original data with draws from the predictive distribution of $Y_{z_1,z_2}$ in Step 6, with ML estimates $\hat{\theta}_{z_1,z_2}^{(b)},\hat{\beta}_{z_1,z_2}^{(b)}$ substituted for the parameters $\theta_{z_1,z_2}^{(b)},\beta_{z_1,z_2}^{(b)}$. Let $\hat{\Delta}_{01,00}^{(b)}=E[Y_{01}-Y_{00}]$, $\hat{\Delta}_{11,00}^{(b)}=E[Y_{11}-Y_{00}]$, and $\hat{\Delta}_{11,01}^{(b)}=E[Y_{11}-Y_{01}]$ denote the average treatment effects, $\hat{\Delta}_{jk,lm}^{(b)}$, with associated pooled variance estimates $W_{jk,lm}^{(b)}$, based on the observed and imputed values of $Y$ for each treatment regimen.

8.  The MI estimate of $\Delta_{jk,lm}$ is then $\bar{\Delta}_{jk,lm,B}=\sum_{b=1}^B\hat{\Delta}_{jk,lm}^{(b)}$, and the MI estimate of the variance of $\bar{\Delta}_{jk,lm}$ is $T_B=\bar{W}_{jk,lm,B}+(1+1/B)D_{jk,lm,B}$, where $\bar{W}_{jk,lm,B}=\sum_{b=1}^BW_{jk,lm}^{(b)}/B$, $D_{jk,lm,B}=\sum_{b=1}^B\frac{(\hat{\Delta}_{jk,lm}^{(b)}-\bar{\Delta}_{jk,lm,B})^2}{B-1}$. The estimate $\Delta_{jk,lm}$ follows a $t$ distribution with degree of freedom $\nu$, $\frac{\Delta_{jk,lm}-\bar{\Delta}_{jk,lm,B}}{\sqrt{T_B}}\sim t_{\nu}$, where $\nu=(B-1)(1+\frac{\bar{W}_{jk,lm,B}}{D_{jk,lm,B}(B+1)})^2$.

Bayesian additive regression trees
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BART [@chipman_bart] is a flexible estimation technique for any arbitrary function. Suppose we have a continuous outcome $Y$ and corresponding $p$ predictors $X=(X_1,\ldots,X_p)$. Suppose $Y$ is related to $X$ via $$Y=f(X)+e$$ where $f(.)$ is any arbitrary function which could involve complicated non-linear and multiple-way interactions and $e\sim N(0,\sigma^2)$. Formally, BART is written as $$\label{chap3:bart_eq}
    Y=\sum_{j=1}^mg(X,T_j,M_j)+e$$ where $(T_j,M_j)$ is the joint distribution of the $j^{\text{th}}$ binary tree structure $T_j$ with its corresponding $b_j$ terminal node parameters $M_j=(\mu_{1j},\ldots,\mu_{b_jj})$. $m$ is the number of regression trees used to estimate $f(X)$ and it is usually fixed at 200.

BART is able to model multiple-way interactions by using regression 