Reconciling design-based and model-based causal inferences for split-plot experiments

The split-plot design arose from agricultural science with experimental units, also known as the subplots, nested within groups whole plots. It assigns different interventions at whole-plot and subplot levels, respectively, providing a convenient way to accommodate hard-to-change factors. By design, subplots same plot receive level of intervention, thereby induce group structure on final treatment assignments. A common strategy is run an ordinary least squares (ols) regression outcome indicators coupled robust standard errors clustered level. does not give consistent estimators for effects interest when sizes vary. Another fit linear mixed-effects model normal random errors. purely model-based approach can be sensitive violations parametric assumptions. In contrast, design-based inference assumes no models relies solely controllable randomization mechanism determined by physical experiment. We first extend existing based Horvitz–Thompson estimator Hajek estimator, establish finite-population central limit theorem both under randomization. then reconcile results those approach, propose two strategies, namely (i) weighted (wls) unit-level data inverse probability weighting (ii) ols aggregate total outcomes, reproduce estimators, respectively. This, together asymptotic conservativeness corresponding cluster-robust covariances estimating true we in process, justifies validity inference. light flexibility formulation covariate adjustment, further theory case covariates, demonstrate efficiency gain regression-based adjustment via simulation. Importantly, all our theories are either numeric or design-based, hold regardless how well equations represent generating process.