Suppose you want to know whether greater engagement with a game feature changes in-game purchases. Regressing purchases on engagement can describe an association, but the engagement coefficient is not automatically a causal effect. First identify an effect using causal assumptions; then choose a model to estimate it. section 10.1.2 section 11.4
Regression fits parameters by minimizing the difference between a function of predictors and an observed response. A linear relationship is convenient to fit, but its shape matters: if the expected outcome under intervention is linear in the treatment, its derivative is constant, so the treatment coefficient can represent a single average treatment effect (ATE). That interpretation depends on modeling the interventional outcome, not merely observing a straight-looking association. section 6.4.7 section 7.4.6
A linear causal mechanism
For an adult male, let x be femur length in centimeters and y be height in centimeters. One proposed mechanism is y = 25 + 3x + nᵧ, where nᵧ represents other causes of height and follows a normal distribution with mean 0 and scale 3.3 cm. The coefficient 3 asserts a proportional change in height with femur length. In a structural causal model, that is an assumption about how a cause acts—not just a convenient fit. section 6.1.1 section 6.1.5
For a backdoor-identified effect, regress the outcome on the treatment and the confounders specified by the adjustment set. With a linear model, the treatment coefficient estimates the ATE; coefficients for included confounders serve the adjustment rather than being the target. Do not add predictors solely because they improve fit: predictors in a structural causal assignment are limited to direct endogenous causes. A regression used for backdoor adjustment instead includes the identified confounders. section 11.4.1 section 6.4.7
Listing 11.5 Estimating the backdoor estimand with linear regression
causal_estimate_reg = model.estimate_effect(
identified_estimand, #1
method_name="backdoor.linear_regression", #2
confidence_intervals=True #3
)
The DoWhy call selects linear regression as the estimator for an already identified backdoor estimand and requests a confidence interval. section 11.4
## Realized estimand b: In-game Purchases~Side-quest Engagement+Guild Membership+Time Spent Playing+Player Skill Level Target units: ate ## Estimate Mean value: 178.08617115757784 95.0% confidence interval: [[168.68114922 187.4911931 ]]
The resulting regression formula includes engagement and three adjustment variables; the displayed ATE estimate is about 178.09, with a 95% confidence interval from about 168.68 to 187.49. section 11.4.1
Linearity can also supply an additional assumption for instrumental-variable estimation when graphical assumptions alone do not identify the effect. That extra reach brings risk: look for evidence of nonlinearity, and do not treat a well-fitting regression as validation of a causal mechanism. Generalized linear models offer other response relationships, but their link functions are not, by themselves, causal mechanisms. section 11.3.2 section 4.5.4 section 6.4.7
