estimating ATE is straightforward as the confounders are balanced between treatment groups. For non-randomized studies, under the RCM framework, the focus is to mimic randomization when randomization is actually not available. RCM emphasizes the equal importance on both the design and analysis stage of non-randomized studies. The idea of being “outcome free” at the design stage of a study before analysis is an important component of RCM. This means that the researchers should not have access to data on the outcome variables before they finalize all aspects of the design including ensuring that balance in distribution of potential baseline confounders between treatments can be achieved.
Since only pre-treatment confounders would be used in this design phase, this approach is similar to the “intention-to-treat” analysis in RCTs. RCM requires three key assumptions.
1. Stable Unit Treatment Value Assumption (SUTVA): the potential outcomes for any subject do not vary with the treatment assigned to other subjects, and, for each subject, there are no different forms or versions of each treatment level, which lead to different potential outcomes.
2. Positivity: the probability of assignment to either intervention for each subject is strictly between 0 and 1.
3. Unconfoundedness: the assignment to treatment for each subject is independent of the potential outcomes, given a set of pre-treatment covariates. In practice, it means all potential confounders should be observed in order to properly assess the causal effect.
If those assumptions hold in a non-randomized study, methods under RCM, such as propensity score-based methods, are able to provide unbiased estimate of the causal effect of the estimand of interest. We will further discuss estimand later in this chapter and provide case examples of the use of propensity score based methods in Chapters 4, 6, 7, and 8 later in this book.
2.3.4 Pearl’s Causal Model
In contrast to the RCM, Pearl advocates a different approach to interpret causation, which combines aspects of structure equations models and path diagrams (Halpern and Pearl 2005a, Halpern and Pearl 2005b, Pearl 2009a, Pearl 2009b). The direct acyclic graph (DAG) approach, which is part of the PCM, is another method commonly used in the field of epidemiology. Figure 2.1 presents a classical causal DAG, that is, a graph whose nodes (vertices) are random variables with directed edges (arrows) and no directed cycles. In situations described in this book, V denotes a (set of) measured pre-treatment patient characteristic(s) (or confounders), A the treatment/intervention of interest, Y the outcome of interest, and U a (set of) unmeasured confounders.
Figure 2.1: Example Directed Acyclic Graph (DAG)
Causal graphs are graphical models that are used to encode assumptions about the data-generating process. All common causes of any pair of variables in the DAG are also included in the DAG. In a DAG, the nodes correspond to random variables and the edges represent the relationships between random variable. The assumptions on the relation of the variables are encoded if there are no arrows. An arrow from node A to node Y may or may not be interpreted as a direct causal effect of A on Y. The absence of an arrow between U and A in the DAG means that U does not affect A. From the DAG, a series of conditional independence will then be induced, so that the joint distribution or probability of (L, A, Y) can be factorized as a series of conditional probabilities. Like RCM, PCM also has several key assumptions, in which the same SUTVA and positivity assumptions are included. For other more complicated assumptions like v-separation, we refer you to the literature of Pearl/Robins and their colleagues. (See above.)
The timing of obtaining the information about V, A, and Y can also be included in DAGs. Longitudinal data that may change over time are therefore shown as a sequence of data points as shown in Figure 2.2. Note: to be consistent with literature on causal inference with longitudinal data, we use L to represent time varying covariates and V for non-time varying covariates (thus is V in Figure 2.1). Time-dependent confounding occurs when a confounder (a variable that influences intervention and outcome) is also affected by the intervention (is an intermediate step on the path from intervention to outcome), as shown in Figure 2.2. In those cases, g-methods, such as inverse probability of treatment weighting (IPTW) (Chapter 11), need to be applied.
Figure 2.2: Example DAG with Time Varying Confounding
Causal graphs can be used to visualize and understand the data availability and data structure as well as to communicate data relations and correlations. DAGs are used to identify:
1. Potential biases
2. Variables that need to be adjusted for
3. Methods that need to be applied to obtain unbiased causal effects
Potential biases might be time-independent confounding, time-dependent confounding, unmeasured confounding, and controlling for a collider.
There are a few notable differences between RCM and PCM that deserve mention:
● PCM can provide understanding of the underlying data-generating system, that is, the relationship between confounders themselves and between confounders and outcomes, while the focus of RCM is on re-creating the balance in the distribution of confounders in non-randomized studies.
● The idea of “outcome-free” analysis is not applicable in PCM.
● The estimand under PCM does not apply to some types of estimands, for instance, the compliers average treatment effect.
2.4 Estimands
As stated before, the individual causal treatment effect is NOT estimable. Thus, we need to carefully consider the other types of causal effect that we would like to estimate, or the estimand. An estimand defines the causal effect of interest that corresponds a particular study objective, or simply speaking, what is to be estimated. In recent drafted ICH E9 Addendum (https://www.fda.gov/downloads/Drugs/ GuidanceComplianceRegulatoryInformation/Guidances/UCM582738.pdf), regulators clearly separate the concept of estimands and estimators. From the addendum, an estimand includes the following key attributes:
● The population, in other words, the patients targeted by the specific study objective
● The variable (or endpoint) to be obtained for each patient that is required to address the scientific question
● The specification of how to account for intercurrent events (events occurring after treatment initiation, such as concomitant treatment or medication switching, and so on) to reflect the scientific question of interest
● The population-level summary for the variable that provides, as required, a basis for a comparison between treatment conditions
Once the estimand of the study is specified, appropriate methods can then be selected. This is of particular importance in the study design stage because different methods may yield different causal interpretations. For example, if the study objective is to estimate the causal treatment effect of drug A versus drug B on the entire study population, then matching might not be appropriate because the matched population might not be representative of the original overall study population.
Below are a few examples of popular estimands, with ATE and ATT often used in comparative analysis of observational data in health care applications.
● Average treatment effect (ATE): ATE is a commonly used estimand in comparative observational research and is defined as the average difference in the pairs of potential outcomes, averaged over the entire population. The ATE can be interpreted as the difference in the outcome of interest had every subject taken treatment A versus had every subject taken treatment B.
● Average treatment effect of treated (ATT): Sometimes we are interested in the causal effect only among those who received one intervention of interest (“treated”). In this case the estimand is the average treatment effect of treated (ATT), which is the average difference of the pairs of potential outcomes, averaged