L4: Selection Bias

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# L4: Selection Bias
### Jean Morrison
### University of Michigan
### 2022-24-10 (updated: 2022-01-26)

---

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

# Example

- We are interested in the effect of drug `$A$`, an HIV treatment on preventing disease progression.

- We will use CD4 count as a measurement of disease progression. Our outcome `$Y$` is 1 if CD4 count falls below a threshold within one year of starting treatment.

- 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. This could occur if their disease progresses and their CD4 count drops or if they are experiencing side effects ( `$L$` ) from the treatment.

- Use a variable `$C$` (for censoring) to represent if a patient drops out of the study before one year ( `$C = 1$` ) or not ( `$C = 0$`).

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

---

# HIV Treatment Example

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---

# HIV Treatment Example

- In this study, if we had been able to measure `$Y$` for all patients, could we identify the effect of `$A$` on `$Y$` -- i.e. would we have `$Y(a) \ci A$`?

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

- No. We have *selection bias*.

- `$C$` is a collider on a path between `$Y$` and `$L$`. Conditioning on `$C$` induces a correlation between `$Y$` and `$L$` and therefore a non-causal association between `$Y$` and `$A$`.

---

# HIV Treatment Example

- Suppose now 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?

---

# Example Continued

- Suppose that `$A$` is effective and 
- Patients with `$Y = 0$` are more likely to remain in the study than patients with `$Y = 1$`.

`$$P[Y = 1 \vert A = 0] = 0.8 \qquad P[Y = 1 \vert A = 1] = 0.1\\\
  P[C = 0 \vert Y = 0] = 1 \qquad P[C = 0 \vert Y = 1] = 0.5$$`

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

- Repeat your calculations assuming that there is no effect of `$A$` on `$Y$` and `$P[Y = 1 \vert A =0 ] = P[Y = 1 \vert A =1] = p$`.

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---

# Example Continued

- In the scenario with a treatment effect:

- The causal risk difference is 0.1-0.8 = -0.7.

- The associational risk difference among those with `$C = 0$` is `$0.05 - 0.67 = -0.62$`
---

# Example Continued

- In the scenario with no effect:

- The causal risk difference is 0.

- The associational risk difernce in the observed data is also 0. So there is no selection bias for the measure of association.

- However, in the observed data, `$E[Y \vert A= a, C = 0] \neq E[Y(a)]$`.
---

# 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*).

- 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 `$Y$`.

- 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.

- If the effect of the exposure on the selection variable is not entirely mediated by the outcome, we will have selection bias under the null.

- Recall that conditioning on the child of a collider will also open a path.

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---

# 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 `$Y$`.

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---
# Selection Bias Definition

- Selection bias occurs when we condition on a variable which is a common effect of two variables.

- One must be either the treatment or *associated* with the treatment.

- The other must be the exposure or *associated* with the exposure.

---

# Selection Could Happen Before the Outcome

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---

# Selection Could Happen Before the Exposure

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---
# Augmenting DAGs

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

- The 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 `$C$`.

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---
# Adjusting for Selection

- Under selection, we are interested in estimating the average of a joint counterfactual `$E[Y(a, C = 0)]$` -- the expected value of `$Y$` if everyone received treatment `$A$` and was not censored.

- We can use a two stage weighting procedure.

- First we re-weight the data to look like the world in which nobody was censored. For this we need a set of variables `$L_1$` such that
`$$Y( C = 0) \ci C \vert L_1$$`

- We use a second set of weights to account for any confounding. So we need a set of variables `$L_2$` such that

`$$Y(a, C = 0) \ci A \vert L_2$$`

- We will also need a modification of the positivity condition.

---
# Adjusting for Selection

- If `$Y(C = 0) \ci C \vert L_1$`, we can weight the data by `$W^C = 1/P[ C = 0 \vert L_1, A]$`

- For example, if half the participants with `$L = l$` and `$A = a$` were censored, the remaining individuals will receive weight 2.

- If we need to do further adjustment for confounding, we can compute weights `$W^A = 1/P[A = a \vert L_2]$`

- Our total weights will be `$W^C\cdot W^A$`.

---

# IP Weighting Example 1

- Using the example from HR (modified fig 8.3), data follow the DAG below:

<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.

- There is no effect of `$A$` on `$Y$`, though this is unknown to investigators.

---
# IP Weighting Example 1

</center>

- 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 `$P[C = 0 \vert L, A]$` for all levels of `$L$` and `$A$`.

---
# 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;"> $Y=0$ </td>
   <td style="text-align:right;"> 1.0 </td>
   <td style="text-align:right;"> 0.5 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> $Y=1$ </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 only 20\% of this strata were uncensored.

---

# 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 also could have stratified on `$L$`.

- This doesn't always work.

<center>
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</center>

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

- But weighting by `$1/P[C = 0 \vert A, L]$` works.
---

# 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 id="htmlwidget-3c70cf0ce9b8ac989c71" style="width:504px;height:108px;" class="grViz html-widget"></div>
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</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>
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</center>

---
# Case-Control Studies

- If `$Y$` is the only cause of selection, we can recover `$E[Y(a)]$` by using outside information.

- 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.

$$
P[Y(a)=1] = P[Y =1 \vert A] = \frac{P[A \vert Y]P[Y = 1]}{P[A]} \\\
= \frac{\alpha P[A \vert Y = 1]}{\alpha P[A \vert Y = 1] + (1-\alpha)P[A \vert Y = 0]}
$$

<center>
<div id="htmlwidget-702622bd8bb74bd61d3d" style="width:504px;height:108px;" class="grViz html-widget"></div>
<script type="application/json" data-for="htmlwidget-702622bd8bb74bd61d3d">{"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>
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</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>
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</center>

---
# Hazard Ratios Example

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

- 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. The control group contains a mix of low and high-risk individuals.

- So the hazard ratio at time 2 will be less than 1 even though the treatment is not beneficial for any patients.

<center>
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</center>

---
# 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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---

# 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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---
# "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.

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---

# 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 inaccurate.

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

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---

# Sources and Types of Measurement Error

- Measurement error is *independent* if `$U_A$` is independent of `$U_Y$`

- Measurement error might be non-independent if `$A$` and `$Y$` are collected in the same survey.
  
- Measurement error is *non-differential* if `$U_A$` and `$U_Y$` are independent of `$A$` and `$Y$`. 
- Differential error could occur in self-reported variables if participants tend to mis-report high or low values of `$A$` or `$Y$`.

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

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