2025_PCI_causal

class: title-slide, center, middle, inverse
background-image: url(./figs/logo.png)
background-position: 95% 95%
background-size: 28%, 25%

# Conceptualizing .yellow[survival] analysis through a .yellow[causal inference] framework

.center[

L. Paloma Rojas-Saunero MD, PhD
<br>
Postdoctoral scholar
<br>
Department of Epidemiology, UCLA

]

???

---
<style>.xe__progress-bar__container {
  top:0;
  opacity: 1;
  position:absolute;
  right:0;
  left: 0;
}
.xe__progress-bar {
  height: 0.25em;
  background-color: #ffb81c;
  width: calc(var(--slide-current) / var(--slide-total) * 100%);
}
.remark-visible .xe__progress-bar {
  animation: xe__progress-bar__wipe 200ms forwards;
  animation-timing-function: cubic-bezier(.86,0,.07,1);
}
@keyframes xe__progress-bar__wipe {
  0% { width: calc(var(--slide-previous) / var(--slide-total) * 100%); }
  100% { width: calc(var(--slide-current) / var(--slide-total) * 100%); }
}</style>

.middle[
.center[<img src=./figs/survival_meme.jpg width="70%"/>]
]

---
## Risk / Incidence proportion

.pull-left[
- The probability of having an incident event, `$Y$`, in a **fixed period of time **, `$k$`

- `$Pr[Y_k =1]$` = `$A/N$`, where:

+ `$N$` = people free of event at start of follow-up

+ `$A$` = people experiencing the event
]

.pull-right[

]

---
## Cumulative incidence

.pull-left[

- In the ideal scenario (no drop out)

- `$Pr[T ≤ t]$`, where:

- `$T$` = time to event

- `$t$` = specific time point of interest

- At baseline, cumulative incidence is 0, `$Pr[T ≤ 0] = 0$`

- Increases monotonically (or remains constant)
]

.pull-right[

]

---
## What makes a valid estimation of _risk_?

- Target and study population clearly specified

- Defined observation period at risk

→ e.g., lifetime, 10-year, 2-week

- Complete follow-up over the observation period

- New (incident) cases only

- Starting point (baseline) is the same for all participants

---
## Examples from other fields

- **Loan default** <br><br> Risk of defaulting on a loan within the first year of the credit

- **Student dropout** <br><br> Risk of student drop out within the first 4 semesters

- **Customer churn** <br><br> Risk of subscription canceling after the 30-days free trial

- **Soccer** <br><br> Risk of a player scoring a goal within in the first half

---
## Censoring

.pull-left[

- By design, we want to prevent participant's drop out

- But in real-world data, people are loss to follow-up and drop out during the study period

- *A censoring event* makes the event of interest _unknown_ at all future time points

- Censoring is the *key* feature of time-to-event analysis

]

.pull-right[

]

- *Time of end of study* is often defined as *administrative censoring* but I will refrain from using this jargon for now

---
## Incidence rates

.pull-left[

Frequency with which incident events occur within a given amount of follow-up time.
<br> 
`$A/PT$`
<br>
where:
  + `$A$` = Number of incident cases

+ `$PT$` = "Person-time", total amount of time that all individuals were at risk of the outcome
]

.pull-right[

]

---
## Incidence rates

.pull-left[

Frequency with which incident events occur within a given amount of follow-up time.
<br> 
`$A/PT$`
<br>
where:
  + `$A$` = Number of incident cases

+ `$PT$` = "Person-time", total amount of time that all individuals were at risk of the outcome
]

.pull-right[

**Comparison of incidence rates across groups is tricky**

<img src=./figs/person_time_westreich.jpg width="100%"/>
]

---
## Hazards

.pull-left[

- At any given time, the probability of experiencing the event of interest, `$Y$`, in the next interval among individuals who had not yet experienced the event by the start of the interval

- **Discrete hazards**

- `$Pr[T = t|T > t − 1]$`
  
  - `$Pr [Y_k = 1|Y_{k−1} = 0]$`

- They can rise, fall, or vary non-monotonically across time points

]

.pull-right[

]
???

---
## Key differences between risks, rates, hazards

**Risks**
- Denominator: Number of individuals at baseline
- Numerator: Number of events ocurring over a fixed period of time

**Rates**
- Denominator: Total person-time at risk (accumulated time contributed by all individuals)
- Numerator: Number of events occurring over the total person-time

**Hazard**
- Denominator: Number of individuals at risk by the start of a time interval
- Numerator: Number of events ocurring in the next time interval

---
## Cox proportional hazard models

.pull-left[
- In 1972, Sir David Cox introduced a semi-parametric regression method for "right-censored survival data"

- It estimates the ratio of the hazards of the event between two groups.

- "The" hazard ratio (HR) from the model output represents the weighted average of the time-varying hazard ratios over the entire follow-up

- Relies on the proportional hazards assumption, the HR is constant from start until end of follow-up

]

.pull-right[

]

---
## The hazard of hazard ratios

.pull-left[
<img src=./figs/hazard_dag.jpg width="100%"/>
]

.pull-right[

- Hazard ratios have an inherent issue of selection bias

- The proportional hazard assumption is often unrealistic

- When the treatment effect changes over time
  
  - When disease susceptibility varies between individuals
]

---
## The hazard of hazard ratios

.center[<img src=./figs/hazard.png width="70%"/>]

.footnote[Stensrud. European Heart Journal. 2019]

???
Figure 1People have different disease risks. Given an effective treatment, high-risk individuals are likely to be depleted from the study population, and the depletion is expected to be larger in the placebo group than in the treatment group. This schematic drawing illustrates the built-in selection bias in population-level hazard ratios: by definition, the population-level hazard ratio at a given time point is based on individuals who survived up to that time point, thereby it is a comparison between the unbalanced groups.

---

.center[
<img src=./figs/poole_quote.jpg width="80%"/>
]

.footnote[Poole. Epidemiologic Perspectives & Innovations. 2010]

---
## From hazard to risks

- Hazard models are useful to estimate risks when censoring is present

- In **discrete time**, the risk at time `$k+1$` is the probability of not having the event up to time `$k$`, and then having the event at time `$k+1$`:

`$$\Pr[Y_{k+1} = 1] = \left( \prod_{j=1}^{k} (1 - h_j) \right) \cdot h_{k+1}$$`

Where

`$$h_j = \Pr[Y_j = 1 \mid Y_{j-1} = 0]$$`

---
## Implications of censoring

.pull-left[

- Statistical literature

- Censoring events are independent of the event of interest
  
  - Censoring events are uninformative
  
- Causal/Counterfactual literature

- Counterfactual scenario where censoring events were eliminated
  
  - `$Pr[Y_{k+1}] = Pr[Y^{\overline{\text{LTFU}} = 0}_{k+1}]$`
  
  -  `$Pr[Y^{a = 1, \overline{\text{LTFU}} = 0}_{k+1}]$` - `$Pr[Y^{a = 0, \overline{\text{LTFU}} = 0}_{k+1}]$`

]

.pull-right[

.center[

]]

---
## Competing events

.pull-left[

- Events that _preclude*_ the outcome of interest.

*to prevent the existence, or occurrence of; make impossible

<img src=./figs/compevents_dag.jpg width="100%"/>
]

.pull-right[

]

---
## Competing events happen everywhere

- **Loan default**: Risk of defaulting on a loan within a given time period
  - Refinancing, paying in full

- **Student dropout**: Risk of student drop out within a specific number of semesters
  - Transfer to another institution

- **Customer churn** Risk of subscription canceling after the 30-days free trial
  - Switching to a different plan

- **Soccer** Risk of a player scoring a goal within in the first half
  - Receiving a red card without scoring

---
## Competing events & estimands, _Young 2020_

<div id="bsphqrvcgb" style="padding-left:0px;padding-right:0px;padding-top:10px;padding-bottom:10px;overflow-x:auto;overflow-y:auto;width:auto;height:auto;">
<style>#bsphqrvcgb table {
  font-family: system-ui, 'Segoe UI', Roboto, Helvetica, Arial, sans-serif, 'Apple Color Emoji', 'Segoe UI Emoji', 'Segoe UI Symbol', 'Noto Color Emoji';
  -webkit-font-smoothing: antialiased;
  -moz-osx-font-smoothing: grayscale;
}

#bsphqrvcgb thead, #bsphqrvcgb tbody, #bsphqrvcgb tfoot, #bsphqrvcgb tr, #bsphqrvcgb td, #bsphqrvcgb th {
  border-style: none;
}

#bsphqrvcgb p {
  margin: 0;
  padding: 0;
}

#bsphqrvcgb .gt_table {
  display: table;
  border-collapse: collapse;
  line-height: normal;
  margin-left: auto;
  margin-right: auto;
  color: #333333;
  font-size: 20px;
  font-weight: normal;
  font-style: normal;
  background-color: #FFFFFF;
  width: auto;
  border-top-style: solid;
  border-top-width: 2px;
  border-top-color: #A8A8A8;
  border-right-style: none;
  border-right-width: 2px;
  border-right-color: #D3D3D3;
  border-bottom-style: solid;
  border-bottom-width: 2px;
  border-bottom-color: #A8A8A8;
  border-left-style: none;
  border-left-width: 2px;
  border-left-color: #D3D3D3;
}

#bsphqrvcgb .gt_caption {
  padding-top: 4px;
  padding-bottom: 4px;
}

#bsphqrvcgb .gt_title {
  color: #333333;
  font-size: 125%;
  font-weight: initial;
  padding-top: 4px;
  padding-bottom: 4px;
  padding-left: 5px;
  padding-right: 5px;
  border-bottom-color: #FFFFFF;
  border-bottom-width: 0;
}

#bsphqrvcgb .gt_subtitle {
  color: #333333;
  font-size: 85%;
  font-weight: initial;
  padding-top: 3px;
  padding-bottom: 5px;
  padding-left: 5px;
  padding-right: 5px;
  border-top-color: #FFFFFF;
  border-top-width: 0;
}

#bsphqrvcgb .gt_heading {
  background-color: #FFFFFF;
  text-align: center;
  border-bottom-color: #FFFFFF;
  border-left-style: none;
  border-left-width: 1px;
  border-left-color: #D3D3D3;
  border-right-style: none;
  border-right-width: 1px;
  border-right-color: #D3D3D3;
}

#bsphqrvcgb .gt_bottom_border {
  border-bottom-style: solid;
  border-bottom-width: 2px;
  border-bottom-color: #D3D3D3;
}

#bsphqrvcgb .gt_col_headings {
  border-top-style: solid;
  border-top-width: 2px;
  border-top-color: #D3D3D3;
  border-bottom-style: solid;
  border-bottom-width: 2px;
  border-bottom-color: #D3D3D3;
  border-left-style: none;
  border-left-width: 1px;
  border-left-color: #D3D3D3;
  border-right-style: none;
  border-right-width: 1px;
  border-right-color: #D3D3D3;
}

#bsphqrvcgb .gt_col_heading {
  color: #333333;
  background-color: #FFFFFF;
  font-size: 100%;
  font-weight: normal;
  text-transform: inherit;
  border-left-style: none;
  border-left-width: 1px;
  border-left-color: #D3D3D3;
  border-right-style: none;
  border-right-width: 1px;
  border-right-color: #D3D3D3;
  vertical-align: bottom;
  padding-top: 5px;
  padding-bottom: 6px;
  padding-left: 5px;
  padding-right: 5px;
  overflow-x: hidden;
}

#bsphqrvcgb .gt_column_spanner_outer {
  color: #333333;
  background-color: #FFFFFF;
  font-size: 100%;
  font-weight: normal;
  text-transform: inherit;
  padding-top: 0;
  padding-bottom: 0;
  padding-left: 4px;
  padding-right: 4px;
}

#bsphqrvcgb .gt_column_spanner_outer:first-child {
  padding-left: 0;
}

#bsphqrvcgb .gt_column_spanner_outer:last-child {
  padding-right: 0;
}

#bsphqrvcgb .gt_column_spanner {
  border-bottom-style: solid;
  border-bottom-width: 2px;
  border-bottom-color: #D3D3D3;
  vertical-align: bottom;
  padding-top: 5px;
  padding-bottom: 5px;
  overflow-x: hidden;
  display: inline-block;
  width: 100%;
}

#bsphqrvcgb .gt_spanner_row {
  border-bottom-style: hidden;
}

#bsphqrvcgb .gt_group_heading {
  padding-top: 8px;
  padding-bottom: 8px;
  padding-left: 5px;
  padding-right: 5px;
  color: #333333;
  background-color: #FFFFFF;
  font-size: 100%;
  font-weight: initial;
  text-transform: inherit;
  border-top-style: solid;
  border-top-width: 2px;
  border-top-color: #D3D3D3;
  border-bottom-style: solid;
  border-bottom-width: 2px;
  border-bottom-color: #D3D3D3;
  border-left-style: none;
  border-left-width: 1px;
  border-left-color: #D3D3D3;
  border-right-style: none;
  border-right-width: 1px;
  border-right-color: #D3D3D3;
  vertical-align: middle;
  text-align: left;
}

#bsphqrvcgb .gt_empty_group_heading {
  padding: 0.5px;
  color: #333333;
  background-color: #FFFFFF;
  font-size: 100%;
  font-weight: initial;
  border-top-style: solid;
  border-top-width: 2px;
  border-top-color: #D3D3D3;
  border-bottom-style: solid;
  border-bottom-width: 2px;
  border-bottom-color: #D3D3D3;
  vertical-align: middle;
}

#bsphqrvcgb .gt_from_md > :first-child {
  margin-top: 0;
}

#bsphqrvcgb .gt_from_md > :last-child {
  margin-bottom: 0;
}

#bsphqrvcgb .gt_row {
  padding-top: 8px;
  padding-bottom: 8px;
  padding-left: 5px;
  padding-right: 5px;
  margin: 10px;
  border-top-style: solid;
  border-top-width: 1px;
  border-top-color: #D3D3D3;
  border-left-style: none;
  border-left-width: 1px;
  border-left-color: #D3D3D3;
  border-right-style: none;
  border-right-width: 1px;
  border-right-color: #D3D3D3;
  vertical-align: middle;
  overflow-x: hidden;
}

#bsphqrvcgb .gt_stub {
  color: #333333;
  background-color: #FFFFFF;
  font-size: 100%;
  font-weight: initial;
  text-transform: inherit;
  border-right-style: solid;
  border-right-width: 2px;
  border-right-color: #D3D3D3;
  padding-left: 5px;
  padding-right: 5px;
}

#bsphqrvcgb .gt_stub_row_group {
  color: #333333;
  background-color: #FFFFFF;
  font-size: 100%;
  font-weight: initial;
  text-transform: inherit;
  border-right-style: solid;
  border-right-width: 2px;
  border-right-color: #D3D3D3;
  padding-left: 5px;
  padding-right: 5px;
  vertical-align: top;
}

#bsphqrvcgb .gt_row_group_first td {
  border-top-width: 2px;
}

#bsphqrvcgb .gt_row_group_first th {
  border-top-width: 2px;
}

#bsphqrvcgb .gt_summary_row {
  color: #333333;
  background-color: #FFFFFF;
  text-transform: inherit;
  padding-top: 8px;
  padding-bottom: 8px;
  padding-left: 5px;
  padding-right: 5px;
}

#bsphqrvcgb .gt_first_summary_row {
  border-top-style: solid;
  border-top-color: #D3D3D3;
}

#bsphqrvcgb .gt_first_summary_row.thick {
  border-top-width: 2px;
}

#bsphqrvcgb .gt_last_summary_row {
  padding-top: 8px;
  padding-bottom: 8px;
  padding-left: 5px;
  padding-right: 5px;
  border-bottom-style: solid;
  border-bottom-width: 2px;
  border-bottom-color: #D3D3D3;
}

#bsphqrvcgb .gt_grand_summary_row {
  color: #333333;
  background-color: #FFFFFF;
  text-transform: inherit;
  padding-top: 8px;
  padding-bottom: 8px;
  padding-left: 5px;
  padding-right: 5px;
}

#bsphqrvcgb .gt_first_grand_summary_row {
  padding-top: 8px;
  padding-bottom: 8px;
  padding-left: 5px;
  padding-right: 5px;
  border-top-style: double;
  border-top-width: 6px;
  border-top-color: #D3D3D3;
}

#bsphqrvcgb .gt_last_grand_summary_row_top {
  padding-top: 8px;
  padding-bottom: 8px;
  padding-left: 5px;
  padding-right: 5px;
  border-bottom-style: double;
  border-bottom-width: 6px;
  border-bottom-color: #D3D3D3;
}

#bsphqrvcgb .gt_striped {
  background-color: rgba(128, 128, 128, 0.05);
}

#bsphqrvcgb .gt_table_body {
  border-top-style: solid;
  border-top-width: 2px;
  border-top-color: #D3D3D3;
  border-bottom-style: solid;
  border-bottom-width: 2px;
  border-bottom-color: #D3D3D3;
}

#bsphqrvcgb .gt_footnotes {
  color: #333333;
  background-color: #FFFFFF;
  border-bottom-style: none;
  border-bottom-width: 2px;
  border-bottom-color: #D3D3D3;
  border-left-style: none;
  border-left-width: 2px;
  border-left-color: #D3D3D3;
  border-right-style: none;
  border-right-width: 2px;
  border-right-color: #D3D3D3;
}

#bsphqrvcgb .gt_footnote {
  margin: 0px;
  font-size: 90%;
  padding-top: 4px;
  padding-bottom: 4px;
  padding-left: 5px;
  padding-right: 5px;
}

#bsphqrvcgb .gt_sourcenotes {
  color: #333333;
  background-color: #FFFFFF;
  border-bottom-style: none;
  border-bottom-width: 2px;
  border-bottom-color: #D3D3D3;
  border-left-style: none;
  border-left-width: 2px;
  border-left-color: #D3D3D3;
  border-right-style: none;
  border-right-width: 2px;
  border-right-color: #D3D3D3;
}

#bsphqrvcgb .gt_sourcenote {
  font-size: 90%;
  padding-top: 4px;
  padding-bottom: 4px;
  padding-left: 5px;
  padding-right: 5px;
}

#bsphqrvcgb .gt_left {
  text-align: left;
}

#bsphqrvcgb .gt_center {
  text-align: center;
}

#bsphqrvcgb .gt_right {
  text-align: right;
  font-variant-numeric: tabular-nums;
}

#bsphqrvcgb .gt_font_normal {
  font-weight: normal;
}

#bsphqrvcgb .gt_font_bold {
  font-weight: bold;
}

#bsphqrvcgb .gt_font_italic {
  font-style: italic;
}

#bsphqrvcgb .gt_super {
  font-size: 65%;
}

#bsphqrvcgb .gt_footnote_marks {
  font-size: 75%;
  vertical-align: 0.4em;
  position: initial;
}

#bsphqrvcgb .gt_asterisk {
  font-size: 100%;
  vertical-align: 0;
}

#bsphqrvcgb .gt_indent_1 {
  text-indent: 5px;
}

#bsphqrvcgb .gt_indent_2 {
  text-indent: 10px;
}

#bsphqrvcgb .gt_indent_3 {
  text-indent: 15px;
}

#bsphqrvcgb .gt_indent_4 {
  text-indent: 20px;
}

#bsphqrvcgb .gt_indent_5 {
  text-indent: 25px;
}
</style>
<table class="gt_table" style="table-layout: fixed;; width: 0px" data-quarto-disable-processing="false" data-quarto-bootstrap="false">
  <colgroup>
    <col style="width:350px;"/>
    <col style="width:400px;"/>
    <col style="width:400px;"/>
  </colgroup>
  <thead>
    <tr class="gt_col_headings">
      <th class="gt_col_heading gt_columns_bottom_border gt_left" rowspan="1" colspan="1" style="text-align: center; font-weight: bold;" scope="col" id="Definition">Definition</th>
      <th class="gt_col_heading gt_columns_bottom_border gt_left" rowspan="1" colspan="1" style="text-align: center; font-weight: bold;" scope="col" id="Description">Description</th>
      <th class="gt_col_heading gt_columns_bottom_border gt_left" rowspan="1" colspan="1" style="text-align: center; font-weight: bold;" scope="col" id="Statistical Literature Terminology">Statistical Literature Terminology</th>
    </tr>
  </thead>
  <tbody class="gt_table_body">
    <tr><td headers="Definition" class="gt_row gt_left">$$\Pr[Y^{\bar{d} = \bar{0}}_{k+1} = 1]$$</td>
<td headers="Description" class="gt_row gt_left">Risk under elimination of competing events</td>
<td headers="Terminology" class="gt_row gt_left">Marginal cumulative incidence, net risk</td></tr>
    <tr><td headers="Definition" class="gt_row gt_left">$$\Pr[Y_{k+1} = 1]$$</td>
<td headers="Description" class="gt_row gt_left">Risk without elimination of competing events</td>
<td headers="Terminology" class="gt_row gt_left">Subdistribution function, cause-specific cumulative incidence, crude risk</td></tr>
    <tr><td headers="Definition" class="gt_row gt_left">$$\Pr[Y^{\bar{d} = \bar{0}}_{k+1} = 1 \mid Y^{\bar{d} = \bar{0}}_k = 0]$$</td>
<td headers="Description" class="gt_row gt_left">Hazard under elimination of competing events</td>
<td headers="Terminology" class="gt_row gt_left">Marginal hazard</td></tr>
    <tr><td headers="Definition" class="gt_row gt_left">$$\Pr[Y_{k+1} = 1 \mid Y_k = 0]$$</td>
<td headers="Description" class="gt_row gt_left">Hazard without elimination of competing events</td>
<td headers="Terminology" class="gt_row gt_left">Subdistribution hazard</td></tr>
    <tr><td headers="Definition" class="gt_row gt_left">$$\Pr[Y_{k+1} = 1 \mid D_{k+1} = Y_k = 0]$$</td>
<td headers="Description" class="gt_row gt_left">Hazard conditioned on competing events</td>
<td headers="Terminology" class="gt_row gt_left">Cause-specific hazard</td></tr>
  </tbody>
  
  
</table>
</div>

---
## Causal effects

- Contrast of (counterfactual) outcome distributions in the **same individuals** but under **different treatments**.

- The only explanation for a difference is the intervention, not comparing different individuals

---
## Quitting smoking and 20-year dementia risk

.center[

]

.footnote[**C**: Shared risk factors]

---
## Total effect

<br>

.center[
.middle[

What is the risk** of dementia at 20 years of follow-up had all individuals stopped smoking, compared to had all individuals continued smoking?

`$$Pr[Y_{20}^{a = 1} = 1] - Pr[Y_{20}^{a = 0} = 1]$$`

]]

** _Cause-specific cumulative incidence_ or _crude_ risk

---
## Controlled direct effect

<br>

.center[
.middle[

What is the risk** of dementia at 20 years of follow-up had all individuals stopped smoking _and not died_ during the study period, compared to had all individuals continued smoking _and not died_ ?

`$$Pr[Y_{20}^{a = 1, d_{19} = 0} = 1] - Pr[Y_{20}^{a = 0, d_{19} = 0} = 1]$$`

]
]

** _Marginal_ or _net_ risk

---
## Controlled direct effect

<br>

.center[.middle[

What is the risk** of dementia at 20 years of follow-up had all individuals stopped smoking _and not died_ during the study period, compared to had all individuals continued smoking _and not died_ ?

`$$Pr[Y_{20}^{a = 1, d_{19} = 0} = 1] - Pr[Y_{20}^{a = 0, d_{19} = 0} = 1]$$`

]
]

** _Marginal_ or _net_ risk

???
So up to now, we can see that these two questions have very different interpretations and rely on different assumptions

---
background-image: url(./figs/rs_logo.jpg)
background-position: 95% 95%
background-size: 12%

## Quitting smoking on dementia risk at 20 years

.center[
<img src=./figs/smoke_aje.png width="70%"/>]

???
What we see is that had we asked a question about the CDE and use appropiate methods, we would see that smoking cessation prevents dementia risk.
Had we asked a question about the total effect, smoking cessation increases your risk of dementia, but again this is not bias, but rather the unbiased result for a question that does not satisfy our interest.

---
## Estimand choice is context- and field-specific.

.center[
<img src=./figs/ventilator_mortality.jpg width="70%"/>]

---
## Identifiability assumptions for death

<div id="mnwnbqlpzy" style="padding-left:0px;padding-right:0px;padding-top:10px;padding-bottom:10px;overflow-x:auto;overflow-y:auto;width:auto;height:auto;">
<style>#mnwnbqlpzy table {
  font-family: system-ui, 'Segoe UI', Roboto, Helvetica, Arial, sans-serif, 'Apple Color Emoji', 'Segoe UI Emoji', 'Segoe UI Symbol', 'Noto Color Emoji';
  -webkit-font-smoothing: antialiased;
  -moz-osx-font-smoothing: grayscale;
}

#mnwnbqlpzy thead, #mnwnbqlpzy tbody, #mnwnbqlpzy tfoot, #mnwnbqlpzy tr, #mnwnbqlpzy td, #mnwnbqlpzy th {
  border-style: none;
}

#mnwnbqlpzy p {
  margin: 0;
  padding: 0;
}

#mnwnbqlpzy .gt_table {
  display: table;
  border-collapse: collapse;
  line-height: normal;
  margin-left: auto;
  margin-right: auto;
  color: #333333;
  font-size: 22px;
  font-weight: normal;
  font-style: normal;
  background-color: #FFFFFF;
  width: auto;
  border-top-style: solid;
  border-top-width: 2px;
  border-top-color: #A8A8A8;
  border-right-style: none;
  border-right-width: 2px;
  border-right-color: #D3D3D3;
  border-bottom-style: solid;
  border-bottom-width: 2px;
  border-bottom-color: #A8A8A8;
  border-left-style: none;
  border-left-width: 2px;
  border-left-color: #D3D3D3;
}

#mnwnbqlpzy .gt_caption {
  padding-top: 4px;
  padding-bottom: 4px;
}

#mnwnbqlpzy .gt_title {
  color: #333333;
  font-size: 125%;
  font-weight: initial;
  padding-top: 4px;
  padding-bottom: 4px;
  padding-left: 5px;
  padding-right: 5px;
  border-bottom-color: #FFFFFF;
  border-bottom-width: 0;
}

#mnwnbqlpzy .gt_subtitle {
  color: #333333;
  font-size: 85%;
  font-weight: initial;
  padding-top: 3px;
  padding-bottom: 5px;
  padding-left: 5px;
  padding-right: 5px;
  border-top-color: #FFFFFF;
  border-top-width: 0;
}

#mnwnbqlpzy .gt_heading {
  background-color: #FFFFFF;
  text-align: center;
  border-bottom-color: #FFFFFF;
  border-left-style: none;
  border-left-width: 1px;
  border-left-color: #D3D3D3;
  border-right-style: none;
  border-right-width: 1px;
  border-right-color: #D3D3D3;
}

#mnwnbqlpzy .gt_bottom_border {
  border-bottom-style: solid;
  border-bottom-width: 2px;
  border-bottom-color: #D3D3D3;
}

#mnwnbqlpzy .gt_col_headings {
  border-top-style: solid;
  border-top-width: 2px;
  border-top-color: #D3D3D3;
  border-bottom-style: solid;
  border-bottom-width: 2px;
  border-bottom-color: #D3D3D3;
  border-left-style: none;
  border-left-width: 1px;
  border-left-color: #D3D3D3;
  border-right-style: none;
  border-right-width: 1px;
  border-right-color: #D3D3D3;
}

#mnwnbqlpzy .gt_col_heading {
  color: #333333;
  background-color: #FFFFFF;
  font-size: 100%;
  font-weight: normal;
  text-transform: inherit;
  border-left-style: none;
  border-left-width: 1px;
  border-left-color: #D3D3D3;
  border-right-style: none;
  border-right-width: 1px;
  border-right-color: #D3D3D3;
  vertical-align: bottom;
  padding-top: 5px;
  padding-bottom: 6px;
  padding-left: 5px;
  padding-right: 5px;
  overflow-x: hidden;
}

#mnwnbqlpzy .gt_column_spanner_outer {
  color: #333333;
  background-color: #FFFFFF;
  font-size: 100%;
  font-weight: normal;
  text-transform: inherit;
  padding-top: 0;
  padding-bottom: 0;
  padding-left: 4px;
  padding-right: 4px;
}

#mnwnbqlpzy .gt_column_spanner_outer:first-child {
  padding-left: 0;
}

#mnwnbqlpzy .gt_column_spanner_outer:last-child {
  padding-right: 0;
}

#mnwnbqlpzy .gt_column_spanner {
  border-bottom-style: solid;
  border-bottom-width: 2px;
  border-bottom-color: #D3D3D3;
  vertical-align: bottom;
  padding-top: 5px;
  padding-bottom: 5px;
  overflow-x: hidden;
  display: inline-block;
  width: 100%;
}

#mnwnbqlpzy .gt_spanner_row {
  border-bottom-style: hidden;
}

#mnwnbqlpzy .gt_group_heading {
  padding-top: 8px;
  padding-bottom: 8px;
  padding-left: 5px;
  padding-right: 5px;
  color: #333333;
  background-color: #FFFFFF;
  font-size: 100%;
  font-weight: initial;
  text-transform: inherit;
  border-top-style: solid;
  border-top-width: 2px;
  border-top-color: #D3D3D3;
  border-bottom-style: solid;
  border-bottom-width: 2px;
  border-bottom-color: #D3D3D3;
  border-left-style: none;
  border-left-width: 1px;
  border-left-color: #D3D3D3;
  border-right-style: none;
  border-right-width: 1px;
  border-right-color: #D3D3D3;
  vertical-align: middle;
  text-align: left;
}

#mnwnbqlpzy .gt_empty_group_heading {
  padding: 0.5px;
  color: #333333;
  background-color: #FFFFFF;
  font-size: 100%;
  font-weight: initial;
  border-top-style: solid;
  border-top-width: 2px;
  border-top-color: #D3D3D3;
  border-bottom-style: solid;
  border-bottom-width: 2px;
  border-bottom-color: #D3D3D3;
  vertical-align: middle;
}

#mnwnbqlpzy .gt_from_md > :first-child {
  margin-top: 0;
}

#mnwnbqlpzy .gt_from_md > :last-child {
  margin-bottom: 0;
}

#mnwnbqlpzy .gt_row {
  padding-top: 8px;
  padding-bottom: 8px;
  padding-left: 5px;
  padding-right: 5px;
  margin: 10px;
  border-top-style: solid;
  border-top-width: 1px;
  border-top-color: #D3D3D3;
  border-left-style: none;
  border-left-width: 1px;
  border-left-color: #D3D3D3;
  border-right-style: none;
  border-right-width: 1px;
  border-right-color: #D3D3D3;
  vertical-align: middle;
  overflow-x: hidden;
}

#mnwnbqlpzy .gt_stub {
  color: #333333;
  background-color: #FFFFFF;
  font-size: 100%;
  font-weight: initial;
  text-transform: inherit;
  border-right-style: solid;
  border-right-width: 2px;
  border-right-color: #D3D3D3;
  padding-left: 5px;
  padding-right: 5px;
}

#mnwnbqlpzy .gt_stub_row_group {
  color: #333333;
  background-color: #FFFFFF;
  font-size: 100%;
  font-weight: initial;
  text-transform: inherit;
  border-right-style: solid;
  border-right-width: 2px;
  border-right-color: #D3D3D3;
  padding-left: 5px;
  padding-right: 5px;
  vertical-align: top;
}

#mnwnbqlpzy .gt_row_group_first td {
  border-top-width: 2px;
}

#mnwnbqlpzy .gt_row_group_first th {
  border-top-width: 2px;
}

#mnwnbqlpzy .gt_summary_row {
  color: #333333;
  background-color: #FFFFFF;
  text-transform: inherit;
  padding-top: 8px;
  padding-bottom: 8px;
  padding-left: 5px;
  padding-right: 5px;
}

#mnwnbqlpzy .gt_first_summary_row {
  border-top-style: solid;
  border-top-color: #D3D3D3;
}

#mnwnbqlpzy .gt_first_summary_row.thick {
  border-top-width: 2px;
}

#mnwnbqlpzy .gt_last_summary_row {
  padding-top: 8px;
  padding-bottom: 8px;
  padding-left: 5px;
  padding-right: 5px;
  border-bottom-style: solid;
  border-bottom-width: 2px;
  border-bottom-color: #D3D3D3;
}

#mnwnbqlpzy .gt_grand_summary_row {
  color: #333333;
  background-color: #FFFFFF;
  text-transform: inherit;
  padding-top: 8px;
  padding-bottom: 8px;
  padding-left: 5px;
  padding-right: 5px;
}

#mnwnbqlpzy .gt_first_grand_summary_row {
  padding-top: 8px;
  padding-bottom: 8px;
  padding-left: 5px;
  padding-right: 5px;
  border-top-style: double;
  border-top-width: 6px;
  border-top-color: #D3D3D3;
}

#mnwnbqlpzy .gt_last_grand_summary_row_top {
  padding-top: 8px;
  padding-bottom: 8px;
  padding-left: 5px;
  padding-right: 5px;
  border-bottom-style: double;
  border-bottom-width: 6px;
  border-bottom-color: #D3D3D3;
}

#mnwnbqlpzy .gt_striped {
  background-color: rgba(128, 128, 128, 0.05);
}

#mnwnbqlpzy .gt_table_body {
  border-top-style: solid;
  border-top-width: 2px;
  border-top-color: #D3D3D3;
  border-bottom-style: solid;
  border-bottom-width: 2px;
  border-bottom-color: #D3D3D3;
}

#mnwnbqlpzy .gt_footnotes {
  color: #333333;
  background-color: #FFFFFF;
  border-bottom-style: none;
  border-bottom-width: 2px;
  border-bottom-color: #D3D3D3;
  border-left-style: none;
  border-left-width: 2px;
  border-left-color: #D3D3D3;
  border-right-style: none;
  border-right-width: 2px;
  border-right-color: #D3D3D3;
}

#mnwnbqlpzy .gt_footnote {
  margin: 0px;
  font-size: 90%;
  padding-top: 4px;
  padding-bottom: 4px;
  padding-left: 5px;
  padding-right: 5px;
}

#mnwnbqlpzy .gt_sourcenotes {
  color: #333333;
  background-color: #FFFFFF;
  border-bottom-style: none;
  border-bottom-width: 2px;
  border-bottom-color: #D3D3D3;
  border-left-style: none;
  border-left-width: 2px;
  border-left-color: #D3D3D3;
  border-right-style: none;
  border-right-width: 2px;
  border-right-color: #D3D3D3;
}

#mnwnbqlpzy .gt_sourcenote {
  font-size: 90%;
  padding-top: 4px;
  padding-bottom: 4px;
  padding-left: 5px;
  padding-right: 5px;
}

#mnwnbqlpzy .gt_left {
  text-align: left;
}

#mnwnbqlpzy .gt_center {
  text-align: center;
}

#mnwnbqlpzy .gt_right {
  text-align: right;
  font-variant-numeric: tabular-nums;
}

#mnwnbqlpzy .gt_font_normal {
  font-weight: normal;
}

#mnwnbqlpzy .gt_font_bold {
  font-weight: bold;
}

#mnwnbqlpzy .gt_font_italic {
  font-style: italic;
}

#mnwnbqlpzy .gt_super {
  font-size: 65%;
}

#mnwnbqlpzy .gt_footnote_marks {
  font-size: 75%;
  vertical-align: 0.4em;
  position: initial;
}

#mnwnbqlpzy .gt_asterisk {
  font-size: 100%;
  vertical-align: 0;
}

#mnwnbqlpzy .gt_indent_1 {
  text-indent: 5px;
}

#mnwnbqlpzy .gt_indent_2 {
  text-indent: 10px;
}

#mnwnbqlpzy .gt_indent_3 {
  text-indent: 15px;
}

#mnwnbqlpzy .gt_indent_4 {
  text-indent: 20px;
}

#mnwnbqlpzy .gt_indent_5 {
  text-indent: 25px;
}
</style>
<table class="gt_table" style="table-layout: fixed;; width: 0px" data-quarto-disable-processing="false" data-quarto-bootstrap="false">
  <colgroup>
    <col style="width:200px;"/>
    <col style="width:150px;"/>
    <col style="width:350px;"/>
  </colgroup>
  <thead>
    <tr class="gt_col_headings">
      <th class="gt_col_heading gt_columns_bottom_border gt_left" rowspan="1" colspan="1" scope="col" id="&lt;b&gt; Assumption &lt;/b&gt;"><b> Assumption </b></th>
      <th class="gt_col_heading gt_columns_bottom_border gt_left" rowspan="1" colspan="1" scope="col" id="&lt;b&gt; Total Effect &lt;/b&gt;"><b> Total Effect </b></th>
      <th class="gt_col_heading gt_columns_bottom_border gt_left" rowspan="1" colspan="1" scope="col" id="&lt;b&gt; Controlled direct effect &lt;/b&gt;"><b> Controlled direct effect </b></th>
    </tr>
  </thead>
  <tbody class="gt_table_body">
    <tr><td headers="<b> Assumption </b>" class="gt_row gt_left"><b> Exchangeability </b></td>
<td headers="<b> Total Effect </b>" class="gt_row gt_left">Not needed</td>
<td headers="<b> Controlled direct effect </b>" class="gt_row gt_left">Death is independent of future outcomes had everyone followed A = a and death was eliminated, conditional on covariates</td></tr>
  </tbody>
  
  
</table>
</div>

---
## Identifiability assumptions for death

<div id="wmyaeptldo" style="padding-left:0px;padding-right:0px;padding-top:10px;padding-bottom:10px;overflow-x:auto;overflow-y:auto;width:auto;height:auto;">
<style>#wmyaeptldo table {
  font-family: system-ui, 'Segoe UI', Roboto, Helvetica, Arial, sans-serif, 'Apple Color Emoji', 'Segoe UI Emoji', 'Segoe UI Symbol', 'Noto Color Emoji';
  -webkit-font-smoothing: antialiased;
  -moz-osx-font-smoothing: grayscale;
}

#wmyaeptldo thead, #wmyaeptldo tbody, #wmyaeptldo tfoot, #wmyaeptldo tr, #wmyaeptldo td, #wmyaeptldo th {
  border-style: none;
}

#wmyaeptldo p {
  margin: 0;
  padding: 0;
}

#wmyaeptldo .gt_table {
  display: table;
  border-collapse: collapse;
  line-height: normal;
  margin-left: auto;
  margin-right: auto;
  color: #333333;
  font-size: 22px;
  font-weight: normal;
  font-style: normal;
  background-color: #FFFFFF;
  width: auto;
  border-top-style: solid;
  border-top-width: 2px;
  border-top-color: #A8A8A8;
  border-right-style: none;
  border-right-width: 2px;
  border-right-color: #D3D3D3;
  border-bottom-style: solid;
  border-bottom-width: 2px;
  border-bottom-color: #A8A8A8;
  border-left-style: none;
  border-left-width: 2px;
  border-left-color: #D3D3D3;
}

#wmyaeptldo .gt_caption {
  padding-top: 4px;
  padding-bottom: 4px;
}

#wmyaeptldo .gt_title {
  color: #333333;
  font-size: 125%;
  font-weight: initial;
  padding-top: 4px;
  padding-bottom: 4px;
  padding-left: 5px;
  padding-right: 5px;
  border-bottom-color: #FFFFFF;
  border-bottom-width: 0;
}

#wmyaeptldo .gt_subtitle {
  color: #333333;
  font-size: 85%;
  font-weight: initial;
  padding-top: 3px;
  padding-bottom: 5px;
  padding-left: 5px;
  padding-right: 5px;
  border-top-color: #FFFFFF;
  border-top-width: 0;
}

#wmyaeptldo .gt_heading {
  background-color: #FFFFFF;
  text-align: center;
  border-bottom-color: #FFFFFF;
  border-left-style: none;
  border-left-width: 1px;
  border-left-color: #D3D3D3;
  border-right-style: none;
  border-right-width: 1px;
  border-right-color: #D3D3D3;
}

#wmyaeptldo .gt_bottom_border {
  border-bottom-style: solid;
  border-bottom-width: 2px;
  border-bottom-color: #D3D3D3;
}

#wmyaeptldo .gt_col_headings {
  border-top-style: solid;
  border-top-width: 2px;
  border-top-color: #D3D3D3;
  border-bottom-style: solid;
  border-bottom-width: 2px;
  border-bottom-color: #D3D3D3;
  border-left-style: none;
  border-left-width: 1px;
  border-left-color: #D3D3D3;
  border-right-style: none;
  border-right-width: 1px;
  border-right-color: #D3D3D3;
}

#wmyaeptldo .gt_col_heading {
  color: #333333;
  background-color: #FFFFFF;
  font-size: 100%;
  font-weight: normal;
  text-transform: inherit;
  border-left-style: none;
  border-left-width: 1px;
  border-left-color: #D3D3D3;
  border-right-style: none;
  border-right-width: 1px;
  border-right-color: #D3D3D3;
  vertical-align: bottom;
  padding-top: 5px;
  padding-bottom: 6px;
  padding-left: 5px;
  padding-right: 5px;
  overflow-x: hidden;
}

#wmyaeptldo .gt_column_spanner_outer {
  color: #333333;
  background-color: #FFFFFF;
  font-size: 100%;
  font-weight: normal;
  text-transform: inherit;
  padding-top: 0;
  padding-bottom: 0;
  padding-left: 4px;
  padding-right: 4px;
}

#wmyaeptldo .gt_column_spanner_outer:first-child {
  padding-left: 0;
}

#wmyaeptldo .gt_column_spanner_outer:last-child {
  padding-right: 0;
}

#wmyaeptldo .gt_column_spanner {
  border-bottom-style: solid;
  border-bottom-width: 2px;
  border-bottom-color: #D3D3D3;
  vertical-align: bottom;
  padding-top: 5px;
  padding-bottom: 5px;
  overflow-x: hidden;
  display: inline-block;
  width: 100%;
}

#wmyaeptldo .gt_spanner_row {
  border-bottom-style: hidden;
}

#wmyaeptldo .gt_group_heading {
  padding-top: 8px;
  padding-bottom: 8px;
  padding-left: 5px;
  padding-right: 5px;
  color: #333333;
  background-color: #FFFFFF;
  font-size: 100%;
  font-weight: initial;
  text-transform: inherit;
  border-top-style: solid;
  border-top-width: 2px;
  border-top-color: #D3D3D3;
  border-bottom-style: solid;
  border-bottom-width: 2px;
  border-bottom-color: #D3D3D3;
  border-left-style: none;
  border-left-width: 1px;
  border-left-color: #D3D3D3;
  border-right-style: none;
  border-right-width: 1px;
  border-right-color: #D3D3D3;
  vertical-align: middle;
  text-align: left;
}

#wmyaeptldo .gt_empty_group_heading {
  padding: 0.5px;
  color: #333333;
  background-color: #FFFFFF;
  font-size: 100%;
  font-weight: initial;
  border-top-style: solid;
  border-top-width: 2px;
  border-top-color: #D3D3D3;
  border-bottom-style: solid;
  border-bottom-width: 2px;
  border-bottom-color: #D3D3D3;
  vertical-align: middle;
}

#wmyaeptldo .gt_from_md > :first-child {
  margin-top: 0;
}

#wmyaeptldo .gt_from_md > :last-child {
  margin-bottom: 0;
}

#wmyaeptldo .gt_row {
  padding-top: 8px;
  padding-bottom: 8px;
  padding-left: 5px;
  padding-right: 5px;
  margin: 10px;
  border-top-style: solid;
  border-top-width: 1px;
  border-top-color: #D3D3D3;
  border-left-style: none;
  border-left-width: 1px;
  border-left-color: #D3D3D3;
  border-right-style: none;
  border-right-width: 1px;
  border-right-color: #D3D3D3;
  vertical-align: middle;
  overflow-x: hidden;
}

#wmyaeptldo .gt_stub {
  color: #333333;
  background-color: #FFFFFF;
  font-size: 100%;
  font-weight: initial;
  text-transform: inherit;
  border-right-style: solid;
  border-right-width: 2px;
  border-right-color: #D3D3D3;
  padding-left: 5px;
  padding-right: 5px;
}

#wmyaeptldo .gt_stub_row_group {
  color: #333333;
  background-color: #FFFFFF;
  font-size: 100%;
  font-weight: initial;
  text-transform: inherit;
  border-right-style: solid;
  border-right-width: 2px;
  border-right-color: #D3D3D3;
  padding-left: 5px;
  padding-right: 5px;
  vertical-align: top;
}

#wmyaeptldo .gt_row_group_first td {
  border-top-width: 2px;
}

#wmyaeptldo .gt_row_group_first th {
  border-top-width: 2px;
}

#wmyaeptldo .gt_summary_row {
  color: #333333;
  background-color: #FFFFFF;
  text-transform: inherit;
  padding-top: 8px;
  padding-bottom: 8px;
  padding-left: 5px;
  padding-right: 5px;
}

#wmyaeptldo .gt_first_summary_row {
  border-top-style: solid;
  border-top-color: #D3D3D3;
}

#wmyaeptldo .gt_first_summary_row.thick {
  border-top-width: 2px;
}

#wmyaeptldo .gt_last_summary_row {
  padding-top: 8px;
  padding-bottom: 8px;
  padding-left: 5px;
  padding-right: 5px;
  border-bottom-style: solid;
  border-bottom-width: 2px;
  border-bottom-color: #D3D3D3;
}

#wmyaeptldo .gt_grand_summary_row {
  color: #333333;
  background-color: #FFFFFF;
  text-transform: inherit;
  padding-top: 8px;
  padding-bottom: 8px;
  padding-left: 5px;
  padding-right: 5px;
}

#wmyaeptldo .gt_first_grand_summary_row {
  padding-top: 8px;
  padding-bottom: 8px;
  padding-left: 5px;
  padding-right: 5px;
  border-top-style: double;
  border-top-width: 6px;
  border-top-color: #D3D3D3;
}

#wmyaeptldo .gt_last_grand_summary_row_top {
  padding-top: 8px;
  padding-bottom: 8px;
  padding-left: 5px;
  padding-right: 5px;
  border-bottom-style: double;
  border-bottom-width: 6px;
  border-bottom-color: #D3D3D3;
}

#wmyaeptldo .gt_striped {
  background-color: rgba(128, 128, 128, 0.05);
}

#wmyaeptldo .gt_table_body {
  border-top-style: solid;
  border-top-width: 2px;
  border-top-color: #D3D3D3;
  border-bottom-style: solid;
  border-bottom-width: 2px;
  border-bottom-color: #D3D3D3;
}

#wmyaeptldo .gt_footnotes {
  color: #333333;
  background-color: #FFFFFF;
  border-bottom-style: none;
  border-bottom-width: 2px;
  border-bottom-color: #D3D3D3;
  border-left-style: none;
  border-left-width: 2px;
  border-left-color: #D3D3D3;
  border-right-style: none;
  border-right-width: 2px;
  border-right-color: #D3D3D3;
}

#wmyaeptldo .gt_footnote {
  margin: 0px;
  font-size: 90%;
  padding-top: 4px;
  padding-bottom: 4px;
  padding-left: 5px;
  padding-right: 5px;
}

#wmyaeptldo .gt_sourcenotes {
  color: #333333;
  background-color: #FFFFFF;
  border-bottom-style: none;
  border-bottom-width: 2px;
  border-bottom-color: #D3D3D3;
  border-left-style: none;
  border-left-width: 2px;
  border-left-color: #D3D3D3;
  border-right-style: none;
  border-right-width: 2px;
  border-right-color: #D3D3D3;
}

#wmyaeptldo .gt_sourcenote {
  font-size: 90%;
  padding-top: 4px;
  padding-bottom: 4px;
  padding-left: 5px;
  padding-right: 5px;
}

#wmyaeptldo .gt_left {
  text-align: left;
}

#wmyaeptldo .gt_center {
  text-align: center;
}

#wmyaeptldo .gt_right {
  text-align: right;
  font-variant-numeric: tabular-nums;
}

#wmyaeptldo .gt_font_normal {
  font-weight: normal;
}

#wmyaeptldo .gt_font_bold {
  font-weight: bold;
}

#wmyaeptldo .gt_font_italic {
  font-style: italic;
}

#wmyaeptldo .gt_super {
  font-size: 65%;
}

#wmyaeptldo .gt_footnote_marks {
  font-size: 75%;
  vertical-align: 0.4em;
  position: initial;
}

#wmyaeptldo .gt_asterisk {
  font-size: 100%;
  vertical-align: 0;
}

#wmyaeptldo .gt_indent_1 {
  text-indent: 5px;
}

#wmyaeptldo .gt_indent_2 {
  text-indent: 10px;
}

#wmyaeptldo .gt_indent_3 {
  text-indent: 15px;
}

#wmyaeptldo .gt_indent_4 {
  text-indent: 20px;
}

#wmyaeptldo .gt_indent_5 {
  text-indent: 25px;
}
</style>
<table class="gt_table" style="table-layout: fixed;; width: 0px" data-quarto-disable-processing="false" data-quarto-bootstrap="false">
  <colgroup>
    <col style="width:200px;"/>
    <col style="width:150px;"/>
    <col style="width:350px;"/>
  </colgroup>
  <thead>
    <tr class="gt_col_headings">
      <th class="gt_col_heading gt_columns_bottom_border gt_left" rowspan="1" colspan="1" scope="col" id="&lt;b&gt; Assumption &lt;/b&gt;"><b> Assumption </b></th>
      <th class="gt_col_heading gt_columns_bottom_border gt_left" rowspan="1" colspan="1" scope="col" id="&lt;b&gt; Total Effect &lt;/b&gt;"><b> Total Effect </b></th>
      <th class="gt_col_heading gt_columns_bottom_border gt_left" rowspan="1" colspan="1" scope="col" id="&lt;b&gt; Controlled direct effect &lt;/b&gt;"><b> Controlled direct effect </b></th>
    </tr>
  </thead>
  <tbody class="gt_table_body">
    <tr><td headers="<b> Assumption </b>" class="gt_row gt_left"><b> Exchangeability </b></td>
<td headers="<b> Total Effect </b>" class="gt_row gt_left">Not needed</td>
<td headers="<b> Controlled direct effect </b>" class="gt_row gt_left">Death is independent of future outcomes had everyone followed A = a and death was eliminated, conditional on covariates</td></tr>
    <tr><td headers="<b> Assumption </b>" class="gt_row gt_left"><b> Positivity </b></td>
<td headers="<b> Total Effect </b>" class="gt_row gt_left">Not needed</td>
<td headers="<b> Controlled direct effect </b>" class="gt_row gt_left">At every follow-up time, there are individuals with any possibly observed level A = a and covariate history who remain alive and free of dementia diagnosis.</td></tr>
  </tbody>
  
  
</table>
</div>

---
## Identifiability assumptions for death

<div id="giannutgie" style="padding-left:0px;padding-right:0px;padding-top:10px;padding-bottom:10px;overflow-x:auto;overflow-y:auto;width:auto;height:auto;">
<style>#giannutgie table {
  font-family: system-ui, 'Segoe UI', Roboto, Helvetica, Arial, sans-serif, 'Apple Color Emoji', 'Segoe UI Emoji', 'Segoe UI Symbol', 'Noto Color Emoji';
  -webkit-font-smoothing: antialiased;
  -moz-osx-font-smoothing: grayscale;
}

#giannutgie thead, #giannutgie tbody, #giannutgie tfoot, #giannutgie tr, #giannutgie td, #giannutgie th {
  border-style: none;
}

#giannutgie p {
  margin: 0;
  padding: 0;
}

#giannutgie .gt_table {
  display: table;
  border-collapse: collapse;
  line-height: normal;
  margin-left: auto;
  margin-right: auto;
  color: #333333;
  font-size: 22px;
  font-weight: normal;
  font-style: normal;
  background-color: #FFFFFF;
  width: auto;
  border-top-style: solid;
  border-top-width: 2px;
  border-top-color: #A8A8A8;
  border-right-style: none;
  border-right-width: 2px;
  border-right-color: #D3D3D3;
  border-bottom-style: solid;
  border-bottom-width: 2px;
  border-bottom-color: #A8A8A8;
  border-left-style: none;
  border-left-width: 2px;
  border-left-color: #D3D3D3;
}

#giannutgie .gt_caption {
  padding-top: 4px;
  padding-bottom: 4px;
}

#giannutgie .gt_title {
  color: #333333;
  font-size: 125%;
  font-weight: initial;
  padding-top: 4px;
  padding-bottom: 4px;
  padding-left: 5px;
  padding-right: 5px;
  border-bottom-color: #FFFFFF;
  border-bottom-width: 0;
}

#giannutgie .gt_subtitle {
  color: #333333;
  font-size: 85%;
  font-weight: initial;
  padding-top: 3px;
  padding-bottom: 5px;
  padding-left: 5px;
  padding-right: 5px;
  border-top-color: #FFFFFF;
  border-top-width: 0;
}

#giannutgie .gt_heading {
  background-color: #FFFFFF;
  text-align: center;
  border-bottom-color: #FFFFFF;
  border-left-style: none;
  border-left-width: 1px;
  border-left-color: #D3D3D3;
  border-right-style: none;
  border-right-width: 1px;
  border-right-color: #D3D3D3;
}

#giannutgie .gt_bottom_border {
  border-bottom-style: solid;
  border-bottom-width: 2px;
  border-bottom-color: #D3D3D3;
}

#giannutgie .gt_col_headings {
  border-top-style: solid;
  border-top-width: 2px;
  border-top-color: #D3D3D3;
  border-bottom-style: solid;
  border-bottom-width: 2px;
  border-bottom-color: #D3D3D3;
  border-left-style: none;
  border-left-width: 1px;
  border-left-color: #D3D3D3;
  border-right-style: none;
  border-right-width: 1px;
  border-right-color: #D3D3D3;
}

#giannutgie .gt_col_heading {
  color: #333333;
  background-color: #FFFFFF;
  font-size: 100%;
  font-weight: normal;
  text-transform: inherit;
  border-left-style: none;
  border-left-width: 1px;
  border-left-color: #D3D3D3;
  border-right-style: none;
  border-right-width: 1px;
  border-right-color: #D3D3D3;
  vertical-align: bottom;
  padding-top: 5px;
  padding-bottom: 6px;
  padding-left: 5px;
  padding-right: 5px;
  overflow-x: hidden;
}

#giannutgie .gt_column_spanner_outer {
  color: #333333;
  background-color: #FFFFFF;
  font-size: 100%;
  font-weight: normal;
  text-transform: inherit;
  padding-top: 0;
  padding-bottom: 0;
  padding-left: 4px;
  padding-right: 4px;
}

#giannutgie .gt_column_spanner_outer:first-child {
  padding-left: 0;
}

#giannutgie .gt_column_spanner_outer:last-child {
  padding-right: 0;
}

#giannutgie .gt_column_spanner {
  border-bottom-style: solid;
  border-bottom-width: 2px;
  border-bottom-color: #D3D3D3;
  vertical-align: bottom;
  padding-top: 5px;
  padding-bottom: 5px;
  overflow-x: hidden;
  display: inline-block;
  width: 100%;
}

#giannutgie .gt_spanner_row {
  border-bottom-style: hidden;
}

#giannutgie .gt_group_heading {
  padding-top: 8px;
  padding-bottom: 8px;
  padding-left: 5px;
  padding-right: 5px;
  color: #333333;
  background-color: #FFFFFF;
  font-size: 100%;
  font-weight: initial;
  text-transform: inherit;
  border-top-style: solid;
  border-top-width: 2px;
  border-top-color: #D3D3D3;
  border-bottom-style: solid;
  border-bottom-width: 2px;
  border-bottom-color: #D3D3D3;
  border-left-style: none;
  border-left-width: 1px;
  border-left-color: #D3D3D3;
  border-right-style: none;
  border-right-width: 1px;
  border-right-color: #D3D3D3;
  vertical-align: middle;
  text-align: left;
}

#giannutgie .gt_empty_group_heading {
  padding: 0.5px;
  color: #333333;
  background-color: #FFFFFF;
  font-size: 100%;
  font-weight: initial;
  border-top-style: solid;
  border-top-width: 2px;
  border-top-color: #D3D3D3;
  border-bottom-style: solid;
  border-bottom-width: 2px;
  border-bottom-color: #D3D3D3;
  vertical-align: middle;
}

#giannutgie .gt_from_md > :first-child {
  margin-top: 0;
}

#giannutgie .gt_from_md > :last-child {
  margin-bottom: 0;
}

#giannutgie .gt_row {
  padding-top: 8px;
  padding-bottom: 8px;
  padding-left: 5px;
  padding-right: 5px;
  margin: 10px;
  border-top-style: solid;
  border-top-width: 1px;
  border-top-color: #D3D3D3;
  border-left-style: none;
  border-left-width: 1px;
  border-left-color: #D3D3D3;
  border-right-style: none;
  border-right-width: 1px;
  border-right-color: #D3D3D3;
  vertical-align: middle;
  overflow-x: hidden;
}

#giannutgie .gt_stub {
  color: #333333;
  background-color: #FFFFFF;
  font-size: 100%;
  font-weight: initial;
  text-transform: inherit;
  border-right-style: solid;
  border-right-width: 2px;
  border-right-color: #D3D3D3;
  padding-left: 5px;
  padding-right: 5px;
}

#giannutgie .gt_stub_row_group {
  color: #333333;
  background-color: #FFFFFF;
  font-size: 100%;
  font-weight: initial;
  text-transform: inherit;
  border-right-style: solid;
  border-right-width: 2px;
  border-right-color: #D3D3D3;
  padding-left: 5px;
  padding-right: 5px;
  vertical-align: top;
}

#giannutgie .gt_row_group_first td {
  border-top-width: 2px;
}

#giannutgie .gt_row_group_first th {
  border-top-width: 2px;
}

#giannutgie .gt_summary_row {
  color: #333333;
  background-color: #FFFFFF;
  text-transform: inherit;
  padding-top: 8px;
  padding-bottom: 8px;
  padding-left: 5px;
  padding-right: 5px;
}

#giannutgie .gt_first_summary_row {
  border-top-style: solid;
  border-top-color: #D3D3D3;
}

#giannutgie .gt_first_summary_row.thick {
  border-top-width: 2px;
}

#giannutgie .gt_last_summary_row {
  padding-top: 8px;
  padding-bottom: 8px;
  padding-left: 5px;
  padding-right: 5px;
  border-bottom-style: solid;
  border-bottom-width: 2px;
  border-bottom-color: #D3D3D3;
}

#giannutgie .gt_grand_summary_row {
  color: #333333;
  background-color: #FFFFFF;
  text-transform: inherit;
  padding-top: 8px;
  padding-bottom: 8px;
  padding-left: 5px;
  padding-right: 5px;
}

#giannutgie .gt_first_grand_summary_row {
  padding-top: 8px;
  padding-bottom: 8px;
  padding-left: 5px;
  padding-right: 5px;
  border-top-style: double;
  border-top-width: 6px;
  border-top-color: #D3D3D3;
}

#giannutgie .gt_last_grand_summary_row_top {
  padding-top: 8px;
  padding-bottom: 8px;
  padding-left: 5px;
  padding-right: 5px;
  border-bottom-style: double;
  border-bottom-width: 6px;
  border-bottom-color: #D3D3D3;
}

#giannutgie .gt_striped {
  background-color: rgba(128, 128, 128, 0.05);
}

#giannutgie .gt_table_body {
  border-top-style: solid;
  border-top-width: 2px;
  border-top-color: #D3D3D3;
  border-bottom-style: solid;
  border-bottom-width: 2px;
  border-bottom-color: #D3D3D3;
}

#giannutgie .gt_footnotes {
  color: #333333;
  background-color: #FFFFFF;
  border-bottom-style: none;
  border-bottom-width: 2px;
  border-bottom-color: #D3D3D3;
  border-left-style: none;
  border-left-width: 2px;
  border-left-color: #D3D3D3;
  border-right-style: none;
  border-right-width: 2px;
  border-right-color: #D3D3D3;
}

#giannutgie .gt_footnote {
  margin: 0px;
  font-size: 90%;
  padding-top: 4px;
  padding-bottom: 4px;
  padding-left: 5px;
  padding-right: 5px;
}

#giannutgie .gt_sourcenotes {
  color: #333333;
  background-color: #FFFFFF;
  border-bottom-style: none;
  border-bottom-width: 2px;
  border-bottom-color: #D3D3D3;
  border-left-style: none;
  border-left-width: 2px;
  border-left-color: #D3D3D3;
  border-right-style: none;
  border-right-width: 2px;
  border-right-color: #D3D3D3;
}

#giannutgie .gt_sourcenote {
  font-size: 90%;
  padding-top: 4px;
  padding-bottom: 4px;
  padding-left: 5px;
  padding-right: 5px;
}

#giannutgie .gt_left {
  text-align: left;
}

#giannutgie .gt_center {
  text-align: center;
}

#giannutgie .gt_right {
  text-align: right;
  font-variant-numeric: tabular-nums;
}

#giannutgie .gt_font_normal {
  font-weight: normal;
}

#giannutgie .gt_font_bold {
  font-weight: bold;
}

#giannutgie .gt_font_italic {
  font-style: italic;
}

#giannutgie .gt_super {
  font-size: 65%;
}

#giannutgie .gt_footnote_marks {
  font-size: 75%;
  vertical-align: 0.4em;
  position: initial;
}

#giannutgie .gt_asterisk {
  font-size: 100%;
  vertical-align: 0;
}

#giannutgie .gt_indent_1 {
  text-indent: 5px;
}

#giannutgie .gt_indent_2 {
  text-indent: 10px;
}

#giannutgie .gt_indent_3 {
  text-indent: 15px;
}

#giannutgie .gt_indent_4 {
  text-indent: 20px;
}

#giannutgie .gt_indent_5 {
  text-indent: 25px;
}
</style>
<table class="gt_table" style="table-layout: fixed;; width: 0px" data-quarto-disable-processing="false" data-quarto-bootstrap="false">
  <colgroup>
    <col style="width:200px;"/>
    <col style="width:150px;"/>
    <col style="width:350px;"/>
  </colgroup>
  <thead>
    <tr class="gt_col_headings">
      <th class="gt_col_heading gt_columns_bottom_border gt_left" rowspan="1" colspan="1" scope="col" id="&lt;b&gt; Assumption &lt;/b&gt;"><b> Assumption </b></th>
      <th class="gt_col_heading gt_columns_bottom_border gt_left" rowspan="1" colspan="1" scope="col" id="&lt;b&gt; Total Effect &lt;/b&gt;"><b> Total Effect </b></th>
      <th class="gt_col_heading gt_columns_bottom_border gt_left" rowspan="1" colspan="1" scope="col" id="&lt;b&gt; Controlled direct effect &lt;/b&gt;"><b> Controlled direct effect </b></th>
    </tr>
  </thead>
  <tbody class="gt_table_body">
    <tr><td headers="<b> Assumption </b>" class="gt_row gt_left"><b> Exchangeability </b></td>
<td headers="<b> Total Effect </b>" class="gt_row gt_left">Not needed</td>
<td headers="<b> Controlled direct effect </b>" class="gt_row gt_left">Death is independent of future outcomes had everyone followed A = a and death was eliminated, conditional on covariates</td></tr>
    <tr><td headers="<b> Assumption </b>" class="gt_row gt_left"><b> Positivity </b></td>
<td headers="<b> Total Effect </b>" class="gt_row gt_left">Not needed</td>
<td headers="<b> Controlled direct effect </b>" class="gt_row gt_left">At every follow-up time, there are individuals with any possibly observed level A = a and covariate history who remain alive and free of dementia diagnosis.</td></tr>
    <tr><td headers="<b> Assumption </b>" class="gt_row gt_left"><b> Consistency </b></td>
<td headers="<b> Total Effect </b>" class="gt_row gt_left">Not needed</td>
<td headers="<b> Controlled direct effect </b>" class="gt_row gt_left">An intervention that “eliminates death” is well-defined.</td></tr>
  </tbody>
  
  
</table>
</div>

---
## Other possible estimands

- **Composite outcome**

`$$Pr[Y_{t}^{a = 1} = 1   \text{ or } {D}_{t}^{a = 1} = 1] - Pr[Y_{t}^{a = 0} = 1  \text{ or } {D}_{t}^{a = 1} = 1]$$`

- **Survivors average causal effect:**

`$$Pr[Y_{t}^{a = 1} = 1 |\bar{D}_{t}^{a = 1} = \bar{D}_{t}^{a = 0} = 0] - Pr[Y_{t}^{a = 0} = 1 |\bar{D}_{t}^{a = 1} = \bar{D}_{t}^{a = 0} = 0]$$`

- **Natural direct effects**:

`$$Pr[Y_{t}^{a = 1, \bar{D}_{t}^{a = 0}} = 1] - Pr[Y_{t}^{a = 0, \bar{D}_{t}^{a = 0}} = 1]$$`

---
## Separable effects

Physical decomposition of the exposure assumed to operate on `$Y$` and `$D$` through separate pathways. (_Stensrud et al. JASA. 2020_)

.center[
<img src=./figs/stensrud.jpg width="30%"/>
]

---
## Take away points

- Incident events form the cornerstone of causal inference in epidemiology

- Risks & cumulative incidence are the only measures that consistently refer back to the target/study population at baseline

- Time is a fundamental component, but that doesn't mean that inherently our primary measure of interest is *time-to-event*

- Censoring and competing events are not exceptions to the rule but rather common occurrences that must be accounted for in analyses

- Estimands for the risk of an event by a given time must explicitly incorporate censoring and competing risks to ensure transparent and interpretable conclusion

---
class: title-slide, center, middle, inverse

# .yellow[Thank You, Gracias!]

.left[
Special thanks to:
- Yixuan Zhou
- Yingyan Wu
- Elizabeth Rose Mayeda
- Onyebuchi Arah
- Jessica G. Young
- Sonja A. Swanson
- Vanessa Didelez
]

.right[
lp.rojassaunero@ucla.edu

]

---
## The trinity to avoid immortal time bias

.pull-left[

- At time zero, the following elements align:

-   Inclusion in the study (based on eligibility criteria)

-   Randomization to intervention

- Start of follow-up for incident events

]

.pull-right[

]

---
## Kaplan-Meier estimator

$$
\hat{S}(t) = \prod_{t_i \leq t} \left( 1 - \frac{d_i}{n_i} \right)
$$

- `$\hat{S}(t)$`: estimated survival probability at time $ t $
- `$t_i$`: the distinct times when events (e.g., deaths) occur
- `$d_i$`: the number of events (e.g., deaths) at time `$t_i$`
- `$n_i$`: the number of individuals at risk just before time `$t_i$`