Propensity score weighting

# Propensity score weighting

**Session 11**

]

---

<br>

---

```r
nhefs_complete_uc <-
  glm(qsmk ~ wt71 + age + marital + race + sex, 
      data = nhefs_complete_uc,
      family = binomial()) %>%
  augment(type.predict = "response",
          data = nhefs_complete_uc) 
```

---
class: title title-1
# Average treatment effect

```r
nhefs_complete_uc <- nhefs_complete_uc %>%
  mutate( 
    w_ate = qsmk / .fitted + 
      (1 - qsmk) / (1 - .fitted)
  )
```
---

```r
nhefs_complete_uc <- nhefs_complete_uc %>%
* mutate(
    w_ate = qsmk / .fitted + 
      (1 - qsmk) / (1 - .fitted)
  )
```
---
class: title title-1
# Average treatment effect

```r
nhefs_complete_uc <- nhefs_complete_uc %>%
  mutate( 
*   w_ate = qsmk / .fitted +
      (1 - qsmk) / (1 - .fitted)
  )
```

---

```r
nhefs_complete_uc <- nhefs_complete_uc %>%
  mutate( 
    w_ate = qsmk / .fitted + 
*     (1 - qsmk) / (1 - .fitted)
  )
```

---

# <svg aria-hidden="true" role="img" viewBox="0 0 640 512" style="height:1em;width:1.25em;vertical-align:-0.125em;margin-left:auto;margin-right:auto;font-size:inherit;fill:currentColor;overflow:visible;position:relative;"><path d="M624 416H381.54c-.74 19.81-14.71 32-32.74 32H288c-18.69 0-33.02-17.47-32.77-32H16c-8.8 0-16 7.2-16 16v16c0 35.2 28.8 64 64 64h512c35.2 0 64-28.8 64-64v-16c0-8.8-7.2-16-16-16zM576 48c0-26.4-21.6-48-48-48H112C85.6 0 64 21.6 64 48v336h512V48zm-64 272H128V64h384v256z"/></svg> Application Exercise

<div class="countdown" id="timer_621bfd2a" style="right:0;bottom:0;" data-warnwhen="0">
<code class="countdown-time"><span class="countdown-digits minutes">05</span><span class="countdown-digits colon">:</span><span class="countdown-digits seconds">00</span></code>
</div>
---

# Mirrored Histogram

```r
df_plot_ps <- nhefs_complete_uc %>%
  tidyr::pivot_wider(names_from = qsmk, 
                     values_from = .fitted, 
                     names_prefix = "qsmk_p")
```
]

---
class: title title-1
# Mirrored Histogram

```r
ggplot(df_plot_ps) +
  geom_histogram(bins = 50, 
                 aes(x = qsmk_p1), 
                 fill = "cornflower blue") + 
  geom_histogram(bins = 50, 
                 alpha = 0.5,
                 aes(x = qsmk_p1, 
                     weight = w_ate), 
                 fill = "cornflower blue") +
  geom_histogram(bins = 50, 
                 aes(x = qsmk_p0, 
                     y = - stat(count))) + 
  geom_histogram(bins = 50, 
                 alpha = 0.5,
                 aes(x = qsmk_p0, 
                     weight = w_ate,
                     y = - stat(count)), 
                 fill = "orange") + 
  scale_y_continuous("count", label = abs) + 
  scale_x_continuous("propensity score") +
  geom_label(
    label = "Quit smoking", 
    x = 0.25,
    y = 3,
  ) + 
  geom_label(
    label = "Didn't quit smoking",
    x = 0.25, 
    y = -3
  ) + 
  theme_minimal()
```
]
]

.right-plot[
<img src="10-propensity-score-weights_files/figure-html/unnamed-chunk-8-1.png" width="504" style="display: block; margin: auto;" />
]
---
class: title title-1
# Mirrored Histogram

```r
ggplot(df_plot_ps) +
  geom_histogram(bins = 50, 
                 aes(x = qsmk_p1), 
                 fill = "cornflower blue") + 
* geom_histogram(bins = 50,
*                alpha = 0.5,
*                aes(x = qsmk_p1,
*                    weight = w_ate),
                 fill = "cornflower blue") +
  geom_histogram(bins = 50, 
                 aes(x = qsmk_p0, 
                     y = - stat(count))) + 
  geom_histogram(bins = 50, 
                 alpha = 0.5,
                 aes(x = qsmk_p0, 
                     weight = w_ate,
                     y = - stat(count)), 
                 fill = "orange") + 
  scale_y_continuous("count", label = abs) + 
  scale_x_continuous("propensity score") +
  geom_label(
    label = "Quit smoking", 
    x = 0.25,
    y = 3,
  ) + 
  geom_label(
    label = "Didn't quit smoking",
    x = 0.25, 
    y = -3
  ) + 
  theme_minimal()
```
]
]

.right-plot[
<img src="10-propensity-score-weights_files/figure-html/unnamed-chunk-10-1.png" width="504" style="display: block; margin: auto;" />
]

---

```r
ggplot(df_plot_ps) +
  geom_histogram(bins = 50, 
                 aes(x = qsmk_p1), 
                 fill = "cornflower blue") + 
  geom_histogram(bins = 50,  
                 alpha = 0.5, 
                 aes(x = qsmk_p1, 
                     weight = w_ate),  
                 fill = "cornflower blue") +
  geom_histogram(bins = 50, 
                 aes(x = qsmk_p0, 
                     y = - stat(count))) + 
* geom_histogram(bins = 50,
*                alpha = 0.5,
*                aes(x = qsmk_p0,
*                    weight = w_ate,
*                    y = - stat(count)),
                 fill = "orange") + 
  scale_y_continuous("count", label = abs) + 
  scale_x_continuous("propensity score") +
  geom_label(
    label = "Quit smoking", 
    x = 0.25,
    y = 3,
  ) + 
  geom_label(
    label = "Didn't quit smoking",
    x = 0.25, 
    y = -3
  ) + 
  theme_minimal()
```
]
]

.right-plot[
<img src="10-propensity-score-weights_files/figure-html/unnamed-chunk-12-1.png" width="504" style="display: block; margin: auto;" />
]
---

---

# Table 1!

```r
library(gtsummary)
nhefs_complete_uc %>%
  tbl_summary(
    by = qsmk, 
    include = c(
      "wt71", 
      "age", 
      "marital",
      "race",
      "sex")) %>%
  add_overall()
```

]]

.right-plot[
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</style>
<table class="gt_table">
  
  <thead class="gt_col_headings">
    <tr>
      <th class="gt_col_heading gt_columns_bottom_border gt_left" rowspan="1" colspan="1"><strong>Characteristic</strong></th>
      <th class="gt_col_heading gt_columns_bottom_border gt_center" rowspan="1" colspan="1"><strong>Overall</strong>, N = 1,566<sup class="gt_footnote_marks">1</sup></th>
      <th class="gt_col_heading gt_columns_bottom_border gt_center" rowspan="1" colspan="1"><strong>0</strong>, N = 1,163<sup class="gt_footnote_marks">1</sup></th>
      <th class="gt_col_heading gt_columns_bottom_border gt_center" rowspan="1" colspan="1"><strong>1</strong>, N = 403<sup class="gt_footnote_marks">1</sup></th>
    </tr>
  </thead>
  <tbody class="gt_table_body">
    <tr><td class="gt_row gt_left">WEIGHT IN KILOGRAMS  IN 1971</td>
<td class="gt_row gt_center">69 (60, 80)</td>
<td class="gt_row gt_center">68 (59, 79)</td>
<td class="gt_row gt_center">71 (61, 81)</td></tr>
    <tr><td class="gt_row gt_left">AGE IN 1971</td>
<td class="gt_row gt_center">43 (33, 53)</td>
<td class="gt_row gt_center">42 (33, 51)</td>
<td class="gt_row gt_center">46 (35, 56)</td></tr>
    <tr><td class="gt_row gt_left">MARITAL STATUS IN 1971  1: Under 17, 2: Married, 3: Widowed, 4: Never married, 5: Divorced, 6: Separated, 8: Unknown</td>
<td class="gt_row gt_center"></td>
<td class="gt_row gt_center"></td>
<td class="gt_row gt_center"></td></tr>
    <tr><td class="gt_row gt_left" style="text-align: left; text-indent: 10px;">2</td>
<td class="gt_row gt_center">1,237 (79%)</td>
<td class="gt_row gt_center">906 (78%)</td>
<td class="gt_row gt_center">331 (82%)</td></tr>
    <tr><td class="gt_row gt_left" style="text-align: left; text-indent: 10px;">3</td>
<td class="gt_row gt_center">88 (5.6%)</td>
<td class="gt_row gt_center">64 (5.5%)</td>
<td class="gt_row gt_center">24 (6.0%)</td></tr>
    <tr><td class="gt_row gt_left" style="text-align: left; text-indent: 10px;">4</td>
<td class="gt_row gt_center">93 (5.9%)</td>
<td class="gt_row gt_center">74 (6.4%)</td>
<td class="gt_row gt_center">19 (4.7%)</td></tr>
    <tr><td class="gt_row gt_left" style="text-align: left; text-indent: 10px;">5</td>
<td class="gt_row gt_center">92 (5.9%)</td>
<td class="gt_row gt_center">74 (6.4%)</td>
<td class="gt_row gt_center">18 (4.5%)</td></tr>
    <tr><td class="gt_row gt_left" style="text-align: left; text-indent: 10px;">6</td>
<td class="gt_row gt_center">55 (3.5%)</td>
<td class="gt_row gt_center">45 (3.9%)</td>
<td class="gt_row gt_center">10 (2.5%)</td></tr>
    <tr><td class="gt_row gt_left" style="text-align: left; text-indent: 10px;">8</td>
<td class="gt_row gt_center">1 (<0.1%)</td>
<td class="gt_row gt_center">0 (0%)</td>
<td class="gt_row gt_center">1 (0.2%)</td></tr>
    <tr><td class="gt_row gt_left">0: WHITE 1: BLACK OR OTHER IN 1971</td>
<td class="gt_row gt_center"></td>
<td class="gt_row gt_center"></td>
<td class="gt_row gt_center"></td></tr>
    <tr><td class="gt_row gt_left" style="text-align: left; text-indent: 10px;">0</td>
<td class="gt_row gt_center">1,360 (87%)</td>
<td class="gt_row gt_center">993 (85%)</td>
<td class="gt_row gt_center">367 (91%)</td></tr>
    <tr><td class="gt_row gt_left" style="text-align: left; text-indent: 10px;">1</td>
<td class="gt_row gt_center">206 (13%)</td>
<td class="gt_row gt_center">170 (15%)</td>
<td class="gt_row gt_center">36 (8.9%)</td></tr>
    <tr><td class="gt_row gt_left">0: MALE 1: FEMALE</td>
<td class="gt_row gt_center"></td>
<td class="gt_row gt_center"></td>
<td class="gt_row gt_center"></td></tr>
    <tr><td class="gt_row gt_left" style="text-align: left; text-indent: 10px;">0</td>
<td class="gt_row gt_center">762 (49%)</td>
<td class="gt_row gt_center">542 (47%)</td>
<td class="gt_row gt_center">220 (55%)</td></tr>
    <tr><td class="gt_row gt_left" style="text-align: left; text-indent: 10px;">1</td>
<td class="gt_row gt_center">804 (51%)</td>
<td class="gt_row gt_center">621 (53%)</td>
<td class="gt_row gt_center">183 (45%)</td></tr>
  </tbody>
  
  <tfoot class="gt_footnotes">
    <tr>
      <td class="gt_footnote" colspan="4"><sup class="gt_footnote_marks">1</sup> Median (IQR); n (%)</td>
    </tr>
  </tfoot>
</table>
</div>
]
]
---

# Table 1!

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<table class="gt_table">
  
  <thead class="gt_col_headings">
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      <th class="gt_col_heading gt_columns_bottom_border gt_left" rowspan="1" colspan="1"><strong>Characteristic</strong></th>
      <th class="gt_col_heading gt_columns_bottom_border gt_center" rowspan="1" colspan="1"><strong>Overall</strong>, N = 1,566<sup class="gt_footnote_marks">1</sup></th>
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    </tr>
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  <tbody class="gt_table_body">
    <tr><td class="gt_row gt_left">WEIGHT IN KILOGRAMS  IN 1971</td>
<td class="gt_row gt_center">69 (60, 80)</td>
<td class="gt_row gt_center">68 (59, 79)</td>
<td class="gt_row gt_center">71 (61, 81)</td></tr>
    <tr><td class="gt_row gt_left">AGE IN 1971</td>
<td class="gt_row gt_center">43 (33, 53)</td>
<td class="gt_row gt_center">42 (33, 51)</td>
<td class="gt_row gt_center">46 (35, 56)</td></tr>
    <tr><td class="gt_row gt_left">MARITAL STATUS IN 1971  1: Under 17, 2: Married, 3: Widowed, 4: Never married, 5: Divorced, 6: Separated, 8: Unknown</td>
<td class="gt_row gt_center"></td>
<td class="gt_row gt_center"></td>
<td class="gt_row gt_center"></td></tr>
    <tr><td class="gt_row gt_left" style="text-align: left; text-indent: 10px;">2</td>
<td class="gt_row gt_center">1,237 (79%)</td>
<td class="gt_row gt_center">906 (78%)</td>
<td class="gt_row gt_center">331 (82%)</td></tr>
    <tr><td class="gt_row gt_left" style="text-align: left; text-indent: 10px;">3</td>
<td class="gt_row gt_center">88 (5.6%)</td>
<td class="gt_row gt_center">64 (5.5%)</td>
<td class="gt_row gt_center">24 (6.0%)</td></tr>
    <tr><td class="gt_row gt_left" style="text-align: left; text-indent: 10px;">4</td>
<td class="gt_row gt_center">93 (5.9%)</td>
<td class="gt_row gt_center">74 (6.4%)</td>
<td class="gt_row gt_center">19 (4.7%)</td></tr>
    <tr><td class="gt_row gt_left" style="text-align: left; text-indent: 10px;">5</td>
<td class="gt_row gt_center">92 (5.9%)</td>
<td class="gt_row gt_center">74 (6.4%)</td>
<td class="gt_row gt_center">18 (4.5%)</td></tr>
    <tr><td class="gt_row gt_left" style="text-align: left; text-indent: 10px;">6</td>
<td class="gt_row gt_center">55 (3.5%)</td>
<td class="gt_row gt_center">45 (3.9%)</td>
<td class="gt_row gt_center">10 (2.5%)</td></tr>
    <tr><td class="gt_row gt_left" style="text-align: left; text-indent: 10px;">8</td>
<td class="gt_row gt_center">1 (<0.1%)</td>
<td class="gt_row gt_center">0 (0%)</td>
<td class="gt_row gt_center">1 (0.2%)</td></tr>
    <tr><td class="gt_row gt_left">0: WHITE 1: BLACK OR OTHER IN 1971</td>
<td class="gt_row gt_center"></td>
<td class="gt_row gt_center"></td>
<td class="gt_row gt_center"></td></tr>
    <tr><td class="gt_row gt_left" style="text-align: left; text-indent: 10px;">0</td>
<td class="gt_row gt_center">1,360 (87%)</td>
<td class="gt_row gt_center">993 (85%)</td>
<td class="gt_row gt_center">367 (91%)</td></tr>
    <tr><td class="gt_row gt_left" style="text-align: left; text-indent: 10px;">1</td>
<td class="gt_row gt_center">206 (13%)</td>
<td class="gt_row gt_center">170 (15%)</td>
<td class="gt_row gt_center">36 (8.9%)</td></tr>
    <tr><td class="gt_row gt_left">0: MALE 1: FEMALE</td>
<td class="gt_row gt_center"></td>
<td class="gt_row gt_center"></td>
<td class="gt_row gt_center"></td></tr>
    <tr><td class="gt_row gt_left" style="text-align: left; text-indent: 10px;">0</td>
<td class="gt_row gt_center">762 (49%)</td>
<td class="gt_row gt_center">542 (47%)</td>
<td class="gt_row gt_center">220 (55%)</td></tr>
    <tr><td class="gt_row gt_left" style="text-align: left; text-indent: 10px;">1</td>
<td class="gt_row gt_center">804 (51%)</td>
<td class="gt_row gt_center">621 (53%)</td>
<td class="gt_row gt_center">183 (45%)</td></tr>
  </tbody>
  
  <tfoot class="gt_footnotes">
    <tr>
      <td class="gt_footnote" colspan="4"><sup class="gt_footnote_marks">1</sup> Median (IQR); n (%)</td>
    </tr>
  </tfoot>
</table>
</div>

---

.box-1[Create Table 1 split by `qsmk`, including an `overall` column and using all variables in your propensity score model]

---

# Weighted Table 1

```r
library(survey)
svy_des <- svydesign(
  ids = ~ 1,
  data = nhefs_complete_uc,
  weights = ~ w_ate
)
```

---

# Weighted Table 1

```r
*library(survey)
svy_des <- svydesign(
  ids = ~ 1,
  data = nhefs_complete_uc,
  weights = ~ w_ate
)
```

---

# Weighted Table 1

```r
library(survey)
*svy_des <- svydesign(
  ids = ~ 1,
  data = nhefs_complete_uc,
  weights = ~ w_ate
)
```

---

# Weighted Table 1

```r
library(survey)
svy_des <- svydesign(
* ids = ~ 1,
  data = nhefs_complete_uc,
  weights = ~ w_ate
)
```

---

# Weighted Table 1

```r
library(survey)
svy_des <- svydesign(
  ids = ~ 1,
* data = nhefs_complete_uc,
  weights = ~ w_ate
)
```

---

# Weighted Table 1

```r
library(survey)
svy_des <- svydesign(
  ids = ~ 1,
  data = nhefs_complete_uc,
* weights = ~ w_ate
)
```

---

# Weighted Table 1

```r
svy_des %>%
  tbl_svysummary(by = qsmk,
              include = c("wt71",
                          "age",
                          "marital",
                          "race",
                          "sex")) %>%
  add_overall()
```

---

# Weighted Table 1

```r
*svy_des %>%
  tbl_svysummary(by = qsmk, 
              include = c("wt71",
                          "age",
                          "marital",
                          "race",
                          "sex")) %>%
  add_overall()
```
---

# Weighted Table 1

```r
svy_des %>%
* tbl_svysummary(by = qsmk,
              include = c("wt71",
                          "age",
                          "marital",
                          "race",
                          "sex")) %>%
  add_overall()
```

---

# Weighted Table 1

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</style>
<table class="gt_table">
  
  <thead class="gt_col_headings">
    <tr>
      <th class="gt_col_heading gt_columns_bottom_border gt_left" rowspan="1" colspan="1"><strong>Characteristic</strong></th>
      <th class="gt_col_heading gt_columns_bottom_border gt_center" rowspan="1" colspan="1"><strong>Overall</strong>, N = 3,138<sup class="gt_footnote_marks">1</sup></th>
      <th class="gt_col_heading gt_columns_bottom_border gt_center" rowspan="1" colspan="1"><strong>0</strong>, N = 1,565<sup class="gt_footnote_marks">1</sup></th>
      <th class="gt_col_heading gt_columns_bottom_border gt_center" rowspan="1" colspan="1"><strong>1</strong>, N = 1,573<sup class="gt_footnote_marks">1</sup></th>
    </tr>
  </thead>
  <tbody class="gt_table_body">
    <tr><td class="gt_row gt_left">WEIGHT IN KILOGRAMS  IN 1971</td>
<td class="gt_row gt_center">69 (59, 79)</td>
<td class="gt_row gt_center">69 (60, 80)</td>
<td class="gt_row gt_center">69 (59, 79)</td></tr>
    <tr><td class="gt_row gt_left">AGE IN 1971</td>
<td class="gt_row gt_center">43 (33, 53)</td>
<td class="gt_row gt_center">43 (33, 52)</td>
<td class="gt_row gt_center">43 (33, 53)</td></tr>
    <tr><td class="gt_row gt_left">MARITAL STATUS IN 1971  1: Under 17, 2: Married, 3: Widowed, 4: Never married, 5: Divorced, 6: Separated, 8: Unknown</td>
<td class="gt_row gt_center"></td>
<td class="gt_row gt_center"></td>
<td class="gt_row gt_center"></td></tr>
    <tr><td class="gt_row gt_left" style="text-align: left; text-indent: 10px;">2</td>
<td class="gt_row gt_center">2,477 (79%)</td>
<td class="gt_row gt_center">1,233 (79%)</td>
<td class="gt_row gt_center">1,244 (79%)</td></tr>
    <tr><td class="gt_row gt_left" style="text-align: left; text-indent: 10px;">3</td>
<td class="gt_row gt_center">167 (5.3%)</td>
<td class="gt_row gt_center">89 (5.7%)</td>
<td class="gt_row gt_center">78 (5.0%)</td></tr>
    <tr><td class="gt_row gt_left" style="text-align: left; text-indent: 10px;">4</td>
<td class="gt_row gt_center">181 (5.8%)</td>
<td class="gt_row gt_center">94 (6.0%)</td>
<td class="gt_row gt_center">87 (5.5%)</td></tr>
    <tr><td class="gt_row gt_left" style="text-align: left; text-indent: 10px;">5</td>
<td class="gt_row gt_center">183 (5.8%)</td>
<td class="gt_row gt_center">94 (6.0%)</td>
<td class="gt_row gt_center">89 (5.7%)</td></tr>
    <tr><td class="gt_row gt_left" style="text-align: left; text-indent: 10px;">6</td>
<td class="gt_row gt_center">121 (3.9%)</td>
<td class="gt_row gt_center">55 (3.5%)</td>
<td class="gt_row gt_center">66 (4.2%)</td></tr>
    <tr><td class="gt_row gt_left" style="text-align: left; text-indent: 10px;">8</td>
<td class="gt_row gt_center">8 (0.3%)</td>
<td class="gt_row gt_center">0 (0%)</td>
<td class="gt_row gt_center">8 (0.5%)</td></tr>
    <tr><td class="gt_row gt_left">0: WHITE 1: BLACK OR OTHER IN 1971</td>
<td class="gt_row gt_center"></td>
<td class="gt_row gt_center"></td>
<td class="gt_row gt_center"></td></tr>
    <tr><td class="gt_row gt_left" style="text-align: left; text-indent: 10px;">0</td>
<td class="gt_row gt_center">2,720 (87%)</td>
<td class="gt_row gt_center">1,359 (87%)</td>
<td class="gt_row gt_center">1,361 (87%)</td></tr>
    <tr><td class="gt_row gt_left" style="text-align: left; text-indent: 10px;">1</td>
<td class="gt_row gt_center">418 (13%)</td>
<td class="gt_row gt_center">206 (13%)</td>
<td class="gt_row gt_center">212 (13%)</td></tr>
    <tr><td class="gt_row gt_left">0: MALE 1: FEMALE</td>
<td class="gt_row gt_center"></td>
<td class="gt_row gt_center"></td>
<td class="gt_row gt_center"></td></tr>
    <tr><td class="gt_row gt_left" style="text-align: left; text-indent: 10px;">0</td>
<td class="gt_row gt_center">1,516 (48%)</td>
<td class="gt_row gt_center">761 (49%)</td>
<td class="gt_row gt_center">755 (48%)</td></tr>
    <tr><td class="gt_row gt_left" style="text-align: left; text-indent: 10px;">1</td>
<td class="gt_row gt_center">1,622 (52%)</td>
<td class="gt_row gt_center">804 (51%)</td>
<td class="gt_row gt_center">818 (52%)</td></tr>
  </tbody>
  
  <tfoot class="gt_footnotes">
    <tr>
      <td class="gt_footnote" colspan="4"><sup class="gt_footnote_marks">1</sup> Median (IQR); n (%)</td>
    </tr>
  </tfoot>
</table>
</div>

---

.box-1[Create weighted Table 1. Compare this to your Table 1 from the previous exercise - which column does it resemble the most?]

---

]

<br>

`$$w_{ATT} = \frac{e_i X_i}{e_i} + \frac{e_i(1-X_i)}{1-e_i}$$`
]

--
<br>
.box-inv-1[What is this when X=1?]
---
class: title title-1

# ATT

---

.box-1[Create weighted Table 1. Compare this to your unweighted Table 1 - which column does it resemble the most?]

---
class: title title-1
.small[
# Average treatment effect among the controls
]

--
<br>

`$$w_{ATC} = \frac{(1 - e_i) X_i}{e_i} + \frac{(1 - e_i)(1-X_i)}{1-e_i}$$`
]

<br>
.box-inv-1[What is this when X=0?]

---

# ATC

---

.box-1[Create weighted Table 1. Compare this to your unweighted Table 1 - which column does it resemble the most?]

---
class: title title-1

--
<br>

`$$w_{ATM} = \frac{\textrm{min}\{e_i, 1-e_i\}}{X_i e_i + (1 - X_i)(1 - e_i)}$$`
]

---

```r
nhefs_complete_uc <- nhefs_complete_uc %>%
  mutate( 
*   w_atm = pmin(.fitted, 1 - .fitted) /
*     (qsmk * .fitted + (1 - qsmk) * (1 - .fitted))
  )
```

---

# ATM

---

.box-1[Create weighted Table 1. Compare this to your unweighted Table 1 - which column does it resemble the most (if any)?]

---
class: title title-1

--
<br>
.box-inv-1[
When the conditional probability of receiving the exposure is small, what does this approximate?
]

--
<br>
.box-1[

`$$e_i\approx 0$$`

`$w_{ATT}$`
]

---

--
<br>
.box-1[

`$$e_i\approx 1$$`

`$w_{ATC}$`
]

---
class: title title-1

# ATO

---

.box-1[Create weighted Table 1. Compare this to your unweighted Table 1 - which column does it resemble the most (if any)?]