Sampling and Simulation

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<br><br>
<div style="align:center; background:black; color:white; font-size: 3em;font-weight: bold;">Sampling</div>

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# Outline

1. Sampling Nature

2. Describing a Sample

3. Using a Sample to Describe a Population

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# Today's quiz:

<br><br><br>
.center[.large[http://tinyurl.com/sampling-pre]]

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# Today's Etherpad
<br><br><br>.center[.large[https://etherpad.wikimedia.org/p/sampling-2020]]

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# What is a population?

**Population** = All Individuals

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# Population
.pull-left[
<img src="03_sampling_lecture_x_files/figure-html/population-1.png" style="display: block; margin: auto;" />
]

.pull-right[
<img src="03_sampling_lecture_x_files/figure-html/normplot-1.png" style="display: block; margin: auto;" />
]
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# What is a sample?

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A **sample** of individuals in a randomly distributed population.

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# Population
.pull-left[
<img src="03_sampling_lecture_x_files/figure-html/sample-1.png" style="display: block; margin: auto;" />
]

.pull-right[
<img src="03_sampling_lecture_x_files/figure-html/normplotsamp-1.png" style="display: block; margin: auto;" />
]

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# Properties of a good sample

1. Validity  
    - Yes, this is a measure of what I am interested in

2. Reliability  
      - If I sample again, I'll get something similar

3. Representative
      - Sample reflects the population
      - Unbiased

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# Validity: Is it measuring what I think it's measuring?

.center[![:scale 75%](./images/03/23andme_health_ancestry_kit.jpg)]

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# Reliability: Is my sample/measure repeatable?

.center[![](./images/03/painscale.jpg)]

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# Bias from Unequal Representation
<img src="03_sampling_lecture_x_files/figure-html/colorSize-1.png" style="display: block; margin: auto;" />
If you only chose one color, you would only get one range of sizes.

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# Unrepresentative Bias from Unequal Change of Sampling
<img src="03_sampling_lecture_x_files/figure-html/spatialBias-1.png" style="display: block; margin: auto;" />
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Spatial gradient in size

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# Unrepresentative bias from Unequal Choice of Sampling
<img src="03_sampling_lecture_x_files/figure-html/spatialSample-1.png" style="display: block; margin: auto;" />
Oh, I'll just grab those individuals closest to me...

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# Solutions

1. Validity: Can you connect your measurement and question?

2. Reliability: Sample a standard or known population

3. Representative: Sampling design!

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# A more representative sample from Random Sampling
<img src="03_sampling_lecture_x_files/figure-html/spatialSample2-1.png" style="display: block; margin: auto;" />
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Two sampling schemes

- Random - samples chosen using random numbers

- Haphazard - samples chosen without any system (careful!)

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# A more representative sample from **Stratified** Sampling
<img src="03_sampling_lecture_x_files/figure-html/stratified-1.png" style="display: block; margin: auto;" />
Sample over a known gradient, aka **cluster sampling**

Can incorporate multiple gradients

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# Stratified or Random?

- How is your population defined?

- What is the scale of your inference?

- What might influence the inclusion of a replicate?

- How important are external factors you know about?

- How important are external factors you cannot assess?

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# Stratified Random Sampling
  
.center[.middle[![](./images/03/stratified_random_sampling_meme.jpg)]]

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# Exercise:

### 1. What is a population you sample?  
  
### 2. How do you ensure validity, reliability, and representativeness of a sample?

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# Outline

1. Sampling Nature

2. .red[Describing a Sample]

3. Using a Sample to Describe a Population

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# Taking a Descriptive
<img src="03_sampling_lecture_x_files/figure-html/sample-1.png" style="display: block; margin: auto;" />

<center>How big are individuals in this population?

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# Our 'Sample'

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# Visualizing Our Sample as Counts with 20 Bins

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# Visualizing Our Sample as Frequencies

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Frequency = % of Sample with that Value

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# Visualizing Our Sample as Frequencies

Frequency = Probability of Drawing that Value from a Sample

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# Visualizing Our Sample as Frequencies

Frequency = Probability Density, Sums to One.

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# Ways to Describe Our Sample

- Describing it as a unique thing
     - What is the value smack in the middle?
     - What are large and small values like?
     - Is our sample clustered, or waaaay spread out?
     
- Describing it with reference to a population
      - If I were to draw from this sample, or it's like, what value would I most likely get?
      - If I assume a normal distribution, what's the range of 66% or 95% of the variability?
      - Is this sample peaked, flat, shifted one way or another?

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# The Empirical Cummulative Distribution Plot of our Sample

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# The Box Plot (or Box-and-Whisker Plot)

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# Two ways of seeing the same data

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# Median: Middle of the Sample
<img src="03_sampling_lecture_x_files/figure-html/unnamed-chunk-4-1.png" style="display: block; margin: auto;" />
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# Quartiles: Upper and Lower Quarter

1<sup>st</sup> and 3<sup>rd</sup> **Quartile** = 25<sup>th</sup> and 75<sup>th</sup> **Percentile**

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# Reasonable Range of Data (Beyond which are Outliers)
<img src="03_sampling_lecture_x_files/figure-html/unnamed-chunk-6-1.png" style="display: block; margin: auto;" />

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# Where's the action in R?

```r
# Quantiles
quantile(samp)
```

```
      0%      25%      50%      75%     100% 
31.41510 41.11041 46.56152 48.15706 60.69390 
```

```r
# Interquartile range
IQR(samp)
```

```
[1] 7.046652
```

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# Outline

1. Sampling Nature

2. Describing a Sample

3. .red[Using a Sample to Describe a Population]

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# This is just a sample

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# We chose it to be representative of the population.

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# But - how does my sample compare to a population?

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# Expected Value: the Mean

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# Sample Properties: **Mean**

`$$\bar{x} = \frac{ \displaystyle \sum_{i=1}^{n}{x_{i}} }{n}$$`

`$\large \bar{x}$` - The average value of a sample  
`$x_{i}$` - The value of a measurement for a single individual   
n - The number of individuals in a sample  
&nbsp;  
`$\mu$` - The average value of a population  
(Greek = population, Latin = Sample)

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# Sample versus Population

- Latin characters (e.g., `$\bar{x}$`) for **sample **
  
- Greek chracters (e.g., `$\mu$`) for **population**

.center[https://istats.shinyapps.io/sampdist_cont/]

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# Mean versus Median

.center[https://istats.shinyapps.io/MeanvsMedian/]

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# How Variable is the Population

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# Sample Properties: **Variance**
How variable was that population?
`$$\large s^2=  \frac{\displaystyle \sum_{i=1}^{n}{(X_i - \bar{X})^2}} {n-1}$$`

* Sums of Squares over n-1  
* n-1 corrects for both sample size and sample bias  
* `$\sigma^2$` if describing the population
* Units in square of measurement...

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# Sample Properties: Standard Deviation
$$ \large s = \sqrt{s^2}$$

* Units the same as the measurement
* If distribution is normal, 67% of data within 1 SD
* 95% within 2 SD
* `$\sigma$` if describing the population

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# Post-Quiz
http://tinyurl.com/sampling-post