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The split-plot design assigns different interventions at the whole-plot andsub-plot levels, respectively, and induces a group structure on the finaltreatment assignments. A common strategy is to use the OLS fit of the outcomeon the treatment indicators coupled with the robust standard errors clusteredat the whole-plot level. It does not give consistent estimator for the causaleffects of interest when the whole-plot sizes vary. Another common strategy isto fit the linear mixed-effects model of the outcome with Normal random effectsand errors. It is a purely model-based approach and can be sensitive toviolations of parametric assumptions. In contrast, the design-based inferenceassumes no outcome models and relies solely on the controllable randomizationmechanism determined by the physical experiment. We first extend the existingdesign-based inference based on the {\htf} estimator to the Hajek estimator,and establish the finite-population central limit theorem for both undersplit-plot randomization. We then reconcile the results with those under themodel-based approach, and propose two regression strategies, namely (i) the WLSfit of the unit-level data based on the inverse probability weighting and (ii)the OLS fit of the aggregate data based on whole-plot total outcomes, toreproduce the Hajek and {\htf} estimators from least squares, respectively.This, together with the asymptotic conservativeness of the correspondingcluster-robust covariances for estimating the true design-based covariances aswe establish in the process, justifies the validity of regression-basedestimators for design-based inference. In light of the flexibility ofregression formulation with covariate adjustment, we further extend the theoryto the case with covariates and demonstrate the efficiency gain byregression-based covariate adjustment via both asymptotic theory andsimulation.

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