Understanding cause and effect is often of interest when conducting research, but in practice it is rarely straightforward. Even seemingly simple questions can quickly become complicated once we consider the broader context in which data is generated.
Take a light-hearted example. Does ice cream cause happiness? At first glance, the relationship seems obvious. However, if we look a little closer another factor emerges: sunshine. Warmer, sunnier days are likely to affect the probability of getting ice cream and happiness. This makes sunshine a classic confounder. If we do not adjust for sunshine, we risk a substantially biased effect which may not reflect the true effect.
This is why causal inference begins with careful thinking, often using by tools like directed acyclic graphs. These help us think about relationships and identify which variables we need to adjust for in, and which ones we should not (i.e., colliders), to obtain the effect we care about. In this case, sunshine creates a backdoor path between ice cream and happiness. This is something we need to block to estimate a valid causal effect.
Traditionally, there are two main ways to address this kind of confounding. One approach focuses on modelling the treatment. Propensity score methods estimate the probability of receiving treatment, such as ice cream, based on observed confounders like sunshine. Another approach is to focus on modelling the outcome, for example using regression models for the outcome, to estimate the relationship between treatment and outcome while adjusting for those same confounders.
Both approaches can be effective, but they share a common limitation. They rely on the assumption that the model has been correctly specified. Even when the right variables are included, the exact nature of the relationship is rarely known with certainty. Some degree of model misspecification is almost inevitable.
This is where doubly robust methods can be helpful. Rather than relying on a single model, they combine both approaches by modelling the treatment and the outcome. [1] The robustness in doubly robust refers specifically to this issue of model misspecification.
The key idea is simple. The bias from a doubly robust estimator is the product of the bias from the treatment model and the bias from the outcome model. In theory, this means that if either model is correctly specified, the overall bias is eliminated. This is unrealistic, however it also means even if neither model is perfect, having two imperfect models can still reduce bias compared to using only one of the models. Put simply, you have two chances to get it right.
However, this does not remove the need for careful modelling. Doubly robust methods are not a safeguard against poor data or missing confounders. If important variables are unmeasured, or if both models are severely misspecified, bias can still persist or be exacerbated if both models are grossly misspecified. However, they are a useful tool when working with complex, imperfect data.
From a practical perspective, implementing a doubly robust approach can be a simple extension of other methods. A common strategy combines inverse probability weighting with an outcome model. First, the causal estimand of interest is decided. Next, a propensity score model is used to estimate the probability of treatment given the observed confounders. Propensity scores are then used to derive weights, depending upon the causal estimand, which adjust the sample to better reflect the target population. An outcome model is fitted using these weights, while also including the same confounders used in the propensity score model. Finally, an appropriate variance estimator is used to account for the inverse probability weights.
As with any causal analysis, careful variable selection remains critical. Confounders should be included, but adjusting for mediators, colliders, or descendants of colliders can introduce bias rather than reduce it. It is also essential to be clear about the causal estimand being targeted. Different methods estimate different effects. Failing to align these can lead to misleading comparisons.
For many researchers, the initial reaction to doubly robust methods is skepticism. If all models are, to some extent, wrong, why combine two of them? The answer is reducing bias. By using two models rather than one, we essentially get two chances that a model will have low misspecification. It is not about achieving perfection, but about reducing bias the best we can. In an environment where decisions are increasingly driven by complex data, this kind of robustness is valuable. Doubly robust methods provide a practical way to strengthen causal inference and improve confidence in results.
References
1.Funk, M. J., Westreich, D., Wiesen, C., Stürmer, T., Brookhart, M. A., & Davidian, M. (2011). Doubly robust estimation of causal effects. American Journal of Epidemiology, 173(7), 761–767. https://doi.org/10.1093/aje/kwq439
