Confounding II

.title[
# Confounding II
]
.author[
### Mabel Carabali
]
.institute[
### EBOH, McGill University
]
.date[
### 01-08-2022 Updated: ( 2024-10-01)
]

---

---
class: middle
###Conditions that allow a variable to be a confounder:
📣 **Modern Epidemiology 4th, page 268** 💡

> <span style="color:darkblue"> _The developments in causal inference over the past decades, summarized in Chapter 3, have made clear that this definition [ ...the traditional criteria described from ME3... ] of a “confounder” is inadequate. It is inadequate because there can be a pre-exposure variable associated with the exposure and the outcome, the control of which introduces, rather than eliminates, bias_ [ME4;p268] </span>

---
class: middle
### .red[The Structure of Confounding??]
<img src="images/dogmath.png" width="60%" style="display: block; margin: auto 0 auto auto;" />
---
class: middle

### The Structure of Confounding

`$$A \leftarrow  L \to Y$$`

This diagram shows two sources of association between treatment and outcome:

1. The path `$A → Y$` that represents the causal effect of A on Y , and

2. The path `$A ← L → Y$` between A and Y that includes the common cause `$L$`

- The path `$A ← L → Y$` links A and Y through the common cause `$L$`, is the **_"backdoor path"_**

---
class: middle

### The structure of Confounding

- In a causal DAG, a backdoor path is a non-causal path between treatment and outcome that remains even if all arrows pointing from treatment to other variables (i.e., the descendants of treatment) are removed. 
- That is, the path has an arrow pointing into treatment.

---
class: middle
### Confounding and exchangeability
- The backdoor criterion, **does not** answer questions regarding the magnitude or direction of confounding.

- It is possible that some unblocked backdoor paths are weak and thus induce little bias, or that several strong backdoor paths induce bias in opposite directions and thus result in a weak net bias.

- Because unmeasured confounding is not an “all or nothing” issue, in practice, it is important to consider the expected direction and magnitude of the bias.

---
class: middle
## Confounders ( `$Y$` &larr; `$L$` &rarr; `$A$` )
.pull-left[
<img src="L8_EPIB704_Confounding_II_files/figure-html/unnamed-chunk-3-1.png" width="80%" />
]
.pull-right[
**Simulated Example**

```r
set.seed(704); N <- 100; 
L <- rbinom(N,1,0.5)
A <- ifelse(L==0,rbinom(N,1,0.25),
            rbinom(N,1,0.75))
Y <- ifelse(L==0,rbinom(N,1,0.20),
            rbinom(N,1,0.8))
#summary(L)
data <- data.frame(N, A, L, Y)
tab <- table(data$A, data$Y)
#tab; tab/margin.table(tab)
l6conf1<-epi.2by2(tab, 
                  method = "cohort.count")
```
]

```
##              Outcome +    Outcome -      Total                 Inc risk *
## Exposed +           39           13         52     75.00 (61.05 to 85.97)
## Exposed -           14           34         48     29.17 (16.95 to 44.06)
## Total               53           47        100     53.00 (42.76 to 63.06)
```
*Outcomes per 100 population units

---

```r
tabl6conf1 <- data.table::as.data.table(l6conf1$massoc.summary)
kable(tabl6conf1, digits = 2) %>% 
  kable_paper()
```

<table class=" lightable-paper" style='font-family: "Arial Narrow", arial, helvetica, sans-serif; margin-left: auto; margin-right: auto;'>
 <thead>
  <tr>
   <th style="text-align:left;"> var </th>
   <th style="text-align:right;"> est </th>
   <th style="text-align:right;"> lower </th>
   <th style="text-align:right;"> upper </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:left;"> Inc risk ratio </td>
   <td style="text-align:right;"> 2.57 </td>
   <td style="text-align:right;"> 1.61 </td>
   <td style="text-align:right;"> 4.11 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Inc odds ratio </td>
   <td style="text-align:right;"> 7.29 </td>
   <td style="text-align:right;"> 3.01 </td>
   <td style="text-align:right;"> 17.63 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Attrib inc risk * </td>
   <td style="text-align:right;"> 45.83 </td>
   <td style="text-align:right;"> 28.40 </td>
   <td style="text-align:right;"> 63.26 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Attrib fraction in exposed (%) </td>
   <td style="text-align:right;"> 61.11 </td>
   <td style="text-align:right;"> 37.90 </td>
   <td style="text-align:right;"> 75.64 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Attrib inc risk in population * </td>
   <td style="text-align:right;"> 23.83 </td>
   <td style="text-align:right;"> 7.68 </td>
   <td style="text-align:right;"> 39.99 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Attrib fraction in population (%) </td>
   <td style="text-align:right;"> 44.97 </td>
   <td style="text-align:right;"> 21.55 </td>
   <td style="text-align:right;"> 61.40 </td>
  </tr>
</tbody>
</table>
]
--
.pull-right[

```r
l6strtab1<- data %>% 
  tbl_summary(by= L,
  label=list(Y ="Outcome",  A ="Exposure"),
  #type = all_continuous() ~ "continuous1",
  statistic =  all_categorical() ~ c( "{n} / {N} ({p}%)"), 
  missing = "no") %>% 
  modify_spanning_header(c("stat_1", "stat_2") ~ "**L=0/L=1**") %>%
  modify_caption("**Summary of covars distribution**") %>% bold_labels()
l6strtab1
```

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<table class="gt_table" data-quarto-disable-processing="false" data-quarto-bootstrap="false">
  <caption class='gt_caption'><strong>Summary of covars distribution</strong></caption>
  <thead>
    <tr class="gt_col_headings gt_spanner_row">
      <th class="gt_col_heading gt_columns_bottom_border gt_left" rowspan="2" colspan="1" scope="col" id="<strong>Characteristic</strong>"><strong>Characteristic</strong></th>
      <th class="gt_center gt_columns_top_border gt_column_spanner_outer" rowspan="1" colspan="2" scope="colgroup" id="<strong>L=0/L=1</strong>">
        <span class="gt_column_spanner"><strong>L=0/L=1</strong></span>
      </th>
    </tr>
    <tr class="gt_col_headings">
      <th class="gt_col_heading gt_columns_bottom_border gt_center" rowspan="1" colspan="1" scope="col" id="<strong>0</strong><br />
N = 51<span class="gt_footnote_marks" style="white-space:nowrap;font-style:italic;font-weight:normal;"><sup>1</sup></span>"><strong>0</strong><br />
N = 51<span class="gt_footnote_marks" style="white-space:nowrap;font-style:italic;font-weight:normal;"><sup>1</sup></span></th>
      <th class="gt_col_heading gt_columns_bottom_border gt_center" rowspan="1" colspan="1" scope="col" id="<strong>1</strong><br />
N = 49<span class="gt_footnote_marks" style="white-space:nowrap;font-style:italic;font-weight:normal;"><sup>1</sup></span>"><strong>1</strong><br />
N = 49<span class="gt_footnote_marks" style="white-space:nowrap;font-style:italic;font-weight:normal;"><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;">N</td>
<td headers="stat_1" class="gt_row gt_center"><br /></td>
<td headers="stat_2" class="gt_row gt_center"><br /></td></tr>
    <tr><td headers="label" class="gt_row gt_left">    100</td>
<td headers="stat_1" class="gt_row gt_center">51 / 51 (100%)</td>
<td headers="stat_2" class="gt_row gt_center">49 / 49 (100%)</td></tr>
    <tr><td headers="label" class="gt_row gt_left" style="font-weight: bold;">Exposure</td>
<td headers="stat_1" class="gt_row gt_center">10 / 51 (20%)</td>
<td headers="stat_2" class="gt_row gt_center">38 / 49 (78%)</td></tr>
    <tr><td headers="label" class="gt_row gt_left" style="font-weight: bold;">Outcome</td>
<td headers="stat_1" class="gt_row gt_center">9 / 51 (18%)</td>
<td headers="stat_2" class="gt_row gt_center">38 / 49 (78%)</td></tr>
  </tbody>
  
  <tfoot class="gt_footnotes">
    <tr>
      <td class="gt_footnote" colspan="3"><span class="gt_footnote_marks" style="white-space:nowrap;font-style:italic;font-weight:normal;"><sup>1</sup></span> n / N (%)</td>
    </tr>
  </tfoot>
</table>
</div>
]

---
class: middle
## Confounders
.pull-left[
**L=0**
<table class=" lightable-paper" style='font-family: "Arial Narrow", arial, helvetica, sans-serif; margin-left: auto; margin-right: auto;'>
 <thead>
<tr>
<th style="padding-bottom:0; padding-left:3px;padding-right:3px;text-align: center; " colspan="1"><div style="border-bottom: 1px solid #00000020; padding-bottom: 5px; ">Exposure</div></th>
<th style="padding-bottom:0; padding-left:3px;padding-right:3px;text-align: center; " colspan="2"><div style="border-bottom: 1px solid #00000020; padding-bottom: 5px; ">Outcome; L=0</div></th>
</tr>
  <tr>
   <th style="text-align:left;">   </th>
   <th style="text-align:right;"> 0 </th>
   <th style="text-align:right;"> 1 </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:left;"> 0 </td>
   <td style="text-align:right;"> 36 </td>
   <td style="text-align:right;"> 5 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> 1 </td>
   <td style="text-align:right;"> 6 </td>
   <td style="text-align:right;"> 4 </td>
  </tr>
</tbody>
</table>
]
.pull-right[
**L=1**
<table class=" lightable-paper" style='font-family: "Arial Narrow", arial, helvetica, sans-serif; margin-left: auto; margin-right: auto;'>
 <thead>
<tr>
<th style="padding-bottom:0; padding-left:3px;padding-right:3px;text-align: center; " colspan="1"><div style="border-bottom: 1px solid #00000020; padding-bottom: 5px; ">Exposure</div></th>
<th style="padding-bottom:0; padding-left:3px;padding-right:3px;text-align: center; " colspan="2"><div style="border-bottom: 1px solid #00000020; padding-bottom: 5px; ">Outcome; L=1</div></th>
</tr>
  <tr>
   <th style="text-align:left;">   </th>
   <th style="text-align:right;"> 0 </th>
   <th style="text-align:right;"> 1 </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:left;"> 0 </td>
   <td style="text-align:right;"> 3 </td>
   <td style="text-align:right;"> 8 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> 1 </td>
   <td style="text-align:right;"> 8 </td>
   <td style="text-align:right;"> 30 </td>
  </tr>
</tbody>
</table>
]

```r
tab1 <- table(data$A, data$Y, data$L)
#tab1
l6conf2<-epi.2by2(tab1, method = "cohort.count")
tabl6conf2 <- data.table::as.data.table(l6conf2$massoc.summary)
```

---
class: middle
## Confounders

**Adjusted**
<table class=" lightable-paper" style='font-family: "Arial Narrow", arial, helvetica, sans-serif; margin-left: auto; margin-right: auto;'>
 <thead>
<tr>
<th style="padding-bottom:0; padding-left:3px;padding-right:3px;text-align: center; " colspan="1"><div style="border-bottom: 1px solid #00000020; padding-bottom: 5px; ">Measure</div></th>
<th style="padding-bottom:0; padding-left:3px;padding-right:3px;text-align: center; " colspan="3"><div style="border-bottom: 1px solid #00000020; padding-bottom: 5px; ">Estimate 95%CIs</div></th>
</tr>
  <tr>
   <th style="text-align:left;"> Measure </th>
   <th style="text-align:right;"> Est. </th>
   <th style="text-align:right;"> LB </th>
   <th style="text-align:right;"> UB </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:left;"> Inc risk ratio (crude) </td>
   <td style="text-align:right;"> 2.57 </td>
   <td style="text-align:right;"> 1.61 </td>
   <td style="text-align:right;"> 4.11 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Inc risk ratio (M-H) </td>
   <td style="text-align:right;"> 1.42 </td>
   <td style="text-align:right;"> 0.87 </td>
   <td style="text-align:right;"> 2.30 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Inc risk ratio (crude:M-H) </td>
   <td style="text-align:right;"> 1.81 </td>
   <td style="text-align:right;">  </td>
   <td style="text-align:right;">  </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Inc odds ratio (crude) </td>
   <td style="text-align:right;"> 7.29 </td>
   <td style="text-align:right;"> 3.01 </td>
   <td style="text-align:right;"> 17.63 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Inc odds ratio (M-H) </td>
   <td style="text-align:right;"> 2.46 </td>
   <td style="text-align:right;"> 0.84 </td>
   <td style="text-align:right;"> 7.21 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Inc odds ratio (crude:M-H) </td>
   <td style="text-align:right;"> 2.96 </td>
   <td style="text-align:right;">  </td>
   <td style="text-align:right;">  </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Attrib inc risk (crude) * </td>
   <td style="text-align:right;"> 45.83 </td>
   <td style="text-align:right;"> 28.40 </td>
   <td style="text-align:right;"> 63.26 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Attrib inc risk (M-H) * </td>
   <td style="text-align:right;"> 16.69 </td>
   <td style="text-align:right;"> -16.33 </td>
   <td style="text-align:right;"> 49.71 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Attrib inc risk (crude:M-H) </td>
   <td style="text-align:right;"> 2.75 </td>
   <td style="text-align:right;">  </td>
   <td style="text-align:right;">  </td>
  </tr>
</tbody>
</table>

---

class: middle
## Confounders?
Consider this  DAG: `$$C \to  E \to  Y$$`
- In this case, C is not a confounder because it does not have an independent effect on Y. 
  - But there will be an observed association between C and Y, by virtue of their common association with E.
  - But it is not an independent association.

**That’s why we should assess this criterion within levels of exposure.**
  - Stratified by E, the association between C and Y is null if there is no direct effect (as shown in the DAG).
  
---
class: middle
## Confounders ?
.pull-left[
`$C \to  E \to  Y$`

```r
set.seed(704)
N <- 100
C <- rbinom(N,1,0.5)
E <- ifelse(C==0,rbinom(N,1,0.8),
            rbinom(N,1,0.5))
Y <- ifelse(E==0,rbinom(N,1,0.2),
            rbinom(N,1,0.5))
#summary(C)
data1 <- data.frame(N, C, E,Y)
tab1C <- table(data1$E, data1$Y, data1$C)
```
]
.pull-right[
<table class=" lightable-paper" style='font-family: "Arial Narrow", arial, helvetica, sans-serif; margin-left: auto; margin-right: auto;'>
 <thead>
<tr>
<th style="padding-bottom:0; padding-left:3px;padding-right:3px;text-align: center; " colspan="1"><div style="border-bottom: 1px solid #00000020; padding-bottom: 5px; ">Measure</div></th>
<th style="padding-bottom:0; padding-left:3px;padding-right:3px;text-align: center; " colspan="3"><div style="border-bottom: 1px solid #00000020; padding-bottom: 5px; ">Estimate 95%CIs</div></th>
</tr>
  <tr>
   <th style="text-align:left;"> Measure </th>
   <th style="text-align:right;"> Est. </th>
   <th style="text-align:right;"> LB </th>
   <th style="text-align:right;"> UB </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:left;"> Inc risk ratio (crude) </td>
   <td style="text-align:right;"> 1.91 </td>
   <td style="text-align:right;"> 1.36 </td>
   <td style="text-align:right;"> 2.70 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Inc risk ratio (M-H) </td>
   <td style="text-align:right;"> 1.75 </td>
   <td style="text-align:right;"> 1.14 </td>
   <td style="text-align:right;"> 2.69 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Inc risk ratio (crude:M-H) </td>
   <td style="text-align:right;"> 1.09 </td>
   <td style="text-align:right;">  </td>
   <td style="text-align:right;">  </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Inc odds ratio (crude) </td>
   <td style="text-align:right;"> 5.12 </td>
   <td style="text-align:right;"> 2.02 </td>
   <td style="text-align:right;"> 12.97 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Inc odds ratio (M-H) </td>
   <td style="text-align:right;"> 3.83 </td>
   <td style="text-align:right;"> 1.45 </td>
   <td style="text-align:right;"> 10.09 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Inc odds ratio (crude:M-H) </td>
   <td style="text-align:right;"> 1.34 </td>
   <td style="text-align:right;">  </td>
   <td style="text-align:right;">  </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Attrib inc risk (crude) * </td>
   <td style="text-align:right;"> 37.15 </td>
   <td style="text-align:right;"> 19.01 </td>
   <td style="text-align:right;"> 55.30 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Attrib inc risk (M-H) * </td>
   <td style="text-align:right;"> 31.20 </td>
   <td style="text-align:right;"> -6.71 </td>
   <td style="text-align:right;"> 69.12 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Attrib inc risk (crude:M-H) </td>
   <td style="text-align:right;"> 1.19 </td>
   <td style="text-align:right;">  </td>
   <td style="text-align:right;">  </td>
  </tr>
</tbody>
</table>
]

---
class: middle
## Confounders ?
Figure 7.4 A version of the famous M-diagram again. No confounding, despite backdoor paths.

Here there are no common causes of treatment A and outcome Y, and therefore there is no confounding.

The back door path between A &larr; U2 &rarr; L &larr; U1 &rarr; Y is locked because `$L$` is a collider on that path.

---

```r
set.seed(704)
N <- 100
U1 <- rbinom(N,1,0.5)
U2 <- rbinom(N,1,0.5)
L <- ifelse(U1==1, rbinom(N,1,0.6),
     ifelse(U2==1, rbinom(N,1,0.6), 
     rbinom(N,1,0.5))) #L is affected by U1 and U2
A <- ifelse(U2==1, rbinom(N,1,0.5), 
     rbinom(N,1,0.5)) #A is affected by U2
Y <- ifelse(A==1, rbinom(N,1,0.6), 
     ifelse( U1==1,  rbinom(N,1,0.6),  
     rbinom(N,1,0.5))) # Y is affected by A and U1

#summary(C)
datanoconf2 <- data.frame(N, U1, U2, L, A,Y)
tab.noconf2<- table(datanoconf2$A, 
              datanoconf2$Y, 
              datanoconf2$L)
```
]

.pull-right[
<table class=" lightable-paper" style='font-family: "Arial Narrow", arial, helvetica, sans-serif; margin-left: auto; margin-right: auto;'>
 <thead>
<tr>
<th style="padding-bottom:0; padding-left:3px;padding-right:3px;text-align: center; " colspan="1"><div style="border-bottom: 1px solid #00000020; padding-bottom: 5px; ">Measure</div></th>
<th style="padding-bottom:0; padding-left:3px;padding-right:3px;text-align: center; " colspan="3"><div style="border-bottom: 1px solid #00000020; padding-bottom: 5px; ">Estimate 95%CIs</div></th>
</tr>
  <tr>
   <th style="text-align:left;"> Measure </th>
   <th style="text-align:right;"> Est. </th>
   <th style="text-align:right;"> LB </th>
   <th style="text-align:right;"> UB </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:left;"> Inc risk ratio (crude) </td>
   <td style="text-align:right;"> 1.73 </td>
   <td style="text-align:right;"> 1.05 </td>
   <td style="text-align:right;"> 2.86 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Inc risk ratio (M-H) </td>
   <td style="text-align:right;"> 1.70 </td>
   <td style="text-align:right;"> 1.04 </td>
   <td style="text-align:right;"> 2.78 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Inc risk ratio (crude:M-H) </td>
   <td style="text-align:right;"> 1.02 </td>
   <td style="text-align:right;">  </td>
   <td style="text-align:right;">  </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Inc odds ratio (crude) </td>
   <td style="text-align:right;"> 2.53 </td>
   <td style="text-align:right;"> 1.11 </td>
   <td style="text-align:right;"> 5.74 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Inc odds ratio (M-H) </td>
   <td style="text-align:right;"> 2.46 </td>
   <td style="text-align:right;"> 1.08 </td>
   <td style="text-align:right;"> 5.61 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Inc odds ratio (crude:M-H) </td>
   <td style="text-align:right;"> 1.03 </td>
   <td style="text-align:right;">  </td>
   <td style="text-align:right;">  </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Attrib inc risk (crude) * </td>
   <td style="text-align:right;"> 22.00 </td>
   <td style="text-align:right;"> 3.21 </td>
   <td style="text-align:right;"> 40.79 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Attrib inc risk (M-H) * </td>
   <td style="text-align:right;"> 21.33 </td>
   <td style="text-align:right;"> 0.74 </td>
   <td style="text-align:right;"> 41.92 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Attrib inc risk (crude:M-H) </td>
   <td style="text-align:right;"> 1.03 </td>
   <td style="text-align:right;">  </td>
   <td style="text-align:right;">  </td>
  </tr>
</tbody>
</table>
]

---
class: middle
## Confounders
<img src="images/L6fig5HR.png" width="30%" />

<span style="color:purple"> There is an arrow `$L \to A$`. </span> The presence of this arrow creates an open backdoor path:

- A &larr;  L &larr; U1 &rarr; Y, because U1 is a common cause of A and Y, and so **confounding exists.**
- Conditioning on L would block that backdoor path but would simultaneously open a backdoor path on which L is a collider (A &larr; U2 &rarr; L &larr; U1 &rarr; Y)

The bias is **intractable:** attempting to block the confounding path opens a selection bias path.

---
class: middle
## Confounding ? Colliders?
.pull-left[

```r
dag <- ggdag::dagify(Y ~ A + U1,
       A ~ L + U2,
       L ~ U1 + U2,
       exposure = "A",
       outcome = "Y",
*      latent = c("U1", "U2"),
       coords = list(x = c(L = 3.2, Y = 3.8, A = 3.5, U1=3, U2=3),
       y = c(U2 = 1, L = 1.3,  A=1.3, Y=1.3, U1 = 1.5)))
dag_plot <- dag %>% 
  ggdag::tidy_dagitty(layout = "manual", seed = 704) %>%
  arrange(name) %>% 
  ggplot(aes(x = x, y = y, xend = xend, yend = yend)) +
  #xlim()+
  geom_dag_point() +
  geom_dag_edges() +
  theme_dag() +
  geom_dag_node(color="darkmagenta") + geom_dag_text(color="white") 
```
]
--
.pull-right[
<img src="L8_EPIB704_Confounding_II_files/figure-html/unnamed-chunk-19-1.png" width="90%" style="display: block; margin: auto auto auto 0;" />

]

---
class: middle
### Confounding ? Colliders?

```r
#control_for(dag, var = "L")
#ggdag_paths(dag) +theme_dag()
ggdag_adjust(dag, var = "L", stylized = T, collider_lines = T) + 
  theme_dag() 
```

---
class: middle
##R can help ...

```r
g <- dagitty::paths(dag, "A", "Y")
a <- paste0("There are ", length(g$paths), 
            " pathways from A to Y")  
b <- paste0("Of these backdoor pathways ", 
            sum(g$open=="TRUE"), " are open")  
c <- paste0("The adjustment sets are ", 
            adjustmentSets(dag, "A", "Y", type = "canonical"))

print(c(a,b,c))
```

```
## [1] "There are 3 pathways from A to Y"     
## [2] "Of these backdoor pathways 2 are open"
## [3] "The adjustment sets are "
```

The bias is **.red[intractable:]** attempting to block the confounding path opens a selection bias path.

---

```r
set.seed(704)
N <- 100
U1 <- rbinom(N,1,0.5)
U2 <- rbinom(N,1,0.5)
L <- ifelse(U1==1, rbinom(N,1,0.65), 
     ifelse(U2==1,  rbinom(N,1,0.65), 
     rbinom(N,1,0.15))) #L is affected by U1 and U2
A <- ifelse(L==1, rbinom(N,1,0.65), 
     ifelse(U2==1, rbinom(N,1,0.65), 
     rbinom(N,1,0.45))) #A is affected by L and U2
Y <- ifelse(A==1,  rbinom(N,1,0.65), 
      ifelse(U1==1, rbinom(N,1,0.6), 
      rbinom(N,1,0.3))) # Y is affected by A and U1

#summary(C)
data2 <- data.frame(N, U1, U2, L, A,Y)
tabL.intract <- table(data2$A, data2$Y, 
                data2$L)
```
]
.pull-right[
<table class=" lightable-paper" style='font-family: "Arial Narrow", arial, helvetica, sans-serif; margin-left: auto; margin-right: auto;'>
 <thead>
<tr>
<th style="padding-bottom:0; padding-left:3px;padding-right:3px;text-align: center; " colspan="1"><div style="border-bottom: 1px solid #00000020; padding-bottom: 5px; ">Measure</div></th>
<th style="padding-bottom:0; padding-left:3px;padding-right:3px;text-align: center; " colspan="3"><div style="border-bottom: 1px solid #00000020; padding-bottom: 5px; ">Estimate 95%CIs</div></th>
</tr>
  <tr>
   <th style="text-align:left;"> Measure </th>
   <th style="text-align:right;"> Est. </th>
   <th style="text-align:right;"> LB </th>
   <th style="text-align:right;"> UB </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:left;"> Inc risk ratio (crude) </td>
   <td style="text-align:right;"> 1.91 </td>
   <td style="text-align:right;"> 1.16 </td>
   <td style="text-align:right;"> 3.13 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Inc risk ratio (M-H) </td>
   <td style="text-align:right;"> 2.12 </td>
   <td style="text-align:right;"> 1.18 </td>
   <td style="text-align:right;"> 3.80 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Inc risk ratio (crude:M-H) </td>
   <td style="text-align:right;"> 0.90 </td>
   <td style="text-align:right;">  </td>
   <td style="text-align:right;">  </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Inc odds ratio (crude) </td>
   <td style="text-align:right;"> 3.00 </td>
   <td style="text-align:right;"> 1.31 </td>
   <td style="text-align:right;"> 6.88 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Inc odds ratio (M-H) </td>
   <td style="text-align:right;"> 3.13 </td>
   <td style="text-align:right;"> 1.35 </td>
   <td style="text-align:right;"> 7.28 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Inc odds ratio (crude:M-H) </td>
   <td style="text-align:right;"> 0.96 </td>
   <td style="text-align:right;">  </td>
   <td style="text-align:right;">  </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Attrib inc risk (crude) * </td>
   <td style="text-align:right;"> 25.97 </td>
   <td style="text-align:right;"> 7.09 </td>
   <td style="text-align:right;"> 44.85 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Attrib inc risk (M-H) * </td>
   <td style="text-align:right;"> 28.01 </td>
   <td style="text-align:right;"> 5.35 </td>
   <td style="text-align:right;"> 50.67 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Attrib inc risk (crude:M-H) </td>
   <td style="text-align:right;"> 0.93 </td>
   <td style="text-align:right;">  </td>
   <td style="text-align:right;">  </td>
  </tr>
</tbody>
</table>
]

---

class: middle
## Confounders
Figure 7.7 is another non confounding example in which the traditional criteria lead to selection bias due to adjustment for L.

- The traditional criteria would not have resulted in bias had condition (3) been replaced by the condition that L is not caused by treatment.
 - .red[ _(3) it does not lie on a causal pathway between treatment and outcome._]
 
> Replace condition (3) by the condition that “there exist variables A and Y such that there is conditional exchangeability within their joint levels `$Y^a \perp A| L, U$`". .blue[H&R, Technical Point 7.2]

---

```r
set.seed(704)
N <- 100
U <- rbinom(N,1,0.5)
A <-rbinom(N,1,0.55) #A affects L 
L <- ifelse(U==1, rbinom(N,1,0.65), 
     ifelse(A==1, rbinom(N,1,0.65), 
     rbinom(N,1,0.25))) #L is affected by U and A
Y <- ifelse(U==1, rbinom(N,1,0.6), 
     rbinom(N,1,0.25)) # Y is affected by U

datanoconf3 <- data.frame(N, U, L, A,Y)
tabL.noconf3 <- table(datanoconf3$A,
               datanoconf3$Y,
               datanoconf3$L)
```
]
.pull-right[
<table class=" lightable-paper" style='font-family: "Arial Narrow", arial, helvetica, sans-serif; margin-left: auto; margin-right: auto;'>
 <thead>
<tr>
<th style="padding-bottom:0; padding-left:3px;padding-right:3px;text-align: center; " colspan="1"><div style="border-bottom: 1px solid #00000020; padding-bottom: 5px; ">Measure</div></th>
<th style="padding-bottom:0; padding-left:3px;padding-right:3px;text-align: center; " colspan="3"><div style="border-bottom: 1px solid #00000020; padding-bottom: 5px; ">Estimate 95%CIs</div></th>
</tr>
  <tr>
   <th style="text-align:left;"> Measure </th>
   <th style="text-align:right;"> Est. </th>
   <th style="text-align:right;"> LB </th>
   <th style="text-align:right;"> UB </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:left;"> Inc risk ratio (crude) </td>
   <td style="text-align:right;"> 0.98 </td>
   <td style="text-align:right;"> 0.67 </td>
   <td style="text-align:right;"> 1.42 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Inc risk ratio (M-H) </td>
   <td style="text-align:right;"> 0.98 </td>
   <td style="text-align:right;"> 0.61 </td>
   <td style="text-align:right;"> 1.57 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Inc risk ratio (crude:M-H) </td>
   <td style="text-align:right;"> 1.00 </td>
   <td style="text-align:right;">  </td>
   <td style="text-align:right;">  </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Inc odds ratio (crude) </td>
   <td style="text-align:right;"> 0.96 </td>
   <td style="text-align:right;"> 0.42 </td>
   <td style="text-align:right;"> 2.18 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Inc odds ratio (M-H) </td>
   <td style="text-align:right;"> 0.97 </td>
   <td style="text-align:right;"> 0.42 </td>
   <td style="text-align:right;"> 2.23 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Inc odds ratio (crude:M-H) </td>
   <td style="text-align:right;"> 0.99 </td>
   <td style="text-align:right;">  </td>
   <td style="text-align:right;">  </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Attrib inc risk (crude) * </td>
   <td style="text-align:right;"> -1.10 </td>
   <td style="text-align:right;"> -21.55 </td>
   <td style="text-align:right;"> 19.36 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Attrib inc risk (M-H) * </td>
   <td style="text-align:right;"> -0.85 </td>
   <td style="text-align:right;"> -25.69 </td>
   <td style="text-align:right;"> 23.99 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Attrib inc risk (crude:M-H) </td>
   <td style="text-align:right;"> 1.30 </td>
   <td style="text-align:right;">  </td>
   <td style="text-align:right;">  </td>
  </tr>
</tbody>
</table>
]

---
###Surrogate confounders (Is L a confounder?)
In Figure 7.8, confounding of A on Y via unmeasured common cause U .

- Measured variable L is a proxy or surrogate for U . Adjust for the variable L? 
- On the one hand, L is not a confounder because it does not lie on a backdoor path between A and Y .

---

```r
set.seed(704)
N <- 100
U <- rbinom(N,1,0.8)
A <- ifelse(U==1, rbinom(N,1,0.65), 
     rbinom(N,1,0.5)) #A is affected by  U
L <- ifelse(U==1, rbinom(N,1,0.65), 
     rbinom(N,1,0.5)) #L is affected by U 
Y <- ifelse(A==1,  rbinom(N,1,0.65),
     ifelse(U==1, rbinom(N,1,0.65), 
     rbinom(N,1,0.15))) # Y is affected by U and A
dataconf4 <- data.frame(N, U, L, A,Y)
tabL.conf4 <- table(dataconf4$A, 
                    dataconf4$Y,
                    dataconf4$L)
```
]
.pull-right[
<table class=" lightable-paper" style='font-family: "Arial Narrow", arial, helvetica, sans-serif; margin-left: auto; margin-right: auto;'>
 <thead>
<tr>
<th style="padding-bottom:0; padding-left:3px;padding-right:3px;text-align: center; " colspan="1"><div style="border-bottom: 1px solid #00000020; padding-bottom: 5px; ">Measure</div></th>
<th style="padding-bottom:0; padding-left:3px;padding-right:3px;text-align: center; " colspan="3"><div style="border-bottom: 1px solid #00000020; padding-bottom: 5px; ">Estimate 95%CIs</div></th>
</tr>
  <tr>
   <th style="text-align:left;"> Measure </th>
   <th style="text-align:right;"> Est. </th>
   <th style="text-align:right;"> LB </th>
   <th style="text-align:right;"> UB </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:left;"> Inc risk ratio (crude) </td>
   <td style="text-align:right;"> 1.94 </td>
   <td style="text-align:right;"> 1.22 </td>
   <td style="text-align:right;"> 3.08 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Inc risk ratio (M-H) </td>
   <td style="text-align:right;"> 2.01 </td>
   <td style="text-align:right;"> 1.23 </td>
   <td style="text-align:right;"> 3.30 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Inc risk ratio (crude:M-H) </td>
   <td style="text-align:right;"> 0.97 </td>
   <td style="text-align:right;">  </td>
   <td style="text-align:right;">  </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Inc odds ratio (crude) </td>
   <td style="text-align:right;"> 3.29 </td>
   <td style="text-align:right;"> 1.39 </td>
   <td style="text-align:right;"> 7.78 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Inc odds ratio (M-H) </td>
   <td style="text-align:right;"> 3.46 </td>
   <td style="text-align:right;"> 1.41 </td>
   <td style="text-align:right;"> 8.50 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Inc odds ratio (crude:M-H) </td>
   <td style="text-align:right;"> 0.95 </td>
   <td style="text-align:right;">  </td>
   <td style="text-align:right;">  </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Attrib inc risk (crude) * </td>
   <td style="text-align:right;"> 28.52 </td>
   <td style="text-align:right;"> 8.61 </td>
   <td style="text-align:right;"> 48.44 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Attrib inc risk (M-H) * </td>
   <td style="text-align:right;"> 29.58 </td>
   <td style="text-align:right;"> -5.76 </td>
   <td style="text-align:right;"> 64.92 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Attrib inc risk (crude:M-H) </td>
   <td style="text-align:right;"> 0.96 </td>
   <td style="text-align:right;">  </td>
   <td style="text-align:right;">  </td>
  </tr>
</tbody>
</table>
]

---
###Surrogate confounders (Is L a confounder?)
- On the other hand, adjusting for L, which is associated with U , will indirectly adjust for some of the confounding caused by U .

- In the extreme case that L were perfectly correlated with U then adjusting for L = adjusting for U.
- Therefore we will typically prefer to adjust, rather than not to adjust, for L.

---

```r
set.seed(704)
N <- 100
U <- rbinom(N,1,0.8)
A <- ifelse(U==1, rbinom(N,1,0.65), 
rbinom(N,1,0.5)) #A is affected by U
L <- ifelse(U==1, rbinom(N,1,0.95), 
rbinom(N,1,0.5)) #L is affected by U 
Y <- ifelse(A==1,  rbinom(N,1, 0.65),
ifelse(U==1, rbinom(N,1,0.65), 
rbinom(N,1,0.15))) # Y is affected by U and A

dataconf5 <- data.frame(N, U, L, A,Y)
tabL.conf5 <- table(dataconf5$A, 
              dataconf5$Y, 
              dataconf5$L)
```
]
.pull-right[
<table class=" lightable-paper" style='font-family: "Arial Narrow", arial, helvetica, sans-serif; margin-left: auto; margin-right: auto;'>
 <thead>
<tr>
<th style="padding-bottom:0; padding-left:3px;padding-right:3px;text-align: center; " colspan="1"><div style="border-bottom: 1px solid #00000020; padding-bottom: 5px; ">Measure</div></th>
<th style="padding-bottom:0; padding-left:3px;padding-right:3px;text-align: center; " colspan="3"><div style="border-bottom: 1px solid #00000020; padding-bottom: 5px; ">Estimate 95%CIs</div></th>
</tr>
  <tr>
   <th style="text-align:left;"> Measure </th>
   <th style="text-align:right;"> Est. </th>
   <th style="text-align:right;"> LB </th>
   <th style="text-align:right;"> UB </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:left;"> Inc risk ratio (crude) </td>
   <td style="text-align:right;"> 1.94 </td>
   <td style="text-align:right;"> 1.22 </td>
   <td style="text-align:right;"> 3.08 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Inc risk ratio (M-H) </td>
   <td style="text-align:right;"> 1.74 </td>
   <td style="text-align:right;"> 1.06 </td>
   <td style="text-align:right;"> 2.88 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Inc risk ratio (crude:M-H) </td>
   <td style="text-align:right;"> 1.11 </td>
   <td style="text-align:right;">  </td>
   <td style="text-align:right;">  </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Inc odds ratio (crude) </td>
   <td style="text-align:right;"> 3.29 </td>
   <td style="text-align:right;"> 1.39 </td>
   <td style="text-align:right;"> 7.78 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Inc odds ratio (M-H) </td>
   <td style="text-align:right;"> 2.69 </td>
   <td style="text-align:right;"> 1.11 </td>
   <td style="text-align:right;"> 6.53 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Inc odds ratio (crude:M-H) </td>
   <td style="text-align:right;"> 1.22 </td>
   <td style="text-align:right;">  </td>
   <td style="text-align:right;">  </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Attrib inc risk (crude) * </td>
   <td style="text-align:right;"> 28.52 </td>
   <td style="text-align:right;"> 8.61 </td>
   <td style="text-align:right;"> 48.44 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Attrib inc risk (M-H) * </td>
   <td style="text-align:right;"> 23.16 </td>
   <td style="text-align:right;"> -8.90 </td>
   <td style="text-align:right;"> 55.21 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Attrib inc risk (crude:M-H) </td>
   <td style="text-align:right;"> 1.23 </td>
   <td style="text-align:right;">  </td>
   <td style="text-align:right;">  </td>
  </tr>
</tbody>
</table>
]

---
class: middle
###Confounders cannot be descendants of treatment, but can be in the future of treatment
In Figure 7.11. L is a descendant of treatment A that blocks all backdoor paths from A to Y. 
<img src="images/L6fig11HR.png" width="30%" style="display: block; margin: auto;" />

- Conditioning on L does not cause selection bias because no collider path is opened. 
- Since the causal effect of A on Y is only through L, conditioning on L completely blocks this pathway. 
- This shows that adjusting for a variable L that blocks all backdoor paths does not eliminate bias when L is a descendant of A. 
- Since `$Y^a \perp \!\!\! \perp A|L$` implies adjustment for L eliminates all bias, there must not be conditional exchangeability, 
- **and thus `$E[Ya=1] − E[Ya=0]$` is not identified.**

---

---
class: middle
## Confounding ? Colliders?
.pull-left[

```r
dag1 <- ggdag::dagify(Y ~ L,
       A ~ U,
       L ~ U + A,
       exposure = "A",
       outcome = "Y",
*      latent = "U",
       coords = list(x = c(L = 2, Y = 2.5, A = 1.5, U=1),
       y = c(U = 1, L = 1.5,  A=1.5, Y=1.5)))
dag_plot1 <- dag1 %>% 
  ggdag::tidy_dagitty(layout = "manual", seed = 704) %>%
  arrange(name) %>% 
  ggplot(aes(x = x, y = y, xend = xend, yend = yend)) +
  #xlim()+
  geom_dag_point() +
  geom_dag_edges() +
  theme_dag() +
  geom_dag_node(color="darkmagenta") + geom_dag_text(color="white") 
```
]
--
.pull-right[
<img src="L8_EPIB704_Confounding_II_files/figure-html/unnamed-chunk-35-1.png" width="90%" style="display: block; margin: auto auto auto 0;" />

]

---
class: middle
### Confounders as descendants ? Colliders?

```r
#control_for(dag1, var = "L")
#ggdag_paths(dag1) +theme_dag()
ggdag_adjust(dag1, var = "L", stylized = T, collider_lines = T) + 
  theme_dag() 
```

---
class: middle
##R can help ...

```r
g1 <- dagitty::paths(dag1, "A", "Y")
a1 <- paste0("There are ", length(g$paths), 
            " pathways from A to Y")  
b1 <- paste0("Of these backdoor pathways ", 
            sum(g1$open=="TRUE"), " are open")  
c1 <- paste0("The adjustment sets are ", 
            adjustmentSets(dag1, "A", "Y", type = "canonical"))

print(c(a1,b1,c1))
```

```
## [1] "There are 3 pathways from A to Y"     
## [2] "Of these backdoor pathways 2 are open"
## [3] "The adjustment sets are "
```

The bias is **.red[ and thus `$E[Ya=1] − E[Ya=0]$` is not identified.]** attempting to block the confounding path opens a selection bias path.

---
class: middle

### Do we know what a confounder is?

[Confounding Variable Joke](http://www.bazinganomics.com/bazinganomics//confounding-variable-joke)

---
class: middle
## How to adjust for confounding

---
class: middle
## How to adjust for confounding

- **Randomization is the best method.**
 - In conditionally randomized experiments given covariates `$L$`, the common causes (i.e., the covariates L) are measured and thus the adjusted (standardization or IP weighting) association measure is expected to equal the effect measure.

- Subject-matter knowledge to identify adjustment variables is **_discretionary_ in "ideal" randomized experiments**.

- On the other hand, **subject-matter knowledge is key (a must!) in observational studies** in order to identify and measure adjustment variables (e.g., for regression adjustment).

---
class: middle
## How to adjust for confounding

- Causal inference from observational data relies on the **uncheckable assumption** that we have used our knowledge to identify and measure a set of variables `$L$` that is a sufficient set for confounding adjustment:

- The set of non-descendants of treatment that includes enough variables to block all backdoor paths.
 
- Under this assumption of no unmeasured confounding or of conditional exchangeability given `$L$`, standardization and Inverse Probability (IP) weighting can be used to compute the average causal effect in the population.

---
class: middle
### Standardization 
**Why standardize?**
 - To control for confounding
 - To summarize many estimates into one

- Is a weighted average of measures of occurrence across a distribution (say, age). 
- Can be applied to any measure of occurrence or measure of effect
- Weights are chosen based on the population of interest

(ME3, pg. 49)
---
### Standardized measures of association and effect

- Let `$I_k$` represent strata specific incidence rates and 
- let `$I_k^*$` represent another schedule of such rates (perhaps based on a different exposure distribution)
- Let `$Tk$` represent person-time at risk in each strata

`$I_s = \left(\frac{\sum_{k=1}^K T_k I_k}{\sum_{k=1}^K T_k }\right)$`

`$I_s^* = \left(\frac{\sum_{k=1}^K T_k I_k^*}{\sum_{k=1}^K T_k }\right)$`

- Then the standardized rate ratio is: `$IR_s = I_s/I_s^*$`
- The standardized rate difference is: `$IR_s = I_s - I_s^* = \sum T_k(I_k - I_k^*)$`

(ME3, pg. 67)

---
class: middle
## Standardized measures of association and effect
- Note that the standardized rate difference is a weighted average of stratum-specific rate differences

**Interpretation of both measures:** 
- Effects of exposure on this population. 
  - For the standardized rate ratio we need to assume that the relative distribution of person-time would be unaffected by exposure.
  
  – Standardized <span style="color:purple"> risk ratios </span> do not require this assumption because the denominators do not use person-time.

---
### Example: [COVID-19 vaccine effectiveness in the UK](https://assets.publishing.service.gov.uk/government/uploads/system/uploads/attachment_data/file/1025358/Vaccine-surveillance-report-week-41.pdf) 
UK Health Security Agency "COVID-19 vaccine surveillance report", Week 41

**Rates (per 100,000) by vaccination status from week 37 to week 40 2021**

---

#### Example: COVID-19 vaccine effectiveness in the UK (2)
[Numbers by variant are reported by Public Health England](https://assets.publishing.service.gov.uk/government/uploads/system/uploads/attachment_data/file/1018547/Technical_Briefing_23_21_09_16.pdf).

From: Table 5. Attendance to emergency care and deaths of sequenced and genotyped Delta cases in England by vaccination status
(1 February 2021 to 12 September 2021) [here]((https://assets.publishing.service.gov.uk/government/uploads/system/uploads/attachment_data/file/1018547/Technical_Briefing_23_21_09_16.pdf).)

---

#### Example: COVID-19 vaccine effectiveness in the UK 
**Let's play with the numbers (1):** <span style="color:blue"> check the risk difference (RD) </span>

```r
#157400 - 2361 #exposed without outcome
#257357 - 30801 #unexposed without outcome
l6UKdata<-c(2361,155039, 3080, 254277)
l6UKest<- epi.2by2(l6UKdata, method = "cohort.count")
l6UKest
```

```
##              Outcome +    Outcome -      Total                 Inc risk *
## Exposed +         2361       155039     157400        1.50 (1.44 to 1.56)
## Exposed -         3080       254277     257357        1.20 (1.16 to 1.24)
## Total             5441       409316     414757        1.31 (1.28 to 1.35)
## 
## Point estimates and 95% CIs:
## -------------------------------------------------------------------
## Inc risk ratio                                 1.25 (1.19, 1.32)
## Inc odds ratio                                 1.26 (1.19, 1.33)
## Attrib risk in the exposed *                   0.30 (0.23, 0.38)
## Attrib fraction in the exposed (%)            20.21 (15.85, 24.35)
## Attrib risk in the population *                0.12 (0.06, 0.17)
## Attrib fraction in the population (%)         8.77 (6.64, 10.86)
## -------------------------------------------------------------------
## Uncorrected chi2 test that OR = 1: chi2(1) = 69.360 Pr>chi2 = <0.001
## Fisher exact test that OR = 1: Pr>chi2 = <0.001
##  Wald confidence limits
##  CI: confidence interval
##  * Outcomes per 100 population units
```

** what would we conclude?**

---
### Example: COVID-19 vaccine effectiveness in the UK 
 - **<span style="color:blue"> Missing something? </span>**
 
<img src="images/L6reportUKage1.png" width="75%" style="display: block; margin: auto;" />

From: Table 5. Attendance to emergency care and deaths of sequenced and genotyped Delta cases in England by vaccination status (1 February 2021 to 12 September 2021) [here]((https://assets.publishing.service.gov.uk/government/uploads/system/uploads/attachment_data/file/1018547/Technical_Briefing_23_21_09_16.pdf).) .small[Note: The totals do not exactly sum up to the previous table, as age was missing in a few cases.]

---
class: middle
#### Example: COVID-19 vaccine effectiveness in the UK 
**Let's play with the numbers (2) - Standardization**

```r
l6UKdatu50<-c(453,84954, 2416,  246387)
l6UKt1u50<- epi.2by2(l6UKdatu50, method = "cohort.count")
l6UKt1u50$tab
```

```
##              Outcome +    Outcome -      Total                 Inc risk *
## Exposed +          453        84954      85407        0.53 (0.48 to 0.58)
## Exposed -         2416       246387     248803        0.97 (0.93 to 1.01)
## Total             2869       331341     334210        0.86 (0.83 to 0.89)
```
]
.pull-right-narrow[
**Outcomes among people `$\geq$` 50 years**

```r
l6UKdatm50<-c(1908 , 70083, 664,   7887)
l6UKt1m50<- epi.2by2(l6UKdatm50, method = "cohort.count")
l6UKt1m50$tab
```

```
##              Outcome +    Outcome -      Total                 Inc risk *
## Exposed +         1908        70083      71991        2.65 (2.53 to 2.77)
## Exposed -          664         7887       8551        7.77 (7.21 to 8.35)
## Total             2572        77970      80542        3.19 (3.07 to 3.32)
```
]

---

#### Example: COVID-19 vaccine effectiveness in the UK 
**Let's play with the numbers (3)** <span style="color:blue"> check the risk differences (RD)</span>

.pull-left[
**Outcomes among people under 50 years**
<table class=" lightable-paper" style='font-family: "Arial Narrow", arial, helvetica, sans-serif; margin-left: auto; margin-right: auto;'>
 <thead>
<tr>
<th style="padding-bottom:0; padding-left:3px;padding-right:3px;text-align: center; " colspan="1"><div style="border-bottom: 1px solid #00000020; padding-bottom: 5px; ">Measure</div></th>
<th style="padding-bottom:0; padding-left:3px;padding-right:3px;text-align: center; " colspan="3"><div style="border-bottom: 1px solid #00000020; padding-bottom: 5px; ">Estimate 95%CIs</div></th>
</tr>
  <tr>
   <th style="text-align:left;"> Measure </th>
   <th style="text-align:right;"> Est. </th>
   <th style="text-align:right;"> LB </th>
   <th style="text-align:right;"> UB </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:left;"> Inc risk ratio </td>
   <td style="text-align:right;"> 0.55 </td>
   <td style="text-align:right;"> 0.49 </td>
   <td style="text-align:right;"> 0.60 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Inc odds ratio </td>
   <td style="text-align:right;"> 0.54 </td>
   <td style="text-align:right;"> 0.49 </td>
   <td style="text-align:right;"> 0.60 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Attrib inc risk * </td>
   <td style="text-align:right;"> -0.44 </td>
   <td style="text-align:right;"> -0.50 </td>
   <td style="text-align:right;"> -0.38 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Attrib fraction in exposed (%) </td>
   <td style="text-align:right;"> -83.08 </td>
   <td style="text-align:right;"> -102.34 </td>
   <td style="text-align:right;"> -65.65 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Attrib inc risk in population * </td>
   <td style="text-align:right;"> -0.11 </td>
   <td style="text-align:right;"> -0.16 </td>
   <td style="text-align:right;"> -0.06 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Attrib fraction in population (%) </td>
   <td style="text-align:right;"> -13.12 </td>
   <td style="text-align:right;"> -14.92 </td>
   <td style="text-align:right;"> -11.34 </td>
  </tr>
</tbody>
</table>
]
.pull-right[
**Outcomes among people `$\geq$` 50 years**
<table class=" lightable-paper" style='font-family: "Arial Narrow", arial, helvetica, sans-serif; margin-left: auto; margin-right: auto;'>
 <thead>
<tr>
<th style="padding-bottom:0; padding-left:3px;padding-right:3px;text-align: center; " colspan="1"><div style="border-bottom: 1px solid #00000020; padding-bottom: 5px; ">Measure</div></th>
<th style="padding-bottom:0; padding-left:3px;padding-right:3px;text-align: center; " colspan="3"><div style="border-bottom: 1px solid #00000020; padding-bottom: 5px; ">Estimate 95%CIs</div></th>
</tr>
  <tr>
   <th style="text-align:left;"> Measure </th>
   <th style="text-align:right;"> Est. </th>
   <th style="text-align:right;"> LB </th>
   <th style="text-align:right;"> UB </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:left;"> Inc risk ratio </td>
   <td style="text-align:right;"> 0.34 </td>
   <td style="text-align:right;"> 0.31 </td>
   <td style="text-align:right;"> 0.37 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Inc odds ratio </td>
   <td style="text-align:right;"> 0.32 </td>
   <td style="text-align:right;"> 0.30 </td>
   <td style="text-align:right;"> 0.35 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Attrib inc risk * </td>
   <td style="text-align:right;"> -5.11 </td>
   <td style="text-align:right;"> -5.69 </td>
   <td style="text-align:right;"> -4.54 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Attrib fraction in exposed (%) </td>
   <td style="text-align:right;"> -192.99 </td>
   <td style="text-align:right;"> -219.11 </td>
   <td style="text-align:right;"> -169.00 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Attrib inc risk in population * </td>
   <td style="text-align:right;"> -4.57 </td>
   <td style="text-align:right;"> -5.15 </td>
   <td style="text-align:right;"> -3.99 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Attrib fraction in population (%) </td>
   <td style="text-align:right;"> -143.17 </td>
   <td style="text-align:right;"> -159.10 </td>
   <td style="text-align:right;"> -128.21 </td>
  </tr>
</tbody>
</table>
]

---
class: middle
#### Example: COVID-19 vaccine effectiveness in the UK 
**.red[Confounding?]**

** <span style="color:blue"> We know that IRL the "L" includes a vector / set of potential covariates that could be considered as Confounders... this is an illustration only! </span>**

---
class: middle
### Direct standardization
Suppose we want to estimate `$E[Y^a=1] - E[Y^a=0] = RD$`.

The conditional exchangeability allows us to say `$Y^a \perp \!\!\! \perp A | L$`

[According to the law of total expectation](https://en.wikipedia.org/wiki/Law_of_total_expectation):

<span style="color:purple"> `$E[Y^a =1] = \sum_{x} E[Y^a=1 | X=x]Pr(x)$` </span>;

</span> <span style="color:blue"> `$E[Y^a =0] = \sum_{x} E[Y^a=0 | X=x]Pr(x)$` </span>
- `$\sum_{x}$` means sum over all values x that occur in the study population.
- `$Pr(x)$` refers to the distribution of x in that population.

** `$RD=$` ** `$E[Y^a =1] - E[Y^a =0]  =$`

<span style="color:purple"> `$\sum_{x} E[Y^a=1 | X=x]P(x)$` </span> `$-$` <span style="color:blue"> `$\sum_{x} E[Y^a=0 | X=x]P(x)$` </span>
  
  
  
---
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#### Example: COVID-19 vaccine effectiveness in the UK 
**Let's play with the numbers  - Standardization**

---
#### Example: COVID-19 vaccine effectiveness in the UK 
**Let's play with the numbers (4) - Standardization**

To compute the PO using observed data, we need the consistency assumption

`$RD =$` <span style="color:purple"> `$\sum_{x} E[Y |A=1, X=x]P(x)$` </span> `$-$` <span style="color:blue"> `$\sum_{x} E[Y |A=0, X=x]Pr(x)$` </span>

<span style="color:purple"> Standardized risk in the vaccinated : </span>
`$(453/85,407 × 334,210/414,752 +  1,908/71,991 × 80,542/414,752)≈ 0.94\%$`

`$R_{vax}=$` 0.94

<span style="color:blue"> Standardized risk in the unvaccinated : </span>
`$(2,416/248,803 × 334,210/414,752 +  664/8,551 × 80,542/414,752)≈ 2.29\%$`

`$R_{unvax}=$` 2.29

**Standardized RD = -1.35 **
  from (` 0.94% − 2.29% = −1.35%`) `$\neq 0.3$` in the crude estimates.

**Standardized RR = 0.41 **   from (` 0.0094 / 0.0229`) `$\neq 1.25$` in the crude estimates.

---
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### Example: [COVID-19 vaccine effectiveness in the UK](https://assets.publishing.service.gov.uk/government/uploads/system/uploads/attachment_data/file/1025358/Vaccine-surveillance-report-week-41.pdf) 
UK Health Security Agency "COVID-19 vaccine surveillance report", Week 41

Cases presenting to emergency care (within 28 days of a positive test) resulting in overnight inpatient admission.[here](https://assets.publishing.service.gov.uk/government/uploads/system/uploads/attachment_data/file/1025358/Vaccine-surveillance-report-week-41.pdf).

---
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## What about the Mantel-Haenzel Methods?
- _Cochran-Mantel-Haenzel_ methods are useful for associations, when only few covariates are involved in the calculation.
- Takes the effect in each strata of `$L$` or `$Z$` (our third variable), 
- Combines these measures across `$L$` using calculated weights `$^1$`, for example example:

$$ RD_{M-H} = \left(\frac{\sum_l (RD_l w_l)} {\sum_l w_l}\right) = \left(\frac{RD_0 w_0 + RD_1 w_1}{ w_0 + w_1}\right)$$

- <span style="color:blue"> Are expected to work in closed cohorts and **.red[assumes homogeneity across strata!!]** </span>

- Limited use in a set of covariates `$L$` and in presence of Effect measure modification and or interaction.
  
`$^1$` There are specific formulas for RD,  RR and ORs as well

---

- No assumption of homogeneity, "agnostic of the distribution",
Model-based direct standardization ** `$^1$`** are used when `$L (X, E, A)$` consists of a large vectors of covariates.

Involves two steps: 
* Fitting a regression model for the outcome given exposure and covariates

* Averaging the exposure effect over the covariate distribution of the standard population.

** `$^1$` ** More on "advanced" techniques to address confounding empirically after we deal with regressions.

---

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## Standardized Morbidity Ratio (SMR) 
- A generalization to standardization when the standard population is the exposed sub-population.
- In this case, the standardized rate ratio
becomes:

`$$I_s = \left(\frac{\sum_{k=1}^K T_k I_k}{\sum_{k=1}^K T_k I_k^* }\right) =  \left(\frac{\sum_{k=1}^K A_k}{\sum_{k=1}^K T_k I_k^*}\right)$$`

**<span style="color:purple"> [Numerator] </span>** cases occurring in exposed (**Observed**)

**<span style="color:blue">[Denominator] </span>** cases **expected** to occur in absence of exposure if exposure doesn’t affect person time at risk

(ME3, pg. 68-69)

---

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### How to adjust for confounding
Standardization and Inverse Probability (IP) weighting are not the only methods.

`$$IPW_z = \left(\frac{1}{Pr(A=a|L=z)}\right)$$`

Often using regression models, **assuming the model specification is correct!** 😬

**<span style="color:purple"> IPW removes the arrow from `$L \to A$`: </span>**

---

Two categories of methods for confounding adjustment:

**<span style="color:blue"> 1) G-methods (including G-formula, IP weighting, and G-estimation).</span>** These exploit conditional exchangeability in subsets defined by L to estimate the causal effect of A on Y in the entire population or in any subset of the population.

- Under the assumption of conditional exchangeability given `$L$`, g-methods simulate `$A-Y$` associations in the population if backdoor paths involving variables `$L$` did not exist; simulated `$A-Y$` associations can then be attributed to the effect of `$A$` on `$Y$`.
  
  - IP weighting achieves this by creating a pseudo-population in which `$A$` is independent of measured confounders `$L$`, by “deleting” the arrow from `$L \to A$`.

---
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# How to adjust for confounding

**<span style="color:blue"> 2) Stratification-based methods (including Stratification, Restriction, Matching). </span>**

Methods that exploit conditional exchangeability in subsets defined by L to estimate the association between A and Y in those subsets only.

Stratification-based methods estimate the association between A and Y in one or more subsets of the population in which the treated and the untreated are assumed to be exchangeable.

- Hence the `$A \to Y$` association in each subset is entirely attributed to the effect of `$A$` on `$Y$` . 
 
 - Stratification/restriction do not delete the arrow from `$L \to A$`, but instead calculate the association within strata of `$L$`, since within each level of `$L$`, there is no `$L \to A$` association to cause confounding.

---
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## How to adjust for confounding

All these methods require conditional exchangeability given the measured covariates `$L$` to identify the effect of treatment `$A$` on outcome `$Y$`. 
  - When interested in the effect in the entire population, conditional exchangeability is required in all strata defined by `$L$`; 
  
  - When interested in the effect in a subset of the population, conditional exchangeability is required in that subset only.

- Achieving conditional exchangeability may be an unrealistic goal in many observational studies but expert knowledge can be used to get as close as possible to that goal.

- At the very least, investigators should generally avoid adjustment for variables affected by either the treatment or the outcome.

---
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## How to adjust for confounding

Thoughtful and knowledgeable investigators could believe that various causal structures, possibly leading to different conclusions regarding confounding, are equally plausible. 
  - DAGs simply allow us to have that discussion.

- Existence of common causes of treatment and outcome does not depend on the adjustment method (although it does depend on the target population).
- Adjustment for measured confounding will generally imply a change in the estimate, but not necessarily the other way around.
- Changes in estimates may occur for reasons other than confounding, 
  - **including selection bias when adjusting for non-confounders and the use of non-collapsible effect measures.**
  
**H & R write:** 
> _"Attempts to define confounding based on change in estimates have been long abandoned because of these problems.” This is overstated. When using a DAG and collapsible measures, the method is a reasonable and practical strategy._"

---
class: middle
**A note on stratification and non-collapsibility**

Comparing crude to adjusted estimates is reliable for RR and RD, but not for OR unless:
 a) rare outcome or b) OR ≈ RR due to design (e.g. case-cohort).
 
**Recall the case of `$C \to E \to Y$`**
<table class=" lightable-paper" style='font-family: "Arial Narrow", arial, helvetica, sans-serif; margin-left: auto; margin-right: auto;'>
 <thead>
<tr>
<th style="padding-bottom:0; padding-left:3px;padding-right:3px;text-align: center; " colspan="1"><div style="border-bottom: 1px solid #00000020; padding-bottom: 5px; ">Measure</div></th>
<th style="padding-bottom:0; padding-left:3px;padding-right:3px;text-align: center; " colspan="3"><div style="border-bottom: 1px solid #00000020; padding-bottom: 5px; ">Estimate 95%CIs</div></th>
</tr>
  <tr>
   <th style="text-align:left;"> Measure </th>
   <th style="text-align:right;"> Est. </th>
   <th style="text-align:right;"> LB </th>
   <th style="text-align:right;"> UB </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:left;"> Inc risk ratio (crude) </td>
   <td style="text-align:right;"> 1.91 </td>
   <td style="text-align:right;"> 1.36 </td>
   <td style="text-align:right;"> 2.70 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Inc risk ratio (M-H) </td>
   <td style="text-align:right;"> 1.75 </td>
   <td style="text-align:right;"> 1.14 </td>
   <td style="text-align:right;"> 2.69 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Inc risk ratio (crude:M-H) </td>
   <td style="text-align:right;"> 1.09 </td>
   <td style="text-align:right;">  </td>
   <td style="text-align:right;">  </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Inc odds ratio (crude) </td>
   <td style="text-align:right;"> 5.12 </td>
   <td style="text-align:right;"> 2.02 </td>
   <td style="text-align:right;"> 12.97 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Inc odds ratio (M-H) </td>
   <td style="text-align:right;"> 3.83 </td>
   <td style="text-align:right;"> 1.45 </td>
   <td style="text-align:right;"> 10.09 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Inc odds ratio (crude:M-H) </td>
   <td style="text-align:right;"> 1.34 </td>
   <td style="text-align:right;">  </td>
   <td style="text-align:right;">  </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Attrib inc risk (crude) * </td>
   <td style="text-align:right;"> 37.15 </td>
   <td style="text-align:right;"> 19.01 </td>
   <td style="text-align:right;"> 55.30 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Attrib inc risk (M-H) * </td>
   <td style="text-align:right;"> 31.20 </td>
   <td style="text-align:right;"> -6.71 </td>
   <td style="text-align:right;"> 69.12 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Attrib inc risk (crude:M-H) </td>
   <td style="text-align:right;"> 1.19 </td>
   <td style="text-align:right;">  </td>
   <td style="text-align:right;">  </td>
  </tr>
</tbody>
</table>

---

### A note on stratification and non-collapsibility
-  We can say a measure of the association between A and Y is collapsible across L if the adjusted association, `$RR_{AY}|L$`, is equal to the crude association, `$RR_{AY}$`, where L is not a confounder
   — This means that a crude measure of association will not change if we adjust for a variable that is not a confounder `$(L)$`

-  The odds (OR) and incidence density ratios (IDR) fail this property and are considered non collapsible effect measures

-  For the OR, the crude measure may be closer to the null than the pooled/adjusted OR, particularly with a common outcome

-  Therefore, for some measures, our simple crude vs. adjusted comparison **may suggest confounding when there really isn’t!**

---
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### Change in estimate?? 
## .red[Not Really!!!]

<img src="images/changeest.png" width="65%" style="display: block; margin: auto;" />
---
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### Structural confounding, violation of Positivity
High correlations between confounder and exposure: violation of the “positivity assumption”.
When this is “structural” (in the sense of a high correlation that exists because of causal relations in the source population), Oakes calls this “structural confounding”.
<img src="images/L6strconfounding.png" width="35%" />

Oakes JM. Advancing neighbourhood-effects research selection, inferential support,
    and structural confounding. Int J Epidemiol. 2006 Jun;35(3):643-7.
    Messer et al. Effects of Socioeconomic and Racial Residential Segregation on Preterm Birth: 
    A Cautionary Tale of Structural Confounding AJE 2010; Mar 15;171(6):664-73.

---
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### Structural confounding, violation of Positivity

```r
set.seed(704); n=500
ses1 <- sample(1:12, n, replace = TRUE); 
ses1[ses1>=10]<-0
ses2 <- cut(ses1, breaks = c(0, 5, 10, 15), 
        labels = c("0", "1", "2"))
ses2[is.na(ses2 )]<- "2"
exposure<- ifelse(ses2=="1", 
    rbinom(n,1,0.45), 
    ifelse(ses2=="0", rbinom(n,1,0.5),
    ifelse(ses2=="2", rbinom(n,1,0.0001),
    rbinom(n,1,0.2))))
outcome<- ifelse(ses2=="0", rbinom(n,1,0.75), 
    ifelse(ses2=="1", rbinom(n,1,0.25),
    rbinom(n,1,0.25)))
data.strconf <- data.frame(outcome, exposure, ses2)
table(exposure, ses2)
```

```
##         ses2
## exposure   0   1   2
##        0 101 107 123
##        1 104  65   0
```

```r
strconf2 <- glm(outcome ~ exposure, 
           data= data.strconf, 
           family = "binomial") 
strconf2a <- glm(outcome ~ exposure + 
            as.factor(ses2), data= data.strconf, family = "binomial")     
```
]
--

.pull-right[
**Regression Results**
**Crude/Unadjusted**
<table class=" lightable-paper" style='font-family: "Arial Narrow", arial, helvetica, sans-serif; margin-left: auto; margin-right: auto;'>
 <thead>
  <tr>
   <th style="text-align:left;">   </th>
   <th style="text-align:right;"> exp(Est.) </th>
   <th style="text-align:right;"> 2.5% </th>
   <th style="text-align:right;"> 97.5% </th>
   <th style="text-align:right;"> z val. </th>
   <th style="text-align:right;"> p </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:left;"> (Intercept) </td>
   <td style="text-align:right;"> 0.663 </td>
   <td style="text-align:right;"> 0.532 </td>
   <td style="text-align:right;"> 0.827 </td>
   <td style="text-align:right;"> -3.657 </td>
   <td style="text-align:right;"> 0.000 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> exposure </td>
   <td style="text-align:right;"> 1.677 </td>
   <td style="text-align:right;"> 1.154 </td>
   <td style="text-align:right;"> 2.437 </td>
   <td style="text-align:right;"> 2.713 </td>
   <td style="text-align:right;"> 0.007 </td>
  </tr>
</tbody>
</table>

**Adjusted**
<table class=" lightable-paper" style='font-family: "Arial Narrow", arial, helvetica, sans-serif; margin-left: auto; margin-right: auto;'>
 <thead>
  <tr>
   <th style="text-align:left;">   </th>
   <th style="text-align:right;"> exp(Est.) </th>
   <th style="text-align:right;"> 2.5% </th>
   <th style="text-align:right;"> 97.5% </th>
   <th style="text-align:right;"> z val. </th>
   <th style="text-align:right;"> p </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:left;"> (Intercept) </td>
   <td style="text-align:right;"> 2.461 </td>
   <td style="text-align:right;"> 1.676 </td>
   <td style="text-align:right;"> 3.612 </td>
   <td style="text-align:right;"> 4.598 </td>
   <td style="text-align:right;"> 0.000 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> exposure </td>
   <td style="text-align:right;"> 1.011 </td>
   <td style="text-align:right;"> 0.634 </td>
   <td style="text-align:right;"> 1.613 </td>
   <td style="text-align:right;"> 0.046 </td>
   <td style="text-align:right;"> 0.963 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> as.factor(ses2)1 </td>
   <td style="text-align:right;"> 0.119 </td>
   <td style="text-align:right;"> 0.074 </td>
   <td style="text-align:right;"> 0.190 </td>
   <td style="text-align:right;"> -8.860 </td>
   <td style="text-align:right;"> 0.000 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> as.factor(ses2)2 </td>
   <td style="text-align:right;"> 0.168 </td>
   <td style="text-align:right;"> 0.097 </td>
   <td style="text-align:right;"> 0.290 </td>
   <td style="text-align:right;"> -6.399 </td>
   <td style="text-align:right;"> 0.000 </td>
  </tr>
</tbody>
</table>
]

---
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##Which way will the confounding go?
<img src="images/L6_dag_conf2.png" width="50%" />

`Vander Stoep A, et al. A didactic device for teaching epidemiology students how to anticipate the effect of a third factor on an exposure-outcome relation. AJE 1999; 15;150(2):221.`

---
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##Which way will the confounding go?

<img src="images/L6_dag_conf1.png" width="50%" />
 
 
**These schematics are just illustrations, it depends on the strength (degree of correlation) of the covariates!!, <span style="color:blue"> simulations works better than "blanket" type of statements </span>**
 
---

## Positive, negative, and “qualitative” confounding

- Confounding may lead to an overestimation or an underestimation of the true magnitude of an effect.

- **<span style="color:blue"> Positive confounding:</span>** the magnitude of the unadjusted vis-à- vis the adjusted association is exaggerated.

- **<span style="color:blue"> Negative confounding:</span>** the magnitude of the unadjusted vis-à- vis the adjusted association is underestimated.

- **<span style="color:blue"> Qualitative confounding:</span>** An extreme case when confounding results in an inversion of the direction of the association.

---
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## Magnitude of confounding

- The magnitude of confounding will depend on the strength of the confounder-exposure AND confounder-outcome associations.

- Conversely, if there is no association between the confounder - exposure OR no association between the confounder-outcome then no confounding of the main effect could be present.

- The strength of the confounder-exposure and confounder- outcome associations bounds the confounding effect
  - e.g., if RRcrude = 2 and the confounder-outcome relation is 2 (a doubling of risk), then the confounder would have to be perfectly correlated with the exposure in order to fully explain the main effect of RR=2

---
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**How strong the the _unmeasured confounding_ should be to explain away my estimated association?**

**<span style="color:blue"> E values:</span>** respond to this question for ratio ** `$^1$`**  measures, how?

$$E-value = RR + \sqrt{RR \times (RR-1)} $$
--
- E-value is the minimum value of the association between `$U \to A$` and `$U \to Y$` that will be capable of attenuating the observed association towards the null.

- Example: RR=1.33; `$1.33 + \sqrt{1.33 \times (1.33-1)} =$` 1.99 then, if there was an `$U$`, it should:

1) double the risk among unexposed and/or exposed ( `$RR_{UY} =2$` ), AND
  
  2) be twice as prevalent among exposed than among unexposed ( `$RR_{AU} =2$` )
  
To completely explain away the observed association, but a weaker confounder (given the E-value), say 1.5 or 1.3, would not.

** `$^1$`**  <span style="color:blue"> E values are debatable for some but still a straightforward calculation and useful information to have. </span> [Versions of the E-value exists for ORs and HRs. E-value calculator.](https://www.evalue-calculator.com/)

---

###Statistical significance?

##In general, NO! 
- But if you MUST use p-values, set the criteria on the high side
  (e.g. p < 0.30). This way you adjust for some non-confounders, but you don’t miss many true confounders.
 
`Mickey RM, Greenland S. The impact of confounder selection criteria on effect estimation. AJE 1989;129(1):125-37.`

- Residual confounding (unmeasured L’s (U1, U2, etc), categorization, measurement error, etc):
 
`Kaufman JS, et al. Socioeconomic status and health in blacks and whites: the problem of residual confounding and the resiliency of race. Epidemiology 1997; 8(6):621-8.`
`Ogburn EL, Vanderweele TJ. Bias attenuation results for nondifferentially mismeasured ordinal and coarsened confounders. Biometrika. 2013;100(1):241- 248. PMID: 24014285`

---

## Residual confounding
Residual confounding occurs when adjustment does not completely remove the confounding effect of a given variable(s):

**1) Misclassification of confounding variables**
- (e.g., the variable is an imperfect proxy for the characteristic we want to adjust for)

**2) Improper modeling of the confounding variable**
- (e.g., if we are studying air pollution and lung cancer and want to control for smoking, we should measure smoking in a way that best predicts lung cancer—i.e., pack-years not ever-never)

**3) Other important confounders are not included (also known as unmeasured confounding or omitted variable bias)**

---
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## Validity and Bias:
- The epidemiologist’s goal: the most **VALID and PRECISE** estimate possible of the causal effect of exposure on disease.

- Error comes from sampling variability (lack of precision) and bias (lack of validity).
<img src="images/L6validity.png" width="50%" />
 
---
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## Confounded `$^1$` ?

<img src="images/L6confounded.png" width="50%" />
 
`$^1$` We all are!! We will have more on this and empirical examples after we deal with regressions.

---
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###  QUESTIONS?

## COMMENTS?

# RECOMMENDATIONS?

---