add subheaders

vignettes/practical_causality.Rmd

@@ -115,7 +115,9 @@ d$predictions <- predict(m_ipw, newdata = d, type = "response")
 d$ipw <- ifelse(d$treatment == 1, 1 / d$predictions, 1 / (1 - d$predictions))
 ```
 
-Next, we run the same model again, including the weights. We see that the treatment effect, after weighting, is about `9` points, i.e. the treatment on average results in a 9-point higher score for QoL.
+## ATE calculated from IPW in simpler models
+
+Now we run the same model again, including the weights. We see that the treatment effect, after weighting, is about `9` points, i.e. the treatment on average results in a 9-point higher score for QoL.
 
 ```{r}
 m2 <- glmmTMB::glmmTMB(
@@ -132,6 +134,8 @@ Given the simplicity of the model, g-computation offers no significant advantage
 estimate_contrasts(m2, "treatment", estimate = "population")
 ```
 
+## ATE calculated from IPW and g-computation in more complex models
+
  However, we have not properly modelled the longitudinal nature of our data. Since patients' quality of life (QoL) was measured at three distinct time points, allowing for an examination of treatment effects over time, we need to include an interaction between `treatment` and `time`. This revealed a substantially greater increase in QoL over time within the treatment group. In such more complex modeling scenarios, g-computation becomes particularly advantageous.
 
 ```{r fig.width = 6, fig.height = 4.5, out.width = "90%", out.height = "90%"}
@@ -178,6 +182,8 @@ estimate_contrasts(m4, "treatment")
 estimate_contrasts(m4, "treatment", estimate = "population")
 ```
 
+## Average treatment effect on the treated and untreated
+
 The last example completes the "causal inference" topic by showing how to calculate the average treatment effect on the treated (ATT), and average treatment effect on the untreated (ATU).
 
 The ATT measures how much the treatment changes the outcome for those who already received it. It can help to decide whether a program or treatment should be discontinued for the people currently benefiting from it. Likewise, the ATU estimates how much the treatment would change the outcome if it were given to those who didn't already receive it. It helps us decide whether a program or treatment should be expanded to include those who haven't yet benefited from it. It helps to assess the potential benefits of extending a program or treatment to a wider population.