Inspired by a free online course titled Complier Average Causal Effects (CACE) Analysis and taught by Booil Jo and Elizabeth Stuart (through Johns Hopkins University), I’ve decided to explore the topic a little bit. My goal here isn’t to explain CACE analysis in extensive detail (you should definitely go take the course for that), but to describe the problem generally and then (of course) simulate some data. A plot of the simulated data gives a sense of what we are estimating and assuming. And I end by describing two simple methods to estimate the CACE, which we can compare to the truth (since this is a simulation); next time, I will describe a third way.
Non-compliance in randomized trials
Here’s the problem. In a randomized trial, investigators control the randomization process; they determine if an individual is assigned to the treatment group or control group (I am talking about randomized trials here, but many of these issues can apply in the context of observed or quasi-experimental settings, but require more data and assumptions). However, those investigators may not have as much control over the actual treatments that study participants receive. For example, an individual randomized to some type of behavioral intervention may opt not to take advantage of the intervention. Likewise, someone assigned to control may, under some circumstances, figure out a way to get services that are quite similar to the intervention. In all cases, the investigator is able to collect outcome data on all of these patients, regardless of whether or not they followed directions. (This is different from drop-out or loss-to-followup, where outcome data may be missing.)
CACE
Typically, studies analyze data based on treatment assignment rather than treatment received. This focus on assignment is called an intention-to-treat (ITT) analysis. In a policy environment, the ITT may make a lot of sense; we are answering this specific question: “What is the overall effect in the real world where the intervention is made available yet some people take advantage of it while others do not?” Alternatively, researchers may be interested in different question: “What is the causal effect of actually receiving the treatment?”
Now, to answer the second question, there are numerous subtle issues that you need to wrestle with (again, go take the course). But, long story short, we need to (1) identify the folks in the intervention group who actually do what they have been encouraged to do (receive the intervention) but only because they were encouraged, and not because they would have received the intervention anyways had they not been randomized, and compare their outcomes with (2) the folks in the control group who did not seek out the intervention on their own initiative but would have received the intervention had they been encouraged. These two groups are considered to be compliers - they would always do what they are told in the context of the study. And the effect of the intervention that is based on outcomes from this type of patient is called the complier average causal effect (CACE).
The biggest challenge in estimating the CACE is that we cannot actually identify if people are compliers or not. Some of those receiving the treatment in the intervention group are compliers, but the rest are always-takers. Some of those not receiving the treatment in the control arm are also compliers, but the others are never-takers. There are several methods available to overcome this challenge, two of which I will briefly mention here: method of moments and instrumental variables.
Using potential outcomes to define CACE
In an earlier post, I briefly introduced the idea of potential outcomes. Since we are talking about causal relationships, they are useful here. If \(Z\) is the randomization indicator, \(Z=1\) for those randomized to the intervention, \(Z=0\) for those in control. \(M\) is the indicator of whether or not the individual received the intervention. Since \(M\) is an outcome, we can imagine the potential outcomes \(M_{0i}\) and \(M_{1i}\), or what the value of \(M_i\) would be for an individual if \(Z_i=0\) or \(Z_i=1\), respectively. And let us say \(Y\) is the outcome, so we have potential outcomes that can be written as \(Y_{0,M_0}\) and \(Y_{1,M_1}\). Think about that for a bit.
Using these potential outcomes, we can define the compliers and the CACE. Compliers are people for whom \(M_0 = 0\) and \(M_1 = 1\). (Never-takers look like this: \(M_0 = 0\) and \(M_1 = 0\). Always-takers: \(M_0 = 1\) and \(M_1 = 1\)). Now, the average causal effect is the average difference between potential outcomes. In this case, the CACE is \(E[Y_{1,M_1} - Y_{0,M_0}|M_0 = 0 \ \&amp; \ M_1 = 1]\). The patients for whom \(M_0 = 0\) and \(M_1 = 1\) are the compliers.
Simulating data
The data simulation will be based on generating potential outcomes. Observed outcomes will be a function of randomization group and complier status.
options(digits = 3)
library(data.table)
library(simstudy)
library(ggplot2)
# Status :
# 1 = A(lways taker)
# 2 = N(ever taker)
# 3 = C(omplier)
def