Introduction to causal estimation

If trials are carried out perfectly, with perfect randomisation and blinding, full follow up and complete adherence on the part of all patients and treatment administrators, analysis according to randomisation will provide an unbiased estimate of the causal effect of treatment as received compared to control. However, as discussed in this chapter, complications often arise in trials as a result of nonadherence to treatment or follow up protocols, leading to changes to treatment or incomplete data. Before considering, in subsequent chapters, some examples of real-life trials exhibiting such compliance issues, this chapter will conclude with an explanation of why such deviations may cause problems when aiming to estimate efficacy of treatment.

In particular, this chapter ends with a discussion of the definition and difference between associational and causal effects, and how causal effects are most easily defined using a potential outcomes (counterfactual) framework. A brief introduction to the use of causal diagrams, which provide a pictorial aid to clarify the relationship between the treatment, outcome and other covariates associated with the particular causal scenario, will help to clarify the issues of confounding and selection bias and reveal how naïve methods (such as PP or AT analyses) introduce such biases.

Initially, therefore, it is necessary to provide an introduction to causal estimation and the framework within which such parameters may be estimated, beginning with the definition of a cause and an explanation of the difference between associational and causal inference.

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3.8.1. Definition of a cause

The notion of cause dates back at least as far as Aristotle, who defined four types of causes (the material cause (that out of which the thing is made), the formal cause (that into

which the thing is made), the final cause (that for which the thing is made) and the efficient

cause (that which makes the thing), the last of these being that most relevant to statistical inference, while Locke in 1690 defined both the cause (“that which produces any simple or complex idea”) and its effect (“that which is produced”) (84).

There are two necessary conditions for the definition and estimation of the effects produced by a certain cause (logically referred to as “causal effects”).

Firstly, the effect of a particular cause can only be meaningfully defined in relative terms to another cause. In other words, it takes at least two causes (or two versions or levels of a cause) to define an effect; thus in experimental studies, the treatment under study must always be compared to a relevant control condition, such that “experiments without control conditions are simply not experiments”. Indeed, stating that “A causes B” inherently implies a comparison of the effect of A on B to some condition not involving A (84).

Secondly, the key notion that distinguishes a “cause” from an attribute or characteristic is the potential for all units in the population of interest to be exposed to all levels of the cause being compared. In other words, before a unit has been assigned a certain level of the cause, it must be technically possible to define and observe, in principle, every level 𝑎 of the causal factor 𝐴 under consideration; no outcome 𝑌_𝑖(𝑎) can be a priori “counterfactual” for any individual 𝑖 (85). Thus Holland (84) explains how statements regarding the effect of an individual’s traits (such as sex, race, eye colour) can only describe observational associations rather than be given causal interpretation. To

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understand why this is so, it is necessary to consider the fundamental difference between causal and associational analyses.

3.8.2. Distinction between causal and associational analysis

In order to understand the difference between associational and causal analyses, it is vital to appreciate the difference between the underlying causal model and the process of observation (84). Thus consider a comparison of exposure to experimental (𝐴 = 1) versus control treatment (𝐴 = 0) on outcome 𝑌. An individual’s observed outcome 𝑌_𝐴𝑖 may differ from their underlying unconfounded potential outcome under each potential treatment assignment (𝑌0𝑖 or 𝑌1𝑖), as their observed outcome may be influenced by factors other than just treatment received. Therefore, the association between treatment received and observed outcome may be contaminated by selection and confounding factors and thus may not reflect the true underlying causal relationship between treatment received and true potential outcome, the observed data (𝐴𝑖, 𝑌𝐴𝑖) may therefore differ from the underlying causal variables (𝐴𝑖, 𝑌0𝑖, 𝑌1𝑖).

As such, in providing information only on the observed association between variables, results from standard statistical analysis methods (for example, regression or stratification methodologies) can only be interpreted in terms of descriptive statistics rather than providing any evidence of causality (84).

Such methods are used to estimate population parameters from which study samples are selected, and thus may be used to provide information on the observed relationship between variables by considering their joint distribution. For example, if 𝑃(𝑌 = 𝑦, 𝐴 = 𝑎) denotes the proportion of individuals 𝑖 in the population for which 𝑌𝑖 = 𝑦 and 𝐴𝑖 = 𝑎, parameters estimated from this joint distribution simply describe the observed

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relationship between variables at a single point in time; for example, the conditional distribution of 𝑌 given 𝐴

𝑃(𝑌 = 𝑦|𝐴 = 𝑎) = 𝑃(𝑌 = 𝑦, 𝐴 = 𝑎) 𝑃(𝐴 = 𝑎)

describes how the distribution of 𝑌 changes with 𝐴. A typical parameter from this distribution may be obtained by regressing 𝑌 on 𝐴 , providing the conditional expectation (or average) of 𝑌 given 𝐴, 𝐸(𝑌|𝐴 = 𝑎), i.e. the expected value of 𝑌 given a specific value of 𝐴.

In contrast, statements regarding causality cannot be defined from a joint distribution alone. This is due to the potential bias arising when comparing different treatment effects observed in distinct sections of the population, as underlying inherent differences between individuals in each treatment group may distort the comparison. The definitions of, and reasons for, these forms of bias are most easily depicted and defined through the use of causal diagrams.