Table 1.

Summary of model structures used in comparative analysis of treatment effectiveness.

Sources of bias considered	Modeling approach	Model Structure	Sample size
None	1. “Naïve” model (Difference in means)	$y_i~\mathrm{Negative\; binomial}\left({\mu }_i,\theta \right)$ (1a)	n = 20,000 observations of sagebrush cover 10 years post-fire, in 1539 fires
		$\log \left({\mu }_i\right)=\alpha +\beta \left({g\mathrm{roup}}_i\right)+{\epsilon }_i$ (1b)
Selection bias associated with measured, time-invariant site characteristics	2. Regression following propensity score matching	$y_i~\mathrm{Negative\; binomial}\left({\mu }_i,\theta \right)$ (2a)	n = 11,012 “matched” observations of sagebrush cover 10 years post-fire, in 940 fires
		$\log \left({\mu }_i\right)=\alpha +\beta \left({\mathrm{group}}_i\right)+{\epsilon }_i$ (2b)
	3. Regression with environmental covariates (with varying intercept for fire identity)	$y_{ik}~\mathrm{Negative\; binomial}\left({\mu }_{ik},\theta \right)$ (3a)	n = 20,000 observations of sagebrush cover 10 years post-fire, in 1539 fires
		$\log \left({\mu }_{ik}\right)={\alpha }_k+\beta \left({\mathrm{group}}_i\right)+\mathit{\omega }\left(\mathrm{X}\right)+{\epsilon }_{ik}\left(3\mathrm{b}\right)$
		${\alpha }_k~\mathrm{Normal}\left(\alpha ,\sigma \right)$ (3c)
Selection bias associated with unmeasured time-invariant group characteristics	4. Difference-in-differences regression model (with varying intercepts for location and fire identity)	$y_{ijk}~\mathrm{Negative\; binomial}\left({\mu }_{ijk},\theta \right)$ (4a)	n = 40,000 observations pre-treatment (year 0 post-fire) and 10 years following treatment, in 1539 fires
		$\log \left({\mu }_{ijk}\right)={\alpha }_{jk}+\tau \left({\mathrm{time}}_{ijk}\right)+\gamma \left({\mathrm{group}}_{ijk}\right)+\beta \left({\mathrm{time}}_{ijk}*{\mathrm{group}}_{ijk}\right)+{\epsilon }_{ijk}$ (4b)
		${\alpha }_{jk}~\mathrm{Normal}\left({\alpha }_k,{\sigma }_j\right)$ (4c)
		${\alpha }_k~\mathrm{Normal}\left(\alpha ,{\sigma }_k\right)$ (4d)
Selection bias associated with: 1) unobserved characteristics of timepoints and groups; and 2) measured time-varying and group-varying factors (e.g. weather).	5. Within-estimator panel regression model (with varying intercepts for location and fire identity)	$y_{ijk}~\mathrm{Negative\; binomial}\left({\mu }_{ijk},\theta \right)$ (5a)	n = 220,000 observations from year 0 (pre-treatment) to year 10 following treatment, in 1539 fires
		$\log \left({\mu }_{ijk}\right)={\alpha }_{jk}+\tau \left({\mathrm{time}\mathrm{since}\mathrm{treatment}}_{ijk}\right)+\gamma \left({\mathrm{group}}_{ijk}\right)$
		$+\beta \left({\mathrm{treatment}\mathrm{indicator}}_{ijk}\right)$
		+ $\mathit{\omega }\left(\mathbf{W}\right)+{\epsilon }_{ijk}$ (5b)
		${\alpha }_{jk}~\mathrm{Normal}\left({\alpha }_k,{\sigma }_j\right)$ (5c)
		${\alpha }_k~\mathrm{Normal}\left(\alpha ,{\sigma }_k\right)$ (5d)

y represents individual observations (i) of sagebrush percent cover, which are nested within locations (j, where repeated measures are used) and fires (k). μ represents the expectation for y, α represents global intercepts (with varying intercept components α_j and α_k for location and fire identity, which are normally distributed with standard deviations (σ) associated with each). Group is a categorical variable indicating whether an observation was in treated or untreated groups. Time indicates whether the observation is pre-treatment (year 0 postfire) or post-treatment application (year 10 postfire) in DiD models. In the within-estimator panel regression model, time since treatment is a categorical variable for the observed timepoint (0–10 years post-treatment) and treatment indicator represents whether the treatment has occurred at a site by the observed timepoint. X and W represent matrices of either time-invariant biophysical covariates or time-varying weather variables (described in the text) with an associated vector of parameters ω. Time-invariant biophysical characteristics were selected based on their inclusion in frameworks for prioritization of sagebrush restoration sites or in past studies of sagebrush recovery as “control” variables. In all models, the parameter associated with treatment application is indicated by β.