If health status is the only confounding covariate-that is, the only variable that predicts both the treatment and the outcome-and if the regression model is properly specified, then the coefficient of the treatment indicator corresponds to the average causal effect in the sample. Another way to estimate the causal effect in this scenario is to regress the outcome on two inputs: the treatment indicator and previous health status. Intuitively, the simplest way to do this is to compare the averages of the current health status measurements across treatment groups only within each previous health status category we discuss this kind of subclassification strategy later. In these simple examples, however, there is a simple solution, which is to compare treated and control units conditional on previous health status. The preceding theoretical examples illustrate how a simple predictive comparison is not necessarily an appropriate estimate of a causal effect. For this reason, such predictors are sometimes called confounding covariates. If a causal estimate is desired, simple comparisons of average outcomes across groups that ignore this variable will be misleading because the effect of the treatment will be “confounded” with the effect of previous health status. Previous health status plays an important role in both these scenarios because it is related both to treatment assignment and future health status. It is then possible to see equal average outcomes of patients in the two groups, with sick patients who received the treatment canceling out healthy patients who received the control. However, suppose that this time, sicker patients are given the treatment and healthier patients are assigned to the control condition. So, for any given unit, we would expect the outcome to be better under treatment than control. In this scenario, the treatment has a positive effect for all patients, whatever their previous health status. This sort of discrepancy between the predictive comparison and the causal effect is sometimes called self-selection bias, or simply selection bias, because participants are selecting themselves into different treatments.ġ.2 Hypothetical example of positive causal effect but zero positive predictive comparisonĬonversely, it is possible for a truly nonzero treatment effect to not show up in the predictive comparison. This scenario leads to a positive predictive comparison between the treatment and control groups, even though the causal effect is zero. However, let us further suppose that treated and control groups systematically differ, with healthier patients receiving the treatment and sicker patients receiving the control.
That is, the causal effect of the treatment is zero. We first suppose that the treatment would have no effect on the health status of any given patient, compared with what would have happened under the control. In this scenario, the causal effect represents a comparison between what would have happened to a given patient had he or she received the treatment compared to what would have happened under control. 167, Gelman and Hill, 2007)ġ.1 Hypothetical example of zero causal effect but positive predictive comparisonĬonsider a hypothetical medical experiment in which 100 patients receive the treatment and 100 receive the control condition. More generally, causal inference can be viewed as a special case of prediction in which the goal is to predict what would have happened under different treatment options.
In the usual regression context, predictive inference relates to comparisons between units, whereas causal inference addresses comparisons of different treatments if applied to the same units. Three basic concepts are used to define causal effects (Rubin, 2007).Ī unit is a physical object, for example, a patient, at a particular place and point of time, say time \(t\).Ī treatment is an action or intervention that can be initiated or withheld from that unit at t (e.g., an anti-hypertensive drug, a statin) if the active treatment is withheld, we will say that the unit has been exposed to the control treatment.Īssociated with that unit are two potential outcomes at a future point in time, say, \(t^
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