Analysing the rate of change in a longitudinal study with missing data, taking into account the number of contact attempts

In longitudinal and multivariate settings incomplete data, due to missed visits, dropouts or non-return of questionnaires are quite common. A longitudinal trial in which potentially informative missingness occurs is the Collaborative Ankle Support Trial (CAST). The aim of this study is to estimate the clinical effectiveness of four different methods of mechanical support after severe ankle sprain. The clinical status of multiple subjects was measured at four points in time via a questionnaire and, based on this, a continuous and bounded outcome score was calculated. Motivated by this study, a model is proposed for continuous longitudinal data with non-ignorable or informative missingness, taking into account the number of attempts made to contact initial non-responders. The model combines a non-linear mixed model for the underlying response model with a logistic regression model for the reminder process. The outcome model enables us to analyze the rate of improvement including the dependence on explanatory variables. The non-linear mixed model is derived under the assumption that the rate of improvement in a given time interval is proportional to the current score and the still achievable score. Based on this assumption a differential equation is solved in order to obtain the model of interest. The response model relates the probability of response at each contact attempt and point in time to covariates and to observed and missing outcomes. Using this model the impact of missingness on the rate of improvement is evaluated for different missingness processes.