POL90: Statistics

# POL90: Statistics
## <span class="math inline"><em>t</em></span>-tests Continued
### Prof Wasow<br/>Assistant Professor, Politics<br/>Pomona College
### 2021-02-15

---

# Announcements

+ PS03 due <mark>Friday, 2/11</mark>
    
    + Report 1

]

---
# 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;"> 2 </td>
   <td style="text-align:left;"> Jan 24 </td>
   <td style="text-align:left;"> Mon </td>
   <td style="text-align:left;"> Drawing Statistical Conclusions </td>
   <td style="text-align:right;"> 1 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 2 </td>
   <td style="text-align:left;"> Jan 26 </td>
   <td style="text-align:left;"> Wed </td>
   <td style="text-align:left;"> Drawing Statistical Conclusions </td>
   <td style="text-align:right;"> 1 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 3 </td>
   <td style="text-align:left;"> Jan 31 </td>
   <td style="text-align:left;"> Mon </td>
   <td style="text-align:left;"> Inference Using t-Distributions </td>
   <td style="text-align:right;"> 2 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 3 </td>
   <td style="text-align:left;"> Feb 2 </td>
   <td style="text-align:left;"> Wed </td>
   <td style="text-align:left;"> Inference Using t-Distributions </td>
   <td style="text-align:right;"> 2 </td>
  </tr>
  <tr>
   <td style="text-align:right;color: black !important;background-color: yellow !important;"> 4 </td>
   <td style="text-align:left;color: black !important;background-color: yellow !important;"> Feb 7 </td>
   <td style="text-align:left;color: black !important;background-color: yellow !important;"> Mon </td>
   <td style="text-align:left;color: black !important;background-color: yellow !important;"> A Closer Look at Assumptions </td>
   <td style="text-align:right;color: black !important;background-color: yellow !important;"> 3 </td>
  </tr>
  <tr>
   <td style="text-align:right;color: black !important;background-color: yellow !important;"> 4 </td>
   <td style="text-align:left;color: black !important;background-color: yellow !important;"> Feb 9 </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;"> A Closer Look at Assumptions </td>
   <td style="text-align:right;color: black !important;background-color: yellow !important;"> 3 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 5 </td>
   <td style="text-align:left;"> Feb 14 </td>
   <td style="text-align:left;"> Mon </td>
   <td style="text-align:left;"> Alternatives to the t-Tools </td>
   <td style="text-align:right;"> 4 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 5 </td>
   <td style="text-align:left;"> Feb 16 </td>
   <td style="text-align:left;"> Wed </td>
   <td style="text-align:left;"> Alternatives to the t-Tools </td>
   <td style="text-align:right;"> 4 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 6 </td>
   <td style="text-align:left;"> Feb 21 </td>
   <td style="text-align:left;"> Mon </td>
   <td style="text-align:left;"> Comparison Among Several Samples </td>
   <td style="text-align:right;"> 5 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 6 </td>
   <td style="text-align:left;"> Feb 23 </td>
   <td style="text-align:left;"> Wed </td>
   <td style="text-align:left;"> Comparison Among Several Samples </td>
   <td style="text-align:right;"> 5 </td>
  </tr>
</tbody>
</table>

---
## Assignment 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;"> Assignment </th>
   <th style="text-align:right;"> Percent </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:right;"> 3 </td>
   <td style="text-align:left;"> Feb 4 </td>
   <td style="text-align:left;"> Fri </td>
   <td style="text-align:left;"> PS02 </td>
   <td style="text-align:right;"> 3 </td>
  </tr>
  <tr>
   <td style="text-align:right;color: black !important;background-color: yellow !important;"> 4 </td>
   <td style="text-align:left;color: black !important;background-color: yellow !important;"> Feb 11 </td>
   <td style="text-align:left;color: black !important;background-color: yellow !important;"> Fri </td>
   <td style="text-align:left;color: black !important;background-color: yellow !important;"> PS03 </td>
   <td style="text-align:right;color: black !important;background-color: yellow !important;"> 3 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 5 </td>
   <td style="text-align:left;"> Feb 18 </td>
   <td style="text-align:left;"> Fri </td>
   <td style="text-align:left;"> PS04 </td>
   <td style="text-align:right;"> 3 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 6 </td>
   <td style="text-align:left;"> Feb 25 </td>
   <td style="text-align:left;"> Fri </td>
   <td style="text-align:left;"> PS05 </td>
   <td style="text-align:right;"> 3 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 7 </td>
   <td style="text-align:left;"> Mar 4 </td>
   <td style="text-align:left;"> Fri </td>
   <td style="text-align:left;"> Report1 </td>
   <td style="text-align:right;"> 6 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 8 </td>
   <td style="text-align:left;"> Mar 11 </td>
   <td style="text-align:left;"> Fri </td>
   <td style="text-align:left;"> PS06 </td>
   <td style="text-align:right;"> 3 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 9 </td>
   <td style="text-align:left;"> Mar 18 </td>
   <td style="text-align:left;"> Fri </td>
   <td style="text-align:left;"> Spring break </td>
   <td style="text-align:right;"> NA </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 10 </td>
   <td style="text-align:left;"> Mar 25 </td>
   <td style="text-align:left;"> Fri </td>
   <td style="text-align:left;"> PS07 </td>
   <td style="text-align:right;"> 3 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 11 </td>
   <td style="text-align:left;"> Apr 1 </td>
   <td style="text-align:left;"> Fri </td>
   <td style="text-align:left;"> PS08 </td>
   <td style="text-align:right;"> 3 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 12 </td>
   <td style="text-align:left;"> Apr 8 </td>
   <td style="text-align:left;"> Fri </td>
   <td style="text-align:left;"> Report2 </td>
   <td style="text-align:right;"> 8 </td>
  </tr>
</tbody>
</table>

---

## Coding tips

- Based on ED & other questions

- Only `install.packages()` about once per year
  - `install.packages()` is like installing an app, done rarely
  - `library()` is like opening an app, done every time used

- In addition to knitting Rmd, run chunks interactively, confirm each step works
  
]

---

# Report 1

---
class: center
## Report 1: Test a theory, elites vs masses

.pull-left[<img src="images/Zaller_James_big_square.jpg" alt="drawing" style="width:200px;"/><img src="images/lenz_gabriel2.jpg" alt="drawing" style="width:200px;"/>]

.pull-right[<img src="images/lee_taeku_big.jpg" alt="drawing" style="width:200px;"/><img src="images/erica-chenoweth-maria-stephan-nsc-briefing.jpg" alt="drawing" style="width:200px;"/><img src="images/Daniel-Gillion3.jpg" alt="drawing" style="width:200px;"/>]

---
class: center
## Report 1:  Engage with texts, not people

<img src="images/lenz_book_cover.jpg" alt="drawing" style="width:150px;"/>]

<img src="images/gillion_book_cover.jpg" alt="drawing" style="width:100px;"/>]

---
## Pew: US opinion on same-sex marriage, 2001-2017

---

## Report 1: Goals

- A contest or horserace of theories
    - "Three cornered fight" or a court proceeding
]

.large[
* Use data as rhetoric
    - Report a statistical test
    - Summarize data in a table
    - Convey trends and relationships with visualization
]
--

.large[
* Produce replicable research
    - See how "literate programming" like R + R Markdown + Latex + knitr contributes to replication
    - Practice good programming and statistical style
]

---
## Report Questions & Suggestions

- Plan to use iPoll
  - Register on iPoll it makes downloading many polls much easier

- You will need to do some data cleaning, that's part of the assignment
  - See Data Scrubbing handout: http://appliedstats.org/data_scrub_handout.html

- Collaboration may be easier with RStudio.cloud. See link on Canvas & handout: http://appliedstats.org/rstudio_cloud_guide.html

]

---
class: center, middle, inverse

# Simulation vs Analytic methods

---
## Downey: Simplifying hypothesis testing

1. A test statistic,

2. A model of a null hypothesis, and usually,

3. A method that computes or approximates the *p*-value.
]
]
---
## Calculate Test Statistic

- First calculate a test statistic 
<br><br>

---
## Simulation methods

- With randomization, compare test statistic to simulated null distribution 
<br><br>

.footnote[Source: http://allendowney.blogspot.com/2016/06/there-is-still-only-one-test.html]
  
---
## Analytical methods, example *t*-test

- With analytical methods, we transform the test statistic (such as by dividing by the standard error), and then compare to a theoretical distribution

---
## With Randomization, No Transformations of Estimate

```r
library(infer)

creativity_null_distribution <- creativity %>%
  
  infer::specify(score ~ treatment) %>%
  
  infer::hypothesize(null = "independence") %>%
  
  infer::generate(reps = 1000, type = "permute") %>%
  
  infer::calculate(stat  = "diff in means",              
                   order = c("Intrinsic", "Extrinsic"))

head(creativity_null_distribution)
```

```
Response: score (numeric)
Explanatory: treatment (factor)
Null Hypothesis: independence
# A tibble: 6 × 2
  replicate   stat
      <int>  <dbl>
1         1  2.67 
2         2 -0.973
3         3  0.892
4         4  0.364
5         5 -0.471
6         6  1.02 
```

---
## Estimate on Same Scale as Null Distribution

```r
library(ggplot2)
infer::visualize(creativity_null_distribution) +
  geom_vline(xintercept = c(-4.14, 4.14), col = "red")
```

---
## With `$t$`-test, Estimate Must be Transformed
.vertical-center[
$$
t\text{-ratio} = \dfrac{\text{Estimate}-\text{Parameter}}{\text{SE(Estimate)}}
$$

$$
t-\text{ratio}(\text{if } \mu \text{ is zero}) = \dfrac{0.199-0}{0.0615} = 3.236
$$
]

---
## Visualizing *t*-Ratio on *t*-distribution

```r
visualize::visualize.t(stat = c(-3.23, 3.23), df = 14, section = "tails")
```

---

## Data as Storytelling

---
<br/><br/><br/>

---
.center[![](images/hans_rosling.jpg)]

.footnote[Source: https://www.ted.com/talks/hans_rosling_shows_the_best_stats_you_ve_ever_seen]
---
class: middle, center
background-color: #000000

---
class: center, middle

# Questions?