Using a shared parameter mixture model to estimate change during treatment when termination is related to recovery speed

It is important for clinicians, patients, policy-makers, and researchers to know how much time in therapy patients need in order to accrue clinically significant benefits. Such information is essential for planning and allocating resources, and it can be used as a benchmark against which an individual patient’s progress is tracked (cf. Finch, Lambert, & Schaalje, 2001). Characterizing the typical rate of improvement in response to a particular dose of therapy is complicated in naturalistic settings in which the dose of therapy is determined by the patient and therapist rather than by a researcher or other independent influence (Feaster, Newman, & Rice, 2003). In naturalistic settings, the amount of time that one spends in therapy is inversely related to speed of recovery; patients who recover most quickly leave therapy earliest (Baldwin, Berkeljon, Atkins, & Nielsen, 2009; Barkham, Connell, Stiles, Miles, Margison, Evans, et al. 2006). Moreover, data collection often only occurs when patients are in therapy. Consequently, we will have far more observations on patients who remain in therapy (i.e., those who change the slowest) than we will on patients who terminate. This type of missing patient data can be considered non-ignorable or non-random because the longitudinal process which governs change in psychological functioning is related to the process governing termination. This is patient data from sessions following termination are considered missing in situations for which it would be desirable to know what the symptomatology of a patient would have looked like had they continued therapy. In particular, this type of non-random missingness is random coefficient-dependent because the latent trajectory underlying an individual’s rate of change is directly related to missingness.1 In contrast, many common approaches to accommodating missing data, such as those implemented in latent growth and multilevel growth models, assume that missing data are missing at random (MAR; e.g., when maximum likelihood or multiple imputation is implemented) or missing completely at random (MCAR). MCAR is a subtype of MAR that is rarely observed in practice unless data are missing by design. Violating the assumption of random missingness can lead to biased results and inaccurate information about expected rates of change. In this manuscript we illustrate a method for incorporating information about random coefficient-dependent missing data into a growth model in order to obtain results that are directly comparable to those obtained using a standard growth model. We apply this model, referred to as a shared parameter mixture model (SPMM), to a naturalistic dataset where dose of therapy was not controlled by the researcher. In previous work, Gottfredson et al. (in press) presented technical details on the SPMM, along with results from a Monte Carlo study of the model’s relative performance under a variety of data conditions. Here, we minimize the presentation of technical details and instead aim to provide an accessible overview of the SPMM and an illustration of how it can be applied to strengthen research in clinical psychology. The outline of the manuscript is as follows. We begin by orienting the reader to our motivating data analysis problem – we desire to measure psychological change in a naturalistic study of psychotherapy treatment. Second, we briefly review the MAR assumption inherent in standard methods for modeling change and discuss how violations of this assumption lead to biased results in the context of a naturalistic treatment study. Third, we introduce the SPMM as a statistical method that is useful when individual differences in change (e.g., differences in rate of improvement during treatment) may be related to the presence, pattern, or amount of missing data. Fourth, we apply the SPMM to the naturalistic psychotherapy dataset and contrast the results with a traditional latent growth curve model. Finally, we discuss implications of our results for the study of naturalistic change during therapy.