Multiple Imputation Methods For Estimating Causal Effects In Randomized Controlled Trials With Noncompliance And Non-response

True causal effect can only be inferred by comparing the outcome for an individual under treatment to its counterfactual outcome; i.e. the outcome that would have occurred without treatment. However, in reality, these counterfactual effects of individuals cannot be directly observed and are instead approximated under various assumptions. The gold standard of causal effect estimation is the randomized controlled trial which can approximate the counterfactual effect under randomization. Under perfect randomization subjects are assigned to say two groups with equal probability, which means that the two groups differ only in the treatment assigned. Theoretically, the subjects in the two groups should be mirror-images of one another, and we can therefore estimate the causal effect. In practice, theoretical randomization is difficult to achieve, and the noncompliance of subjects seriously distorts the randomizatio.Noncompliant subjects are also usually not a random sample of subjects, and complicated relationships and influences can cause a subject is compliant and lead to erroneous estimation of the causal effect even in a randomized trial.The last half century witnessed a rapid increase in the use of formal methods for the analysis of causal effects. We will focus only on causal effect estimation under all-or-none noncompliance with a continuous outcome variable in a parallel design. The method and concept of analyzing data with missing values is applied to causal effect estimation under noncompliance. Specifically, multiple imputation, a convenient method in handling missing values, is coupled with intention-to-treat (ITT) estimation.First of all, the noncompliant subject’s outcome is treated as missing value since its outcome does not represent the real effect under treatment. Multiple imputation is then applied to generate a series of complete data sets, and the ITT effect estimated for every imputed data set. The ITT estimates are then combined using Rubin’s rule (1987). We use the MI and MIAnalyse procedures of the SAS9.2 software to implement the multiple imputation steps. The main purpose is to compare the proposed method (MIITT) with other4existing methods, which are AT (As-treated), PP (Per-protocol), ITT (Intention-to-treat) and IV (Instrumental Variable Estimation). Through simulations, we compare the bias, root mean square error (RMSE), coverage rates of the95%confidence intervals, and average confidence interval length of the five methods in estimating the causal effect. Finally, we apply the proposed MIITT method and the other4methods to a resilience intervention trial among middle school students.In the simulation study, sample size and the rate of noncompliance both influenced the efficiency of the estimates. Smaller sample sizes tend to give larger RMSE estimates, while larger sample sizes are more likely to yield smaller RMSE estimates. And the difference between the methods is more obvious when the sample size is small. Moreover, there is an interaction between these two factors on the efficiency of estimation. The coverage rates of the95%confidence intervals for MIITT are always near to95%as well as those for PP and AT, while the coverage rate for ITT is not stable. The average confidence interval length decreases as the sample size increases and widens with increasing noncompliance rate. This suggests that the increase in RMSE due to noncompliance and missing values can be offset by increasing the number of subjects, although this needs to be done in consideration of the tradeoff between sample size and cost efficiency.The simulation results are similar when outcomes were MCAR (Missing Completely At Random) in addition to having noncompliance. For MAR (Missing At Random) mechanism, we modeled two conditions according to what variables the missing value was dependent on.MIITT seems more appropriate when the missing value was dependent on an observed variable Xm that was uncorrelated with the other variables. With a small treatment effect, for example0or1, PP yields the largest RMSE and AT is second. MIITT has a bigger RMSE than ITT when the treatment effect is0, but ITT has a bigger RMSE when the effect is1. If the sample size is small and treatment effect is0, IV gives the smallest RMSE. Additionally, the results show that the larger the rate of missingness, the larger the RMSE. When missingness is dependent on baseline measurements, the difference between MIITT with AT and PP is not very large. The RMSE of IV and ITT are biggest when few subjects comply with the treatment assigned, the second one is MIITT, and PP and AT are the smallest. The RMSE of AT, PP, and MIITT tend to be close to each other when the treatment effect is3. Sample size and the rate of missingness have no effect on the coverage rate. The coverage rates of MIITT are always near95%. Compared with the condition when missingness is dependent on the baseline variable, the coverage rates are more approximately95%when missingness is dependent on variable Xm. The average length of intervals can be wider as the rate of missingness increases. Even if the missing dependent variable is Xm, the length becomes narrow. The average lengths of the intervals for ATs and PPs vary wider than those for MIITT.With simple noncompliance or MCAR missing mixed with noncompliance, MIITT is not the best estimator but is second to PP and AT. There is also a wide gap between MIITTand IV and ITT. But MIITT can provide a more reasonable estimation error when the missing mechanism is missing at random as its RMSE and bias both are smaller than the other estimators. The advantage of MIITT decreases for large sample sizes since the difference between estimators decreases but it is still the best estimator of all the study methods. Moreover, the coverage rates of its95%confidence intervals and the average length of the intervals are nearly unchanged and are more stable than AT and PP.