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I am wondering if there is a direct relation between a Randomized Control Trial (RCT) and an Observation Study in terms of the treatment assignment mechanism or the propensity score. It seems in an RCT, certain conditions are listed, such as the propensity score/assignment mechanism being between 0 and 1:

$$ 0 < P(Z_i|X_i) <1 $$

where $Z_i$ is the treatment indicator taking on $1$ or $0$ and $X_i$ are the covariates.

Is there a succint relation between these and the RCT or observational study? Thanks.

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  • $\begingroup$ If it is RCT already why you use propensity score ? since in RCT you randomize subjects to treat and control group. You use it because you don't believe your randomization? I cannot understand the logic for a RCT to use propensity score adjustment(or matching) $\endgroup$
    – Deep North
    Jul 12 '17 at 0:51
  • $\begingroup$ The reason I mention it is because I would like to understand if the RCT is a special case of the Observational Study and vice versa. For example, ideally I am hoping to see if there is something like, "Relax this assumption, then the observational study becomes an RCT" $\endgroup$
    – user321627
    Jul 12 '17 at 1:46
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You can look at it that way. In a RCT you know the exact propensity score for each patient in the sense that you know the exact probability of each patient to get each treatment (and as you say, it is >0 and <1), because you know everything about the treatment assignment mechanism. In the nice simple case of a single fixed randomization scheme for all patients, you have a single set of patients that are perfectly matched in terms of (not just estimated, but exactly known) propensity scores and can just ignore the propensity score in the analysis. In case of stratified randomization, you end up with strata of perfectly matched patients (so you stratify the analysis correspondingly, but again the exact known value of the probabilty to get each treatment does not need to be used explicitly).

The case, where an analysis of a RCT comes closest to a propensity score analysis is with adaptive randomization (i.e. probability of each treatment changes between patients based on the previously observed data and is usually also influenced by patient covariates), where you really need to take the known probabiltity of getting each treatment into account. However, the big difference is that the probability of each patient getting each treatment is still completely known. In contrast, in an observational study, it is estimated based on observed covariates and there is invariably a concern that there could be unobserved confounders that result in a mis-estimation of the propensity score. This is a concern that is completely absent in RCTs as far as the initial treatment assignment is concerned. Some issues like this come back in for RCTs when we need to deal with patients that prematurely discontinue from a RCT so that we do not observe all the data we wanted or when other treatment decisions are taken after the start of the randomized treatments. The mechanism by which these things happen may differ between treatments and is not typically known to us. Even if it is known, it still makes things difficult (one classical example is analyses of survival in oncology trials when patients switch treatment after disease progression).

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