Sample. A sample of 291 females and 212 males between 17 and 77 years of age (mean age: 31.2 years) filled in a couple of questionnaires on four occasions of measurement, each of them three weeks apart. The subjects were paid DM 50 for completing the tests on all four occasions of measurement. About half of the subjects were assessed in group sessions in a lecture room at the University of Trier. The other half of the subjects were recruited via a snowball system and filled in their questionnaires at home. (For a more detailed description of the sample and design, see Steyer, Schwenkmezger, Eid, & Notz, 1991 [20].) The sample analyzed consists of those 503 among the 548 original subjects who delivered their questionnaires on all four occasions. Among some others, a mood state questionnaire, a mood trait questionnaire, and a daily hassles and uplifts scale were administered.
Variables. The mood state questionnaire MDBF (Steyer, Schwenkmezger, Notz & Eid, 1994 [21], 1997 [22]) consists of three mood state scales, only one of which, the well-being state scale, is analyzed in this example. The well-being state scale (WS) consists of four positively ("In this moment I feel ... well") and four negatively ("... not well") formulated items each of which has to be rated on a five point Likert scale. Two negatively formulated items were recoded and aggregated together with two positively formulated items to a first score () and the same procedure was applied to the other four items yielding a second (parallel) score (), both indicating the well-being state on occasion k of measurement.
The well-being trait scale (WT) has been adopted from the German version of the "Mood Survey" (Bohner, Schwarz, & Hormuth, 1989 [2]) originally developed by Underwood and Froming. It consists of nine positively ("most of the time I feel happy") or negatively ("I often feel blue") formulated items each of which has to be rated on a five point Likert scale, too. Negatively formulated items were recoded and aggregated together with the positively formulated items to one scale value for each of the four occasions of measurement. Finally, these four scale values were aggregated into two parallel forms by aggregating the scores of occasions one and three () as well as the scores of occasions two and four ().
The daily hassles and uplifts scale (HU) consists of 60 dichotomous items asking whether or not a daily hassle ("I missed a bus or a train") or a daily uplift ("I had a good conversation") occured. Two parallel scales ( and ), each consisting of an equal number of uplifts and hassles, were constructed for each of the four occasions k of measurement such that a high score indicates few hassles and/or many uplifts. The 60 items of this scale were extracted from the German translation of the Lazarus and Cohen (1977) [4] scales published by Filipp, Ahammer, Angleitner, and Olbrich (1980) [3]. (For a complete documentation of all questionnaires mentioned see Steyer et al., 1991 [20]). Table 1 displays the correlations, standard deviations, and means of these variables for all four occasions of measurement.
Table 1:
Means, standard deviations, and correlations of the observables.
WS_{11} | WS_{21} | WS_{12} | WS_{22} | WS_{13} | WS_{23} | WS_{14} | WS_{24} | WT_{1} | WT_{2} | HU_{11} | HU_{21} | HU_{12} | HU_{22} | HU_{13} | HU_{23} | HU_{14} | HU_{24} | |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Mean | 3.751 | 3.649 | 3.773 | 3.715 | 3.753 | 3.669 | 3.871 | 3.773 | 3.436 | 3.416 | 3.496 | 3.532 | 3.487 | 3.479 | 3.501 | 3.418 | 3.483 | 3.515 |
Std | 0.878 | 0.918 | 0.911 | 0.939 | 0.922 | 0.915 | 0.866 | 0.880 | 0.750 | 0.752 | 0.411 | 0.530 | 0.456 | 0.516 | 0.465 | 0.521 | 0.461 | 0.527 |
WS_{11} | 1.0000 | |||||||||||||||||
WS_{21} | .8191 | 1.0000 | ||||||||||||||||
WS_{12} | .2197 | .2272 | 1.0000 | |||||||||||||||
WS_{22} | .2235 | .2864 | .8907 | 1.0000 | ||||||||||||||
WS_{13} | .2303 | .2209 | .2777 | .2893 | 1.0000 | |||||||||||||
WS_{23} | .2484 | .2865 | .3007 | .3468 | .8895 | 1.0000 | ||||||||||||
WS_{14} | .2579 | .2182 | .3393 | .3293 | .3338 | .3571 | 1.0000 | |||||||||||
WS_{24} | .2304 | .2438 | .3233 | .3501 | .3664 | .4279 | .8772 | 1.0000 | ||||||||||
WT_{1} | .3851 | .4648 | .3342 | .3728 | .3602 | .4302 | .3350 | .4189 | 1.0000 | |||||||||
WT_{2} | .3480 | .4209 | .3830 | .4313 | .3628 | .4290 | .3979 | .4749 | .9285 | 1.0000 | ||||||||
HU_{11} | .2845 | .3302 | .1393 | .1574 | .1374 | .1533 | .1774 | .1867 | .2889 | .2598 | 1.0000 | |||||||
HU_{21} | .4021 | .3790 | .1393 | .1397 | .2094 | .2396 | .2042 | .1729 | .2969 | .2802 | .4724 | 1.0000 | ||||||
HU_{12} | .1371 | .1432 | .3694 | .3820 | .1388 | .1691 | .2104 | .1761 | .2631 | .2929 | .3625 | .3613 | 1.0000 | |||||
HU_{22} | .1565 | .1543 | .4310 | .4163 | .1774 | .1942 | .2014 | .1643 | .2533 | .2866 | .3099 | .4011 | .5358 | 1.0000 | ||||
HU_{13} | .1380 | .1369 | .1834 | .1842 | .3471 | .3611 | .2491 | .2426 | .2808 | .2884 | .3354 | .3058 | .3648 | .2514 | 1.0000 | |||
HU_{23} | .1482 | .1298 | .1929 | .1877 | .4623 | .4392 | .2936 | .2790 | .3093 | .3192 | .2348 | .3943 | .2899 | .4072 | .5009 | 1.0000 | ||
HU_{14} | .0991 | .1242 | .1579 | .1704 | .1452 | .2045 | .3896 | .4025 | .3013 | .3129 | .3111 | .2621 | .3459 | .2809 | .4006 | .3391 | 1.0000 | |
HU_{24} | .1315 | .1281 | .1939 | .1731 | .1793 | .2244 | .4360 | .4306 | .2601 | .2905 | .2220 | .3123 | .2642 | .3443 | .3253 | .3755 | .5797 | 1.0000 |
Model 1. Figure 3 displays a TIC model with across time correlations among the error variables pertaining to the same "parallel" form. All loadings are fixed to one. The measurement error variances are .14 for the errors pertaining to and and .09 for the other six measurement error variables. (There are equality constraints.) The variances and correlations of the latent variables are shown in Table 2. WS denotes the true well-being state variable at occasion 1, whereas WS is an abbreviation for the difference WSWS between the true well-being state variables WS and WS. The other latent variables such as WS and WS are defined correspondingly. Allowing for correlated measurement errors considerably increases the fit of the model. The -difference is 107.22-23.93 = 83.29 with 27-25=2 degrees of freedom.
Table 2: Variances and correlations of the latent variables for Model 1
WS_{1} | WS_{2-1} | WS_{3-2} | WS_{4-3} | |
---|---|---|---|---|
Var | 0.661 | 1.055 | 0.999 | 0.831 |
WS_{1} | 1.000 | |||
WS_{2-1} | -.573 | 1.000 | ||
WS_{3-2} | .000 | -.494 | 1.000 | |
WS_{4-3} | .000 | .000 | -.495 | 1.000 |
A caveat concerns the interpretation of the latent variables. As soon as there are correlated measurement errors, it is doubtful if the variables WS should still be interpreted as true score variables. However, if in fact the residuals of the observables are interpreted as measurement error variables, then the latent variables such as WS must be interpreted as the difference between the true score variables WS and WS. Note that this conclusion does not rely on any plausibility argument. It is also not based on the correlation structure of the latent variables. Instead this conclusion is a logical derivation from the equations defining the TIC model.
Figure 3: A single construct true intraindividual change model for two
measures on each of four occasions of measurement. The measurement errors are
allowed to correlate across time. However, their correlation structure is
restricted by and
for all All paths are fixed to one. Goodness of fit statistics:
with 25 degrees of freedom: = 23.93 (p = 0.52),
RMSEA = 0.0, adjusted goodness of fit index AGFI = 0.98.
Model 2. Which are important correlates of the true intraindividual change variables? Model 2 gives an answer. This model consists of:
Figure 4 describes the correlation structure between the true score change variables and the effect of the well-being trait (WT) on the true well-being state at occasion 1 (WS) and on the true daily hassle state at occasion 1 (HU). The estimated correlations are given in Table 3. Whereas there is a considerable effect of the well-being trait on the well-being state and on the daily hassle state on occasion 1, there is neither an effect of the well-being trait on the true change variables, nor are there correlations between nonneighboured true change variables. In fact, the corresponding effects and correlations could be fixed to be zero. (The -difference between the restricted and the nonrestricted model is 225.00-214.07 = 10.93 with 135-117=18 degrees of freedom.)
The correlations between the true change variables within each occasion of measurement range from .538 to .630. This shows that the true change in daily hassles is in fact an important correlate of the true change in the state of well-being.
Table 3: Variances and correlations of the latent variables for Model 1
WS_{1} | WS_{2-1} | WS_{3-2} | WS_{4-3} | HU_{1} | HU_{2-1} | HU_{3-2} | HU_{4-3} | WT | |
---|---|---|---|---|---|---|---|---|---|
Var | 0.656 | 1.055 | 0.997 | 0.835 | 0.103 | 0.092 | 0.111 | 0.103 | 0.524 |
WS_{1} | 1.000 | ||||||||
WS_{2-1} | -.568 | 1.000 | |||||||
WS_{3-2} | .000 | -.497 | 1.000 | ||||||
WS_{4-3} | .000 | .000 | -.499 | 1.000 | |||||
HU_{1} | .553 | -.216 | .000 | .000 | 1.000 | ||||
HU_{2-1} | -.358 | .630 | -.357 | .000 | -.398 | 1.000 | |||
HU_{3-2} | .000 | -.286 | .588 | -.214 | .000 | -.524 | 1.000 | ||
HU_{4-3} | .000 | .000 | -.285 | .538 | .000 | .000 | -.402 | 1.000 | |
WT | .454 | .000 | .000 | .000 | .438 | .000 | .000 | .000 | 1.000 |
One might also raise the question whether or not it is meaningful to introduce a true score variable for a daily hassles and uplifts scale. How to interpret such a variable? Although a careful discussion of the pros and cons of this procedure is not the focus of this paper, note that the number of daily hassles and uplifts reported in each of the two "parallel" forms is certainly error prone to a certain extent. Hence it seems better to filter out the measurement error by introducing a true score variable. However, there might be better ways to model the daily hassles and uplifts scales.
Figure 4: The structural model for Model 2. Goodness of fit statistics:
with 135 degrees of freedom: 225.00 ( 0.01), RMSEA =
0.036, adjusted goodness of fit index (AGFI) = 0.94.