Propensity Score Weighting

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# Propensity Score Weighting
### Lucy D’Agostino McGowan
### Wake Forest University
### 2020-07-29 (updated: 2020-07-29)

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## Propensity scores

* Weighting
* Matching
* Stratification
* Direct Adjustment
* ...

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## Propensity scores

* **Weighting**
* Matching
* Stratification
* Direct Adjustment
* ...

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## Target estimands

### Average Treatment Effect (ATE)

`$$\Large w_{ATE} = \frac{Z_i}{p_i} + \frac{1-Z_i}{1 - p_i}$$`

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## Target estimands

### Average Treatment Effect Among the Treated (ATT)
`$$\Large w_{ATT} = \frac{p_i Z_i}{p_i} + \frac{p_i (1-Z_i)}{1-p_i}$$`
--

### Average Treatment Effect Among the Controls (ATC)
`$$\Large w_{ATC} = \frac{(1-p_i)Z_i}{p_i} + \frac{(1-p_i)(1-Z_i)}{(1-p_i)}$$`

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## Target estimands

### Average Treatment Effect Among the Evenly Matchable (ATM)
`$$\Large w_{ATM} = \frac{\min \{p_i, 1-p_i\}}{z_ip_i + (1-Z_i)(1-p_i)}$$`
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### Average Treatment Effect Among the Overlap Population
`$$\Large w_{ATO} = (1-p_i)Z_i + p_i(1-Z_i)$$`

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## Histogram of propensity scores

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## ATE

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## ATT

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## ATC

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## ATM

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## ATO

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## ATE in R

* Average Treatment Effect (ATE)
  * `$w_{ATE} = \frac{Z_i}{p_i} + \frac{1-Z_i}{1 - p_i}$`

```r
df <- propensity_model %>% 
  augment(type.predict = "response", data = nhefs_complete) %>% 
* mutate(w_ate = (qsmk / .fitted) +  ((1 - qsmk) / (1 - .fitted)))
```

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## Your Turn

1. Using the propensity scores you created in the previous exercise, add the ATE weights to your data frame `df`

2. Stretch: Using the same propensity scores, create ATT weights