Appendix

Usually, the ALT model is identified by assuming (similar to a conventional AR model) that the variable at the first measurement point is an exogenous variable not influenced by the estimated trajectory factors or the other measurement points. However, this variable can be correlated with the latent intercept and slope parameters. Furthermore, rules of implementation require several steps (Bollen and Curran 2004): (1) multivariate AR (Fig. 3a), (2) LGMs (Fig. 3b), and (3) ALT (Fig. 3c) models. Although AR, LGM and ALT model were not nested, we compared the fit of the ALT model with that of a more parsimonious model in which the autoregressive and cross-lagged parameters were fixed to zero (Morin et al. 2011). This model allowed us to test the plausibility for the autoregressive and cross-lagged structure of the state-like deviations of both constructs (Bollen and Curran 2004). Then, following procedures recommended by Bollen and Curran (2004) we proceeded with fixing sequentially for each construct at a time by (1) zeroing the variance of the slope and (2) removing the slope. Once done, we proceed to (3) exclude correlated within time residuals, (4) constrain within-time residuals to equality, (5) constrain autoregressive paths to equality, (6) include lagged paths, and (7) constrain cross-lagged paths to equality. Tests 1–3 investigate the relevance of specific part of the model, and thus the worthiness to include them in the model. Finally, constraining cross-lagged paths to equality allowed us to identify if the reciprocal relations among the state-like deviations of both constructs remain the same over the 7-days or if they change at specific time point (e.g., during the week-end, Ryan et al. 2010).

Fig. 3 Diagrammatical representations of the autoregressive model (a) latent growth model (b), and of the autoregressive-latent-growth model (c). PA positive affect, POS positive orientation Full size image

To investigate the fit of different alternative models we followed procedures already presented in the paragraph “Model evaluation”. The results from the various multivariate models are reported in Table 2. The ALT model provided an adequate fit to the data and was preferable to simpler Multivariate AR and LGM models. This model can be further refined by taking out the slope factors from both PA and POS without significantly changing the overall fit of the model. Results also revealed that the inclusion of within-time residuals and autoregressive paths was necessary and that all regression parameters could be constrained to equality without worsening the fit of the model (Table 2, see results Models 9–14). Of importance, all parameters in the model may be constrained to be equal over time without significantly degrading the fit of the model (model 14).

Advantages of the ALT Model Over Classical ANOVA and Multiple Regression

The ALT model have a number of advantages over the standard repeated measures ANOVA or multiple regression designs. Moreover, it can be shown, under certain circumstances, that these latter design is a special case of the ALT model. In particular, although individual differences may be present in the ANOVA or multiple regression, change occurs at the group level; that is, if there is change, everyone is impacted in the same fashion. Moreover, trait and state effects are not represented in the classical ANOVA and multiple regression models. Furthermore, the random error variances are homogeneous (i.e., they are equal at every time of measurement) in the ANOVA. Other obvious merits of the ALT model are to allows one to (1) introduce antecedents and consequences of growth factors (i.e., intercepts and slopes), (2) to consider plausible explanations for individual states (by introducing predictors of observed variables), (3) to control parameter estimates for biases introduced by measurement error.