Chapter 16 Causal biases
Here we give a brief description of each of the main types of bias that you may encounter when performing a causal analysis.
16.1 Confounding bias
Sometimes called omitted variable bias by econometricians. If a pre-treatment variable that is related to both the treatment and the outcome is not included in the analysis, this creates a spurious dependence between the two. This is the most common and problematic form of bias in causal studies using observational data, and perhaps best explains the famous statement ‘correlation does not imply causation’.
16.2 Selection bias
Not far behind confounding, this occurs if one chooses subjects in a manner that is related to their outcome (and perhaps also their treatment status). Sometimes this is deliberate: for example, in the case of a rare diseases it is generally impractical to run a prospective study, so a case-control study in which patients are selected given their disease status is very common.
Figure 16.1: Graphical illustration of selection bias. In this case, if we sample students from US colleges, then we will only interview those who are currently enrolled. In order to reach this state, they must either have good grades or be good enough at sport to win a scholarship. Even if these two attributes are independent in the general population, this means that they will appear to be negatively correlated in our sample.
16.3 Bias due to conditioning on a mediator
As well as omitted-variable bias, it is possible to induce bias by controlling for a variable that actually mediates the causal relationship of interst. In Figure 16.2(i) for example, conditioning on \(C\) would entirely block the relationship between \(A\) and \(Y\).
Conversely, conditioning on a mediator may open a spurious path when there is actually no causal relationship at all. This is illustrated in 16.2(ii), where adjusting for \(C\) opens up the path and induces a non-causal association.
Figure 16.2: Graphical illustrations of the danger of conditioning on a mediator: in (i) conditioning on a mediator blocks the entire causal effect; in (ii) there is no causal effect, but conditioning on a mediator opens up a spurious path.
16.4 Time-related biases
There are a variety of biases that only occur in longitudinal studies. We describe them here separately.
16.4.1 Time-dependent confounding
This occurs in cases where early treatment may affect a covariate, which in turn affects later treatment. Adjustment for this requires use of the g-formula. As an example, in Figure 16.3 the first treatment \(A\) affects the intermediate outcome \(Z\). In order to correctly estimate the joint causal effect of \(A,B\) on \(Y\) we must use the g-formula: \[\begin{align*} p(y \mid do(a,b)) = \sum_{\ell} p(\ell\mid a) \cdot p(y \mid a, \ell, b). \end{align*}\]
Figure 16.3: Illustration of time-dependent confounding. If we are interested in the joint effect of \(A\) and \(B\) on \(Y\), then failing to adjust for \(L\) leaves open a backdoor path from \(B\) to \(Y\); however adjusting for \(L\) also fails because this will block a causal path from \(A\) to \(Y\).
16.4.2 Attrition bias
Also known as drop-out, this occurs if patients leave the study differentially between the treatment and control groups. If this happens in a way which is related to measured covariates, then it can be adjusted for using weights. In other cases, it is sensible to use a sensitivity analysis. An example of how this bias may induce an erroneous estimate of a causal effect is shown in Figure 16.4. Note that this can also be considered to be a form of selection bias, but the selection happens after treatment assignment, and is a result of that decision.
Figure 16.4: Differential attrition may induce a spurious correlation. If the treatment \(A\) causes a side effect \(E\), then people in the treatment arm may be more likely to drop out. If this is more likely to occur among people with worse overall health (\(H\)) then this will make the treatment appear helpful if we fail to account for it.