We use the methodology described in Sections 3-5 to analyze the association of longitudinal quit status and weight change in the CTQ II clinical trial described in Section 2. We removed the observations at week 1 and 2 from the data, because the quit rates in these two weeks were very low (0 and 1.10%) due to the design of the study in which subjects were not supposed to try to quit smoking until week 3. Although participants were encouraged to make up missed sessions, there still existed intermittent missing values in quit status and/or weight gain. In addition, a large number of subjects dropped out before the end of the experiment. In what follows, we assume this missingness is ignorable.
For the mean of the two longitudinal responses as a function of covariates, we set β
to be the vector of means at each time point across treatments; thus, the t
th row of the design matrix is a vector of 0’s, with a 1 in the t
th slot. Exploratory analysis suggested setting a0
to be 4; this implies a slower decrease than the raw lag.
6.1 Comparing the Competing Models
We considered several models arising from restrictions on the association matrixes Bi
, the prior for the association matrices Bi
, the correlation matrices Ri,11
and conditional covariance matrices
. We fit a total of nine models to the CTQ II trial data. Denote the k
th alternative model by Mk
. gives the details of all of the models considered.
For each model in , we computed the DIC using the methodology derived in Section 5. The DICs for all of the models are given in . From , we can see that the DIC is the smallest for model M5 and the largest for model M3. In general, the models using shrinkage priors for association matrices fit better.
6.2 Checking the Goodness of Fit
For the CTQ II data, we defined the discrepancy function to be weekly quit rates or weekly average weight gains. gives the cppp for quit rates and average weight gains at each week across treatments for model 5 (M5). These p values were calculated based on comparison of 2,000 pairs of [ppp(u), ppp(uobs)], with the ppp obtained using 5,000 iterations after burn-in. From this table, we can see that there are no extreme p values (<.05 or >.95). These checks suggest that the joint models (M5) fit the mean structure of the CTQ II clinical trial data well.
Calibrated posterior predictive p value for quit rate (QR) and average weight gain (AWG) at each week across treatments for model 5 (M5)
6.3 Inference on the Quit Rates and Association
Given the results presented in Sections 6.1 and 6.2, we base inference on model M5 (). We ran the MCMC algorithm described in Section 4.1 until convergence (determined by examining trace plots of multiple chains) and based inference on the last 10,000 iterations after burn-in.
Posterior means and 95% credible intervals (CIs) for quit rates across treatments are given in . The quit rates over time were slightly lower in the exercise treatment than in the wellness treatment. The 95% CI for the difference of quit rates between two treatments at the final week was (-.014, .150), marginally significant. This suggests that the exercise treatment does not have a positive effect on smoking cessation (more later).
Posterior means and 95% CIs for quit rates
We now turn our attention to the association between smoking cessation and weight gain. The pi
= 1, 2) defined in (9)
can be viewed as a summary measure of the overall magnitude of the association between quit status and weight gain for the two treatments. The estimates of pi
= .26 (no exercise) and p2
= .18 (exercise); the 95% CI for their difference is (.025, .134). These results support the hypothesis that exercise weakens the association between quit status and weight gain.
This weakening also can be seen by examining the posterior means of association matrixes across treatments, as given in . We have removed from the table those elements of the Bi matrix with probabilities of the corresponding indicators being equal to 1 of <.1. The weakened associations between smoking cessation and weight gain is obvious by noting the presence of more 0’s under the exercise treatment and the larger magnitude of the (standardized) coefficients.
Posterior means of the association matrices with 95% CIs for each treatment
shows the posterior means of pairwise correlations with 95% credible intervals; correlations whose 95% credible intervals covered 0 are excluded. We can see that smoking cessation and weight gain appear to have a lagged correlation structure, and that exercise weakens pairwise correlations. In particular, we point out the 2 × 2 blocks in the upper right corners of pairwise correlation matrixes under both treatments (in bold-type). For the wellness treatment, the four pairwise correlations between weight gain at the beginning of the study and quit status at weeks 5 and 6 are all negative. This means that people who gain weight early in the trial are more likely to be smoking at the end. The corresponding correlations are essentially 0’s (no longer significant) under the exercise arm. In addition, looking at the last row in for the wellness arm, the correlations indicate those who quit early in the study are more likely to gain weight by the end (week 6); the corresponding relationship in the exercise arm is weaker.
Posterior means of pairwise correlations with 95% CIs for each treatment