SELECTION BIAS - Core concepts

class: left, middle, inverse, title-slide

.title[
# SELECTION BIAS - Core concepts
]
.author[
### Mabel Carabali
]
.institute[
### EBOH, McGill University
]
.date[
### Updated: 2024-11-19
]

---

class:middle
**Expected Competencies**
- Knowledge about design mechanisms for Selection Bias.
- Knows what would be the impact of selection bias on study designs

## Objectives
- Identify sources and structure of selection bias.
- Differentiate between selection bias and other biases or statistical artifacts. 
- Identify alternative to address selection bias.

---
class:middle
### Selection Bias
- Can arise in both randomized experiments and observational studies, and in both prospective and retrospective studies.

> _"…the common consequence of selection bias is that the association between exposure and outcome among those selected for analysis differs from the association among those eligible."_

- <span style="color:blue">**Type 1:**</span> people are selected or not-selected into analysis
- <span style="color:magenta">**Type 2:**</span> bias introduced by conditioning on a collider, but no one is left out of analysis.

[Hernán MA, Hernández-Díaz S, Robins JM. A structural approach to selection bias. Epidemiology. 2004 Sep;15(5):615-25.](https://pubmed.ncbi.nlm.nih.gov/15308962/)

- Selection bias as used by [H&R What If?](https://www.hsph.harvard.edu/miguel-hernan/causal-inference-book/) does NOT refer to failure to generalize externally.

- It refers to non-causal (spurious) association that occurs in the data set under analysis.

---

class:middle
### A Nice Resource

- Directed Acyclic Graphs
- A Taxonomy of Selection Bias
- Failure to Generalize
- Conditioning on a Collider
- Design Considerations
- Analytic Strategies for Addressing Selection Bias
- Sensitivity Analysis

[Selection Mechanisms and Their Consequences: Understanding and Addressing Selection Bias](https://link.springer.com/article/10.1007/s40471-020-00241-6)

Smith, L.H. Selection Mechanisms and Their Consequences: Understanding and Addressing Selection Bias. Curr Epidemiol Rep 7, 179–189 (2020). https://doi.org/10.1007/s40471-020-00241-6

---
class:middle
### A Nice Resource for causal inference

> _"Selection bias can be further categorized into two broad types:_ 
- _type 1 selection bias owing to restricting to one or more level(s) of a collider (or a descendant of a collider) and,_
- _type 2 selection bias owing to restricting to one or more level(s) of an effect measure modifier."_

[Toward a Clearer Definition of Selection Bias When Estimating Causal Effects](https://oce.ovid.com/article/00001648-202209000-00013/HTML)

Lu, H. , Cole, S. , Howe, C. & Westreich, D. (2022). Toward a Clearer Definition of Selection Bias When Estimating Causal Effects. Epidemiology, 33 (5), 699-706. doi: 10.1097/EDE.0000000000001516.

---
class:middle
###[Chapter 8: H&R What if?](https://www.hsph.harvard.edu/miguel-hernan/causal-inference-book/)

> _“We will use the term selection bias to refer to all biases that arise from conditioning on a common effect of two variables, one of which is either the treatment or a cause of treatment, and the other is either the outcome or a cause of the outcome.”_

**8.2: H&R What if?** Practical examples of such structures:

> 
- Differential loss to follow-up (a.k.a informative censoring) - Missing data bias, nonresponse bias
- Healthy worker bias
- Self-selection bias, volunteer bias
- Selection affected by treatment received before study entry

---
class:middle
###Berkson bias
Berkson (1946, 1955) noted that two diseases (`A` and `B`) that are not associated in the population could be associated among hospitalized patients when both diseases affect the probability of hospital admission (`H`).

Berkson’s bias can be seen to arise from conditioning on the common effect `H` of diseases `A` and `B`:
<img src="L19_EPIB704_Selection_Bias_files/figure-html/unnamed-chunk-1-1.png" width="50%" style="display: block; margin: auto;" />

---
class:middle
###Berkson bias
In a case-control study where cases were hospitalized patients with disease `B` and controls were hospitalized patients with disease `A`, an exposure `F` that causes disease `A` would appear to be a risk factor for disease `B`.

That is, `$OR_{FB|H =1}$` would differ from 1.0 even if `F` does not cause `B`.

---
class:middle
###Berkson bias

This bias occurs because conditioning on `H` induces a correlation between `A` and `B`, even if these are independent in the population.

`Hernán MA, Hernández-Díaz S, Robins JM. A structural approach to selection bias. Epidemiology 2004 Sep;15(5):615-25.`

---
class:middle
### 8.5. H&R What if? How to adjust for selection bias
- Inverse probability of censoring weights (IPCW) work exactly the same way as inverse of probability of treatment weights (IPTW). In fact, a typical study will have both weights.
- Note that positivity is NOT required for the C=1 stratum, since it is not our target population. Only for the C=0 stratum.
  
**Important point:**

For confounding, one had the choice of adjusting for the confounder by stratification or conditioning (conditional effect) or by standardization or g-methods (marginal effect).

- For collapsible effect measures, these are equivalent.

---
class:middle
### IPCW vs "Adjusting/Conditioning"
Let's consider this DAG (reproduced from H&R What If? Figure 8.6).
.pull-left[
<img src="L19_EPIB704_Selection_Bias_files/figure-html/unnamed-chunk-4-1.png" width="80%" height="110%" style="display: block; margin: auto;" />
]
.pull-right[
The other c-word, **Censoring**
- Bias due to informative censoring.
- Restricting the analysis to individuals with complete data (C = 0) may result in bias.
- Healthy worker bias
- Self-selection bias, volunteer bias
]

---
class:middle
### IPCW vs "Adjusting/Conditioning"

IPCW can be used to appropriately adjust for selection bias and conditioning will not work here and similar circumstances.

---
class:middle
### IPCW vs "Adjusting/Conditioning"

- Because IPCW is not based on estimating effect measures conditional on the measured covariates `L`, but rather on estimating unconditional effect measures after reweighting the subjects according to their treatment `A` and their values of `L`.

<img src="L19_EPIB704_Selection_Bias_files/figure-html/unnamed-chunk-6-1.png" width="30%" style="display: block; margin: auto;" />
>This is the first time `H&R` discuss a situation in which stratification cannot be used to validly compute the causal effect of treatment, even if the three conditions of exchangeability, positivity, and consistency hold. `Modified from Jay Kaufman EPIB-704-2021`

---
class:middle
###Hazard of the Hazards .red[(Recall?)]
[Technical Point 8.1: H&R What if?](https://www.hsph.harvard.edu/miguel-hernan/causal-inference-book/) The built-in selection bias of hazard ratios.
.pull-left[
- In discrete time, the hazard of death at time 1 is the probability of dying at time 1 and thus the associational hazard ratio is the same as `$aRR_{AY_1}$`. 
- However, the hazard at time 2 is the probability of dying at time 2 among those who survived
past time 1.
- Thus, the associational hazard ratio at time 2 is then `$aRR_{AY_2|Y_1=0} = \left(\frac{Pr[Y_2 = 1|A=1, Y_1=0]}{Pr[Y_2 = 1|A=0, Y_1=0]}\right)$`
]
.pull-right[
<img src="L19_EPIB704_Selection_Bias_files/figure-html/unnamed-chunk-7-1.png" width="80%" style="display: block; margin: auto;" />
]

---
class:middle
### Hazard of the Hazards .red[(Recall?)]
Survival Produces an Unavoidable Selection Bias.
- Start out with a randomized trial so that all covariates are balanced at time 0.
- Once events occur, if you condition your estimate on having survived to the next time point, every other cause of disease must now be correlated with exposure.

---
class:middle
###Hazard of the Hazards .red[(Recall?)]
This is exactly why the HAZARD RATIO (the parameter estimated by a Cox Proportional Hazards Model) should not be used (unless the outcome is rare):

- The hazard of death at `$t_1$` is the probability of dying at `$t_1$`. 
- But the hazard at `$t_2$` is the probability of dying at `$t_2$` **among those who survived past `$t_1$` .**

Treated survivors of `$t_1$` differ in their distribution of `U` compared to untreated survivors of `$t_1$`, making this conditional measure confounded by `U` in a way that a marginal measure is not.

**This concern applies to both observational studies and randomized experiments.**

[Hernán MA, The hazards of hazard ratios. Epidemiology 2010;21(1):13-5.])(https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3653612/)

---
class:middle
###Hazard of the Hazards .red[(Recall?)]
.pull-left[
<img src="L19_EPIB704_Selection_Bias_files/figure-html/unnamed-chunk-9-1.png" width="80%" style="display: block; margin: auto;" />
]
--
.pull-right[
<img src="L19_EPIB704_Selection_Bias_files/figure-html/unnamed-chunk-10-1.png" width="80%" style="display: block; margin: auto;" />
]

---
class:middle
###Selection bias and Confounding
To the extent that confounding and selection bias are due to measured covariates `C`, these can be handled by inverse weighting (IPTW, IPCW)
- This is especially convenient for longitudinal data in which the confounder `C` may be effected by previous treatment `$X_t$` and may in turn influence the next dose of treatment `$X_{t+1}$`.
- It is also helpful in the longitudinal setting where the remaining cohort at each time `$t$` becomes increasingly selected. 
- Reweighting the cohort by measured characteristics allows remaining subjects to proxy for the ones that are missing.

---
class:middle
###IPTW vs IPCW

- <span style="color:darkred">Unstable or extreme weights could be problematic in both cases!</span>

---
class:middle
###The strength and direction of selection bias.

Regardless of the direction of selection bias, another key issue is its magnitude. 
- Biases that are not large enough to affect the conclusions of the study may be safely ignored in practice, whether the bias is upwards or downwards. 
- Generally speaking, a large selection bias requires strong associations between the collider and both treatment and outcome. 
- Greenland (2003) studied the magnitude of selection bias under the null, which he referred to as collider-stratification bias, in several scenarios.

---
class:middle
### Avoid Selection Bias
- Know your topic
- Know your population(s)
- Use the "best" design you can (remember "built-in" selection bias, e.g., case-control) `$^1$`
  - Case control: Exposure is missing based on outcome: RD is biased
  - Case control: Outcome missing, based on outcome: RD & RR are biased -OR unbiased
- High participation and response rates
- Complete and objective ascertainment
- Complete follow-up

`$^1$` [Ch. 7. Case-Control Studies. Epidemiology by Design by Daniel Westreich](https://academic.oup.com/book/32358/chapter/268624216) or
https://www.epidemiologybydesign.com/about

.pull-right[ <span style="color:darkred">If none of these works... try analytic solutions BUT nothing could completely save a poor design</span>]

---
class:middle
### How else we can adjust for selection bias?
**.blue[(selection) Relative Odds Ratio]**

If the marginal distribution is known, the estimation of the ratio between the Odds among participants (sub-sample) and the corresponding odds in the source population:  
 `$$ROR = \frac{OR_{SubPop}}{OR_{TotPop}}$$`

- `$ROR=1$`, no bias; `$ROR>1$`, overestimation bias; `$ROR<1$`, underestimation bias. Equivalently: $$OR_{Tot} = OR{Sub}\times ROR $$

- Nohr EA, Liew Z. How to investigate and adjust for selection bias in cohort studies. Acta Obstet Gynecol Scand 2018; 97: 407–416. https://doi.org/10.1111/aogs.13319

---
class:middle
### How else we can adjust for selection bias?
**.blue[(selection) Relative Odds Ratio]**

The relative odds ratio (ROR) computed as the cross product ratio of the participation rates in the four exposure by outcome categories.

.small[Nohr EA, Liew Z. How to investigate and adjust for selection bias in cohort studies. Acta Obstet Gynecol Scand 2018; 97: 407–416. https://doi.org/10.1111/aogs.13319]
---
class:middle
### How else we can to adjust for selection bias?

**.blue[Correction by Projecting the Exposed Proportion Among Nonparticipants]**
- Use contingency table
- Depict/Classify participants and non participants by exposure groups
- Obtain the odds of participation by exposure status
- Estimate the OR among non-participants

.red[
+ Ignore some non-participants characteristics
+ Assumes same prevalence of exposure
]

---
class:middle
### How else we can to adjust for selection bias?
**.blue[Correction Using Selection Proportions]**

|Selection probabilities| Exposure = 1  |  Exposure = 0 |
|----------------------:|:-------------:|:-------------:|
|Cases| `$S_{case, 1}$` | `$S_{case, 0}$` |
|Control| `$S_{control, 1}$` | `$S_{control, 0}$` |

`$$OR_{adj} = OR_{obs}\times \frac{S_{case}, S_{control,1}}{S_{case, 1}, S_{control,0}}$$`

.small[Applying Quantitative Bias Analysis to Epidemiologic Data: [Chapter  4 - Selection Bias Spreadsheet](https://sites.google.com/site/biasanalysis/Home)]

---
class: middle

###  QUESTIONS?

## COMMENTS?

# RECOMMENDATIONS?

---

class:middle
**Example from a vector-borne disease** n=839 individuals followed for 3 months

.pull-left[
<div id="pzveloiefp" style="padding-left:0px;padding-right:0px;padding-top:10px;padding-bottom:10px;overflow-x:auto;overflow-y:auto;width:auto;height:auto;">
<style>#pzveloiefp 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;
}

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

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

#pzveloiefp .gt_table {
  display: table;
  border-collapse: collapse;
  line-height: normal;
  margin-left: auto;
  margin-right: auto;
  color: #333333;
  font-size: 16px;
  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;
}

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

#pzveloiefp .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;
}

#pzveloiefp .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;
}

#pzveloiefp .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;
}

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

#pzveloiefp .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;
}

#pzveloiefp .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;
}

#pzveloiefp .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;
}

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

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

#pzveloiefp .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%;
}

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

#pzveloiefp .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;
}

#pzveloiefp .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;
}

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

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

#pzveloiefp .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;
}

#pzveloiefp .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;
}

#pzveloiefp .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;
}

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

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

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

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

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

#pzveloiefp .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;
}

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

#pzveloiefp .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;
}

#pzveloiefp .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;
}

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

#pzveloiefp .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;
}

#pzveloiefp .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;
}

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

#pzveloiefp .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;
}

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

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

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

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

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

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

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

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

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

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

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

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

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

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

#pzveloiefp .gt_indent_5 {
  text-indent: 25px;
}

#pzveloiefp .katex-display {
  display: inline-flex !important;
  margin-bottom: 0.75em !important;
}

#pzveloiefp div.Reactable > div.rt-table > div.rt-thead > div.rt-tr.rt-tr-group-header > div.rt-th-group:after {
  height: 0px !important;
}
</style>
<table class="gt_table" data-quarto-disable-processing="false" data-quarto-bootstrap="false">
  <caption class='gt_caption'><span class='gt_from_md'><strong>Summary of covars distribution</strong></span></caption>
  <thead>
    <tr class="gt_col_headings">
      <th class="gt_col_heading gt_columns_bottom_border gt_left" rowspan="1" colspan="1" scope="col" id="label"><span class='gt_from_md'><strong>Characteristic</strong></span></th>
      <th class="gt_col_heading gt_columns_bottom_border gt_center" rowspan="1" colspan="1" scope="col" id="stat_0"><span class='gt_from_md'><strong>N = 839</strong></span><span class="gt_footnote_marks" style="white-space:nowrap;font-style:italic;font-weight:normal;line-height:0;"><sup>1</sup></span></th>
    </tr>
  </thead>
  <tbody class="gt_table_body">
    <tr><td headers="label" class="gt_row gt_left" style="font-weight: bold;">agecat_c</td>
<td headers="stat_0" class="gt_row gt_center"><br /></td></tr>
    <tr><td headers="label" class="gt_row gt_left">    0</td>
<td headers="stat_0" class="gt_row gt_center">76 / 839 (9.1%)</td></tr>
    <tr><td headers="label" class="gt_row gt_left">    1</td>
<td headers="stat_0" class="gt_row gt_center">81 / 839 (9.7%)</td></tr>
    <tr><td headers="label" class="gt_row gt_left">    2</td>
<td headers="stat_0" class="gt_row gt_center">215 / 839 (26%)</td></tr>
    <tr><td headers="label" class="gt_row gt_left">    3</td>
<td headers="stat_0" class="gt_row gt_center">343 / 839 (41%)</td></tr>
    <tr><td headers="label" class="gt_row gt_left">    4</td>
<td headers="stat_0" class="gt_row gt_center">124 / 839 (15%)</td></tr>
    <tr><td headers="label" class="gt_row gt_left" style="font-weight: bold;">sex</td>
<td headers="stat_0" class="gt_row gt_center">422 / 839 (50%)</td></tr>
    <tr><td headers="label" class="gt_row gt_left" style="font-weight: bold;">ipd</td>
<td headers="stat_0" class="gt_row gt_center">72 / 839 (8.6%)</td></tr>
    <tr><td headers="label" class="gt_row gt_left" style="font-weight: bold;">insurance2</td>
<td headers="stat_0" class="gt_row gt_center"><br /></td></tr>
    <tr><td headers="label" class="gt_row gt_left">    0</td>
<td headers="stat_0" class="gt_row gt_center">465 / 839 (55%)</td></tr>
    <tr><td headers="label" class="gt_row gt_left">    1</td>
<td headers="stat_0" class="gt_row gt_center">344 / 839 (41%)</td></tr>
    <tr><td headers="label" class="gt_row gt_left">    2</td>
<td headers="stat_0" class="gt_row gt_center">30 / 839 (3.6%)</td></tr>
  </tbody>
  
  <tfoot class="gt_footnotes">
    <tr>
      <td class="gt_footnote" colspan="2"><span class="gt_footnote_marks" style="white-space:nowrap;font-style:italic;font-weight:normal;line-height:0;"><sup>1</sup></span> <span class='gt_from_md'>n / N (%)</span></td>
    </tr>
  </tfoot>
</table>
</div>
]
.pull-right[
<div id="wisvdhjkbu" style="padding-left:0px;padding-right:0px;padding-top:10px;padding-bottom:10px;overflow-x:auto;overflow-y:auto;width:auto;height:auto;">
<style>#wisvdhjkbu 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;
}

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

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

#wisvdhjkbu .gt_table {
  display: table;
  border-collapse: collapse;
  line-height: normal;
  margin-left: auto;
  margin-right: auto;
  color: #333333;
  font-size: 16px;
  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;
}

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

#wisvdhjkbu .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;
}

#wisvdhjkbu .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;
}

#wisvdhjkbu .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;
}

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

#wisvdhjkbu .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;
}

#wisvdhjkbu .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;
}

#wisvdhjkbu .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;
}

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

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

#wisvdhjkbu .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%;
}

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

#wisvdhjkbu .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;
}

#wisvdhjkbu .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;
}

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

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

#wisvdhjkbu .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;
}

#wisvdhjkbu .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;
}

#wisvdhjkbu .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;
}

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

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

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

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

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

#wisvdhjkbu .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;
}

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

#wisvdhjkbu .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;
}

#wisvdhjkbu .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;
}

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

#wisvdhjkbu .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;
}

#wisvdhjkbu .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;
}

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

#wisvdhjkbu .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;
}

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

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

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

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

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

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

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

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

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

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

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

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

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

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

#wisvdhjkbu .gt_indent_5 {
  text-indent: 25px;
}

#wisvdhjkbu .katex-display {
  display: inline-flex !important;
  margin-bottom: 0.75em !important;
}

#wisvdhjkbu div.Reactable > div.rt-table > div.rt-thead > div.rt-tr.rt-tr-group-header > div.rt-th-group:after {
  height: 0px !important;
}
</style>
<table class="gt_table" data-quarto-disable-processing="false" data-quarto-bootstrap="false">
  <thead>
    <tr class="gt_col_headings">
      <th class="gt_col_heading gt_columns_bottom_border gt_left" rowspan="1" colspan="1" scope="col" id="label"><span class='gt_from_md'><strong>Characteristic</strong></span></th>
      <th class="gt_col_heading gt_columns_bottom_border gt_center" rowspan="1" colspan="1" scope="col" id="stat_0"><span class='gt_from_md'><strong>N = 839</strong></span><span class="gt_footnote_marks" style="white-space:nowrap;font-style:italic;font-weight:normal;line-height:0;"><sup>1</sup></span></th>
    </tr>
  </thead>
  <tbody class="gt_table_body">
    <tr><td headers="label" class="gt_row gt_left" style="font-weight: bold;">comorb</td>
<td headers="stat_0" class="gt_row gt_center">109 / 784 (14%)</td></tr>
    <tr><td headers="label" class="gt_row gt_left">    Unknown</td>
<td headers="stat_0" class="gt_row gt_center">55</td></tr>
    <tr><td headers="label" class="gt_row gt_left" style="font-weight: bold;">overfever</td>
<td headers="stat_0" class="gt_row gt_center"><br /></td></tr>
    <tr><td headers="label" class="gt_row gt_left">    Median (Q1, Q3)</td>
<td headers="stat_0" class="gt_row gt_center">7.0 (5.0, 9.0)</td></tr>
    <tr><td headers="label" class="gt_row gt_left" style="font-weight: bold;">myalgiadef</td>
<td headers="stat_0" class="gt_row gt_center">771 / 832 (93%)</td></tr>
    <tr><td headers="label" class="gt_row gt_left">    Unknown</td>
<td headers="stat_0" class="gt_row gt_center">7</td></tr>
    <tr><td headers="label" class="gt_row gt_left" style="font-weight: bold;">arthralgiadef</td>
<td headers="stat_0" class="gt_row gt_center">722 / 831 (87%)</td></tr>
    <tr><td headers="label" class="gt_row gt_left">    Unknown</td>
<td headers="stat_0" class="gt_row gt_center">8</td></tr>
    <tr><td headers="label" class="gt_row gt_left" style="font-weight: bold;">abdominalpain</td>
<td headers="stat_0" class="gt_row gt_center">326 / 834 (39%)</td></tr>
    <tr><td headers="label" class="gt_row gt_left">    Unknown</td>
<td headers="stat_0" class="gt_row gt_center">5</td></tr>
    <tr><td headers="label" class="gt_row gt_left" style="font-weight: bold;">a_leucopenia</td>
<td headers="stat_0" class="gt_row gt_center">179 / 710 (25%)</td></tr>
    <tr><td headers="label" class="gt_row gt_left">    Unknown</td>
<td headers="stat_0" class="gt_row gt_center">129</td></tr>
    <tr><td headers="label" class="gt_row gt_left" style="font-weight: bold;">outc</td>
<td headers="stat_0" class="gt_row gt_center">230 / 686 (34%)</td></tr>
    <tr><td headers="label" class="gt_row gt_left">    Unknown</td>
<td headers="stat_0" class="gt_row gt_center">153</td></tr>
  </tbody>
  
  <tfoot class="gt_footnotes">
    <tr>
      <td class="gt_footnote" colspan="2"><span class="gt_footnote_marks" style="white-space:nowrap;font-style:italic;font-weight:normal;line-height:0;"><sup>1</sup></span> <span class='gt_from_md'>n / N (%)</span></td>
    </tr>
  </tfoot>
</table>
</div>

]

---
class:middle
**Example from a vector-borne disease:** A descriptive exercise, no "treatment" or "intervention", just modeling the outcome as function of covariates (n=520 complete cases (`na.rm=T`), vs n=634 where some vars are imputed), `dataset without missingness, n=839`

``` r
mod0or<- glm(outc ~  factor(agecat_c) + sex +  ipd+ insurance2 + comorb + overfever + 
               myalgiadef+ arthralgiadef + abdominalpain + leucopenia + lowplatel, 
             data=sb.data1a, family= binomial)
```

--
.pull-left[
**Complete Case Analysis**

```
##                     OR 2.5 % 97.5 %
## (Intercept)       0.10 -3.64  -1.05
## factor(agecat_c)1 2.54  0.01   1.89
## factor(agecat_c)2 1.35 -0.51   1.15
## factor(agecat_c)3 0.72 -1.13   0.51
## factor(agecat_c)4 0.44 -1.86   0.18
## sex               0.89 -0.56   0.32
## ipd               3.99  0.53   2.33
## insurance2        0.67 -0.87   0.07
## comorb            1.12 -0.48   0.69
## overfever         1.21  0.10   0.28
## myalgiadef        1.74 -0.35   1.53
## arthralgiadef     0.38 -1.69  -0.26
## abdominalpain     1.56 -0.02   0.91
## leucopenia        2.15  0.26   1.27
## lowplatel         3.59  0.74   1.82
```

]
--
.pull-right[
**Analysis with Imputed data**

```
##                     OR 2.5 % 97.5 %
## (Intercept)       0.12 -3.33  -1.03
## factor(agecat_c)1 2.41  0.02   1.78
## factor(agecat_c)2 1.66 -0.26   1.32
## factor(agecat_c)3 0.83 -0.94   0.61
## factor(agecat_c)4 0.81 -1.12   0.70
## sex               0.90 -0.50   0.29
## ipd               4.01  0.55   2.32
## insurance2        0.77 -0.65   0.12
## comorb            1.06 -0.50   0.60
## overfever         1.16  0.07   0.23
## myalgiadef        1.34 -0.53   1.17
## arthralgiadef     0.44 -1.49  -0.18
## abdominalpain     1.74  0.14   0.97
## leucopenia        2.25  0.33   1.29
## lowplatel         3.45  0.71   1.77
```

]

---
class:middle
**Example from a vector-borne disease** Comparative plots
<img src="L19_EPIB704_Selection_Bias_files/figure-html/unnamed-chunk-18-1.png" width="80%" style="display: block; margin: auto;" />

---
class:middle
**Example from a vector-borne disease, Using (IPCW) to correct for the missingness**

Using the  [weightIt package](https://ngreifer.github.io/WeightIt/articles/WeightIt.html) we can upweigth the population to make up for the entire sample: (n=634 + IPCW), `dataset without missingness, n=839`

``` r
*censor.form= "censor ~  factor(agecat_c) + sex + ipd+ insurance2 + comorb + overfever +
myalgiadef+  arthralgiadef + abdominalpain + leucopenia + lowplatel"
 
 CW1 <- weightit(as.formula(censor.form), data = sb.data2, method = "ps",# PScore weighting
               estimand = "ATE")  # ATE estimand (not ATT)
 #summary(CW1)
```

as in PSTW and IPTW, we check the balance:

---
class:middle
**Example from a vector-borne disease (IPCW)**

Here we use the weights of the censoring in the regression model:

``` r
 mod2or<- glm(outc ~  factor(agecat_c) + sex + ipd+ insurance2 + comorb + overfever +
                myalgiadef+  arthralgiadef + abdominalpain + leucopenia + lowplatel, 
*             weights = CW1$weights, data=sb.data1, family= binomial)
round(jtools::summ(mod2or, digits=2)$"coeftable", 2)  #;summary(mod2or); 
```

```
##                    Est. S.E. z val.    p
## (Intercept)       -2.68 0.53  -5.07 0.00
## factor(agecat_c)1  0.88 0.43   2.05 0.04
## factor(agecat_c)2  0.56 0.38   1.45 0.15
## factor(agecat_c)3 -0.15 0.37  -0.41 0.68
## factor(agecat_c)4 -0.13 0.44  -0.29 0.77
## sex               -0.05 0.19  -0.27 0.79
## ipd                1.64 0.43   3.80 0.00
## insurance2        -0.35 0.18  -1.91 0.06
## comorb             0.19 0.27   0.72 0.47
## overfever          0.17 0.04   4.91 0.00
## myalgiadef         0.30 0.39   0.77 0.44
## arthralgiadef     -0.53 0.30  -1.79 0.07
## abdominalpain      0.62 0.20   3.08 0.00
## leucopenia         0.62 0.23   2.67 0.01
## lowplatel          1.05 0.25   4.16 0.00
```

---
class:middle
###Example from a vector-borne disease  (IPCW)

``` r
 plot_summs(mod0or, mod1or, mod2or, robust = list(FALSE,FALSE, FALSE),
           model.names = c("C.Case", "Some Imputed", "IPCW"))
```

---
class:middle
**Example from a vector-borne disease (IPCW)**
Estimating the PS of censoring (n=634 + IPCW), `dataset without missingness, n=839`
**.red[There are several ways to obtain weights as well]**

``` r
modwt<- glm(censor ~ factor(agecat_c) + sex + ipd+ insurance2 + comorb + overfever +
              myalgiadef+  arthralgiadef + abdominalpain + leucopenia + lowplatel,  
            data=sb.data2, family= binomial)
sb.data2$prob_wt <- predict(modwt, type = "response")
#summary(sb.data2$prob_wt)
*sb.data2$cen_wt0<- 1/sb.data2$prob_wt
#summary(sb.data2$cen_wt0) #unstable weight
sb.data2$cen_wt1<- (mean(sb.data2$prob_wt))/ mean(1-sb.data2$prob_wt)/(1-sb.data2$prob_wt)
#summary(sb.data2$cen_wt1) #stabilize weight opt1
sb.data2$cen_wt2<- ifelse(sb.data2$censor==1, sb.data2$cen_wt0, 1/(1-sb.data2$prob_wt))
summary(sb.data2$cen_wt2)#stabilize weight opt2
```

```
##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##   1.000   1.002   1.030   1.618   1.227  44.950
```

---
class:middle
### Example from a vector-borne disease (IPCW)
Estimating the PS of censoring (n=634 + IPCW), `dataset without missingness, n=839`

``` r
mod4or<- glm(outc ~  factor(agecat_c) + sex + ipd + insurance2 + comorb + overfever +
               myalgiadef+  arthralgiadef + abdominalpain + leucopenia + lowplatel, 
*            weights = sb.data2$cen_wt1, data=sb.data1, family= binomial)
mod4ort<-jtools::summ(mod4or)
round(mod4ort$coeftable,2) #;summary(mod4or); 
```

```
##                    Est. S.E. z val.    p
## (Intercept)       -2.68 1.12  -2.39 0.02
## factor(agecat_c)1  0.88 0.91   0.97 0.33
## factor(agecat_c)2  0.56 0.81   0.69 0.49
## factor(agecat_c)3 -0.15 0.79  -0.19 0.85
## factor(agecat_c)4 -0.13 0.94  -0.14 0.89
## sex               -0.05 0.40  -0.13 0.90
## ipd                1.64 0.91   1.79 0.07
## insurance2        -0.35 0.39  -0.90 0.37
## comorb             0.19 0.56   0.34 0.73
## overfever          0.17 0.08   2.32 0.02
## myalgiadef         0.30 0.83   0.36 0.72
## arthralgiadef     -0.53 0.63  -0.85 0.40
## abdominalpain      0.62 0.42   1.46 0.15
## leucopenia         0.62 0.49   1.26 0.21
## lowplatel          1.05 0.53   1.96 0.05
```

---
class:middle
###Example from a vector-borne disease (IPCW)
.pull-left[

---
class:middle
###Example from a vector-borne disease (IPCW)
**Data set without missing data**, `complete dataset without missingness, n=839`

```
##                    Est. S.E. z val.    p
## (Intercept)       -2.65 0.51  -5.22 0.00
## factor(agecat_c)1  0.90 0.40   2.28 0.02
## factor(agecat_c)2  0.80 0.35   2.27 0.02
## factor(agecat_c)3  0.23 0.34   0.68 0.49
## factor(agecat_c)4 -0.21 0.40  -0.53 0.59
## sex               -0.11 0.17  -0.63 0.53
## ipd                0.85 0.33   2.56 0.01
## insurance2        -0.24 0.17  -1.44 0.15
## comorb             0.16 0.25   0.67 0.51
## overfever          0.14 0.03   4.59 0.00
## myalgiadef         0.13 0.36   0.35 0.73
## arthralgiadef     -0.26 0.27  -0.97 0.33
## abdominalpain      0.56 0.18   3.14 0.00
## leucopenia         0.78 0.21   3.82 0.00
## lowplatel          0.94 0.23   4.03 0.00
```

---
class:middle
###Example from a vector-borne disease (IPCW)
**Comparison Plots**
<img src="L19_EPIB704_Selection_Bias_files/figure-html/unnamed-chunk-28-1.png" width="85%" style="display: block; margin: auto;" />

---
class:middle
###Example from a vector-borne disease (IPCW)
**Comparison Plots**
<img src="L19_EPIB704_Selection_Bias_files/figure-html/unnamed-chunk-29-1.png" width="85%" style="display: block; margin: auto;" />

---
class:middle
###The Bayesian way
.pull-left[

``` r
mod0<- stan_glm(outc ~factor(agecat_c) + 
                sex + sgss + comorb + overfever + 
                myalgiadef+  arthralgiadef + 
                abdominalpain + a_leucopenia + 
                a_lowplatel, data=sb.data1, 
                family= binomial(link = "log"), 
                refresh=0)
#print(mod0, digits=2)#;summary(mod0)
mod1<- stan_glm(outc ~ factor(agecat_c) + sex + 
              ipd + insurance2 + comorb + overfever + 
              myalgiadef+ arthralgiadef + 
              abdominalpain + leucopenia + 
*             lowplatel, weights = cen_wt2,
              data=sb.data2,  
              family= binomial(link = "log"),
              refresh=0) 
```
]
--
.pull-right[

``` r
print(mod1, digits=2, detail = F)#;summary(mod1)
```

```
##                   Median MAD_SD
## (Intercept)       -1.85   0.28 
## factor(agecat_c)1  0.37   0.21 
## factor(agecat_c)2  0.16   0.20 
## factor(agecat_c)3 -0.01   0.21 
## factor(agecat_c)4 -0.12   0.24 
## sex               -0.02   0.08 
## ipd                0.17   0.11 
## insurance2        -0.11   0.07 
## comorb             0.15   0.10 
## overfever          0.03   0.01 
## myalgiadef         0.08   0.20 
## arthralgiadef     -0.14   0.09 
## abdominalpain      0.26   0.11 
## leucopenia         0.13   0.08 
## lowplatel          0.65   0.13
```

]
---
class:middle
###The Bayesian way

``` r
mod3RR<- stan_glm(DENV_DICHOT ~  factor(agecat_c) + sex + ipd +insurance2 + comorb + 
                    overfever + myalgiadef+  arthralgiadef + abdominalpain + leucopenia +
                    lowplatel,  data=sb.data2, family= binomial(link = "log"), refresh=0)
print(mod3RR, digits=2, detail = F)
```

```
##                   Median MAD_SD
## (Intercept)       -1.89   0.30 
## factor(agecat_c)1  0.49   0.25 
## factor(agecat_c)2  0.39   0.23 
## factor(agecat_c)3  0.20   0.22 
## factor(agecat_c)4 -0.11   0.27 
## sex               -0.05   0.08 
## ipd                0.07   0.10 
## insurance2        -0.11   0.08 
## comorb             0.14   0.10 
## overfever          0.03   0.01 
## myalgiadef         0.05   0.19 
## arthralgiadef     -0.14   0.10 
## abdominalpain      0.18   0.10 
## leucopenia         0.23   0.10 
## lowplatel          0.55   0.13
```

---
class: middle
#### Extra DAGs
<img src="L19_EPIB704_Selection_Bias_files/figure-html/unnamed-chunk-33-1.png" width="80%" style="display: block; margin: auto;" />

---
class: middle
#### Extra DAGs
<img src="L19_EPIB704_Selection_Bias_files/figure-html/unnamed-chunk-34-1.png" width="80%" style="display: block; margin: auto;" />