8 covariates are simulated for each scenario, each defined by their odds ratio on treatment and odds ratio on outcome:
**OR_TT^X**: Odds ratio describing the increase in probability of patients being treated when covariate X is positive.
**OR_Y^X**: Odds ratio describing the increase in average outcome when covariate X is positive.
Covariates are simulated as in *Sturmer et al, 2021*:
| **Variables** | **OR_TT^X** | **OR_Y^X** |
| -------------- | ----- | ----- |
| **X_1** | 2.0 | 1.0 |
| **X_2** | 1.5 | 1.0 |
| **X_3** | 1.0 | 2.0 |
| **X_4** | 1.0 | 1.5 |
| **X_5** | Epsilon | Epsilon |
| **X_6** | 1.5 x Epsilon | 1.5 x Epsilon |
| **X_7** | 1.0 / 10 | 1.0 / 10 |
| **X_8** | 1.0 / 0.1 | 1.0 / 10 |
**X_1** and **X_2** are thus *Instrumental variables*, **X_3** and **X_4** are *Risk factors for the outcome* and **X_5** and **X_6** are *Cofounders*.
**X_7** and **X_8** are unobserved tail-end cofounders indicating rare treatment decisions for extreme propensity score values in each group. Untreated patients with very high propensity of treatment will be very likely to have **X_7**=1 (patients that should have been treated but weren't due to frailty). Treated patients with very low propensity of treatment will be very likely to have **X_8**=1 (patients that should not have been treated but were due to severe condition).
For each replication, observed covariates explained treatment with an AUC drawn randomly between 0.65 and 0.85 (average: 0.75).
