Confounding

In causal inference, a confounder is a variable that affects both the dependent variable and the independent variable, creating a spurious relationship.

Confounding is a causal concept rather than a purely statistical one, and therefore cannot be fully described by correlations or associations alone. The presence of confounders helps explain why correlation does not imply causation, and why careful study design and analytical methods (such as randomization, statistical adjustment, or causal diagrams) are required to distinguish causal effects from spurious associations.

Several notation systems and formal frameworks, such as causal directed acyclic graphs (DAGs), have been developed to represent and detect confounding, making it possible to identify when a variable must be controlled for in order to obtain an unbiased estimate of a causal effect.

Confounders are threats to internal validity.

Let's assume that a trucking company owns a fleet of trucks made by two different manufacturers. Trucks made by one manufacturer are called "A Trucks" and trucks made by the other manufacturer are called "B Trucks". We want to find out whether A Trucks or B Trucks get better fuel economy. We measure fuel and miles driven for a month and calculate the MPG for each truck. We then run the appropriate analysis, which determines that there is a statistically significant trend that A Trucks are more fuel efficient than B Trucks. Upon further reflection, however, we also notice that A Trucks are more likely to be assigned highway routes, and B Trucks are more likely to be assigned city routes. This is a confounding variable. The confounding variable makes the results of the analysis unreliable. It is quite likely that we are just measuring the fact that highway driving results in better fuel economy than city driving.

In statistics terms, the make of the truck is the independent variable, the fuel economy (MPG) is the dependent variable and the amount of city driving is the confounding variable. To fix this study, we have several choices. One is to randomize the truck assignments so that A trucks and B Trucks end up with equal amounts of city and highway driving. That eliminates the confounding variable. Another choice is to quantify the amount of city driving and use that as a second independent variable. A third choice is to segment the study, first comparing MPG during city driving for all trucks, and then run a separate study comparing MPG during highway driving.

Confounding is defined in terms of the data generating model. Let X be some independent variable, and Y some dependent variable. To estimate the effect of X on Y, the statistician must suppress the effects of extraneous variables that influence both X and Y. We say that X and Y are confounded by some other variable Z whenever Z causally influences both X and Y.

Let ${\displaystyle P(y\mid {\text{do}}(x))}$ be the probability of event Y = y under the hypothetical intervention X = x. X and Y are not confounded if and only if the following holds: