POL90: Statistics

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# POL90: Statistics
## Matching
### Prof. Wasow, PoliticsPomona College
### 2022-04-20

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# Announcements

.large[

- Assignments
  - PS10 due Tuesday
  - Report 3 teams assigned

- Reading:
  - Skim: Elizabeth Stuart & Donald Rubin (2007), "Matching methods for causal inference: Designing observational studies"

]

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# Schedule

<table>
 <thead>
 <tr>
 <th style="text-align:right;"> Week </th>
 <th style="text-align:left;"> Date </th>
 <th style="text-align:left;"> Day </th>
 <th style="text-align:left;"> Title </th>
 <th style="text-align:right;"> Chapter </th>
 </tr>
 </thead>
<tbody>
 <tr>
 <td style="text-align:right;"> 11 </td>
 <td style="text-align:left;"> Mar 30 </td>
 <td style="text-align:left;"> Wed </td>
 <td style="text-align:left;"> Interaction terms </td>
 <td style="text-align:right;"> 9 </td>
 </tr>
 <tr>
 <td style="text-align:right;"> 12 </td>
 <td style="text-align:left;"> Apr 4 </td>
 <td style="text-align:left;"> Mon </td>
 <td style="text-align:left;"> Logistic regression </td>
 <td style="text-align:right;"> 20 </td>
 </tr>
 <tr>
 <td style="text-align:right;"> 12 </td>
 <td style="text-align:left;"> Apr 6 </td>
 <td style="text-align:left;"> Wed </td>
 <td style="text-align:left;"> Logistic regression </td>
 <td style="text-align:right;"> 20 </td>
 </tr>
 <tr>
 <td style="text-align:right;"> 13 </td>
 <td style="text-align:left;"> Apr 11 </td>
 <td style="text-align:left;"> Mon </td>
 <td style="text-align:left;"> Missing Data </td>
 <td style="text-align:right;"> Handout </td>
 </tr>
 <tr>
 <td style="text-align:right;"> 13 </td>
 <td style="text-align:left;"> Apr 13 </td>
 <td style="text-align:left;"> Wed </td>
 <td style="text-align:left;"> Panel Data </td>
 <td style="text-align:right;"> Handout </td>
 </tr>
 <tr>
 <td style="text-align:right;"> 14 </td>
 <td style="text-align:left;"> Apr 18 </td>
 <td style="text-align:left;"> Mon </td>
 <td style="text-align:left;"> Matching </td>
 <td style="text-align:right;"> Handout </td>
 </tr>
 <tr>
 <td style="text-align:right;color: black !important;background-color: yellow !important;"> 14 </td>
 <td style="text-align:left;color: black !important;background-color: yellow !important;"> Apr 20 </td>
 <td style="text-align:left;color: black !important;background-color: yellow !important;"> Wed </td>
 <td style="text-align:left;color: black !important;background-color: yellow !important;"> Matching </td>
 <td style="text-align:right;color: black !important;background-color: yellow !important;"> Handout </td>
 </tr>
 <tr>
 <td style="text-align:right;"> 15 </td>
 <td style="text-align:left;"> Apr 25 </td>
 <td style="text-align:left;"> Mon </td>
 <td style="text-align:left;"> Causal inference: Panel data </td>
 <td style="text-align:right;"> Handout </td>
 </tr>
 <tr>
 <td style="text-align:right;"> 15 </td>
 <td style="text-align:left;"> Apr 27 </td>
 <td style="text-align:left;"> Wed </td>
 <td style="text-align:left;"> Causal inference: Natural Experiments </td>
 <td style="text-align:right;"> Dunning </td>
 </tr>
 <tr>
 <td style="text-align:right;"> 16 </td>
 <td style="text-align:left;"> May 2 </td>
 <td style="text-align:left;"> Mon </td>
 <td style="text-align:left;"> Causal inference: RDD </td>
 <td style="text-align:right;"> NA </td>
 </tr>
 <tr>
 <td style="text-align:right;"> 16 </td>
 <td style="text-align:left;"> May 4 </td>
 <td style="text-align:left;"> Wed </td>
 <td style="text-align:left;"> No class </td>
 <td style="text-align:right;"> NA </td>
 </tr>
</tbody>
</table>

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## Assignment schedule

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# Revisiting: Homicide and Ice Cream

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## Revisiting: Homicide and Ice Cream

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# Propensity Score Matching

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## 
<img src="images/pscore001.png" width="100%" style="display: block; margin: auto;" />

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## 
<img src="images/pscore002.png" width="100%" style="display: block; margin: auto;" />

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<img src="images/pscore003.png" width="100%" style="display: block; margin: auto;" />

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<img src="images/pscore004.png" width="100%" style="display: block; margin: auto;" />

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## 
<img src="images/pscore005.png" width="100%" style="display: block; margin: auto;" />

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## 
<img src="images/pscore006.png" width="100%" style="display: block; margin: auto;" />

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<img src="images/pscore007.png" width="100%" style="display: block; margin: auto;" />

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## 
<img src="images/pscore008.png" width="100%" style="display: block; margin: auto;" />

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## 
<img src="images/pscore020BestCase.png" width="100%" style="display: block; margin: auto;" />

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## 
<img src="images/pscore022BestCase.png" width="100%" style="display: block; margin: auto;" />

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# Mahalanobis Matching

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## 
<img src="images/mah001.png" width="100%" style="display: block; margin: auto;" />

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## 
<img src="images/mah002.png" width="100%" style="display: block; margin: auto;" />

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## 
<img src="images/mah003.png" width="100%" style="display: block; margin: auto;" />

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<img src="images/mah004.png" width="100%" style="display: block; margin: auto;" />

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## 
<img src="images/mah020BestCase.png" width="100%" style="display: block; margin: auto;" />

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## 
<img src="images/mah021BestCase.png" width="100%" style="display: block; margin: auto;" />

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# Other Matching Methods

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## 
<img src="images/diamond_sekhon_2013.png" width="100%" style="display: block; margin: auto;" />

.footnote[Source: https://www.mitpressjournals.org/doi/pdf/10.1162/REST_a_00318]

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.center[
<img src="images/stuart_matching_abstract.png" width="85%" style="display: block; margin: auto;" />
]

.footnote[Source: https://www.ncbi.nlm.nih.gov/pmc/articles/PMC2943670/]

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.center[
<img src="images/stuart_matching_subclass_full.png" width="85%" style="display: block; margin: auto;" />
]

.footnote[Source: https://www.ncbi.nlm.nih.gov/pmc/articles/PMC2943670/]

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## Birdkeeping + lung cancer data (modified)

```r
birds <- Sleuth3::case2002 %>% clean_names()

head(birds, 10)
```

```
           lc   fm   ss     bk ag yr cd
1  LungCancer Male  Low   Bird 37 19 12
2  LungCancer Male  Low   Bird 41 22 15
3  LungCancer Male High NoBird 43 19 15
4  LungCancer Male  Low   Bird 46 24 15
5  LungCancer Male  Low   Bird 49 31 20
6  LungCancer Male High NoBird 51 24 15
7  LungCancer Male High   Bird 52 31 20
8  LungCancer Male  Low NoBird 53 33 20
9  LungCancer Male  Low   Bird 56 33 10
10 LungCancer Male High NoBird 56 26 25
```

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## What if age is related to birdkeeping?

```r
birds %>% ggplot() + aes(x = ag, color = bk) + geom_density() + theme_bw()
```

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## What if we match via subclass?

```r
birds <- birds %>% 
 mutate(
 lc_bin = ifelse(lc == "LungCancer", 1, 0),
 bk_bin = ifelse(bk == "Bird", 1, 0)
 )

# find "nearest neighbors"
match_out <- MatchIt::matchit(
 formula = bk_bin ~ ag + yr + fm + ss, 
 data = birds, 
 method = "subclass"
 ) 
```

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## Look at Subclassifications

```r
plot(match_out, type = "jitter", interactive = FALSE)
```

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.center[
<img src="images/imai_ratkovic_2014.png" width="85%" style="display: block; margin: auto;" />
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.footnote[Source: https://imai.fas.harvard.edu/research/files/CBPS.pdf]

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## What are Weights?

.large[

- The process of using weights in matching is analogous to the kinds of weights that survey researchers use to try and accurately represent a population with a subsample. 
  
  - For example, imagine pollsters are trying to estimate approval for President Biden but young people answer their phones at about half the rate of older respondents. 
  
  - The pollster would know this by comparing the survey response rates to population-level data like the Census.

]

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## What are Weights?

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- So, for example, if 18 to 23 year olds are found to be 10 percent of the population in sources like the Census but are only one in twenty of our survey respondents, we might weight each 18 to 23 year old respondent as if they were two people. 
  
  - Conversely, if we oversampled Baby Boomers, we might weight them as if each respondent counted for "half" a person. Through weighting, we could make our sample representative of the larger population as measured by the Census. 
  
  - There's obviously a lot more that survey researchers do to ensure the representativeness of their samples but that's the basic idea. 
]

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## How do Weights Work in Matching?

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- With matching weights, we're doing something similar to survey weights. 
  
  - We're using weights to increase or decrease the influence of certain observations so that the CONTROL sample looks as similar as possible to our TREATED sample on a set of characteristics. 
  
  - In essence, we're often using our TREATED sample a bit like the Census as a reference and then re-weighting the CONTROL observations as if they were a sample that needed to be adjusted to be representative.

]

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## Matching Exercise

.large[
http://appliedstats.org/matching_exercise.html
]

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# Questions?