EL SENTIDO DEL ERASMISMO.

Having estimated a model which describes the shape of the population average growth trajectory, and the degree of individual heterogeneity around this group mean, the obvious next step for the substantive analyst is to examine predictors of individual change over time. The way in which predictors of individual change are incorporated depends on whether the predictor is time- constant or time-varying.

Within the SEM framework, the inclusion of time-constant covariates is straightforward; we simply incorporate the covariates of interest into the model and estimate regression paths running from the exogenous predictors to the latent growth parameters. Figure 6.2 show a path diagram for a conditional growth curve model with a time-constant predictor for the example in Section 6.5, where more details about the variables are given. Because the slope growth parameter is now specified as endogenous to the time-constant covariates, this becomes a conditional growth curve model. A regression path running from the time-constant covariate to the intercept growth parameter denotes the expected change in the mean of the developmental variable at the first wave of measurement, for each unit increase in the fixed covariate. Note that the interpretation of the mean of the intercept growth factor depends on the coding of time for the slope growth parameter. It is standard to constrain the first factor loading of the slope parameter to zero, making the mean of the intercept latent variable equivalent to the mean of the trajectory at the first wave of measurement. A regression path from a time-constant covariate to the slope latent variable is interpreted as the expected change in the mean rate of change, for

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each unit increase in the covariate. Time-constant covariates in the LGC framework can be specified as simple observed variables or as latent variables with multiple indicators. Standard z- tests can be used to determine the significance of the regression paths, although the likelihood ratio test can also be used by comparing the model with these parameters estimated, to a model in which they are constrained to zero. The full range of standard SEM indices of model fit can be used to assess the empirical adequacy of the full structural model.

Figure 6.2: Predicting change in economic perceptions by 1992 vote choice.

There are two main ways to incorporate a time-varying covariate into the model. The first approach is to regress the repeated measures on the observed time-varying covariate at each time point, as shown in Figure 6.3, where

Y

denotes the response variable and

x

the time-varying covariate. When the time-varying variables are allowed to covary with the growth parameters, the parameter estimates for the effects of the time-varying variable on the repeated response are equivalent to those from the fixed effects model briefly discussed in Section 5 (Allison, 2005). By using this model specification, we control for unobserved unit heterogeneity but that we cannot

Intercept Slope Slp1 1 0 Slp2 1 ₁ Slp3 1 Slp4 1 3 Slp5 1 4 1

e

e

₂

e

3

e

4

e

5 2 Conservative in 1992 0.323* _0.046*

d

₂ 1

d

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use observed time-constant variables to model the repeated measured directly (Halaby 2004). However, as shown in Figure 6.2, the effect of time-constant variables can be estimated indirectly via their effect on the growth parameters. As with time-constant predictors of growth, time- varying predictors can be specified as directly observed single indicators or as multiple indicator latent variables. The basic model specification in Figure 6.3 can easily be extended to incorporate multiple time-varying predictors and to include a combination of time-varying and time-constant predictors of growth.

Figure 6.3: Conditional 4 wave LGC with time a varying covariate.

Constraining the covariance between time-varying covariates and the growth parameters to zero yields the random effects specification, in which time-constant variables can be used to model the repeated measures directly, as opposed to indirectly via their influence on the growth parameters. Whether the fixed or random effects model should be preferred can be evaluated via the likelihood ratio test, as the fixed effects model is nested within the random effects model.

Intercept Slope 1

Y

Y

2

Y

3

Y

4 1

e

e

2

e

₃

e

2 2 1 1 1 1 0 1 3 1

x

x

₂

x

3

x

4

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A second way of incorporating the effect of time-varying predictors into the model is to simultaneously estimate two or more growth trajectories, the so-called multiple process model. The univariate LGC model may be extended to include the simultaneous estimation of two or more growth trajectories, with correlations and regressions estimated between the slope and intercept growth parameters of each process. For instance, we can estimate the correlation between two slope parameters to examine whether change in one variable is related to change in a second variable over the same time period. Alternatively, we might regress the slope latent variable of one process on to the slope latent variable of a second process. This yields a conditional slope parameter, which can be interpreted in a similar way to the time-varying covariate model in Figure 6.3. Similarly, we might regress the slope factors on to the intercept parameters to test whether rate of change is a function of initial status on the same, or a different variable (Duncan et al. 1999).

Whether the analyst prefers the first or the second of these approaches to modelling time-varying predictors of growth will depend on the specific context of the particular substantive analysis. In general, however, the time-varying covariate approach may be favoured due there being no assumptions required regarding the shape of the trajectory. On the other hand, the multiple process model yields a potentially richer picture of growth on all time-varying variables and allows an assessment of covariance between intra-individual change for these variables.

In this section we have presented a basic overview of some simple LGC models. However, it is important to note that the main benefit of the LGC framework is not to be found in this type of model, for these may generally be equally well estimated within the random effects frameworks. The real benefits of the LGC framework are to be found in the extensions of these basic models to incorporate, inter alia, the use of multiple indicator measurement models with tests of factorial invariance, growth mixture models to examine qualitatively different growth trajectories across population sub-groups (where group membership is unknown), multiple process models with multiple indicators of the repeated observations, modelling distal outcomes with growth parameters, and auto-regressive latent trajectory models which integrate the autoregressive and

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LGC approaches into a single framework. Interested readers are directed toward the excellent book by Bollen and Curren (2006) for an introduction to these more advanced techniques.

In document Joaquím Xirau : una filosofía de ultimidades (página 30-35)