In 1980, an analysis of the Coronary Drug Project (CDP) randomized trial found a greater survival in individuals who adhered to placebo than in those who did not adhere to placebo [1]. Statistical adjustment for multiple prognostic factors could not remove these survival differences. This finding was used to support the thesis that adherence adjustment in randomized trials is not generally possible and, therefore, that only intention-to-treat analyses should be trusted.

In 2016, we reported a re-analysis of the CDP placebo arm that incorporated statistical advances not available in 1980. Our re-analysis showed that most of the survival differences between placebo adherers and non-adherers were actually removed after adjustment for pre- and post-randomization prognostic factors [2]. Specifically, we modified the definition of adherence at missed visits, updated the method of adjustment for baseline predictors of adherence, and added adjustment for post-randomization predictors of adherence via inverse-probability (IP) weighting.

With these changes, the 5-year survival difference between placebo adherers and non-adherers decreased from 10.9 percentage points (95% confidence interval, 7.5, 14.4) to 2.5 percentage points (95% confidence interval, − 2.1, 7.0).

However, our re-analysis did not take full advantage of the recent methodological advances. In an attempt to conduct a re-analysis as comparable as possible to the original 1980 analysis, we used a cumulative incidence model that did not account for the exact time of death during the 5-year period, and a somewhat unintuitive measure of adherence – a binary indicator for cumulative adherence greater than 80% during the entire 5-year period. Although modeling the data this way was useful to demonstrate that methods developed since 1980 can address previously intractable sources of bias, the analytic approach we used is not the recommended one. Here, we demonstrate a better approach which uses a more natural definition of adherence and a survival analysis approach to incorporate time of death.

We restricted our analyses to the 2787 men in the placebo arm. Let t be an index for visit, where t = 0 is the baseline visit and t = 14 is the last visit at year 5, with 4-month intervals between each visit. Let A _t be an indicator for adherence ≥ 80% to the protocol-specified placebo dose between t and t + 1, Y _t an indicator of death between t and t + 1, C_{t + 1} an indicator for loss to follow-up defined as three consecutive missed study visits at t + 1, V a set of 39 variables measured at the time of randomization, and L _t a set of post-randomization variables measured at each t. The baseline covariates V were adherence during the run-in period, demographics (age, race), lifestyle characteristics (cigarette smoking, physical activity), medical history (risk group, weight, New York Heart Association class, comorbidities, blood pressure), use of non-study medications, laboratory findings, and electrocardiogram findings. All of these variables (except age, race, weight, risk group) were also post-randomization variables L _t at each visit t. When an individual missed a study visit, the most recent covariate and adherence values were carried over from the most recent available data, up to three consecutive missed visits. Participants were censored at the expected date of their third consecutive missed study visit.

The choice of 80% as a cut-point for adherence was based on standard practice, as reflected by the use of a run-in period requiring 80% adherence to placebo among all trial participants. Note that, since adherence was assessed as a categorical variable, with the highest category ≥ 80% of prescribed pills taken, higher thresholds were not possible for the binary adherence indicator.

where M _t is an indicator for measurement of adherence at visit t (1 if measured, 0 otherwise), and overbars indicate history of the variable. The weight models were fit in the full population before artificial censoring; in a sensitivity analysis, we restricted the fit of the weight models to person-visits with A _k = A₀ for 0 < k ≤ t.

To estimate the denominator of the weights, we fit the model logit(Pr[M _t = 1|\( {A}_0,{\bar{A}}_{t-1},V,{\overline{L}}_t,{C}_t=0 \)]) = α_0t + α₁A₀ + α₂A_{t − 1} + α₃V + α₄L_{t − 1} to all person-visits, and the model logit(Pr[A _t = 0|\( {A}_0,{\bar{A}}_{t-1},V,{\overline{L}}_t,{C}_t=0 \),M _t = 1]) = β_0t + β₁A₀ + β₂A_{t − 1} + β₃V + β₄L _t to the person-visits with measured adherence. When adherence was not measured at a visit (M _t = 0) but the individual was not yet defined to be lost to follow-up (that is, at the first or second consecutive missed visit), adherence was carried forward from the previous visit and the factor in the denominator of the adherence weight was 1 for that visit.

Similar models that did not include the time-varying covariates were fit to estimate the numerators of the weights. The final weight for each individual at each time was the product of the measurement and adherence weights for that individual up to that time point. As in previous studies, we truncated the estimated IP weights at the 99th percentile to avoid undue influence of outliers. The truncated weight estimates had a mean of 1.00 (SD = 0.29) and a range of 0.02 to 2.55.

We used the parameter estimates from the weighted outcome logistic model to estimate the 5-year survival as previously described [2]. We compared the survival for always vs. never at least 80% adherent, that is, A₀ = 0 vs. A₀ = 1.

where f[·] is a dose-response function and θ₁ is a vector parameter.

In separate analyses, we specified more flexible dose-response functions. Specifically, we considered both a quadratic dose-response function θ₁f[cum(Ā _t )] = θ_1,1cum(Ā _t ) + θ_1,2[cum(Ā _t )]², and a function that allowed for a separate effect of recent adherence θ_1,0A _t + θ_1,1cum(Ā_{t - 1}) + θ_1,2[cum(Ā_{t - 1})]². We also considered functions that included product terms with the time parameters.

To compute 95% confidence intervals, we used non-parametric bootstrapping with 500 samples.

SAS 9.4 was used for all analyses and code is provided in the supplementary online materials (see Additional file 1).

Our analysis confirms that the survival differences between adherers and non-adherers to placebo can be largely adjusted away in the Coronary Drug Project. Adjustment for post-randomization predictors of adherence results in estimated 5-year mortality risk differences comparing adherence and non-adherence to placebo that are approximately null regardless of the analytic approach used. Our previous analysis [2] reached a similar conclusion but, by simply estimating the cumulative mortality at the end of follow-up, ignored the timing of events. In contrast, the current analyses use a survival analysis approach that allows estimation of adjusted survival curves for adherers and non-adherers throughout the follow-up.

We used two methods to compare the survival under continuous high-average adherence vs. low adherence (< 80%). The first method was a relatively inefficient censoring procedure that does not require a dose-response model for the effect of cumulative adherence on mortality. The second method was a theoretically more efficient method that requires a dose-response model and, therefore, will result in bias if the dose-response model is misspecified. We considered a variety of dose-response functions, which made different assumptions about the relationship between adherence and mortality over time. When sufficient data are available, more flexible dose-response functions are generally preferable as they impose fewer a priori constraints. For example, a function that models separately current adherence and prior cumulative adherence (using, say, linear and quadratic terms as in our analysis) requires fewer assumptions than a function that models total cumulative adherence. However, the added flexibility comes at the price of less precise estimates. Unlike in other examples [12], our estimates were not very sensitive to the choice of dose-response model. Further research into dose-response modeling for adherence-adjusted estimates is warranted.

Obtaining a null estimate when comparing adherers and non-adherers in the placebo arm supports the validity of adherence-adjusted effect estimates in randomized trials. However, this comparison relies on the lack of psychobiological effects of placebo on mortality. For other outcomes, such as pain or symptom severity, this assumption may be less reasonable.

Adherence-adjusted analyses of randomized controlled trials have been viewed with some skepticism with some authors suggesting that adherence is intractably confounded [1]. However, we have demonstrated that adjustment for post-randomization predictors of adherence can create comparability between adherers and non-adherers when rich data on pre-and post-randomization confounders exist.