Examining Stability Autoregressive Models

Autoregressive models are used to examine the stability of the relative standing of individuals over time. Figure 21.1 illustrates an autoregressive model for a three-wave data set. In this data set (Biesanz, West, & Millevoi, 2004), 188 college students were assessed at weekly intervals on a measure of the personality trait of conscientiousness (Saucier & Ostendorf, 1999). According to Saucier and Ostendorf, conscientiousness is comprised of four closely related facets: orderliness, decisiveness, reliability, and industriousness. At each time period, we estimated the latent construct of conscientiousness. In the model presented in Figure 21.1, the factor loading of each facet was constrained to be equal over time so that the units of the latent construct would be the same at each measurement wave. Orderliness serves as the marker variable for the construct (A = 1). As for the other facets range from .62 to .67.

In the basic autoregressive model, the scores on the factor at Time t only affect the scores on the factor at Time t +1. If there is perfect stability in the rank order of the students on the factor from one time period to the next, then the correlation will be 1.0, whereas if there is no stability, then the correlation will be 0. In the present example, there is considerable stability in the conscientiousness factor: the unstandardized regression coefficients are .78 (correlation = .85) for Week 1 to Week 2 and .84 (correlation = .88) for Week 2 to Week 3. These stabilities greatly exceed the corresponding simple test-retest correlations of .63 and .65, respectively.

Multiindicator autoregressive models have two distinct advantages over simple test-retest correlations. First, the model partitions the variance associated with the four indicators (facets) at each time into variance associated with the factor of conscientiousness and residual variance so that the stability coefficients are not attenuated by measurement error. Second, part of the residual variance may be due to a systematic feature of the facet (uniqueness) that is not shared with the latent construct of conscientiousness. Correlating the uniquenesses over

'For example, Goldberg's (1992) measure of the Big Five personality traits explicitly instructs informants to rate the participant relative to others of the same age and gender.

FIGURE 21.1. Autoregressive model.

each pair of time periods removes any influence of the stability of these systematic components of the residual. Otherwise, the estimate of the stability for the conscientiousness factor would be confounded by these unique components associated with each of the facets.

We estimated three alternative models to illustrate features of the model depicted in Figure 21.1. First, we investigated the effect of correlat ing the uniquenesses. Model (a), which included the correlated uniquenesses, showed a substantially better fit to the data, ^2(40) = 35.1, ns, RMSEA = .00, than Model (b), in which the correlations between the uniquenesses are deleted, X2(52) = 500.2, p < .0001, RMSEA = .22). An RMSEA of .05 or less is typically taken as evidence of a close-fitting model. This result indicates that the correlated uniquenesses need to be included in the model. Second, we investigated the effect of constraining the factor loadings to be constant over time. Model (c), which is portrayed in Figure 21.1, also resulted in an acceptable fit to the data, *2(46) = 37.0, ns, RMSEA = .00. The difference in fit between Models (a) and (c) may be directly compared based on their respective %2 and df values using the likelihood ratio test (Bender & Bonett, 1980), %2{b) = 1.9, ns. Given that the fit of the two models to the data does not differ, Model (c) is preferred both because it has fewer parameters (parsimony) and more importantly, because it simplifies interpretation by guaranteeing that the conscientiousness construct has the same units at each measurement wave.

Cross-lagged autoregressive models may be used to investigate the ability of one longitudinal series to predict another series. For example, Aneshensel, Frerichs, and Huba (1984) measured several indicators of illness and several indicators of depression every 4 months. The two constructs were modeled as latent factors. Moderate stabilities were found for both the illness and depression constructs. The level of depression at Wave t consistently predicted the level of illness at Wave t + 1, over and above the level of illness at the Wave t. In a similar study, Finch (1998) found that social undermining consistently predicted negative affect 1 week later over and above the level of negative affect the previous week. Such lagged effects show both association and temporal precedence, providing support for hypothesized direction of the causal relationship between the two variables (e.g., depression —» physical illness). Joreskog (1979) and Dwyer (1983) presented several useful variants of the basic autoregressive model for longitudinal data. Of importance, clear interpretation of the findings of these models assumes there is not systematic change in the level of the series of measures (growth or decline) for each individual over time (Willett, 1988). Curran and Bollen (2001) and McArdle (2001) have proposed models that combine growth and autoregressive components to address this issue.

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