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Next: Discussion Up: MPR-online 1997Vol.2, No.1 Previous: True intraindividual change as

An Example

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 (tex2html_wrap_inline659) and the same procedure was applied to the other four items yielding a second (parallel) score (tex2html_wrap_inline661), 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 (tex2html_wrap_inline667) as well as the scores of occasions two and four (tex2html_wrap_inline669).

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 (tex2html_wrap_inline673 and tex2html_wrap_inline675), 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.
WS11WS21WS12WS22WS13 WS23WS14WS24WT1WT2 HU11HU21HU12HU22HU13 HU23HU14HU24
Mean3.7513.6493.7733.715 3.7533.6693.8713.7733.436 3.4163.4963.5323.4873.479 3.5013.4183.4833.515
Std0.8780.9180.9110.939 0.9220.9150.8660.8800.750 0.7520.4110.5300.4560.516 0.4650.5210.4610.527
WS24.2304.2438.3233.3501.3664.4279.8772 1.0000
WT1.3851.4648.3342.3728.3602.4302.3350 .41891.0000
WT2.3480.4209.3830.4313.3628.4290.3979 .4749.92851.0000
HU11.2845.3302.1393.1574.1374.1533.1774 .1867.2889.25981.0000
HU21.4021.3790.1393.1397.2094.2396.2042 .1729.2969.2802.47241.0000
HU12.1371.1432.3694.3820.1388.1691.2104 .1761.2631.2929.3625.36131.0000
HU22.1565.1543.4310.4163.1774.1942.2014 .1643.2533.2866.3099.4011.53581.0000
HU13.1380.1369.1834.1842.3471.3611.2491 .2426.2808.2884.3354.3058.3648.25141.0000
HU23.1482.1298.1929.1877.4623.4392.2936 .2790.3093.3192.2348.3943.2899.4072.5009 1.0000
HU14.0991.1242.1579.1704.1452.2045.3896 .4025.3013.3129.3111.2621.3459.2809.4006 .33911.0000
HU24.1315.1281.1939.1731.1793.2244.4360 .4306.2601.2905.2220.3123.2642.3443.3253 .3755.57971.0000
Note. The labels of the variables are explained in the text.

Model 1. Figure 3 displays a TICtex2html_wrap_inline623 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 tex2html_wrap_inline681 and tex2html_wrap_inline683 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. WStex2html_wrap_inline609 denotes the true well-being state variable at occasion 1, whereas WStex2html_wrap_inline687 is an abbreviation for the difference WStex2html_wrap_inline689WStex2html_wrap_inline609 between the true well-being state variables WStex2html_wrap_inline623 and WStex2html_wrap_inline609. The other latent variables such as WStex2html_wrap_inline697 and WStex2html_wrap_inline699 are defined correspondingly. Allowing for correlated measurement errors considerably increases the fit of the model. The tex2html_wrap_inline701-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

WS1WS2-1 WS3-2WS4-3
Var0.6611.055 0.9990.831
WS2-1-.573 1.000
WS3-2.000 -.4941.000
WS4-3.000 .000-.4951.000
Note. The covariances of the error variables are estimated .00 for the errors pertaining to WS1k and WS1l and .04 for the errors pertaining to WS2k and WS2l. All zeroes in the table denoted .000 are also fixed to zero. According to Figure 3 (and Table 2), there are only nonzero correlations between neighboured latent variables, whereas there are no correlations between the latent variables WStex2html_wrap_inline609 and WStex2html_wrap_inline697, WStex2html_wrap_inline609 and WStex2html_wrap_inline699, and between WStex2html_wrap_inline687 and WStex2html_wrap_inline699. In fact, the corresponding correlations were fixed to zero without significant loss of fit. (The tex2html_wrap_inline701-difference is 23.93 - 23.39 = 0.54 with 25 - 22 = 3 degrees of freedom.) Note that WStex2html_wrap_inline725 WStex2html_wrap_inline689WStex2html_wrap_inline609, for instance, correlates with WStex2html_wrap_inline731 WStex2html_wrap_inline733WStex2html_wrap_inline623 because of the common component WStex2html_wrap_inline623. The same argument holds for the correlation of the difference variables WStex2html_wrap_inline697 and WStex2html_wrap_inline699 because of their common component WStex2html_wrap_inline743.

A caveat concerns the interpretation of the latent variables. As soon as there are correlated measurement errors, it is doubtful if the variables WStex2html_wrap_inline745 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 WStex2html_wrap_inline687 must be interpreted as the difference between the true score variables WStex2html_wrap_inline623 and WStex2html_wrap_inline609. 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 TICtex2html_wrap_inline623 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 tex2html_wrap_inline755 and tex2html_wrap_inline757 for all tex2html_wrap_inline759 All paths are fixed to one. Goodness of fit statistics: tex2html_wrap_inline701 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:

    a measurement model of the type discussed above for the well-being state variables tex2html_wrap_inline765,
    another measurement model of the same type for the daily hassles and uplifts scales tex2html_wrap_inline767,
    a third measurement model for the two well-being trait scales WTtex2html_wrap_inline609 and WTtex2html_wrap_inline623.
Again, for all three measurement models the loadings are all fixed to one and the same correlation structure of the measurement error variables as in Model 1 is assumed. The first measurement model for the variables WStex2html_wrap_inline773 is exactly the same as Model 1. Even the estimates of the variances of the measurement errors are the same: .14 for the first occasion of measurement and .09 for the other occasions. The second measurement model for the daily hassles and uplifts scales tex2html_wrap_inline767 is the same as the previous one, with the exception that the equality constraints now hold for those variances of the measurement errors that pertain to the same scale. Their estimates are .09 and .14, respectively. The third measurement model is a model of parallel tests, i.e., equal loadings (fixed to one) and equality constraints for the two measurement error variances. Their estimates are .04.

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 (WStex2html_wrap_inline609) and on the true daily hassle state at occasion 1 (HUtex2html_wrap_inline609). 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 tex2html_wrap_inline701-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

WS1WS2-1 WS3-2WS4-3 HU1HU2-1 HU3-2HU4-3 WT
Var0.6561.055 0.9970.8350.103 0.0920.1110.103 0.524
WS3-2.000-.497 1.000
WS4-3.000.000 -.4991.000
HU1.553-.216 .000.0001.000
HU2-1-.358.630 -.357.000-.398 1.000
HU3-2.000-.286 .588-.214.000 -.5241.000
HU4-3.000.000 -.285.538.000 .000-.4021.000
WT.454.000 .000.000.438 .000.000.000 1.000
Whereas a causal interpretation of the state-trait regression seems reasonable, there is no safe ground for explaining the true well-being change by the true change of the daily hassles, although such an interpretation might seem natural at first sight. The reason for being cautious with respect to such a causal interpretation is that the daily hassles are self-reported. Hence, it may very well be that the number of daily hassles reported is determined to some degree by the actual mood state.

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: tex2html_wrap_inline701 with 135 degrees of freedom: 225.00 (tex2html_wrap_inline789 0.01), RMSEA = 0.036, adjusted goodness of fit index (AGFI) = 0.94.

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Next: Discussion Up: MPR-online 1997Vol.2, No.1 Previous: True intraindividual change as

Methods of Psychological Research 1997 Vol.2 No.1
© 1997 Pabst Science Publishers