3 Randomized Controlled Trials
To get two similar groups of people, the easiest way is to to randomize the treatment assignment. But first several assumptions on the population must be introduced.
Assumption 3.1: No Interference
This assumption means that the outcome is unaffected by anyone else’s treatment. Rather, the outcome for an individual \(i\) is only a function of its own treatment. This assumption is not met when considering a vaccine as the treatment for example.
Assumption 3.2: Consistency
For an individual \(i\) if the treatment is \(T\), then the observed outcome \(Y\) is the potential outcome under treatment \(T\). In other words, it means that all individuals who received \(T=t\) actually received the exact same treatment (e.g. same dose). Note that this so-called consistency assumption has nothing to do with the consistency of an estimator.
When the treatment is binary \(T \in \{0,1\}\) consistency and no interference translates to the stable unit treatment value assumption (SUTVA)
Assumption 3.3: Stable Unit Treatment Value Assumption (SUTVA)
Note that \(Y\) is a random variable and that is corresponds to the observed outcome defined earlier.
As we just said the easiest way To get two similar groups of people is to use radomization in the treatement assignement. By this manipulation, the treatment is made independent of the potential outcome for each individual, which ensures what is called Exchangeability, ignorability, or internal validity of a trial. This assumption can also be written with mathematical terms:
Assumption 3.4: Exchangeability, ignorability
In randomized controlled trials (RCTs) all these assumptions are verified and calculating the average treatment effect (ATE) is relatively straightforward due to the rigorous experimental design:
\[\begin{align*} \tau &= \mathbb{E}[Y^{(1)} - Y^{(0)}] \\ &= \mathbb{E}[Y^{(1)}|T = 1] - \mathbb{E}[Y^{(0)}|T = 0] && \text{(Exchangeability)} \\ &= \mathbb{E}[Y|T = 1] - \mathbb{E}[Y|T = 0] && \text{(SUTVA)} \end{align*}\]
Ignorability is mainly achieved using two designs of randomized controlled trials : Completely randomized trial and Bernoulli trial.
3.1 Design
3.2 Bernoulli trial
The so-called Bernoulli trial design is the simplest possible design. Assume you have at hand \(n\) units, the treatment allocation follows a Bernoulli law with probability \(\pi\) (for example \(\pi = 0.5\)). The consequences is that the assignment mechanism is independent for all units. A disadvantage of such design is the fact that there is always a small probability that all units receive the treatment or no treatment. This is why other designs are possible, where the treatment allocation depends on previous decision for other units. The interest is to ensure to get a balanced group of treated and controls, and avoid a possible pathological case of high unbalance between the number of treated and control individuals.
3.3 Completely randomized trial
The so-called completely randomized experiment considers a fixed number of subjects is assigned to receive the active treatment. The simplest completely randomized experiment takes an even number of units and divides them at random in two groups, with exactly one-half of the sample receiving the active treatment and the remaining units receiving the control treatment. This is accomplished, for example, by putting labels for the \(\mathrm{n}\) units in an urn and drawing \(n_1 = n / 2\) at random to be treated. Here, the probability of being treated for an individual is also constant for all units, namely \(n_1 / n\).
Since we have different kinds of designs for our study our random variables and their expectancy won’t behave the same way. Hence, we need to define new notations to understand what design we are talking about.
Definition 3.1: Expectancy under a Design
In both designs only the expectancy of the random variables that are not independent from \(T_i\) will be affected. Hence, for instance if we are under a Completely randomized trial the random variables \(T_i\) are not independent since the number of treated and untreated is fixed. \[\begin{align*} \mathbb{E}_{\mathcal{B}}[T] = \mathbb{P}_{\mathcal{B}}[T=1] &= \pi && \text{def. of a Bernoulli trial}\\ \mathbb{E}_{\mathcal{C}}[T] = \mathbb{P}_{\mathcal{B}}[T=1] &= \frac{\left(\array{n-1\\n_1-1}\right)}{\left(\array{n\\n_1}\right)} && \text{def. of a Completely randomized trial}\\ &= \frac{n_1}{n} \\ \end{align*}\] Since the treatment are not I.I.D in a Completely randomized trial we have: \[\begin{align*} \underbrace{\mathbb{E}_{\mathcal{B}}[TT^{'}]}_{\pi^2} &\neq \mathbb{E}_{\mathcal{C}}[TT^{'}] \end{align*}\] However, using Exchangeability we have for \(t \in \{0, 1\}\): \[\begin{align*} \mathbb{E}_{\mathcal{B}}[Y^{(t)}] &= \mathbb{E}_{\mathcal{C}}[Y^{(t)}] \end{align*}\] And this implies that the value of the ATE does change with the disgn.
Now that we have presented all the framework in the case of RCTs. We can introduce our first estimator for the Average Treatment Effect.