De one or more TICs to predict the set of growth factors. There are several variations of the multivariate model that attempt to simultaneously examine bidirectional effects between two constructs both at the level of the growth trajectories and at the level of repeated measures. Two examples include the latent difference score model (McArdle, Ferrer-Caja, Hamagami, Woodcock, 2002) and the autoregressive latent trajectory model (Bollen Curran, 2004; Curran Bollen, 2001), although several other approaches exist as well. The systematic study of the bidirectional relation between two or more constructs is a topic of much ongoing research, so we can expect additional multivariate methods to become available soon.NIH-PA Author Manuscript NIH-PA Author Manuscript NIH-PA Author ManuscriptCAN GROWTH MODELS BE SIMULTANEOUSLY ESTIMATED WITHIN TWO OR MORE GROUPS?It is important to realize that when estimating the growth models described thus far, strong assumptions are made about the equivalence of the model parameters across all individuals within the sample (e.g., Bollen Curran, 2006, chap. 6). As a simple example, consider fitting a model to data that consist of responses from males and females. If an unconditional growth model is 4-Deoxyuridine msds fitted to the pooled sample (i.e., the usual single-group analysis), it is explicitly assumed that all of the parameters that define the growth model are precisely equal for both CPI-455 site gender groups. If gender differences were hypothesized, the growth model can easily be expanded to include gender as a time-invariant predictor; however, this only introduces differences in the conditional means of the growth factors (e.g., on average, males may start higher or lower compared with females and increase more or less steeply). Gender thus serves to shift the conditional means of the intercept and slope to higher or lower values, yet all other parameters that govern the model are assumed to be equal between the two groups. Whereas in many situations these assumptions are perfectly reasonable, in others, they may be distinctly questionable. For example, a potential outcome of a treatment intervention is to decrease variability in the expression of certain behaviors within the treatment group but not the control group over time (e.g., an intervention designed to decrease antisocial behavior in preschool children will also likely decrease the variability of types of disruptive behavior in the children exposed to the treatment). If these estimates of variability are markedly different across groups, yet a model is fitted that assumes these to be the same, then biased parameter estimates are expected. Both the SEM and multilevel approaches address this issue through the simultaneous estimation of growth models across two or more groups in what are called multiple-groups growth models. If all model parameters are set equal across all groups, this is equivalent to estimating a single-group growth model. Alternatively, if all parameters are allowed to freely vary across all groups, this is equivalent to estimating a growth model within each group separately. The typical application will fall somewhere between these two extremes in which some parameters are equated and others are not. This framework provides yet another option for maximally understanding growth processes both within and across groups.WHAT IF THERE IS A POTENTIALLY IMPORTANT GROUPING VARIABLE THAT WAS NOT DIRECTLY OBSERVED?In the multiple-groups growth model descri.De one or more TICs to predict the set of growth factors. There are several variations of the multivariate model that attempt to simultaneously examine bidirectional effects between two constructs both at the level of the growth trajectories and at the level of repeated measures. Two examples include the latent difference score model (McArdle, Ferrer-Caja, Hamagami, Woodcock, 2002) and the autoregressive latent trajectory model (Bollen Curran, 2004; Curran Bollen, 2001), although several other approaches exist as well. The systematic study of the bidirectional relation between two or more constructs is a topic of much ongoing research, so we can expect additional multivariate methods to become available soon.NIH-PA Author Manuscript NIH-PA Author Manuscript NIH-PA Author ManuscriptCAN GROWTH MODELS BE SIMULTANEOUSLY ESTIMATED WITHIN TWO OR MORE GROUPS?It is important to realize that when estimating the growth models described thus far, strong assumptions are made about the equivalence of the model parameters across all individuals within the sample (e.g., Bollen Curran, 2006, chap. 6). As a simple example, consider fitting a model to data that consist of responses from males and females. If an unconditional growth model is fitted to the pooled sample (i.e., the usual single-group analysis), it is explicitly assumed that all of the parameters that define the growth model are precisely equal for both gender groups. If gender differences were hypothesized, the growth model can easily be expanded to include gender as a time-invariant predictor; however, this only introduces differences in the conditional means of the growth factors (e.g., on average, males may start higher or lower compared with females and increase more or less steeply). Gender thus serves to shift the conditional means of the intercept and slope to higher or lower values, yet all other parameters that govern the model are assumed to be equal between the two groups. Whereas in many situations these assumptions are perfectly reasonable, in others, they may be distinctly questionable. For example, a potential outcome of a treatment intervention is to decrease variability in the expression of certain behaviors within the treatment group but not the control group over time (e.g., an intervention designed to decrease antisocial behavior in preschool children will also likely decrease the variability of types of disruptive behavior in the children exposed to the treatment). If these estimates of variability are markedly different across groups, yet a model is fitted that assumes these to be the same, then biased parameter estimates are expected. Both the SEM and multilevel approaches address this issue through the simultaneous estimation of growth models across two or more groups in what are called multiple-groups growth models. If all model parameters are set equal across all groups, this is equivalent to estimating a single-group growth model. Alternatively, if all parameters are allowed to freely vary across all groups, this is equivalent to estimating a growth model within each group separately. The typical application will fall somewhere between these two extremes in which some parameters are equated and others are not. This framework provides yet another option for maximally understanding growth processes both within and across groups.WHAT IF THERE IS A POTENTIALLY IMPORTANT GROUPING VARIABLE THAT WAS NOT DIRECTLY OBSERVED?In the multiple-groups growth model descri.