Alternative Non-Linear Growth Models for Longitudinal Prevention Trials

Randomized controlled trials conducting long-term assessments with repeatedly measured outcomes are widely considered to be the strongest preventive intervention evaluation designs, partially because they allow the examination of putative outcomes in growth trajectories and intervention effects on those trajectories. Two distinct, but interrelated questions associated with the analysis of data from such trials are: “What statistical model best fits the observed data?” and “What is the optimal interpretation of estimated model parameters?”

In preventive intervention research, there is a substantial body of literature discussing polynomial growth trajectories (for example, latent growth curve models with linear or quadratic growth patterns) for data with repeated measurements. Data from our three large-scale trials, however, show evidence of non-linear growth patterns that may not be well described by a polynomial function, such as the S-shaped curves for lifetime drunkenness and marijuana use (Spoth et al., 2004, 2006). Also, a clear interpretation of the substantive or practical meaning of model parameters with higher-order polynomial functions frequently is lacking (Cudeck & Toit, 2002). 

In this presentation, alternative growth models that are better-known in other areas of research, such as business, will be introduced and discussed. These models not only are well-suited to addressing S-shaped growth trajectories, but also provide more readily interpretable results. The application of these models to substance initiation will be addressed. 

To illustrate, Dodd (1955) introduced a basic probability diffusion model as

P_t+1 = gP_t(1-P_t) + P_t ------ (1)

In this model (1), each period’s incremental increase in prevalence (e.g., frequency of use of a given substance) is proportional (g) to the product of the previous period’s user and non-user rates, P(1-P). This model is based on endogenous influences (within population) on growth.

In contrast to the basic diffusion model (1), an alternative model (Lekvall & Wahlbin, 1973) that considers only non-users can be defined as

P_t+1 = k(1-P_t) + P_t ------ (2)

In model (2), each period’s incremental increase is only proportional (k)  to the previous non-user rate, (1-P). This model (2) is based on exogenous influences on growth.

A model (3) combining models (1) and (2) suggested by Bass (1969) and Phillips (2007) will be applied with data from our three trial datasets. The presentation will discuss model fits as well as the proper and practical interpretations of the parameters in model (3).

P_t+1 = gP_t(1-P_t) + k(1-P_t) + P_t ------ (3)