Three observations were dropped as part of data quality control because they occurred either more than 1
year prior or 1
year after the date of randomization and almost certainly represented gross date coding errors. An additional observation was dropped due to a missing value on a baseline regressor. This left a total sample size of 999 participants.
Table provides the baseline characteristics of the participants in the pooled sample. Participants self-reported using the drug of dependence for about 15 out of the 30-days prior to baseline. Average age was ~40
years. African-Americans and Whites together constituted roughly 75% of the sample. 58% of participants were randomly assigned to an active study arm.
Baseline characteristics of pooled sample.
Table demonstrates that the observed sample means declined across the three successive assessments. Sample size also declined post-randomization. By the end of the active-intervention period, only 71% of baseline respondents provided assessments (n
705). Sample size dropped substantially by the third assessment because only two of the seven trials (modafinil–cocaine and tiagabine–cocaine) required a third assessment by protocol.
Table summarizes regression findings for joint modeling of non-response, drug-use subscale total, and sexual-behavior subscale total. Risky sexual behavior and non-response may both decline with increasing age (p
0.0004). Risky sexual behavior (p
0.0039) and non-response (p
0.0001) may also both be associated with individual therapy, although association appears to be in opposite directions per their regression coefficient estimates. How to interpret this finding is not clear since only cocaine trials employed individual therapy and only methamphetamine trials employed group therapy. Table also indicates that whites may demonstrate significantly more risky injection-drug use than other racial groups (p
0.0283); and non-response was lower in active arms (p
Table 3 Fit of trivariate probit regression model simultaneously to the three outcome variables of (1) dichotomized injection-drug-use subscale total, (2) four-category ordinal scale for sexual-behavior subscale total, and (3) non-response. Coefficient estimates (more ...)
The coefficient estimate of the shared random effect for the injection-drug-use subscale total was positive and statistically significant (3.05, p
0.0001), which contrasts with the negative coefficient for the sexual-behavior subscale total (−0.598, p
0.0001). Together, the signs of the three shared random-effect coefficients indicate that one or more latent factors couple an increased tendency for non-response (+ coefficient) with an increased tendency for risky injection-drug use (+ coefficient) and a reduced tendency for risky sexual behavior (− coefficient). Assessments appear to be missing in an informative way. In practical terms, after accounting for the other factors in the model, those who choose not to provide assessments may also be doing worse in terms of injection-drug-use risk-taking and better in terms of sexual-behavior risk-taking.
These findings are consistent with detected trends in means over time. After adjusting for non-response, self-reported risky injection-drug-use appears to increase over time between baseline and end of the active-intervention period (p
0.0011). Those who are leaving assessment have elevated risky injection-drug use and that is causing injection-drug risk-taking to increase on average within the population over time. Due to the formulation of the regression model, this time trend is, strictly speaking, only for those on placebo. However, no differences in slopes were detected between active and placebo during active intervention or follow-up for the injection-drug-use subscale total (p
0.1488). In contrast, because non-response was associated with lower sexual-behavior subscale totals, outcome on this subscale is estimated to decline on average over the active-intervention period (p
0.0001). That is, the population self reports less risky sexual-behavior over time. For this subscale total, no evidence is observed for a difference in slopes between active and placebo during active intervention or follow-up (p
The potential advantages here of joint modeling are bias reduction through correction for informatively missing assessments, increased statistical power, and additional insights into the relationship between the two HRBS subscales. For comparison, univariate probit regression models were fit, with a model for injection-drug-use subscale total as an outcome and a separate model for sexual-behavior subscale total as an outcome. Like the trivariate-outcome model, univariate-outcome analyses found that whites demonstrate more risky injection-drug use than other racial groups (regression coefficient
0.0139); and risky sexual-behavior declines with age (regression coefficient
0.0006) and is negatively associated with individual therapy (regression coefficient
0.0066). Further, as with the trivariate-outcome model, univariate-outcome modeling found a decline in risky sexual-behavior over the active-intervention period (regression coefficient
0.0001). No other effects were detected for the univariate-outcome models (data not shown), which contrasts to the trivariate-outcome model’s finding that risky injection-drug-use increases during active intervention (Table ). Presumably this difference arose because trivariate-outcome modeling allowed the fit for each outcome to leverage information from the others via the shared random effect. Univariate-outcome modeling is not recommended for the HRBS.
Does evidence of positive association exist between non-response and risky injection-drug use in the observed data? Obviously, observed data cannot provide a complete answer because the unobserved responses were unobserved and probit modeling is on the outcomes’ underlying latent scales. Even so, a rough approximation is available from the observed data. Table provides the sample mean for each HRBS subscale total at baseline as stratified by non-response status at the end of the active-intervention period. Compared to respondents, non-respondents’ baseline mean was approximately 13% higher on the injection-drug-use subscale total, which is in the same direction as detected by the trivariate-outcome model. In contrast, on the sexual-behavior subscale total, non-respondents’ baseline mean was 5% higher than respondents’, which is opposite in direction from that detected by the trivariate-outcome model but also nearly threefold smaller than the difference between respondents’ and non-respondents’ means for the injection-drug-use subscale total.
Estimated sample mean (SE) of HRBS subscale totals at baseline stratified by non-response status at the end of the active-intervention period.
Also, the signs of the subscales’ coefficients for the shared random effect were opposite, indicating negative association (Table ). The observed data also contains evidence for a negative association between these two HRBS components. Table reveals that the only detectable association between these two subscale totals was negative in sign (
A look at the distribution of the estimated shared random effect does provide some additional insights into its latent structure (Figure ). Per standard approach, the regression model assumed that the shared random effect has a normal distribution of mean zero. Clearly, neither of those assumptions are met. The histogram and a non-parametric density estimator reveal a distribution that is bimodal with the larger of the two modes below zero. Interestingly, the smaller mode is approximately located at zero. The reason for this departure may be that, while random effects are typically used to estimate the collective impact of many small latent factors, in this population perhaps included among those smaller effects is a dominant larger latent effect. In particular, this appears to be a grouping variable, which would explain the presence of two distinct modes with one at zero. The implication is that the regression model could be more correctly specified if the identity of this large, latent grouping factor were known. A clue to its identity is suggested by the fact that the shared random effect has a particularly strong association with the non-response outcome (r
0.0001) but more modest associations with the HIV risk-taking subscale totals at baseline (|r
Figure 1 Frequency histogram of predicted shared random effect from the fitted trivariate probit regression model. Overlain on the histogram are a fitted normal distribution (solid line) and a non-parametric kernel density estimate (dashed line) of the distribution. (more ...)