Latent mixed Markov modelling of smoking transitions using Monte Carlo bootstrapping

It has been established that measures and reports of smoking behaviours are subject to substantial measurement errors. Thus, the manifest Markov model which does not consider measurement error in observed responses may not be adequate to mathematically model changes in adolescent smoking behaviour over time. For this purpose we fit several Mixed Markov Latent Class (MMLC) models using data sets from two longitudinal panel studies—the third Waterloo Smoking Prevention study and the UWO smoking study, which have varying numbers of measurements on adolescent smoking behaviour. However, the conventional statistics used for testing goodness of fit of these models do not follow the theoretical chi-square distribution when there is data sparsity. The two data sets analysed had varying degrees of sparsity. This problem can be solved by estimating the proper distribution of fit measures using Monte Carlo bootstrap simulation. In this study, we showed that incorporating response uncertainty in smoking behaviour significantly improved the fit of a single Markov chain model. However, the single chain latent Markov model did not adequately fit the two data sets indicating that the smoking process was heterogeneous with regard to latent Markov chains. It was found that a higher percentage of students (except for never smokers) changed their smoking behaviours over time at the manifest level compared to the latent or true level. The smoking process generally accelerated with time. The students had a tendency to underreport their smoking behaviours while response uncertainty was estimated to be considerably less for the Waterloo smoking study which adopted the ‘bogus pipeline’ method for reducing measurement error while the UWO study did not. For the two-chain latent mixed Markov models, incorporating a ‘stayer’ chain to an unrestricted Markov chain led to a significant improvement in model fit for the UWO study only. For both data sets, the assumption for the existence of an independence chain did not lead to significant improvement in model fit. The unrestricted two-chain latent mixed Markov model led to a significant improvement of model fit compared to a simple latent Markov model, but this model was overparameterized when the latent transition probabilities and=or response probabilities were assumed nonstationary. For the other models, the manifest=latent transition probabilities and response probabilities (for the four-wave Waterloo study only) were tested to be nonstationary for both data sets.