Method: Large-N Approach

# Method: Large-N Approach
## Public Policy Analysis: Perspectives and Methods (30700953)
### Yue HU

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## Overview

- What's large-N
- Why large-N
- How large-N

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# What's Large-N

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## Large-N: A Terminology

In lieu of "quantitative"...

???

What's a 定量研究?

Did you really 定量？

N: Number of .red[observations]---not subjects, not trails
 
 Large: Well......

> We don't know......   
> It depends......  
> The larger, the better!

1. Why large-N?
1. How large is large?
]
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# Why Large-N

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## When Large-N?

Experiment is not an option!

- Practical reasons
- Ethical reasons
- Research goal

???

Research goal: Have a requirement of representativeness

&therefore; Analysis based on .red[observed data]

*Goal*:

- **.red[Large] number of observations**
- .red[Sufficient] variance in the observations
- .red[Strong] assumption

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## How Large Is Large

Head or Tail?

<img src="images/large_toss.gif" height = 350 />]

How about 1000 times?

???

https://rpubs.com/pgrosse/545948

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*Why large-N?*

Answer 1: To unearth the probability!

A.k.a.,

.bg-black.golden.ba.shadow-5.ph4.mt3[
.center[Law of Large Numbers (LLN)]
]

`$${\displaystyle \lim _{n\to \infty }\sum _{i=1}^{n}{\frac {X_{i}}{n}}={\overline {X}}}$$`

As the number<sup>*</sup> of experiments (sample) .red[increases], the ratio of outcomes will .blue[converge to] the theoretical (population) average.

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# How to Conduct Large Analysis
## Why large-N hasn't been fully answered yet

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## Large-N Approaches

**Univariate analysis**

- Average
- Deviation
- Other moments

**Bivariate analysis**

- Cross-tabulating
- Covariate
- Difference in means

- Correlation analysis
- Spatial analysis
- Time series
- Causal inference
]

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## Data Whisper

.pull-left[
Surveying seven dorm rooms asking the members did they watch the show.
The right table is the records:
]

.pull-right[
<table class="table" style="font-size: 25px; margin-left: auto; margin-right: auto;">
 <thead>
  <tr>
   <th style="text-align:center;"> Student_ID </th>
   <th style="text-align:left;"> Episodes </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:center;"> 2020996642 </td>
   <td style="text-align:left;"> 1 </td>
  </tr>
  <tr>
   <td style="text-align:center;"> 2020991501 </td>
   <td style="text-align:left;"> 1 </td>
  </tr>
  <tr>
   <td style="text-align:center;"> 2020997812 </td>
   <td style="text-align:left;"> 1 </td>
  </tr>
  <tr>
   <td style="text-align:center;"> 2020992534 </td>
   <td style="text-align:left;"> 2 </td>
  </tr>
  <tr>
   <td style="text-align:center;"> 2020998108 </td>
   <td style="text-align:left;"> 3 </td>
  </tr>
  <tr>
   <td style="text-align:center;"> 2020991166 </td>
   <td style="text-align:left;"> 3 </td>
  </tr>
  <tr>
   <td style="text-align:center;"> 2020993630 </td>
   <td style="text-align:left;"> 4 </td>
  </tr>
</tbody>
</table>
]

???

Let's say we'll test a school policy of Internet access management. 
To do so, researchers surveyed a dorm room, asking the members how many shows they watched in the past week.

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## Univariate Analysis

+ Given the list: (1, 1, 1, 2, 3, 3, 4)

+ Mean: `$\frac{1 + 1 + 1 + 2 + 3 + 3 + 4}{7} = \frac{15}{7} \approx 2.143.$`

+ Median: 1, 1, 1, .magenta[2], 3, 3, 4

+ Mode: three .magenta[1]s, one 2, two 3s, and one 4.

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## Relations among These Ways

Let's expand the data to 1000 dorms.

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## Mean, Median, or Mode?

.pull-left[
<img src="10_largeN_files/figure-html/distribution-1.png" style="display: block; margin: auto;" />
]

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background-image: url("images/large_game.gif")
background-position: center
background-size: contain

???

1. Students standing in a line
1. Toss the coins six times
    - Head: left
    - Tail: right
1. Going down to the bottom line
1. Take a picture

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.pull-left[
<img src="10_largeN_files/figure-html/draws-.gif" style="display: block; margin: auto;" />
]

*Why large-N?*

A.k.a.,

.bg-black.golden.ba.shadow-5.ph4.mt3[
.center[Central Limit Theorem (CLT)]
]

When .red[**independent**(!!!)] random variables are summed up, their properly normalized sum tends toward a .blue[normal distribution], even if the original variables themselves are not normally distributed.
]

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## What Can A Large-N Sample Do?

Show

- The average/general status of the sample
- How individual observations deviate from the average

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Beyond single variables......

## Binary Analysis

Relations between variables:

<div class="container st-container">
<table class="table table-bordered st-table st-table-bordered st-cross-table ">
<thead>
<tr>
<th></th>
<th colspan="2" align="center" class="st-protect-top-border">smoker</th>
</tr>
<tr>
<td align="center">
<strong>gender</strong>
</td>
<th colspan="1" align="center">Yes</th>
<th colspan="1" align="center">No</th>
</tr>
</thead>
<tbody>
<tr>
<td>
<strong align="center">F</strong>
</td>
<td>
<span>147</span>
</td>
<td>
<span>342</span>
</td>
</tr>
<tr>
<td>
<strong align="center">M</strong>
</td>
<td>
<span>143</span>
</td>
<td>
<span>346</span>
</td>
</tr>
<tr>
<td>
<strong align="center"><NA></strong>
</td>
<td>
<span>8</span>
</td>
<td>
<span>14</span>
</td>
</tr>
</tbody>
</table>
</div>

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## Scatter plot

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## Controls

What do controls do?

Bring the Experimental Logic Back

???

How to assign the groups?

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## Randomization = A Control

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## What If You Cannot Randomize?

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## Controls May Not Work

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background-image: url("images/lar_mindmap.png")
background-position: center
background-size: contain

## Take-Home Points

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## Procedure

The presenters talk.

### Challenging (10min)

Drawing the Luck Dog

]

*Process*

1. Presenters query.
    - A question about the paper
1. Challengers respond.
1. Challengers query.
    - A question about the presentation
1. Presenters respond.
]

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background-image: url("images/cha_challenge.gif")
background-position: center
background-size: contain