Eid, Schneider, and Schwenkmezger (1999) proposed an alternative va-riant of the model depicted in Figure 2.1. In this model, there is an indicator-specific factor for the second indicator, representing that part of the second indicator that is not shared with the first one. The observed variables are decomposed into :
Yil =αil+λTilT +λTiilISTiIi+λOilOl+Eil, (2.2.0.6) where λTiil is the loading of indicator i on the indicator specific factor ISTi. Ii is an indicator variable and is equal to zero when i is the first indicator and one otherwise.
This model has several advantages. It allows a more general decomposi-tion of variance (see below), it avoids multicollinearity problems due to highly correlated trait variables, and it represents the idea of a common trait.
The coefficients indicating the amount of observed variance that is de-termined by the different latent variables can be reformulated according to the new parameterization. The definition of these coefficients is based on the decomposition of variance :
V ar(Yil) =λ2TilV ar(T)+λ2TiilV ar(ISTi)+λ2OilV ar(Ol)+V ar(Eil). (2.2.0.7) According to the new parameterization, the consistency coefficient be-comes :
Con(Yil) = λ2TilV ar(T) +λ2TiilV ar(ISTi)
V ar(Yil) . (2.2.0.8) It represents the degree of observed interindividual differences that are neither influenced by the occasion of measurement nor by measurement error.
This coefficient can now be decomposed into acommon consistency coefficient representing trait variance shared with the trait of the standard indicator :
ComCon(Yil) = λ2ilV ar(T)
V ar(Yil) , (2.2.0.9) and anindicator-specific consistency coefficient representing the variance spe-cific to an indicator :
SpeCon(Yil) = λ2iilV ar(ISTi)
V ar(Yil) . (2.2.0.10) If indicators measure exactly the same construct, this coefficient is 0.
The occasion-specificity coefficient indicates the proportion of observed va-riance due to occasion-specific interindividual differences :
OSpe(Yil) = λ2OilV ar(Ol)
V ar(Yil) . (2.2.0.11) The reliability coefficient is the sum of these coefficients and represents the proportion of variance of an observed variable that is not due to measu-rement error :
Rel(Yil) = ComCon(Yil) +SpeCon(Yil) +OSpe(Yil). (2.2.0.12) In order to get an identified LST model, several parameters have to be fixed.
1. As in other structural equation models, at least one loading parameter has to be fixed to a positive value (usually 1) for each factor. If there are only two indicators of a factor and the factor is not correlated with other factors (e.g., occasion-specific variables in Figure 2.1), the two loadings have to be fixed.
2. The mean value of all residuals factors (indicator-specific trait factors ISTj and occasion-specific factors Ok) must be fixed to 0.
3. The common latent trait variable T is the only latent variable whose mean value can be estimated. The mean value of the common trait variable T is identified only if one interceptαik is fixed (usually to 0).
If, for example, the loading of the first indicator on the first occasion of measurement on the common trait factor has been fixed to 1 and its intercept fixed to 0, the mean value of the common trait factor equals the mean value of the first indicator on the first occasion of measurement.
LST models are very useful because they are longitudinal and can there-fore assess change. On the contrary, cross-sectional data, that is data measu-red on one occasion only cannot be used to assess change or evolution. Two examples can best illustrate the problems caused by studying a construct at only one specific time point (i.e., occasion of measurement). The first example concerns clinical psychology. One classic design of studies of depres-sion is to measure depresdepres-sion, provide a treatment and measure depresdepres-sion again. The mean difference between the two measures is supposed to be due to the treatment (see for example Kampf-Sherf et al., 2004; Stewart, Stack, Farrell, Pauls, & Jenike, 2005). This repeated measurement design may lead to biased results because depression is often influenced by the weather (Faust
& Ladewig, 1972; Molin, Mellerup, Bolwig, & Scheike, 1996). Therefore, if the first measure is taken on a rainy day and the second is taken on a sunny day, the difference between scores will be caused by both the treatment, the weather and any possible interaction between treatment and weather, but the proportion of each cause will remain unknown. If the impact of wea-ther, and any other situational influences, had been factored out, the mean difference of scores could then have been more correctly interpreted as a treatment effect (even if it could still be due to spontaneous remission, for example). The second example concerns affective psychology. Studies on tem-porary mood suppose that, by definition, temtem-porary mood is variable from occasion to occasion. This does not mean that temporary mood does not have a stable component. However, it does mean that this stable part is sup-posed to be small. More generally, scores of any construct that is supsup-posed to be fluctuating due to situational influences should not be measured only
once (Eid et al., 1994; Eid & Diener, 1999). If a cross-sectional design is applied to mood states, the risk is that the mean value of the trait will be very dependent on the occasion of measurement. Moreover, other problems may arise. For example, on days when the subject’s temporary mood is very positive, his/her friends will easily evaluate his/her mood correctly while on days when his/her mood is rather neutral, his/her friends may have more difficulties in judging his/her mood. However, LST models still have some limitations.