Causal graphical views of fixed effects and random effects models

Despite the long-standing discussion on fixed effects (FE) and random effects (RE) models, how and under what conditions both methods can eliminate unmeasured confounding bias has not yet been widely understood in practice. Using a simple pretest-posttest design in a linear setting, this paper translates the conventional algebraic formalization of FE and RE models into causal graphs and provides intuitively accessible graphical explanations about their data-generating and bias-removing processes. The proposed causal graphs highlight that FE and RE models consider different data-generating models. RE models presume a data-generating model that is identical to a randomized controlled trial, while FE models allow for unobserved time-invariant treatment-outcome confounding. Augmenting regular causal graphs that describe data-generating processes by adding the computational structures of FE and RE estimators, the paper visualizes how FE estimators (gain score and deviation score estimators) and RE estimators (quasi-deviation score estimators) offset unmeasured confounding bias. In contrast to standard regression or matching estimators that reduce confounding bias by blocking non-causal paths via conditioning, FE and RE estimators offset confounding bias by deliberately creating new non-causal paths and associations of opposite sign. Though FE and RE estimators are similar in their bias-offsetting mechanisms, the augmented graphs reveal their subtle differences that can result in different biases in observational studies.