Target Trials

# Target Trials

**Session 6**

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# Potential Outcome

What *will* happen to you in the future given you have a particular exposure

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.box-inv-1.medium[

**fixed** but **unknown**

]

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# Exposure assignment

.box-inv-1[Understanding the **assignment mechanism** is crucial for understanding the causal effect]

.box-1[Assignment probability must be **individualistic** (your covariates / potential outcomes can't influence my assignment)]

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# What do we observe?

`\(Y_i^{obs}=Y_i(1)\)` if `\(X_i=1\)`
]

`\(Y_i^{obs}=Y_i(0)\)` if `\(X_i=0\)`
]

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# What do we observe?

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## Stable Unit Treatment Value Assumption

.box-inv-1[
The potential outcomes for any unit do not vary with the exposures assigned to other units, and, for each unit, there are no different forms or versions of each exposure level, which lead to different potential outcomes.]

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## Summary of assumptions

.pull-left[
.box-1[**Potential outcomes** for any unit do not vary with the exposure of other units]
]

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## Summary of assumptions

.pull-left[
.box-1[**Potential outcomes** for any unit do not vary with the exposure of other units]
]

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## Summary of assumptions

.pull-left[
.box-1[**Potential outcomes** for any unit do not vary with the exposure of other units]
]

## Summary of assumptions

.pull-left[
.box-1[**Potential outcomes** for any unit do not vary with the exposure of other units]
]

## Summary of assumptions

]

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## Summary of assumptions

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## Summary of assumptions

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## Summary of assumptions

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## Summary of assumptions

.pull-left[
.box-1[**Exposure assignment** probability for any unit does not depend on covariates / potential outcomes of other units]
]

]

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## Summary of assumptions

.pull-left[
.box-1[**Exposure assignment** probability for any unit does not depend on covariates / potential outcomes of other units]
]

]
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## Summary of assumptions

.pull-left[
.box-1[**Exposure assignment** probability for any unit does not depend on covariates / potential outcomes of other units]
]

]

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class: title title-1

## Summary of assumptions

--
.pull-right[

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class: title title-1

## Summary of assumptions

.box-inv-1[👩‍🔬 subject matter knowledge (structural violations), parameteric models (random violations)]

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class: title title-1

## Summary of assumptions

.box-inv-1[👩‍🔬 subject matter knowledge (structural violations), parameteric models (random violations)]

]
---
class: title title-1

## Summary of assumptions

.box-inv-1[👩‍🔬 subject matter knowledge (structural violations), parameteric models (random violations)]

---
class: title title-1

## Summary of assumptions

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## Summary of assumptions

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class: title title-1

## Summary of assumptions

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class: title title-1

## Summary of assumptions

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# Summary of assumptions

Assumption | Classic Randomized Trial| Randomized Trial
------------|-----------------
No Interference | 🤷 | 🤷 
Consistency | 😄 | 😄 
Individualistic | 😄| 🤷
Probabilistic |😄| 😄
Unconfounded |😄| 🤷

---
class: title title-1

# Designing randomized trials
.footer[Hernán & Robins. (2016) American Journal of Epidemiology doi:10.1093/aje/kwv254]

.box-inv-1.medium[consistent exposure]

.box-inv-1.medium[assignment procedures]

]
--
.pull-right[
.box-inv-1.medium[follow-up period]

.box-inv-1.medium[outcome]

.box-inv-1.medium[causal contrasts of interest]

.box-inv-1.medium[analysis plan]

]

---

# Designing randomized trials
.footer[Hernán & Robins. (2016) American Journal of Epidemiology doi:10.1093/aje/kwv254]

.box-inv-1.medium[consistent exposure]

.box-inv-1.medium[assignment procedures]

]

.box-inv-1.medium[outcome]

.box-2.medium[causal contrasts of interest]

.box-2.medium[analysis plan]

]

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# Summary of assumptions

Assumption | Eligibility Criteria | Consistent Exposure | Assignment Procedures | Follow-up Period | Outcome
------------|-----------------
No Interference | | ✔️|✔️ | | ✔️
Consistency | |✔️|||
Individualistic | ||✔️||
Probabilistic |✔️||✔️||
Unconfounded |✔️||✔️|✔️||

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# Summary of assumptions

Assumption | Classic Randomized Trial| Randomized Trial| Observational Study
------------|-----------------|-------------------
No Interference | 🤷 | 🤷 | 🤷
Consistency | 😄 | 😄 | 🤷
Individualistic | 😄| 🤷| 🤷
Probabilistic |😄| 😄| 🤷
Unconfounded |😄| 🤷| 🤷

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# Why not always conduct an experiment?

--
.box-1.medium[⚕️ ethics ]

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# Why not always conduct an experiment?

.box-1.medium[⚕️ ethics   💸 money ]
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# Why not always conduct an experiment?

.box-1.medium[⚕️ ethics   💸 money  🕒 time ]

.box-inv-1.medium[Deferring a decision **is** a decision]

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# Target Trial

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# Algorithm for causal inference

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# Algorithm for causal inference

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# Algorithm for causal inference

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# Compontents of a target trial

.box-inv-1.medium[consistent exposure]

.box-inv-1.medium[assignment procedures]

]
--
.pull-right[
.box-inv-1.medium[follow-up period]

.box-inv-1.medium[outcome]

.box-inv-1.medium[causal contrasts of interest]

.box-inv-1.medium[analysis plan]

]

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# Compontents of a target trial

.box-inv-1.medium[consistent exposure]

.box-inv-1.medium[assignment procedures]

]

.box-inv-1.medium[outcome]

.box-2.medium[causal contrasts of interest]

.box-2.medium[analysis plan]

]

---

---

<figure>
<img src = "img/diagram/diagram/diagram.002.jpeg" width = "80%"></img>
</figure>
---

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# Compontents of a target trial

.box-inv-1.medium[consistent exposure]

.box-inv-1.medium[assignment procedures]

]

.box-2.medium[outcome]

.box-2.medium[causal contrasts of interest]

.box-2.medium[analysis plan]

]

---

---

# Compontents of a target trial

.box-2.medium[consistent exposure]

.box-2.medium[assignment procedures]

]

.box-inv-1.medium[outcome]

.box-2.medium[causal contrasts of interest]

.box-2.medium[analysis plan]

]

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<figure>
<img src = "img/diagram/diagram/diagram.009.jpeg" width = "80%"></img>
</figure>
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class: center middle

<figure>
<img src = "img/diagram/diagram/diagram.010.jpeg" width = "80%"></img>
</figure>
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class: center middle

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# Compontents of a target trial

.box-2.medium[consistent exposure]

.box-2.medium[assignment procedures]

]

.box-2.medium[outcome]

.box-2.medium[causal contrasts of interest]

.box-2.medium[analysis plan]

]

---

---

<figure>
<img src = "img/diagram/diagram/diagram.013.jpeg" width = "80%"></img>
</figure>
---

---

---

# Compontents of a target trial

.box-2.medium[consistent exposure]

.box-2.medium[assignment procedures]

]

.box-2.medium[outcome]

.box-2.medium[causal contrasts of interest]

.box-2.medium[analysis plan]

]

---

---

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class: center middle

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class: center middle

<figure>
<img src = "img/diagram/diagram/diagram.021.jpeg" width = "80%"></img>
</figure>
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class: center middle

<figure>
<img src = "img/diagram/diagram/diagram.022.jpeg" width = "80%"></img>
</figure>
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class: center middle

<figure>
<img src = "img/diagram/diagram/diagram.025.jpeg" width = "80%"></img>
</figure>
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# An example

* Women who used hormone replacement therapy had **lower risk** of heart disease compared to those who didn't
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* Women who were randomized to use hormone replacement therapy had **higher risk** of heart disease compared to those who didn't
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class: title title-1

# Why the difference?

.box-inv-1[The observational trial compared *current* users; people who were susceptible to heart disease were already out of the cohort! So current use became a marker of not being susceptible]

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# The solution?

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# The importance of time zero

.box-inv-1.medium[In a randomized study]

* Eligibility criteria are met
* Treatment is assigned
* Study outcomes begin to be recorded

.box-inv-1.medium[The same should be true for an observational study!]

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# What can go wrong?

.pull-right[.small[ [1] This can bias the effect estimate because the analysis is restricted to those who remained under follow-up by time zero (like the example!)]
]

.footer[Hernán et al (2016) Journal of Clinical Epidemiology doi: 10.1016/j.jclinepi.2016.04.014]
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# What can go wrong?

.pull-right[.small[ [2] This increases the opportunity for selection biasbecause the analysis will only include individuals who remain under follow-up at the reset time zero and who meet these posttreatment criteria]]

.footer[Hernán et al (2016) Journal of Clinical Epidemiology doi: 10.1016/j.jclinepi.2016.04.014]
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# What can go wrong?

.pull-right[.small[[3] Because treatment assignment predates eligibility, selection bias may arise (like 2). In addition, bias may also occur because by definition nobody would have an event between treatment assignment and the completion of the eligibility criteria. This is a period, often labeled as "immortal time", during which the risk is guaranteed to be exactly zero. ]]

.footer[Hernán et al (2016) Journal of Clinical Epidemiology doi: 10.1016/j.jclinepi.2016.04.014]
---

# What can go wrong?

.pull-left[![](img/time-zero.png)]
.pull-right[.small[[4] This bias occurs when information on treatment after time zero is used to assign individuals to a consistent exposure. ]]

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# Observational Study Target Trial

.box-1.medium[eligibility criteria]

.box-1.medium[consistent exposure]

.box-1.medium[follow-up time]

.box-1.medium[outcome]

.box-inv-1.medium[What is left?]

---

# the only difference between an observational study and target randomized study should be the **exposure assignment procedures**

---

# Exposure assignment

.box-inv-1[Understanding the **assignment mechanism** is crucial for understanding the causal effect]

.box-1[Assignment probability must be **individualistic** (your covariates / potential outcomes can't influence my assignment)]

---

# Exposure assignment

.box-inv-1[Understanding the **assignment mechanism** is crucial for understanding the causal effect]

.box-2[Assignment probability must be **individualistic** (your covariates / potential outcomes can't influence my assignment)]

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# Assignment **cannot** depend on the potential outcomes

.box-1.medium[

`\(\mathbf{Y}(1), \mathbf{Y}(0) \perp \mathbf{X}\)`
]

---
class: section-title-1 title-1 middle

# What if we could capture everything that causes this independence assumption to break?
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class: section-title-1 title-1 middle

# confounders
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class: section-title-1 title-1 middle

# confounders: all factors associated with both the potential outcomes and **X**
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# Assignment **cannot** depend on the potential outcomes

.box-1.medium[

`\(\mathbf{Y}(1), \mathbf{Y}(0) \perp \mathbf{X}|\mathbf{Z}\)`
]

---
class: title title-1

# Lucy Land Trial!

```r
set.seed(1)

n <- 1000
meeple <- tibble(
 happy = sample(rep(c(1, 0), each = n / 2)),
 happiness = case_when(
 happy == 1 ~ rbinom(n, 5, 0.7),
 happy == 0 ~ rbinom(n, 3, 0.2)
 ),
 y0 = happiness,
 y1 = happiness
)
```
]
---

# Lucy Land Trial!

```r
set.seed(5)

d_lucy <- meeple %>%
 mutate(x = case_when(
 happy == 1 ~ rbinom(n, 1, 0.9), 
 happy == 0 ~ rbinom(n, 1, 0.1) 
 ), 
 y_obs = ifelse(x == 1, y1, y0))
```

---

# Lucy Land Trial!

```r
set.seed(5)

d_lucy <- meeple %>%
* mutate(x = case_when(
* happy == 1 ~ rbinom(n, 1, 0.9),
* happy == 0 ~ rbinom(n, 1, 0.1)
* ),
 y_obs = ifelse(x == 1, y1, y0))
```

---
class: section-title-1 title-1 middle

# the baseline indicator `happy` is a confounder for the exposure and the outcome

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# Confounding

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# Lucy Land Trial!

```r
d_lucy %>%
  summarise(true_causal_effect = mean(y1) - mean(y0))
```

```
## # A tibble: 1 × 1
## true_causal_effect
## <dbl>
## 1 0
```
]
--

# Lucy Land Trial!

```r
d_lucy %>%
  summarise(observed_causal_effect = 
              sum(y_obs * x) / sum(x) -
              sum(y_obs * (1 - x)) / sum(1 - x))
```

```
## # A tibble: 1 × 1
## observed_causal_effect
## <dbl>
## 1 2.22
```
]
---
class: title title-1

# Lucy Land Trial!

```r
d_lucy %>%
  summarise(observed_causal_effect = 
*             sum(y_obs * x) / sum(x) -
              sum(y_obs * (1 - x)) / sum(1 - x))
```

```
## # A tibble: 1 × 1
## observed_causal_effect
## <dbl>
## 1 2.22
```
]

.box-inv-1.medium[

`\(\bar{Y}^{obs}_e\)`
]
---

# Lucy Land Trial!

```r
d_lucy %>%
  summarise(observed_causal_effect = 
              sum(y_obs * x) / sum(x) - 
*             sum(y_obs * (1 - x)) / sum(1 - x))
```

```
## # A tibble: 1 × 1
## observed_causal_effect
## <dbl>
## 1 2.22
```
]

.box-inv-1.medium[

`\(\bar{Y}^{obs}_c\)`
]

---

# Lucy Land Trial!

```r
d_lucy %>%
  group_by(happy) %>%
  summarise(observed_causal_effect = 
              sum(y_obs * x) / sum(x) - 
              sum(y_obs * (1 - x)) / sum(1 - x))
```

```
## # A tibble: 2 × 2
## happy observed_causal_effect
## <dbl> <dbl>
## 1 0 -0.0347
## 2 1 0.0194
```
]

---

# Lucy Land Trial!

```r
d_lucy %>%
* group_by(happy) %>%
  summarise(observed_causal_effect = 
              sum(y_obs * x) / sum(x) - 
              sum(y_obs * (1 - x)) / sum(1 - x))
```

```
## # A tibble: 2 × 2
## happy observed_causal_effect
## <dbl> <dbl>
## 1 0 -0.0347
## 2 1 0.0194
```
]

---

# Lucy Land Trial!

```r
d_lucy %>%
* group_by(happy) %>%
  summarise(observed_causal_effect = 
              sum(y_obs * x) / sum(x) - 
              sum(y_obs * (1 - x)) / sum(1 - x))
```

```
## # A tibble: 2 × 2
## happy observed_causal_effect
## <dbl> <dbl>
## 1 0 -0.0347
## 2 1 0.0194
```
]

# the only difference between an observational study and target randomized study should be the **exposure assignment procedures**

---

# what if we have several confounders?

---

# Lucy Land Trial!

```r
d_lucy %>%
  group_by(happy) %>%
  summarise(observed_causal_effect = 
              sum(y_obs * x) / sum(x) - 
              sum(y_obs * (1 - x)) / sum(1 - x))
```

```
## # A tibble: 2 × 2
## happy observed_causal_effect
## <dbl> <dbl>
## 1 0 -0.0347
## 2 1 0.0194
```
]

---

# Lucy Land Trial!

```r
p <- glm(x ~ happy, data = d_lucy, family = binomial()) %>%
 predict(type = "response")

d_lucy <- d_lucy %>%
 mutate(prob_of_exposure = p)
```

---

# Lucy Land Trial!

```r
*p <- glm(x ~ happy, data = d_lucy, family = binomial()) %>%
* predict(type = "response")

d_lucy <- d_lucy %>%
 mutate(prob_of_exposure = p)
```

---

# Lucy Land Trial!

```r
p <- glm(x ~ happy, data = d_lucy, family = binomial()) %>%
 predict(type = "response")

*d_lucy <- d_lucy %>%
* mutate(prob_of_exposure = p)
```

---
class: title title-1

# Lucy Land Trial!

```r
d_lucy
```

```
## # A tibble: 1,000 × 7
## happy happiness y0 y1 x y_obs prob_of_exposure
## <dbl> <int> <int> <int> <int> <int> <dbl>
## 1 0 1 1 1 0 1 0.130
## 2 0 1 1 1 0 1 0.130
## 3 1 4 4 4 0 4 0.886
## 4 0 1 1 1 0 1 0.130
## 5 0 0 0 0 0 0 0.130
## 6 1 4 4 4 1 4 0.886
## 7 1 4 4 4 1 4 0.886
## 8 1 2 2 2 1 2 0.886
## 9 0 0 0 0 0 0 0.130
## 10 1 4 4 4 1 4 0.886
## # … with 990 more rows
```

]

---
class: title title-1

# Lucy Land Trial!

```r
d_lucy %>%
  mutate(wt = x / p + (1 - x) / (1 - p)) %>%
  summarise(observed_causal_effect = 
              sum(y_obs * x * wt) / sum(x * wt) - 
              sum(y_obs * (1 - x) * wt) / sum((1 - x) * wt))
```

```
## # A tibble: 1 × 1
## observed_causal_effect
## <dbl>
## 1 -0.00761
```
---
class: title title-1

# Lucy Land Trial!

```r
d_lucy %>%
* mutate(wt = x / p + (1 - x) / (1 - p)) %>%
  summarise(observed_causal_effect = 
              sum(y_obs * x * wt) / sum(x * wt) - 
              sum(y_obs * (1 - x) * wt) / sum((1 - x) * wt)) 
```

```
## # A tibble: 1 × 1
## observed_causal_effect
## <dbl>
## 1 -0.00761
```
---

# Lucy Land Trial!

```r
d_lucy %>%
  mutate(wt = x / p + (1 - x) / (1 - p)) %>%
  summarise(observed_causal_effect = 
*             sum(y_obs * x * wt) / sum(x * wt) -
*             sum(y_obs * (1 - x) * wt) / sum((1 - x) * wt))
```

```
## # A tibble: 1 × 1
## observed_causal_effect
## <dbl>
## 1 -0.00761
```

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# If you can condition on all confounders, you can regain an exposure assignment independent of the potential outcomes 💪

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