R less steep rates of change over time (e.g., Curran, Bauer Willoughby, 2004). Importantly, TICs are assumed to be independent of the passage of time. In other words, the given value of the TIC could in principle be assessed at any time point as this is constant over time. This assumption is sometimes strictly true (e.g., biological sex, country of origin), and at other times, the construct might in principle vary with time but is only assessed at a single time period (e.g., baseline anxiety or initial reaction time). However, growth models can easily be expanded to include the effects of covariates that do vary as a function of time; these are TVC models (Bollen Curran, 2006, pp. 192?98; Raudenbush Bryk, 2002, pp. 179?86; Singer Willett, 2003, pp. 159?88). Whereas TICs directly predict the growth factors (e.g., Bollen Curran, 2006, Figure 5.1), TVCs directly predict the repeated measures while controlling for the influence of the growth factors (e.g., Bollen Curran, 2006, Figure 7.1). Thus, any given repeated measure is jointly determined by the underlying growth factors and the impact of the TVC at that time period. The TVC model can then be expanded to include interactions between the TVCs and time (to T0901317 cancer Thonzonium (bromide) structure assess differences in the magnitude of the TVC effect as a function of time) and interactions between the TVCs and the TICs (to assess differences in the magnitude of the TVC effect as a function of betweenperson characteristics such as gender or ethnicity). Taken together, models can be constructed that simultaneously evaluate within-person influences (via TVCs) and betweenperson influences (via TICs) on stability and change of the outcome over time.NIH-PA Author Manuscript NIH-PA Author Manuscript NIH-PA Author ManuscriptCAN GROWTH IN TWO CONSTRUCTS BE SIMULTANEOUSLY MODELED OVER TIME?Although the TVC model allows for covariates to change in value over time, it is assumed that the covariates themselves are not characterized by a systematic growth process. For example, say that the repeated outcome was reading ability, and the TVC was number of days of instruction that were missed in a given academic year. It would be reasonable to assume that days of instruction may influence reading ability at a given time point but that there is not a systematic growth process underlying days of instruction missed (that is, children would not be expected to show consistent developmental trends in days missed). However, say instead that the outcome was again reading ability, but the TVC is substance use; in this case, developmental theory would predict that the onset and escalation of substance use itself is characterized by some type of systematic growth function. If so, then the TVC model may be mis-specified and result in biased effects. Both the multilevel and SEM growth frameworks can be expanded to allow for the simultaneous growth of two constructs over time, and this is commonly called a multivariate growth model (Bollen Curran, 2006, chap. 7; MacCallum, Kim, Malarkey, KiecoltGlaser, 1997; McArdle, 1988). Each construct can be characterized by a unique functional form (e.g., one may be linear, the other quadratic), and their relation is examined at the level of the growth factors (e.g., direct estimates of the relation between the intercepts and slopes within and across construct). Finally, these multivariate models can themselves be extendedJ Cogn Dev. Author manuscript; available in PMC 2011 July 7.Curran et al.Pageto inclu.R less steep rates of change over time (e.g., Curran, Bauer Willoughby, 2004). Importantly, TICs are assumed to be independent of the passage of time. In other words, the given value of the TIC could in principle be assessed at any time point as this is constant over time. This assumption is sometimes strictly true (e.g., biological sex, country of origin), and at other times, the construct might in principle vary with time but is only assessed at a single time period (e.g., baseline anxiety or initial reaction time). However, growth models can easily be expanded to include the effects of covariates that do vary as a function of time; these are TVC models (Bollen Curran, 2006, pp. 192?98; Raudenbush Bryk, 2002, pp. 179?86; Singer Willett, 2003, pp. 159?88). Whereas TICs directly predict the growth factors (e.g., Bollen Curran, 2006, Figure 5.1), TVCs directly predict the repeated measures while controlling for the influence of the growth factors (e.g., Bollen Curran, 2006, Figure 7.1). Thus, any given repeated measure is jointly determined by the underlying growth factors and the impact of the TVC at that time period. The TVC model can then be expanded to include interactions between the TVCs and time (to assess differences in the magnitude of the TVC effect as a function of time) and interactions between the TVCs and the TICs (to assess differences in the magnitude of the TVC effect as a function of betweenperson characteristics such as gender or ethnicity). Taken together, models can be constructed that simultaneously evaluate within-person influences (via TVCs) and betweenperson influences (via TICs) on stability and change of the outcome over time.NIH-PA Author Manuscript NIH-PA Author Manuscript NIH-PA Author ManuscriptCAN GROWTH IN TWO CONSTRUCTS BE SIMULTANEOUSLY MODELED OVER TIME?Although the TVC model allows for covariates to change in value over time, it is assumed that the covariates themselves are not characterized by a systematic growth process. For example, say that the repeated outcome was reading ability, and the TVC was number of days of instruction that were missed in a given academic year. It would be reasonable to assume that days of instruction may influence reading ability at a given time point but that there is not a systematic growth process underlying days of instruction missed (that is, children would not be expected to show consistent developmental trends in days missed). However, say instead that the outcome was again reading ability, but the TVC is substance use; in this case, developmental theory would predict that the onset and escalation of substance use itself is characterized by some type of systematic growth function. If so, then the TVC model may be mis-specified and result in biased effects. Both the multilevel and SEM growth frameworks can be expanded to allow for the simultaneous growth of two constructs over time, and this is commonly called a multivariate growth model (Bollen Curran, 2006, chap. 7; MacCallum, Kim, Malarkey, KiecoltGlaser, 1997; McArdle, 1988). Each construct can be characterized by a unique functional form (e.g., one may be linear, the other quadratic), and their relation is examined at the level of the growth factors (e.g., direct estimates of the relation between the intercepts and slopes within and across construct). Finally, these multivariate models can themselves be extendedJ Cogn Dev. Author manuscript; available in PMC 2011 July 7.Curran et al.Pageto inclu.