Interpret the estimand framework from a causal inference perspective

The estimand framework was drafted in 2017 by ICH in Efficacy Guideline E9(R1) as addendum to Efficacy Guideline E9 and later published in 2019 [2, 3]. It has gained increasing attention in the pharmaceutical industry. Many clinical trials have used the estimand framework to develop new drugs for both oncology and non-oncology diseases [4], and professional working groups such as the Oncology Estimand Working Group (https://oncoestimand.github.io/oncowg_webpage/docs/) have been initiated to study how the estimand framework should be incorporated in pharmaceutical practices.

The estimand framework aims to clearly define a clinical question. This means to clearly define a treatment effect in this clinical question. The estimand framework introduces and compares estimands, estimators and estimates with regard to statistical roles in estimation of treatment effects. An estimand is a precise definition of a treatment effect in a clinical question, an estimator is a statistical method that estimates this estimand, and an estimate is a result from this estimator [3]. Five attributes are proposed for an estimand. They are treatments, endpoints, a target population, intercurrent events, population-level summary [3]. Treatments are medical interventions that patients would take in clinical trials. Endpoints are outcomes used to assess efficacy and safety of treatments. A target population is a group of patients with medical conditions of clinical interest. Intercurrent events are events that happen after treatment initiation and affect the definition of a treatment effect. Population-level summary is an approach to estimate the treatment effect in a target population.

Intercurrent events are frequent in practice but conceptually novel. E9(R1) listed many examples for intercurrent events, such as use of concomitant therapies, treatment switching and death before endpoint measurement. To suit different study objectives, five strategies have been proposed to reduce bias from intercurrent events. They are the treatment policy strategy, the hypothetical strategy, the composite variable strategy, the while on treatment strategy, the principal stratum strategy [3].

The estimand framework is still a relatively new concept, where different interpretations may exist. Any contribution to further clarification of the estimand framework would be beneficial for clinical development. The original ideas in the estimand framework are mainly conveyed through text, but a statistical description of the estimand framework may be more intuitive, precise and holistic. The core of the estimand framework is treatment effects, and attributes of an estimand are developed to define a treatment effect. It would be helpful to understand a treatment effect through a statistical causal inference framework and see how these attributes and strategies are related to underlying theories.

A causal inference framework is based on the potential outcome framework [1, 5]. Suppose there is an ideal two-arm randomized controlled clinical trial, with full compliance to treatment and no intercurrent events. This trial has a sample size of $N$, a binary treatment $X$, a continuous endpoint $Y$, a randomization scheme $R$ and some confounders $C$ that affect both $X$ and $Y$. $X$, $Y$, $R$ are random vectors of length $N$, and $C$ is a random matrix of row dimension $N$. $X_{i}$, $Y_{i}$, $R_{i}$, $C_{i}$ represent random variables or vectors for a participant $i$, where $i\in\{1,2,3,\ldots,N\}$. $R_{i}=0$ means that the participant is assigned to the control arm and $R_{i}=1$ means that the participant is assigned to the treatment arm. $X_{i}(R_{i}=0)=0$ means that the participant would take the control treatment if assigned to the control arm and $X_{i}(R_{i}=1)=1$ means that the participant would take the experimental treatment that is of primary clinical interest if assigned to the treatment arm. $Y_{i}(X_{i}(R_{i}=0)=0)$ would be the endpoint if the participant took the control treatment and $Y_{i}(X_{i}(R_{i}=1)=1)$ would be the endpoint if the participant took the experimental treatment. $R_{i}$, $X_{i}$ and $Y_{i}$ are potential outcomes. They are not real because we hypothesize what would happen if the participant took the control treatment or instead took the experimental treatment. Figure 1 is a causal directed acyclic graph that shows the causal relationships between $X$, $Y$, $R$, $C$. $R$ affects $Y$ only through $X$.

A clinical question is to estimate the overall treatment effect of taking the experimental treatment versus taking the control treatment on the endpoint over all participants. For the participant $i$, if he had $X_{i}(R_{i}=0)=0$ then he would have $Y_{i}(X_{i}(R_{i}=0)=0)$, and if had $X_{i}(R_{i}=1)=1$ then he would have $Y_{i}(X_{i}(R_{i}=1)=1)$. Hence, for this participant, an individual treatment effect is $Y_{i}(X_{i}(R_{i}=1)=1)-Y_{i}(X_{i}(R_{i}=0)=0)$. This individual treatment effect controls confounders on the endpoint within the same participant and means how the endpoint would change when only the treatment condition changes. There are $N$ individual treatment effects. They can be assumed to be equal or unequal. Generally, we are interested in an average treatment effect (ATE) among all participants. The ATE is an average of all individual treatment effects and can be described as $E(Y_{i}(X_{i}(R_{i}=1)=1)-Y_{i}(X_{i}(R_{i}=0)=0)~{}|~{}C)$. And the estimand in this clinical question is the ATE. If there is no confounder, the ATE would become $E(Y_{i}(X_{i}(R_{i}=1)=1)-Y_{i}(X_{i}(R_{i}=0)=0))$. Without adjusting for existing confounding effects, the ATE estimation may be biased.

Further, $E(Y_{i}(X_{i}(R_{i}=1)=1)-Y_{i}(X_{i}(R_{i}=0)=0)~{}|~{}C)=E(Y_{i}(X_{i}(R_{i}=1)=1)~{}|~{}C)-E(Y_{i}(X_{i}(R_{i}=0)=0)~{}|~{}C)$. This also means that the estimand is the difference between the ATE of taking the experimental treatment among all participants and the ATE of taking the control treatment among all participants. The problem is that, in real world, each participant only takes one kind of treatment, and only one of the two hypothetical situations with regard to two arms can happen. In this ideal trial, the observed randomized assignment $R_{i}^{o}$ is either 0 and 1, the observed treatment $X_{i}^{o}$ is either $X_{i}(R_{i}=0)$ or $X_{i}(R_{i}=1)$, and the observed outcome $Y_{i}^{o}(X_{i}^{o})$ is either $Y_{i}(X_{i}(R_{i}=0))$ or $Y_{i}(X_{i}(R_{i}=1))$. Hence, the relationship between potential outcomes and observed variables can be described as

What we can estimate from the actual trial data is the difference between the ATE from participants who take the experimental treatment in the treatment arm and the ATE from participants who take the control treatment in the control arm, that is, $E(Y^{o}~{}|~{}X^{o}=1,R^{o}=1,C)-E(Y^{o}~{}|~{}X^{o}=0,R^{o}=0,C)$. With certain assumptions including the stable unit treatment value assumption, the randomization assumption and the exclusion restriction assumption, it can proved that $E(Y^{o}~{}|~{}X^{o}=1,R^{o}=1,C)-E(Y^{o}~{}|~{}X^{o}=0,R^{o}=0,C)=E(Y^{o}~{}|~{}X^{o}=1,C)-E(Y^{o}~{}|~{}X^{o}=0,C)=E(Y(X_{i}(R_{i}=1)=1)~{}|~{}C)-E(Y(X_{i}(R_{i}=0)=0)~{}|~{}C)$. Without these assumptions, this equality chain will not hold.

Suppose we choose a causal linear model to estimate the estimand. The linear model is

where $E(Y^{o}~{}|~{}X^{o},C)=\beta_{0}+\beta_{1}X^{o}+\beta_{2}C$. Then, $E(Y^{o}~{}|~{}X^{o}=0,C)=\beta_{0}+\beta_{2}C$ and $E(Y^{o}~{}|~{}X^{o}=1,C)=\beta_{0}+\beta_{1}+\beta_{2}C$, which implies that $E(Y^{o}~{}|~{}X^{o}=1,C)-E(Y^{o}~{}|~{}X^{o}=0,C)=\beta_{1}$. Hence, $\beta_{1}$ equals to the ATE and thus is an estimator to the estimand from this linear model. After the linear model is built with the actual trial data, an estimate $\hat{\beta}_{1}$ on $\beta_{1}$ would be obtained, and $\hat{\beta}_{1}$ would be an estimate to the estimand.

The statistical formula of the estimand as $E(Y_{i}(X_{i}(R_{i}=1)=1)-Y_{i}(X_{i}(R_{i}=0)=0)~{}|~{}C)$ has already contained five attributes proposed in the estimand framework. The attribute of treatments is represented by $X$. The attribute of endpoints is represented by $Y$. The attribute of population-level summary is represented by the difference between two ATEs. The attribute of intercurrent events is not necessary here since no intercurrent events are assumed. The attribute of a target population is implicitly stated as the trial population and can be made explicit in the definition of the estimand. We can introduce a selection variable $S$ that indicates how the target population is selected from the general public and update the estimand to $E(Y_{i}(X_{i}(R_{i}=1)=1)-Y_{i}(X_{i}(R_{i}=0)=0)~{}|~{}C,S)$.

The causal inference framework is not restricted to the above situation. It is applicable to multivalued treatments, discrete outcomes and Bayesian methods. It is also the underlying theory that applies to any effort of estimating treatment effects. We may have used it, even if the causal inference framework does not appear explicitly in inference, because potential outcomes have been transformed to observed variables in statistical models and we only see observed variables. Causal inference is a mature, continuing research area in Statistics. Methods may be available for difficult practical problems. Hence, it would be recommended that we try to interpret the estimand from a causal inference perspective based on our study objectives, and understand or choose relevant methods to estimate estimands.

The attribute of intercurrent events can also be incorporated in the statistical formula of an estimand. Following the trial example in the last section, suppose now the trial has intercurrent events shown in figure 2. Also suppose the endpoint is not related to death and the second assessment on the endpoint is used for primary analysis. We use these intercurrent events to discuss the statistical formula of an estimand in different strategies.

Analysis sets are a group of participants that meets specific criteria and is used for specific analytical objectives. ICH E9 proposed different analysis sets, including full analysis set (FAS) and per protocol set (PPS) [2]. There are also other analysis sets, including intention to treat set (ITTS) and safety set (SS). ITTS is usually all participants enrolled in study. SS comes from ITTS and is usually a group of participants who take treatment at least once. FAS comes from SS and excludes some participants whose data are not suitable for analysis, such as those who take no treatment after randomization. PPS comes from FAS and only includes participants whose data have no or little protocol deviation.

Different analysis sets can lead to different treatment effect estimates, because from the estimand framework, they are different target populations. In the estimand framework, analysis sets are actually not required. We can define analysis sets into target populations or use the principal stratum strategy to estimate treatment effects in subpopulations equal to analysis sets. However, analysis sets are usually subsets of all trial data. If we build statistical models through the estimand framework on data subsets, we may introduce selection bias and break randomization of the assignment scheme, which means that the participants in an analysis set may not be considered randomized any more. Without the randomization assumption, statistical inference on estimands may be unviable or biased. Hence, we should carefully examine randomization in analysis sets if they are used as target populations.

The estimand framework is advantageously useful for estimation of causal treatment effects. Learning and strengthening statistical causal inference theories behind the estimand framework would facilitate implementation and improvement of the estimand framework in industry.

There is no funding for this work.