L4: Selection Bias

class: center, middle, inverse, title-slide

.title[
# L4: Selection Bias
]
.author[
### Jean Morrison
]
.institute[
### University of Michigan
]
.date[
### 2024-01-31 (updated: 2024-03-05)
]

---

`$\newcommand{\ci}{\perp\!\!\!\perp}$`

## Lecture Outline

1. Selection Bias and Censoring
1. Non-Compliance
1. Measurement Error
---

# 1. Selection Bias and Censoring

---
## Example

- Drug `$A$`, is an HIV treatment. We are interested in measuring its effect on disease progression.

- We assess disease progression using CD4 count. The outcome, `$Y$`, is binary and is 1 if CD4 count falls below a threshold within one year of starting treatment (bad) or is 0 if CD4 count stays above the threshold.

- Some patients drop out of the study before the one year mark and we cannot observe their outcome.

- Patients may drop out if they are in poor health due to disease progression.

- They also might drop out if they are in poor health due to side effects of treatment, `$L$`.

- Use a variable `$C$` (for censoring) to represent whether a patient drops out of the study before one year ( `$C = 1$` ) or stays in the study ( `$C = 0$`).

- With a partner, draw a DAG representing this scenario.

---

## HIV Treatment Example

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---

## HIV Treatment Example

- If there had been no censoring, could we identify `$E[Y(a)]$`? (i.e. is there a set of variables `$\mathcal{V}$` such that `$Y(a) \ci A \vert\ \mathcal{V}$`)?

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- With some data unobserved, can we identify the effect of `$A$` on `$Y$`?
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</center>

---

## HIV Treatment Example

- Once we condition on the collider, `$C$`, we open the path `$A \to L \to C \leftarrow Y$`, inducing non-causal association between `$A$` and `$Y$`.

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- This is one type of **selection bias**

---

## HIV Treatment Example

- Suppose that treatment has no side effects.

- Treatment can only influence selection *through* its effect on `$Y$`.

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- Do we still have selection bias if only the outcome affects censoring?

---

## Example Continued

- Suppose that `$A$` is effective:

`$$E[Y(1)] = E[Y \vert A = 1] = 0.1 \qquad E[Y(0)] = E[Y \vert A = 0] =  0.8$$`

- And patients with `$Y = 0$` are more likely to remain in the study than patients with `$Y = 1$`.

`$$P[C = 0 \vert Y = 0] = 1 \qquad P[C = 0 \vert Y = 1] = 0.5$$`

- With your partner, compute the average causal effect of `$A$` on `$Y$` and compute the associational 
effect in the sub-population with `$C = 0$`, 
`$E[Y \vert A = 1, C = 0] - E[Y \vert A = 0, C = 0]$`.

<center>
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---

## Example Continued

Use Bayes' Theorem:

$$
`\begin{split}
P( Y = 1 \vert A = 1, C = 0) = & \frac{P(C = 0 \vert Y = 1,  A = 1)P(Y = 1 \vert A = 1)}{P(C = 0 \vert A = 1)}\\
P( Y = 1 \vert A = 0, C = 0) = & \frac{P(C = 0 \vert Y = 1,  A = 0)P(Y = 1 \vert A = 0)}{P(C = 0 \vert A = 0)}
\end{split}`
$$

$$
`\begin{split}
P(C = 0 \vert A = 1) = & P(C =  0 \vert A = 1, Y = 0) P(Y = 0 \vert A = 1) + \\
                       & P(C = 0 \vert A = 1, Y = 1) P(Y = 1 \vert A = 1)\\
 = & 1 \cdot 0.9 + 0.5 \cdot 0.1 = 0.95\\
 P(C = 0 \vert A = 0) = & P(C =  0 \vert A = 0, Y = 0) P(Y = 0 \vert A = 0) + \\ 
                       & P(C = 0 \vert A = 0, Y = 1) P(Y = 1 \vert A = 0)\\
 = & 1 \cdot 0.2 + 0.5 \cdot 0.8 = 0.6
\end{split}`
$$
 
---

## Example Continued

$$
`\begin{split}
P( Y = 1 \vert A = 1, C = 0) = & \frac{0.5 \cdot 0.1 }{0.95} \approx 0.053 \\
P( Y = 1 \vert A = 0, C = 0) = & \frac{0.5 \cdot 0.8 }{0.6} \approx 0.67
\end{split}`
$$
`$$E[Y \vert A = 1, C  =0] - E[Y \vert A = 0, C = 0] \approx -0.61$$`

`$$E[Y(1)] - E[Y(0)] = -0.7$$`

- The associational value is not equal to the ATE so there is selection bias.

---

## Example Continued

- Now assume there is no effect of `$A$` on `$Y$`:

`$$E[Y(1)] = E[Y \vert A = 1] = p \qquad E[Y(0)] = E[Y \vert A = 0] =  p$$`

- And patients with `$Y = 0$` are more likely to remain in the study than patients with `$Y = 1$`.

`$$P[C = 0 \vert Y = 0] = 1 \qquad P[C = 0 \vert Y = 1] = 0.5$$`
- Repeat your calculation of  `$E[Y \vert A = 1, C = 0] - E[Y \vert A = 0, C = 0]$`.

---

## Example Continued

$$
`\begin{split}
P(C = 0 \vert A = 1) = P(C = 0 \vert A = 0) = (1-p) + 0.5 p = \frac{2-p}{2}\\
\end{split}`
$$

$$
`\begin{split}
P( Y = 1 \vert A = 1, C = 0) = & \frac{0.5 p}{\frac{2-p}{2}} = \frac{p}{2-p}\\
P( Y = 1 \vert A = 0, C = 0) = & \frac{0.5 p}{\frac{2-p}{2}} = \frac{p}{2-p}
\end{split}`
$$

- `$E[Y \vert A =a, C = 0] \neq E[Y(a)]$`, however, there is no bias in the estimate of the ATE.

---

## Selection Bias Under the Null

- In both examples we have selection bias because `$C$` is a common effect of both `$A$` and `$Y$`.

- In the first case, this condition is true whether or not `$A$` has a non-zero effect on `$Y$` (*selection bias under the null*).

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- In the second case (all of the effect of `$A$` on `$C$` is mediated by `$Y$`), this condition *only* occurs when there is a non-zero causal effect of `$A$` on `$Y$`.

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- Selection bias under the null always implies selection bias in non-null settings.

- The reverse is not true (as we have seen).

---
## Colliding Creates Selection Bias

- Conditioning on a variable that is a child of both the outcome and the exposure creates selection bias.

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</center>

---

## Children of Colliders are Colliders

- Conditioning on a child of a confounder opens the path the confounder is on.

- In the graph below, conditioning on `$C$` opens the `$A \to L \to U \leftarrow Y$` path.

- Conditioning on `$C$` is the same as conditioning on a noisy measurement of `$U$`.

<center>
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</center>
---

## Selection Bias without Colliding

- In our previous HIV treatment example, suppose that low CD4 count does not directly cause censoring.

- Instead there is a variable `$U$` representing health which is a common cause of both `$Y$` and `$S$`.

- We still have selection bias in this case, but `$C$` is not a descendant of `$Y$` and is not a collider.

<center>
<div class="grViz html-widget html-fill-item-overflow-hidden html-fill-item" id="htmlwidget-b93ad92f08d972847a2d" style="width:504px;height:252px;"></div>
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</center>

---
## Selection Bias Definition

- **Selection bias** is bias that occurs due the presence of selection. 
  - An estimator that would be unbiased if all data were observed is biased due to conditioning on `$C = 0$`.

- Selection bias occurs when we condition on a variable, `$C$`, which is a common effect of two variables, `$X_1$` and `$X_2$` and

- `$X_1$` is either the treatment or *associated* with the treatment.

- `$X_2$` is either the exposure or *associated* with the exposure.

- Equivalently, conditioning on `$C$` leads to selection bias **unless** `$Y \ci C \vert A$` (i.e. `$Y$` is `$d$`-separated from `$C$` by `$A$`).

---

## Selection Could Happen Before the Outcome

<center>
<div class="grViz html-widget html-fill-item-overflow-hidden html-fill-item" id="htmlwidget-e2cd9363b5769dcee817" style="width:504px;height:360px;"></div>
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</center>

---

## Selection Could Happen Before the Exposure

<center>
<div class="grViz html-widget html-fill-item-overflow-hidden html-fill-item" id="htmlwidget-0203a41752541426dab0" style="width:504px;height:360px;"></div>
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</center>

---
# Augmenting DAGs

- The DAGs we have been drawing are harboring a hidden counterfactual.

- If there is no effect of selection on `$Y$`, he node `$Y$` could be written as `$Y(C = 0)$`, the value of `$Y$` that we would observe if nobody was censored.

- Our DAGs are missing the value of `$Y$` that we actually observe, `$Y^{obs}$` which is determined by `$Y(C = 0)$` and `$C$`.

<div class="grViz html-widget html-fill-item-overflow-hidden html-fill-item" id="htmlwidget-d0e8f8e183c6d1adedee" style="width:504px;height:252px;"></div>
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</center>

---
## Adjusting for Selection

- In some cases we can recover from selection bias.

- This will generally require information about the distribution of some variables without selection.

---
## Example

- In this graph, if we were did not have censoring (no conditioning on `$C$`), `$Y(a) \ci A$` (unconditionally)

- With no censoring, it is also true that `$Y(a) \ci A \vert L$` so `$E[Y(a) \vert L = l] = E[ Y \vert L = l, A = a]$`.

- From the causal markov property, `$Y \ci C \vert L, A$`, so `$E[Y \vert L = l, A = a] = E[Y \vert L = l, A = a, C = 0]$`.

<center>
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</center>

---
## Example

- We can identify the causal effect using the formula

$$
`\begin{split}
E[ Y(a) ] = \sum_l E[Y(a) \vert L = l] P[L = l]\\
= \sum_{l} E[ Y \vert L = l, A = a, C = 0] P[L = l]
\end{split}`
$$
- But this requires an estimate of `$P[L = l]$`, not `$P[L = l \vert C = 0]$`.

<center>
<div class="grViz html-widget html-fill-item-overflow-hidden html-fill-item" id="htmlwidget-24f6e64734dfef6ebbcf" style="width:504px;height:216px;"></div>
<script type="application/json" data-for="htmlwidget-24f6e64734dfef6ebbcf">{"x":{"diagram":"digraph {\n\ngraph [layout = \"neato\",\n       outputorder = \"edgesfirst\",\n       bgcolor = \"white\"]\n\nnode [fontname = \"Helvetica\",\n      fontsize = \"10\",\n      shape = \"circle\",\n      fixedsize = \"true\",\n      width = \"0.5\",\n      style = \"filled\",\n      fillcolor = \"aliceblue\",\n      color = \"gray70\",\n      fontcolor = \"gray50\"]\n\nedge [fontname = \"Helvetica\",\n     fontsize = \"8\",\n     len = \"1.5\",\n     color = \"gray80\",\n     arrowsize = \"0.5\"]\n\n  \"1\" [label = \"A\", fontname = \"Helvetica\", fontsize = \"10\", width = \"0.3\", fontcolor = \"black\", color = \"black\", fixedsize = \"TRUE\", shape = \"circle\", fillcolor = \"#FFFFFF\", pos = \"0,0!\"] \n  \"2\" [label = \"Y\", fontname = \"Helvetica\", fontsize = \"10\", width = \"0.3\", fontcolor = \"black\", color = \"black\", fixedsize = \"TRUE\", shape = \"circle\", fillcolor = \"#FFFFFF\", pos = \"1.8,0!\"] \n  \"3\" [label = \"L\", fontname = \"Helvetica\", fontsize = \"10\", width = \"0.3\", fontcolor = \"black\", color = \"black\", fixedsize = \"TRUE\", shape = \"circle\", fillcolor = \"#FFFFFF\", pos = \"1.2,1!\"] \n  \"4\" [label = \"C\", fontname = \"Helvetica\", fontsize = \"10\", width = \"0.3\", fontcolor = \"black\", color = \"black\", fixedsize = \"TRUE\", shape = \"square\", fillcolor = \"#FFFFFF\", pos = \"0.6,0.5!\"] \n\"1\"->\"2\" [color = \"black\"] \n\"3\"->\"2\" [color = \"black\"] \n\"3\"->\"4\" [color = \"black\"] \n\"1\"->\"4\" [color = \"black\"] \n}","config":{"engine":"dot","options":null}},"evals":[],"jsHooks":[]}</script>
</center>

---

## Selection Backdoor Criterion

- Barenboim, Tian, and Pearl (2014) give an extension of the backdoor criterion for settings with selection.

- Let `$\mathbf{Z}$` be a set of conditioning variables with `$\mathbf{Z}^{+}$` non-descendents of `$A$` and `$\mathbf{Z}^{-}$` descendents of `$A$`.

- `$\mathbf{Z}$` satisfies the s-backdoor criterion relative to `$A$` and `$Y$` if:

1. `$\mathbf{Z}^{+}$` blocks all backdoor paths from `$A$` to `$Y$`.
1. `$Y \ci \mathbf{Z}^{-} \vert A, \mathbf{Z}^{+}$` ( `$Y$` is `$d$`-separated from `$\mathbf{Z}^{-}$` by `$A$` and `$\mathbf{Z}^{+}$`)
1. `$Y \ci C \vert A, \mathbf{Z}$` ( `$Y$` is `$d$`-separated from `$C$` by `$A$` and `$\mathbf{Z}$`)
1. `$P(\mathbf{Z})$` can be measured without selection.

---

## Selection Backdoor Criterion

- If `$\mathbf{Z}$` satisfies the s-backdoor criterion, then `$E[Y(a)]$` can be identified by

`$$P[Y(a)] = \sum_{z} P(Y \vert A, \mathbf{Z}, C = 0) P(\mathbf{Z})$$`

- We need `$P(A = a, Z = z, C = 0) > 0$` for all `$a$` and `$z$`.
  - Or equivalently, `$P[C = 0 \vert A = a, Z = z] > 0$`. 
---
## Example

- In this example, we can satisfy the s-backdoor criterion with `$\mathbf{Z} = \left\lbrace L \right \rbrace$` with `$\mathbf{Z}^{+} = \left\lbrace L \right \rbrace$` and `$\mathbf{Z}^{-} = \left\lbrace  \right \rbrace$`:

1. There are no backdoor paths from `$A$` to `$Y$`.
1. `$\mathbf{Z}^{-}$` is the empty set so the second condition is satisfied.
1.  `$Y \ci C \vert A, L$`
1. So we must be able to observe `$L$` without censoring.

<center>
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</center>

---
## IP Weighting for Selection Bias

- There is an alternative IP weighting approach to selection bias.

- First, we want to weight the data to look like a population with no selection. We need a set of variables `$L_1$` such that

`$$Y( C = 0) \ci C \vert\ L_1, A$$`

- We then weight the data by `$W^{c} = \frac{1}{P(C = 0 \vert L_1, A)}$`

- Second, we need a set of variables `$L_2$` such that

`$$Y(a, C = 0) \ci A \vert\ L_2$$`
- The second stage weights are `$W^{A} = \frac{1}{f(A \vert L_2)}$`

- The total weights are `$W = W^{C} W^{A}$`

---

## IP Weighting Example 1

- The graph we saw earlier is a modified verison of  HR Fig 8.3:

<center>
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</center>

- `$L$` represents pre-existing heart disease.

- `$A$` is random assignment to a diet containing wasabi.

- `$Y$` indicates death by the end of the trial.

- Some participants are lost to follow-up ( `$C = 1$` ) due either to heart disease or the treatment assignment.

---
## IP Weighting Example 1

- We must condition on `$L$` to block the path between `$Y(C = 0)$` and `$C.$`

- `$Y(a, C = 0)$` is independent of `$A$` unconditionally.

- Since there is no confounding, we only need to compute `$W^{C} = 1/P[C = 0 \vert L, A]$` for all levels of `$L$` and `$A$`.

- We can do this as long as only `$Y$` is censored.

- To use the stratification strategy, we only needed uncensored estimates of `$P(L)$`.

<center>
<div class="grViz html-widget html-fill-item-overflow-hidden html-fill-item" id="htmlwidget-51e168f0cf571b3827ee" style="width:504px;height:216px;"></div>
<script type="application/json" data-for="htmlwidget-51e168f0cf571b3827ee">{"x":{"diagram":"digraph {\n\ngraph [layout = \"neato\",\n       outputorder = \"edgesfirst\",\n       bgcolor = \"white\"]\n\nnode [fontname = \"Helvetica\",\n      fontsize = \"10\",\n      shape = \"circle\",\n      fixedsize = \"true\",\n      width = \"0.5\",\n      style = \"filled\",\n      fillcolor = \"aliceblue\",\n      color = \"gray70\",\n      fontcolor = \"gray50\"]\n\nedge [fontname = \"Helvetica\",\n     fontsize = \"8\",\n     len = \"1.5\",\n     color = \"gray80\",\n     arrowsize = \"0.5\"]\n\n  \"1\" [label = \"A\", fontname = \"Helvetica\", fontsize = \"10\", width = \"0.3\", fontcolor = \"black\", color = \"black\", fixedsize = \"TRUE\", shape = \"circle\", fillcolor = \"#FFFFFF\", pos = \"0,0!\"] \n  \"2\" [label = \"Y(C = 0)\", fontname = \"Helvetica\", fontsize = \"10\", width = \"0.3\", fontcolor = \"black\", color = \"black\", fixedsize = \"FALSE\", shape = \"ellipse\", fillcolor = \"#FFFFFF\", pos = \"1.8,0!\"] \n  \"3\" [label = \"L\", fontname = \"Helvetica\", fontsize = \"10\", width = \"0.3\", fontcolor = \"black\", color = \"black\", fixedsize = \"TRUE\", shape = \"circle\", fillcolor = \"#FFFFFF\", pos = \"1.2,1!\"] \n  \"4\" [label = \"C\", fontname = \"Helvetica\", fontsize = \"10\", width = \"0.3\", fontcolor = \"black\", color = \"black\", fixedsize = \"TRUE\", shape = \"square\", fillcolor = \"#FFFFFF\", pos = \"0.6,0.5!\"] \n\"1\"->\"2\" [color = \"black\"] \n\"3\"->\"2\" [color = \"black\"] \n\"3\"->\"4\" [color = \"black\"] \n\"1\"->\"4\" [color = \"black\"] \n}","config":{"engine":"dot","options":null}},"evals":[],"jsHooks":[]}</script>
</center>

---
## IP Weighting Example 1

- The table shows `$P[C = 0 \vert A, L]$` from the HR example.
<center> 
<table>
 <thead>
  <tr>
   <th style="text-align:left;">   </th>
   <th style="text-align:right;"> $A=0$ </th>
   <th style="text-align:right;"> $A=1$ </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:left;"> $L=0$ </td>
   <td style="text-align:right;"> 1.0 </td>
   <td style="text-align:right;"> 0.5 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> $L=1$ </td>
   <td style="text-align:right;"> 0.6 </td>
   <td style="text-align:right;"> 0.2 </td>
  </tr>
</tbody>
</table>
</center>

- Individuals with `$A = 0$` and `$L = 0$` get weight 1 because they were never censored.

- Individuals with `$A = 1$` and `$L =1$` get weight 5 because `$4/5$` of this stratum is censored.

---

## Positivity and Consistency

- In order to use IP weighting, we need `$P[C = 0 \vert A, L] > 0$` in all strata of `$A$` and `$L$`.

- We do not need `$P[C = 1 \vert A, L] > 0$`.

- We also need the the counterfactual outcome `$Y(a, C = 0)$` to be well-defined. 
  + If `$C$` is loss to follow-up, it makes sense to suppose that all patients were followed.

- Suppose that `$C$` is censoring due to death resulting from causes other than `$A$`. 
  + HR argue that it doesn't make sense to propose an intervention that that eliminates all other causes of death.
  
---
## IP Weighting Example 2

- In the previous example, we saw that both IP weighting and stratification on `$L$` could be used to identify the treatment effect.

<center>
<div class="grViz html-widget html-fill-item-overflow-hidden html-fill-item" id="htmlwidget-8df18c277611a6c2e213" style="width:504px;height:180px;"></div>
<script type="application/json" data-for="htmlwidget-8df18c277611a6c2e213">{"x":{"diagram":"digraph {\n\ngraph [layout = \"neato\",\n       outputorder = \"edgesfirst\",\n       bgcolor = \"white\"]\n\nnode [fontname = \"Helvetica\",\n      fontsize = \"10\",\n      shape = \"circle\",\n      fixedsize = \"true\",\n      width = \"0.5\",\n      style = \"filled\",\n      fillcolor = \"aliceblue\",\n      color = \"gray70\",\n      fontcolor = \"gray50\"]\n\nedge [fontname = \"Helvetica\",\n     fontsize = \"8\",\n     len = \"1.5\",\n     color = \"gray80\",\n     arrowsize = \"0.5\"]\n\n  \"1\" [label = \"A\", fontname = \"Helvetica\", fontsize = \"10\", width = \"0.3\", fontcolor = \"black\", color = \"black\", fixedsize = \"FALSE\", shape = \"circle\", fillcolor = \"#FFFFFF\", pos = \"0,0!\"] \n  \"2\" [label = \"Y(C = 0)\", fontname = \"Helvetica\", fontsize = \"10\", width = \"0.3\", fontcolor = \"black\", color = \"black\", fixedsize = \"FALSE\", shape = \"ellipse\", fillcolor = \"#FFFFFF\", pos = \"2.4,0!\"] \n  \"3\" [label = \"L\", fontname = \"Helvetica\", fontsize = \"10\", width = \"0.3\", fontcolor = \"black\", color = \"black\", fixedsize = \"FALSE\", shape = \"circle\", fillcolor = \"#FFFFFF\", pos = \"0.8,0!\"] \n  \"4\" [label = \"C\", fontname = \"Helvetica\", fontsize = \"10\", width = \"0.3\", fontcolor = \"black\", color = \"black\", fixedsize = \"FALSE\", shape = \"square\", fillcolor = \"#FFFFFF\", pos = \"1.6,0!\"] \n  \"5\" [label = \"U\", fontname = \"Helvetica\", fontsize = \"10\", width = \"0.3\", fontcolor = \"black\", color = \"black\", fixedsize = \"FALSE\", shape = \"circle\", fillcolor = \"#FFFFFF\", pos = \"1.6,0.8!\"] \n\"1\"->\"3\" [color = \"black\"] \n\"3\"->\"4\" [color = \"black\"] \n\"5\"->\"3\" [color = \"black\"] \n\"5\"->\"2\" [color = \"black\"] \n}","config":{"engine":"dot","options":null}},"evals":[],"jsHooks":[]}</script>
</center>

- In this case, stratifying by `$L$` induces confounding through the path `$A \rightarrow L \leftarrow U \rightarrow Y$`.

- In this graph, there is no way to satisfy the s-backdoor criterion without observing `$U$`:
  - To `$d$`-separate `$Y$` from `$C$`, we must condition on `$L$`. 
  - But `$L$` is a child of `$A$`, so in `$\mathbf{Z}^{-}$` and there is no way to `$d$`-separate `$L$` from `$Y$` without `$U$`.

---
## IP Weighting Example 2

<center>
<div class="grViz html-widget html-fill-item-overflow-hidden html-fill-item" id="htmlwidget-63aa34f09de31094051e" style="width:504px;height:180px;"></div>
<script type="application/json" data-for="htmlwidget-63aa34f09de31094051e">{"x":{"diagram":"digraph {\n\ngraph [layout = \"neato\",\n       outputorder = \"edgesfirst\",\n       bgcolor = \"white\"]\n\nnode [fontname = \"Helvetica\",\n      fontsize = \"10\",\n      shape = \"circle\",\n      fixedsize = \"true\",\n      width = \"0.5\",\n      style = \"filled\",\n      fillcolor = \"aliceblue\",\n      color = \"gray70\",\n      fontcolor = \"gray50\"]\n\nedge [fontname = \"Helvetica\",\n     fontsize = \"8\",\n     len = \"1.5\",\n     color = \"gray80\",\n     arrowsize = \"0.5\"]\n\n  \"1\" [label = \"A\", fontname = \"Helvetica\", fontsize = \"10\", width = \"0.3\", fontcolor = \"black\", color = \"black\", fixedsize = \"FALSE\", shape = \"circle\", fillcolor = \"#FFFFFF\", pos = \"0,0!\"] \n  \"2\" [label = \"Y(C = 0)\", fontname = \"Helvetica\", fontsize = \"10\", width = \"0.3\", fontcolor = \"black\", color = \"black\", fixedsize = \"FALSE\", shape = \"ellipse\", fillcolor = \"#FFFFFF\", pos = \"2.4,0!\"] \n  \"3\" [label = \"L\", fontname = \"Helvetica\", fontsize = \"10\", width = \"0.3\", fontcolor = \"black\", color = \"black\", fixedsize = \"FALSE\", shape = \"circle\", fillcolor = \"#FFFFFF\", pos = \"0.8,0!\"] \n  \"4\" [label = \"C\", fontname = \"Helvetica\", fontsize = \"10\", width = \"0.3\", fontcolor = \"black\", color = \"black\", fixedsize = \"FALSE\", shape = \"square\", fillcolor = \"#FFFFFF\", pos = \"1.6,0!\"] \n  \"5\" [label = \"U\", fontname = \"Helvetica\", fontsize = \"10\", width = \"0.3\", fontcolor = \"black\", color = \"black\", fixedsize = \"FALSE\", shape = \"circle\", fillcolor = \"#FFFFFF\", pos = \"1.6,0.8!\"] \n\"1\"->\"3\" [color = \"black\"] \n\"3\"->\"4\" [color = \"black\"] \n\"5\"->\"3\" [color = \"black\"] \n\"5\"->\"2\" [color = \"black\"] \n}","config":{"engine":"dot","options":null}},"evals":[],"jsHooks":[]}</script>
</center>

- Without `$U$`, we cannot apply the selection backdoor criterion to estimate `$E[Y(a)]$`, even with uncensored observations of `$A$` and `$L$`.

- However, weighting by `$1/P[C = 0 \vert A, L]$` works. 
  - We must be able to observe `$A$`, and `$L$` without censoring.

---

## Sources of Selection Bias

- Differential loss to follow-up: Participants may drop out of the study for reasons related to the treatment or outcome.

- Non-response: Social stigmas may make people more likely to omit some kinds of information than others.
  
- Self-selection/volunteer bias: Some individuals may be more likely to volunteer for a study than others. 
  
  
  
- Healthy worker bias: Participants for a study of an occupational exposure on an outcome are recruited from among those who are at work on the day the exposure is measured.
  + People may be more likely to miss work for reasons directly related to the outcome or for 
  reasons that are associated with both outcome and exposure (e.g. SES).

---

## Case-Control Studies

- The graph from our first example could have described a case-control study.

<center>
<div class="grViz html-widget html-fill-item-overflow-hidden html-fill-item" id="htmlwidget-0dfc6ccdc2cc6553989a" style="width:504px;height:108px;"></div>
<script type="application/json" data-for="htmlwidget-0dfc6ccdc2cc6553989a">{"x":{"diagram":"digraph {\n\ngraph [layout = \"neato\",\n       outputorder = \"edgesfirst\",\n       bgcolor = \"white\"]\n\nnode [fontname = \"Helvetica\",\n      fontsize = \"10\",\n      shape = \"circle\",\n      fixedsize = \"true\",\n      width = \"0.5\",\n      style = \"filled\",\n      fillcolor = \"aliceblue\",\n      color = \"gray70\",\n      fontcolor = \"gray50\"]\n\nedge [fontname = \"Helvetica\",\n     fontsize = \"8\",\n     len = \"1.5\",\n     color = \"gray80\",\n     arrowsize = \"0.5\"]\n\n  \"1\" [label = \"A\", fontname = \"Helvetica\", fontsize = \"10\", width = \"0.3\", fontcolor = \"black\", color = \"black\", shape = \"circle\", fillcolor = \"#FFFFFF\", pos = \"0,0!\"] \n  \"2\" [label = \"Y\", fontname = \"Helvetica\", fontsize = \"10\", width = \"0.3\", fontcolor = \"black\", color = \"black\", shape = \"circle\", fillcolor = \"#FFFFFF\", pos = \"0.8,0!\"] \n  \"3\" [label = \"C\", fontname = \"Helvetica\", fontsize = \"10\", width = \"0.3\", fontcolor = \"black\", color = \"black\", shape = \"square\", fillcolor = \"#FFFFFF\", pos = \"1.6,0!\"] \n\"1\"->\"2\" [color = \"black\"] \n\"2\"->\"3\" [color = \"black\"] \n}","config":{"engine":"dot","options":null}},"evals":[],"jsHooks":[]}</script>
</center>

- Individuals are selected into the study based on their value of `$Y$`.

- In this case, we are no longer able estimate the average counterfactuals or the causal risk ratio. 
- However, in this DAG, we can estimate the causal odds ratio due to cancellation. 
---
## Case-Control Studies

- Without censoring, `$Y(a)$` and `$A$` are exchangeable so `$E[Y(a)]= E[Y \vert A]$`.

- We only get to observe `$E[Y \vert A, C = 0]$`
$$
\text{OR} = \frac{P[Y = 1 \vert A ]}{P[Y = 0 \vert A ]}\\\
= \frac{P[Y = 1\vert A, C = 0]P[C = 0 \vert A]}{P[Y = 0\vert A, C = 0]P[C = 0 \vert A]} \\\
= \frac{P[Y = 1\vert A, C = 0]}{P[Y = 0\vert A, C = 0]}
$$
- So the association odds ratio is equal to the causal odds ratio.

<center>
<div class="grViz html-widget html-fill-item-overflow-hidden html-fill-item" id="htmlwidget-06897d8de67961a17b0e" style="width:504px;height:108px;"></div>
<script type="application/json" data-for="htmlwidget-06897d8de67961a17b0e">{"x":{"diagram":"digraph {\n\ngraph [layout = \"neato\",\n       outputorder = \"edgesfirst\",\n       bgcolor = \"white\"]\n\nnode [fontname = \"Helvetica\",\n      fontsize = \"10\",\n      shape = \"circle\",\n      fixedsize = \"true\",\n      width = \"0.5\",\n      style = \"filled\",\n      fillcolor = \"aliceblue\",\n      color = \"gray70\",\n      fontcolor = \"gray50\"]\n\nedge [fontname = \"Helvetica\",\n     fontsize = \"8\",\n     len = \"1.5\",\n     color = \"gray80\",\n     arrowsize = \"0.5\"]\n\n  \"1\" [label = \"A\", fontname = \"Helvetica\", fontsize = \"10\", width = \"0.3\", fontcolor = \"black\", color = \"black\", shape = \"circle\", fillcolor = \"#FFFFFF\", pos = \"0,0!\"] \n  \"2\" [label = \"Y\", fontname = \"Helvetica\", fontsize = \"10\", width = \"0.3\", fontcolor = \"black\", color = \"black\", shape = \"circle\", fillcolor = \"#FFFFFF\", pos = \"0.8,0!\"] \n  \"3\" [label = \"C\", fontname = \"Helvetica\", fontsize = \"10\", width = \"0.3\", fontcolor = \"black\", color = \"black\", shape = \"square\", fillcolor = \"#FFFFFF\", pos = \"1.6,0!\"] \n\"1\"->\"2\" [color = \"black\"] \n\"2\"->\"3\" [color = \"black\"] \n}","config":{"engine":"dot","options":null}},"evals":[],"jsHooks":[]}</script>
</center>

---
## Case-Control Studies

- If `$Y$` is the only cause of selection, we can recover `$E[Y(a)]$` by using outside information, even though there is no way to satisfy the s-backdoor criterion.

- If we know `$P[Y = 1] = \alpha$` in the target population, we can compute the value of `$P[Y \vert A]$` that we would have observed in the full population.

$$
`\begin{split}
P[Y(a)=1] = & P[Y =1 \vert A = a] = \frac{P[A = a \vert Y = 1]P[Y = 1]}{P[A = a]} \\
= &\frac{\alpha P[A = a \vert Y = 1]}{\alpha P[A = a \vert Y = 1] + (1-\alpha)P[A = a \vert Y = 0]}\\
= & \frac{\alpha P[A = a \vert Y = 1, C= 0]}{\alpha P[A = a \vert Y = 1, C = 0] + (1-\alpha)P[A = a \vert Y = 0, C = 0]}
\end{split}`
$$
<center>
<div class="grViz html-widget html-fill-item-overflow-hidden html-fill-item" id="htmlwidget-97755cc046aadd44034c" style="width:504px;height:72px;"></div>
<script type="application/json" data-for="htmlwidget-97755cc046aadd44034c">{"x":{"diagram":"digraph {\n\ngraph [layout = \"neato\",\n       outputorder = \"edgesfirst\",\n       bgcolor = \"white\"]\n\nnode [fontname = \"Helvetica\",\n      fontsize = \"10\",\n      shape = \"circle\",\n      fixedsize = \"true\",\n      width = \"0.5\",\n      style = \"filled\",\n      fillcolor = \"aliceblue\",\n      color = \"gray70\",\n      fontcolor = \"gray50\"]\n\nedge [fontname = \"Helvetica\",\n     fontsize = \"8\",\n     len = \"1.5\",\n     color = \"gray80\",\n     arrowsize = \"0.5\"]\n\n  \"1\" [label = \"A\", fontname = \"Helvetica\", fontsize = \"10\", width = \"0.3\", fontcolor = \"black\", color = \"black\", shape = \"circle\", fillcolor = \"#FFFFFF\", pos = \"0,0!\"] \n  \"2\" [label = \"Y\", fontname = \"Helvetica\", fontsize = \"10\", width = \"0.3\", fontcolor = \"black\", color = \"black\", shape = \"circle\", fillcolor = \"#FFFFFF\", pos = \"0.8,0!\"] \n  \"3\" [label = \"C\", fontname = \"Helvetica\", fontsize = \"10\", width = \"0.3\", fontcolor = \"black\", color = \"black\", shape = \"square\", fillcolor = \"#FFFFFF\", pos = \"1.6,0!\"] \n\"1\"->\"2\" [color = \"black\"] \n\"2\"->\"3\" [color = \"black\"] \n}","config":{"engine":"dot","options":null}},"evals":[],"jsHooks":[]}</script>
</center>

---

## Selection Bias and Hazard Ratios

- Suppose we have a single treatment `$A$` and then individuals are followed over time.

- We are interested in estimating the counterfactual risk of death under treatment `$a$` (or the RR comparing treatment `$a$` and `$a^\prime$` ).

- For simplicity, assume we have two discrete time points and know 
  + `$Y_1$`: death  by time point 1
  + `$Y_2$`: death by time point 2
  
<center>
<div class="grViz html-widget html-fill-item-overflow-hidden html-fill-item" id="htmlwidget-f68611e04999048778a5" style="width:504px;height:180px;"></div>
<script type="application/json" data-for="htmlwidget-f68611e04999048778a5">{"x":{"diagram":"digraph {\n\ngraph [layout = \"neato\",\n       outputorder = \"edgesfirst\",\n       bgcolor = \"white\"]\n\nnode [fontname = \"Helvetica\",\n      fontsize = \"10\",\n      shape = \"circle\",\n      fixedsize = \"true\",\n      width = \"0.5\",\n      style = \"filled\",\n      fillcolor = \"aliceblue\",\n      color = \"gray70\",\n      fontcolor = \"gray50\"]\n\nedge [fontname = \"Helvetica\",\n     fontsize = \"8\",\n     len = \"1.5\",\n     color = \"gray80\",\n     arrowsize = \"0.5\"]\n\n  \"1\" [label = \"A\", fontname = \"Helvetica\", fontsize = \"10\", width = \"0.3\", fontcolor = \"black\", color = \"black\", shape = \"circle\", fillcolor = \"#FFFFFF\", pos = \"0,0!\"] \n  \"2\" [label = <Y<FONT POINT-SIZE=\"8\"><SUB>2<\/SUB><\/FONT>>, fontname = \"Helvetica\", fontsize = \"10\", width = \"0.3\", fontcolor = \"black\", color = \"black\", shape = \"circle\", fillcolor = \"#FFFFFF\", pos = \"1.6,0!\"] \n  \"3\" [label = <Y<FONT POINT-SIZE=\"8\"><SUB>1<\/SUB><\/FONT>>, fontname = \"Helvetica\", fontsize = \"10\", width = \"0.3\", fontcolor = \"black\", color = \"black\", shape = \"circle\", fillcolor = \"#FFFFFF\", pos = \"0.8,0!\"] \n  \"4\" [label = \"U\", fontname = \"Helvetica\", fontsize = \"10\", width = \"0.3\", fontcolor = \"black\", color = \"black\", shape = \"circle\", fillcolor = \"#FFFFFF\", pos = \"1.2,0.5!\"] \n\"1\"->\"3\" [color = \"black\"] \n\"4\"->\"3\" [color = \"black\"] \n\"4\"->\"2\" [color = \"black\"] \n\"3\"->\"2\" [color = \"black\"] \n}","config":{"engine":"dot","options":null}},"evals":[],"jsHooks":[]}</script>
</center>

---

## Hazard Ratios

- In this DAG, we can estimate the total causal effect of `$A$` on both `$Y_1$` and `$Y_2$` since we have exchangeability for both. 
  - In both cases, the causal risk ratio is equal to the association risk ratio.

- The *hazard* at time 2 is the probability of dying by time 2 conditional on being alive at time 1 (for discrete time).

- Based on our DAG, conditional on `$Y_1$`, there is no effect of `$A$` on `$Y_2$`.

- However, conditioning on `$Y_1$` induces a non-causal association between `$Y_2$` and `$A$` through `$U$`.

<center>
<div class="grViz html-widget html-fill-item-overflow-hidden html-fill-item" id="htmlwidget-4a4b0eca9c18ff4f73f4" style="width:504px;height:180px;"></div>
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</center>

---
## Hazard Ratios Example

- Suppose that `$U$` is an indicator for being high-risk or low-risk.

- With no treatment, half of the high-risk individuals would die at each time and most of the low-risk individuals would survive.

- Suppose that the treatment kills all high-risk individuals by time 1 and has no effect for low-risk individuals.

- At time 2, the treatment group contains only low-risk individuals, but the control group contains a mix of low and high-risk individuals.

- At time 2, a greater proportion of  individuals in the control group will die than in the treatment group, so the hazard ratio at time 2 will be less than 1.
  - Even though the treatment is not beneficial for any patients at any time point!

---
# 2. Non-Compliance

---
## Non-Compliance

- We perform a randomized trial of smoking cessation.

- A population of current smokers with no immediate plans to quit are recruited.

- Half the participants are assigned to quit smoking for six weeks, the other half are assigned to continue smoking as usual ( `$Z$` ).

- We measure cardiovascular endurance at the beginning and end of the study.

- Our outcome, `$Y$`, represents the change in endurance over 6 weeks.

---
## Non-Compliance

- Suppose that both treatment groups have some rate of non-adherence to the treatment plan.

+ There are some people who are assigned to quit and don't. 
  + Some people assigned to continue smoking are inspired by their study particpation and decide to quit anyway.

- Let `$A$` represent the actual treatment each person receives (quitting or not).

- Let `$U$` be a confounder that affects both adherence and change in endurance.

- Draw a DAG of this scenario.

---
## Non-Compliance

<center>
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- The blue arrow may exist if knowledge of the treatment assignment alters participants behavior. 
  + People who are assigned to quit and don't may exercise more to "make up" for not quitting. 
  
- The blue arrow might be eliminated if it is possible to conceal the treatment from participants (*blinding*).

---

## Non-Compliance

- We would like to estimate `$E[Y(a)]$` and 
   + `$E[Y(A=1)] - E[Y(A = 0)]$`, the *per-protocol (PP) effect*.
   + In this graph, the presence of `$U$` means that `$E[Y(a)]$` is not identifiable.

- We can identify `$E[Y(z)]$`. 
  + `$E[Y(z =1)] - E[Y(z = 0)]$` is the *intention-to-treat (ITT) effect*. 
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</center>
---

## Pros of the ITT

- The ITT can be measured from the data without confounding and is therefore often preferred.

- If we further assume that the blue arrow does not exist, then the following arguments are in favor of the ITT.

- The ITT preserves the null: If there is no effect of `$A$` on `$Y$` then there is no effect of `$Z$` on `$Y$`.

- If we further assume *monotonicity* ( `$Y_i(1) \geq Y_i(0)$` for all individuals `$i$` ), then the ITT effect is closer to zero than the PP effect, making the estimate conservative.

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---

## Cons of the ITT

- Conservativeness is not always good.

+ For example, if we are looking for adverse effects of a medication, a conservative estimate is dangerous. 
  
- If montonicity does not hold, the ITT may be anti-conservative:

+ Suppose individuals who benefit from the treatment are more likely to comply than individuals who would be harmed by it.
  
- In some cases, assuming the blue arrow is not present is unreasonable. 
  + If the blue arrow is present the ITT may differ from the PP in any direction.

---

## "As-Treated" Analysis to Estimate the PP Effect

- If we can measure confounding factors between `$A$` and `$Y$`, we can estimate the PP effect using IP weighting or standardization.

- In this case we are treating our trial data like observational data. 
  
- This is the "as-treated" analysis.

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</center>

---
## "Per-Protocol" Analysis to Estimate the PP Effect

- Another commonly used alternative is to exclude all non-compliers from the analysis.

- This approach introduces selection bias unless the the confounders `$U$` are measured.

- So either way, we need to measure `$U$`.

- Later we will see an alternative method, instrumental variable analysis, which requires some additional assumptions.

<center>
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</center>

---
# 3. Measurment Error

---
## Measurement Error

- The non-compliance problem is similar to a measurement error problem.

+ `$Z$` is like a mis-measured version of `$A$`.

- More generally, measurements of `$A$`, `$Y$`, or other variables could be inaccurate.

- We won't cover methods for accounting for measurement error, but it is important to be aware of.

---

## Measurement Error in DAGs

- To represent measurement error in a DAG, we can use different nodes for measured values ( `$A^*$` and `$Y^*$` below) and true values ( `$A$` and `$Y$`).

- Add in other variables that might affect the measured value.

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</center>

---

## Independent, Non-Differential Measurement Error

- The graph below represents **independent**, **non-differential** measurement error.

- It is **independent** because `$U_A$` is independent of `$U_Y$`.

- It is **non-differential** because `$U_A$` and `$U_Y$` are independent of `$A$` and `$Y$`.

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</center>

---

## Independent, Non-Differential Measurement Error

- Even though `$Y(a) \ci A$` unconditionally, `$E[Y^* \vert A^*] \neq E[Y(a)]$`.

- If the strict null holds, then `$E[Y^*\vert A^* = 1] - E[Y^* \vert A^* = 0]$` is an unbiased estimate of the ATE (which is 0).

- However, if the strict null does not hold, bias could be in any direction. The associational estimate may even be opposite sign from the true value.

- This can occur if `$E[A^* \vert A]$` is not monotonic in `$A$`.

<center>
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</center>

---

## Differential Measurement Error

- `$Y$` might affect `$U_A$` if `$A$` is measured after some effect of `$Y$` has already occurred, creating the appearance of reverse causation.

- `$A$` might affect `$U_Y$` if observation of `$A$` affects measurement of `$Y$`, e.g. closer monitoring of those with `$A = 1$`.

<center>
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---

## Non-Independent Measurement Error

- Non-independent measurement error occurs if measurement errors for `$A$` and `$Y$` are associated.

- For example, if both `$A$` and `$Y$` are measured by patient recall, some patients might have generally bad recall and their memory of `$A$` could affect their memory of `$Y$`.

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</center>

---

## Measurment Error in Confounders

- If a confounder, `$L$`, is measured with error, it will generally not be true that `$Y(a) \ci A \vert L^*$` even if `$Y(a) \ci A \vert L$`.

- Conditioning on `$L^*$` rather than `$L$` will leave residual confounding.

- Dichotomizing or coarsening confounders can introduce measurement error.

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</center>

---

## Dealing with Measurment Error

- Accounting for measurement error generally requiires outisde information.

- For example, with some "gold standard" samples, we could estimate a model for `$E[A^* \vert A]$` and `$E[Y^* \vert Y]$`.

- For the rest of this class, we will generally not worry about measurment error (or optimistically assume there is none).