Merge branch 'main' of https://github.com/StatisticsHealthEconomics/mimR

vignettes/Basic_example.Rmd

@@ -191,19 +191,19 @@ $$
 
 The difference between Bayesian G-computation and its maximum-likelihood counterpart is in the estimated distribution of the predicted outcomes. The Bayesian approach also marginalizes, integrates or standardizes over the joint posterior distribution of the conditional nuisance parameters of the outcome regression, as well as the joint covariate distribution.
 
-Draw a vector of size $N*$ of predicted outcomes $y*z$ under each set intervention $z* \in \{0, 1\}$ from its posterior predictive distribution under the specific treatment. This is defined as $ p(y*_{z*} | \mathcal{D}_{AC}) = \int_{\beta} p(y*_{z*} | \beta) p(\beta | \mathcal{D}_{AC}) d\beta$ where $p(\beta | \mathcal{D}_{AC})$ is the posterior distribution of the outcome regression coefficients $\beta$, which encode the predictor-outcome relationships observed in the _AC_ trial IPD. This is given by:
+Draw a vector of size $N^*$ of predicted outcomes $y^*_z$ under each set intervention $z^* \in \{0, 1\}$ from its posterior predictive distribution under the specific treatment. This is defined as $p(y^*_{z^*} \mid \mathcal{D}_{AC}) = \int_{\beta} p(y^*_{z^*} \mid \beta) p(\beta \mid \mathcal{D}_{AC}) d\beta$ where $p(\beta \mid \mathcal{D}_{AC})$ is the posterior distribution of the outcome regression coefficients $\beta$, which encode the predictor-outcome relationships observed in the _AC_ trial IPD. This is given by:
 
 $$
-p(y*_{z*} \mid \mathcal{D}_{AC}) = \int_{x*} p(y* \mid z*, x*, \mathcal{D}_{AC}) p(x* \mid \mathcal{D}_{AC})\;  dx*
+p(y^*_{^z*} \mid \mathcal{D}_{AC}) = \int_{x^*} p(y^* \mid z^*, x^*, \mathcal{D}_{AC}) p(x^* \mid \mathcal{D}_{AC})\;  dx^*
 $$
 $$
-= \int_{x*} \int_{\beta} p(y* \mid z*, x*, \beta) p(x* \mid \beta) p(\beta \mid \mathcal{D}_{AC})\; d\beta \; dx*
+= \int_{x*} \int_{\beta} p(y^* \mid z^*, x^*, \beta) p(x^* \mid \beta) p(\beta \mid \mathcal{D}_{AC})\; d\beta \; dx^*
 $$
 In practice, the integrals above can be approximated numerically, using full Bayesian estimation via Markov chain Monte Carlo (MCMC) sampling.
 
 The average, variance and interval estimates of the marginal treatment effect can be derived empirically from draws of the posterior density.
 
-We can draw a vector of size $N*$ of predicted outcomes $y^*_z$ under each set intervention $z*$ from its posterior predictive distribution under the specific treatment.
+We can draw a vector of size $N^*$ of predicted outcomes $y^*_z$ under each set intervention $z^*$ from its posterior predictive distribution under the specific treatment.
 
 ```{r hat_Delta_stats_gcomp_stan}
 hat_Delta_stats_gcomp_stan <- hat_Delta_stats(AC.IPD, BC.ALD, strategy = strategy_gcomp_stan())