Causal relationship and confounding in statistical models.

Randomized trials are designed to eliminate systematic error (bias) when estimating the effect of an intervention (usually a treatment) through randomized allocation of patients to treatment with concealment of allocation. In an observational study, no disturbance of the process of care takes place and the outcome is merely ‘observed’ along with other relevant information, such as patients’ pre-intervention characteristics and any treatment received. Unless the group of individuals receiving the intervention and the control group are similar (or such groups can be created from them), selection, information and confounding bias need to be addressed in the statistical analysis. The most commonly used method to adjust a parameter estimate (for instance, the beneficial effect of a treatment or the risk of exposure to a hazardous agent) is the development of a statistical model that includes variables representing not only exposure (for example, intervention or control) and outcome, but also the potential confounding factors to adjust for (for example, disease severity). Data-driven methods are often used to select confounding variables. Screening of candidate variables, typically based on univariable analyses using forward or backward stepwise multiple regression, and model development based on a measure that assesses relative model performance (for example using Akaike’s information criterion) are commonly used. However, confounding bias may undermine these processes as the causal relationships between variables need to be taken into account, something that cannot be achieved by applying the methods blindly. Four simple examples of causal relationships are shown in Fig. 1, which involves an exposure (E), an outcome (O) and a third factor (C). The cause–effect relationships between variables (including direction) are shown as arrows. Depending upon these links, the consequences of including C in a statistical adjustment model will differ. First, if the third factor is independent of the exposure or the outcome, or both of these (C0), adjustment is unnecessary (in terms of bias) (Fig. 1a). It will reduce the number of degrees of freedom, although this is not problematical in most studies. Second, a classical confounder (C1) is E O