# Scenario catalogue
## Scenarios without DIF
| | **J** | | **Epsilon** | |
| -------------- | ----- | ----- | ----- | ----- |
| | **4** | **7** | **0.2** | **0.5** |
| **Scenario 101** | X | | X | |
| **Scenario 102** | X | | | X |
| **Scenario 103** | | X | X | |
| **Scenario 104** | | X | | X |
Each scenario has 5 subscenarios:
- A - H< sub > 0</ sub > true
- B - H< sub > 1</ sub > true, effect size 0.2
- C - H< sub > 1</ sub > true, effect size 0.4
- D - H< sub > 1</ sub > true, effect size -0.2
- E - H< sub > 1</ sub > true, effect size -0.4
## Scenarios with DIF
| | **J** | | **Epsilon** | | **Nb items DIF** | | | **DIF size** | |
| --------------- | ----- | ----- | ----- | ----- | ---------------- | ----- | ----- | -------------- | ------- |
| | **4** | **7** | **0.2** | **0.5** | **1** | **2** | **3** | **0.3** | **0.5** |
| **Scenario 105** | X | | X | | X | | | X | |
| **Scenario 106** | X | | | X | X | | | X | |
| **Scenario 107** | X | | X | | X | | | | X |
| **Scenario 108** | X | | | X | X | | | | X |
| **Scenario 109** | X | | X | | | X | | X | |
| **Scenario 110** | X | | | X | | X | | X | |
| **Scenario 111** | X | | X | | | X | | | X |
| **Scenario 112** | X | | | X | | X | | | X |
| **Scenario 113** | | X | X | | | X | | X | |
| **Scenario 114** | | X | | X | | X | | X | |
| **Scenario 115** | | X | X | | | X | | | X |
| **Scenario 116** | | X | | X | | X | | | X |
| **Scenario 117** | | X | X | | | | X | X | |
| **Scenario 118** | | X | | X | | | X | X | |
| **Scenario 119** | | X | X | | | | X | | X |
| **Scenario 120** | | X | | X | | | X | | X |
Each scenario has 7 subscenarios:
- A - H< sub > 0</ sub > true, DIF -0.3/0.5 on traitement group
- **MPR example**
- B - H< sub > 1</ sub > true, effect size 0.2, DIF +0.3/0.5 on traitement group
- **BDI example**
- C - H< sub > 1</ sub > true, effect size 0.2, DIF -0.3/0.5 on traitement group
- **Dentistry example**
- D - H< sub > 1</ sub > true, effect size 0.4, DIF +0.3/0.5 on traitement group
- **BDI example**
- E - H< sub > 1</ sub > true, effect size 0.4, DIF -0.3/0.5 on traitement group
- **Dentistry example**
- F - H< sub > 1</ sub > true, effect size -0.2, DIF -0.3/0.5 on traitement group
- **Adverse effects example**
- G - H< sub > 1</ sub > true, effect size -0.4, DIF -0.3/0.5 on traitement group
- **Adverse effects example**
## Confusion simulation details
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 *Confounders* .
**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).
