented by the occasion-specific variablesO_ij1l and by the occasion-specific method variables OM_ijkl. This means that this proportion is due to the dif-ferent situations realized on occasion l and/or the interaction between the situations and the person and not due to true to true interindividual diffe-rences. This coefficient can be decomposed into two more specific coefficient :

OM Con(Y_ijkl) = V ar(E(O_ijkl|O_ij1l))

V ar(Yijkl) , (5.4.5.9) OMSpe(Y_ijkl) = V ar(OM_ijkl)

V ar(Y_ijkl) . (5.4.5.10) (5.4.5.11) Theoccasion-specificity coefficient OSpe(Yijkl)is the proportion of variance of the observed variables explained by the latent occasion-specific variableO_ij1l and/or the person-situations interaction of the standard method. This means that this proportion is due to the different situations realized on occasionlas measured by the standard method. The occasion-specific method coefficient OMSpe(Y_ijkl)is the proportion of variance of the observed variables explained by the latent occasion-specific method variable OMijkl. This means that this proportion is due to the occasion-specific influences as measured by method

kthat are not shared with the standard method or to the interaction between the situations and the method.

It is also possible to define a occasion convergency coefficient OConv estimating the convergent validity on the occasion level, that is the proportion of consistency explained by the occasion-specific variable and not by the occasion-specific method deviation :

OConv = V ar(E(O_ijkl|O_ij1l))

V ar(E(O_ijkl|O_ij1l)) +V ar(OM_ijkl). (5.4.5.12) Finally, the unreliability coefficient Unrel(Y_ijkl) is the proportion of va-riance of the observed variables not explained by true interindividual diffe-rences :

Unrel(Yijkl) = V ar(E_ijkl)

V ar(Y_ijkl). (5.4.5.13) If the trait consistency are higher than the occasion consistency coeffi-cients, the observed variables measure mostly a stable component. Whereas if the opposite is true, the observed variables measure mostly the influences of the situations on the construct studied.

If the trait specificity coefficient is higher than the trait-specific me-thod coefficient, true interindividual differences on one occasion of measure-ment depend to a stronger extent on the construct measured with the stan-dard method rather than on the method (non-stanstan-dard) used to measure the construct. By the same reasoning, if the occasion specificity coefficient is higher than the occasion-specific method coefficient, state variables depend more on the occasions as measured by the standard method rather than on the deviation from the occasion due to the method.

Items of a questionnaire for measuring enduring traits should have high trait consistency coefficients. On the contrary, items of a questionnaire as-sessing states should have high occasion consistency coefficients. Moreover, the items of a questionnaire trying to measure an objectivable (i.e., that can be judged in the same way by all) trait should have low trait and occa-sion convergency coefficients because this would mean that the questionnaire has high convergent validity. For example, an easy-to-judge personality trait measured by self-report (k = 1) and peer report (k = 2) should have low trait-specific and occasion-specific method coefficients (and thus high trait and occasion convergency coefficients) because the peer and the subjects would easily perceive the same personality trait.

5.4.6 Definition of the multitrait-multimethod LST