lesson_07_generalization

class: inverse, left, bottom
background-image: url(https://images.unsplash.com/photo-1601039834020-62e7a2a54e07?ixid=MXwxMjA3fDB8MHxzZWFyY2h8NDZ8fGJveGluZyUyMG9seW1waWNzfGVufDB8fDB8&ixlib=rb-1.2.1&auto=format&fit=crop&w=500&q=60)
background-size: cover

# .Large[theory based method and central limit theorem]

## .small[]
#### .tiny[Dr. Evangeline Reynolds | Meeting 6 | 2021-02-08 | Image credit: Jonathan Thomas, Upsplash]

???

Title slide

---

# Agenda

- Housekeeping

- Eye dominance Exploration Exercise #1 due next Tuesday (10 points)
  - project proposal due today... 4 vars...; resubmits by next Friday

---
name: dolphins
class: inverse
background-image: url(https://images.unsplash.com/photo-1611890129309-31e797820019?ixid=MXwxMjA3fDB8MHxzZWFyY2h8OXx8ZG9scGhpbnN8ZW58MHx8MHw%3D&ixlib=rb-1.2.1&auto=format&fit=crop&w=600&q=60)
background-size: cover

# Could getting 15 out of 16 fish have happen just by chance?

p-value - How likely is it to see as extreme or more extreme an outcome as $ \hat{p} $, if chance alone generated the outcome

"this would only happen 1 in 1000 based on a chance model" 
--
$$ p - value = 1/1000 = .001 $$

---

# p-value: How often 
--
(the proportion of the time) 
--
we would see an outcome 
--
as extreme 
--
or more extreme 
--
than what is observed 
--
if the null (chance model) is true.

---

# Notation

## Null Hypothesis (Chance) $$ H_0 $$

## If evidence is strong against, 
--
then we can 
--
**reject the null** 
--
and accept the alternative hypothesis 
--
`$$H_A$$`

---

## Alternatively, we can use the "z score" to talk about being extreme from what would typically be observed under chance.

## How far away from the chance model center?

## Measure...

### in fish?

### in standard deviations; z 'standardized statisic'

---

# Calculating the Z statistic

$$ z = \frac{(observed - average of chance model)} / {standard deviation} $$

---

# General formula for Standard Deviation, used in simulation method:

$$ SD = \sqrt\frac{\sum_{i=1}^{n}(x_i - \bar{x})^2} {n-1} $$

---

![](lesson_07_generalization_validity_files/figure-html/dolphins_plot_chance_20_01_output-1.png)

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![](lesson_07_generalization_validity_files/figure-html/dolphins_plot_2_01_output-1.png)

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![](lesson_07_generalization_validity_files/figure-html/dolphins_plot_2_02_output-1.png)

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![](lesson_07_generalization_validity_files/figure-html/chance_v_observed_user_01_output-1.png)

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![](lesson_07_generalization_validity_files/figure-html/chance_v_observed_100000_2_01_output-1.png)

---
count: false

![](lesson_07_generalization_validity_files/figure-html/chance_v_observed_100000_2_02_output-1.png)

---

## Consistent or not consistent w/ chance model?

## Z score of 1.2?

## Z score of 3?

## Z score of -1?

## Z score of -5?

---

background-image: url(images_for_lecture/normal_histogram.svg)
background-size: 550px
background-position: 50% 80%

#### Interpretation

---
class: center, inverse, middle

# Can I also calculate a p-value with the z-score?

z-score seems imprecise.  I want to know: *exactly* how unlikely to observe such extreme a case just by chance

---

# Theory-based test for a single proportion

# 'one-proportion z-test'

---

# Relies on

the Central Limit Theorem

> If the sample size (n) is large enough, sampling distributions are normally distributed...

---

# The Central Limit Theorem for proportions ->

> The distribution of sample proportions will be bell-shaped (normal), centered at the long-run proportion (pi), with a standard deviation of:

# $$ sd_{prop} = \sqrt\frac{\pi*(1-\pi)}{n} $$

---

---

# Step 1 calculate sample proportion

## Is the proportion of Yuks is .25?

```r
13/15
```

```
[1] 0.8666667
```

---

## Step 2 calc sd for the null:

```r
sqrt((.25 * (1-.25)) / 15 )
```

```
[1] 0.1118034
```

---

# Step 3 calc z stat

## z stat (number of SDs away from chance model mean)

```r
(0.866 - .25) / # observed minus null prop
0.1118034
```

```
[1] 5.509671
```

---

# Step 4

## situate in normal distribution

```r
# pnorm gives us cumulative density function 
# for normal distribution
2 * (1 - pnorm(abs(5.509671)))
```

```
[1] 3.59505e-08
```

```r
# result is p-value 
# the proportion of times we expect 
# to see something as extreme or more extreme
# than observed stat
# just by chance!
```

---

```r
## calculate sample proportion p hat
*13 / 15
```
]
 
.panel2-theory_p_val_calc-auto[

```
[1] 0.8666667
```
]

---
count: false
 
## All together!
.panel1-theory_p_val_calc-auto[

```r
## calculate sample proportion p hat
13 / 15 ->
* sample_proportion
```
]
 
.panel2-theory_p_val_calc-auto[

]

---
count: false
 
## All together!
.panel1-theory_p_val_calc-auto[

```r
## calculate sample proportion p hat
13 / 15 ->
  sample_proportion

## establish pi
*.25
```
]
 
.panel2-theory_p_val_calc-auto[

```
[1] 0.25
```
]

---
count: false
 
## All together!
.panel1-theory_p_val_calc-auto[

```r
## calculate sample proportion p hat
13 / 15 ->
  sample_proportion

## establish pi
.25 ->
* null_prob
```
]
 
.panel2-theory_p_val_calc-auto[

]

---
count: false
 
## All together!
.panel1-theory_p_val_calc-auto[

```r
## calculate sample proportion p hat
13 / 15 ->
  sample_proportion

## establish pi
.25 ->
  null_prob

## calc sd for the null:
*sqrt((null_prob * (1-null_prob)) / 15 )
```
]
 
.panel2-theory_p_val_calc-auto[

```
[1] 0.1118034
```
]

---
count: false
 
## All together!
.panel1-theory_p_val_calc-auto[

```r
## calculate sample proportion p hat
13 / 15 ->
  sample_proportion

## establish pi
.25 ->
  null_prob

## calc sd for the null:
sqrt((null_prob * (1-null_prob)) / 15 ) ->
* sd_chance_model
```
]
 
.panel2-theory_p_val_calc-auto[

]

---
count: false
 
## All together!
.panel1-theory_p_val_calc-auto[

```r
## calculate sample proportion p hat
13 / 15 ->
  sample_proportion

## establish pi
.25 ->
  null_prob

## calc sd for the null:
sqrt((null_prob * (1-null_prob)) / 15 ) ->
  sd_chance_model

## calculate z score
*(sample_proportion - null_prob)
```
]
 
.panel2-theory_p_val_calc-auto[

```
[1] 0.6166667
```
]

---
count: false
 
## All together!
.panel1-theory_p_val_calc-auto[

```r
## calculate sample proportion p hat
13 / 15 ->
  sample_proportion

## establish pi
.25 ->
  null_prob

## calc sd for the null:
sqrt((null_prob * (1-null_prob)) / 15 ) ->
  sd_chance_model

## calculate z score
(sample_proportion - null_prob) /
* sd_chance_model
```
]
 
.panel2-theory_p_val_calc-auto[

```
[1] 5.515634
```
]

---
count: false
 
## All together!
.panel1-theory_p_val_calc-auto[

```r
## calculate sample proportion p hat
13 / 15 ->
  sample_proportion

## establish pi
.25 ->
  null_prob

## calc sd for the null:
sqrt((null_prob * (1-null_prob)) / 15 ) ->
  sd_chance_model

## calculate z score
(sample_proportion - null_prob) /
  sd_chance_model ->
*z_score
```
]
 
.panel2-theory_p_val_calc-auto[

]

---
count: false
 
## All together!
.panel1-theory_p_val_calc-auto[

```r
## calculate sample proportion p hat
13 / 15 ->
  sample_proportion

## establish pi
.25 ->
  null_prob

## calc sd for the null:
sqrt((null_prob * (1-null_prob)) / 15 ) ->
  sd_chance_model

## calculate z score
(sample_proportion - null_prob) /
  sd_chance_model ->
z_score

# situate in normal distribution
*(1 - pnorm(abs(z_score))) * 2
```
]
 
.panel2-theory_p_val_calc-auto[

```
[1] 3.475241e-08
```
]

---
class: inverse, center, middle

# Does this example meet the validity conditions though?

## No

## At least 10 of each case (10 'successes' 10 'failures')

---

# Olympics - where theory holds (10 successes, 10 failures)

---

```r
*options(scipen = 10)
```
]
 
.panel2-theory_p_val_calc_box-auto[

]

---
count: false
 
## All together boxing!
.panel1-theory_p_val_calc_box-auto[

```r
options(scipen = 10)

## calculate sample proportion p hat
*248 / 457
```
]
 
.panel2-theory_p_val_calc_box-auto[

```
[1] 0.5426696
```
]

---
count: false
 
## All together boxing!
.panel1-theory_p_val_calc_box-auto[

```r
options(scipen = 10)

## calculate sample proportion p hat
248 / 457 ->
* sample_proportion
```
]
 
.panel2-theory_p_val_calc_box-auto[

]

---
count: false
 
## All together boxing!
.panel1-theory_p_val_calc_box-auto[

```r
options(scipen = 10)

## calculate sample proportion p hat
248 / 457 ->
  sample_proportion

## establish pi
*.5
```
]
 
.panel2-theory_p_val_calc_box-auto[

```
[1] 0.5
```
]

---
count: false
 
## All together boxing!
.panel1-theory_p_val_calc_box-auto[

```r
options(scipen = 10)

## calculate sample proportion p hat
248 / 457 ->
  sample_proportion

## establish pi
.5 ->
* null_prob
```
]
 
.panel2-theory_p_val_calc_box-auto[

]

---
count: false
 
## All together boxing!
.panel1-theory_p_val_calc_box-auto[

```r
options(scipen = 10)

## calculate sample proportion p hat
248 / 457 ->
  sample_proportion

## establish pi
.5 ->
  null_prob

## calc sd for the null:
*sqrt((null_prob * (1-null_prob)) / 457 )
```
]
 
.panel2-theory_p_val_calc_box-auto[

```
[1] 0.02338901
```
]

---
count: false
 
## All together boxing!
.panel1-theory_p_val_calc_box-auto[

```r
options(scipen = 10)

## calculate sample proportion p hat
248 / 457 ->
  sample_proportion

## establish pi
.5 ->
  null_prob

## calc sd for the null:
sqrt((null_prob * (1-null_prob)) / 457 ) ->
* sd_chance_model
```
]
 
.panel2-theory_p_val_calc_box-auto[

]

---
count: false
 
## All together boxing!
.panel1-theory_p_val_calc_box-auto[

```r
options(scipen = 10)

## calculate sample proportion p hat
248 / 457 ->
  sample_proportion

## establish pi
.5 ->
  null_prob

## calc sd for the null:
sqrt((null_prob * (1-null_prob)) / 457 ) ->
  sd_chance_model

## calculate z score
*(sample_proportion - null_prob)
```
]
 
.panel2-theory_p_val_calc_box-auto[

```
[1] 0.04266958
```
]

---
count: false
 
## All together boxing!
.panel1-theory_p_val_calc_box-auto[

```r
options(scipen = 10)

## calculate sample proportion p hat
248 / 457 ->
  sample_proportion

## establish pi
.5 ->
  null_prob

## calc sd for the null:
sqrt((null_prob * (1-null_prob)) / 457 ) ->
  sd_chance_model

## calculate z score
(sample_proportion - null_prob) /
* sd_chance_model
```
]
 
.panel2-theory_p_val_calc_box-auto[

```
[1] 1.824343
```
]

---
count: false
 
## All together boxing!
.panel1-theory_p_val_calc_box-auto[

```r
options(scipen = 10)

## calculate sample proportion p hat
248 / 457 ->
  sample_proportion

## establish pi
.5 ->
  null_prob

## calc sd for the null:
sqrt((null_prob * (1-null_prob)) / 457 ) ->
  sd_chance_model

## calculate z score
(sample_proportion - null_prob) /
  sd_chance_model ->
*z_score
```
]
 
.panel2-theory_p_val_calc_box-auto[

]

---
count: false
 
## All together boxing!
.panel1-theory_p_val_calc_box-auto[

```r
options(scipen = 10)

## calculate sample proportion p hat
248 / 457 ->
  sample_proportion

## establish pi
.5 ->
  null_prob

## calc sd for the null:
sqrt((null_prob * (1-null_prob)) / 457 ) ->
  sd_chance_model

## calculate z score
(sample_proportion - null_prob) /
  sd_chance_model ->
z_score

# situate in normal distribution
*(1 - pnorm(abs(z_score))) * 2
```
]
 
.panel2-theory_p_val_calc_box-auto[

```
[1] 0.06810023
```
]

---

```
# A tibble: 9 × 3
  some_zstats dnorm_result pnorm_result
        <int>        <dbl>        <dbl>
1          -4     0.000134    0.0000317
2          -3     0.00443     0.00135  
3          -2     0.0540      0.0228   
4          -1     0.242       0.159    
5           0     0.399       0.5      
6           1     0.242       0.841    
7           2     0.0540      0.977    
8           3     0.00443     0.999    
9           4     0.000134    1.00     
```

---

https://evamaerey.github.io/statistics/distributions#1

---

```r
*((-100:100)/20)
```
]
 
.panel2-dnorm_pnorm-auto[

```
  [1] -5.00 -4.95 -4.90 -4.85 -4.80 -4.75 -4.70 -4.65 -4.60 -4.55 -4.50 -4.45
 [13] -4.40 -4.35 -4.30 -4.25 -4.20 -4.15 -4.10 -4.05 -4.00 -3.95 -3.90 -3.85
 [25] -3.80 -3.75 -3.70 -3.65 -3.60 -3.55 -3.50 -3.45 -3.40 -3.35 -3.30 -3.25
 [37] -3.20 -3.15 -3.10 -3.05 -3.00 -2.95 -2.90 -2.85 -2.80 -2.75 -2.70 -2.65
 [49] -2.60 -2.55 -2.50 -2.45 -2.40 -2.35 -2.30 -2.25 -2.20 -2.15 -2.10 -2.05
 [61] -2.00 -1.95 -1.90 -1.85 -1.80 -1.75 -1.70 -1.65 -1.60 -1.55 -1.50 -1.45
 [73] -1.40 -1.35 -1.30 -1.25 -1.20 -1.15 -1.10 -1.05 -1.00 -0.95 -0.90 -0.85
 [85] -0.80 -0.75 -0.70 -0.65 -0.60 -0.55 -0.50 -0.45 -0.40 -0.35 -0.30 -0.25
 [97] -0.20 -0.15 -0.10 -0.05  0.00  0.05  0.10  0.15  0.20  0.25  0.30  0.35
[109]  0.40  0.45  0.50  0.55  0.60  0.65  0.70  0.75  0.80  0.85  0.90  0.95
[121]  1.00  1.05  1.10  1.15  1.20  1.25  1.30  1.35  1.40  1.45  1.50  1.55
[133]  1.60  1.65  1.70  1.75  1.80  1.85  1.90  1.95  2.00  2.05  2.10  2.15
[145]  2.20  2.25  2.30  2.35  2.40  2.45  2.50  2.55  2.60  2.65  2.70  2.75
[157]  2.80  2.85  2.90  2.95  3.00  3.05  3.10  3.15  3.20  3.25  3.30  3.35
[169]  3.40  3.45  3.50  3.55  3.60  3.65  3.70  3.75  3.80  3.85  3.90  3.95
[181]  4.00  4.05  4.10  4.15  4.20  4.25  4.30  4.35  4.40  4.45  4.50  4.55
[193]  4.60  4.65  4.70  4.75  4.80  4.85  4.90  4.95  5.00
```
]

---
count: false
 
## What's pnorm()
.panel1-dnorm_pnorm-auto[

```r
((-100:100)/20) %>%
* tibble(sd = .)
```
]
 
.panel2-dnorm_pnorm-auto[

```
# A tibble: 201 × 1
      sd
   <dbl>
 1 -5   
 2 -4.95
 3 -4.9 
 4 -4.85
 5 -4.8 
 6 -4.75
 7 -4.7 
 8 -4.65
 9 -4.6 
10 -4.55
# … with 191 more rows
```
]

---
count: false
 
## What's pnorm()
.panel1-dnorm_pnorm-auto[

```r
((-100:100)/20) %>%
  tibble(sd = .) %>%
* mutate(prob_from_normal =
*          dnorm(sd))
```
]
 
.panel2-dnorm_pnorm-auto[

```
# A tibble: 201 × 2
      sd prob_from_normal
   <dbl>            <dbl>
 1 -5          0.00000149
 2 -4.95       0.00000191
 3 -4.9        0.00000244
 4 -4.85       0.00000311
 5 -4.8        0.00000396
 6 -4.75       0.00000503
 7 -4.7        0.00000637
 8 -4.65       0.00000805
 9 -4.6        0.0000101 
10 -4.55       0.0000127 
# … with 191 more rows
```
]

---
count: false
 
## What's pnorm()
.panel1-dnorm_pnorm-auto[

```r
((-100:100)/20) %>%
  tibble(sd = .) %>%
  mutate(prob_from_normal =
           dnorm(sd)) %>%
* mutate(cummulative_from_normal =
*          pnorm(sd))
```
]
 
.panel2-dnorm_pnorm-auto[

```
# A tibble: 201 × 3
      sd prob_from_normal cummulative_from_normal
   <dbl>            <dbl>                   <dbl>
 1 -5          0.00000149             0.000000287
 2 -4.95       0.00000191             0.000000371
 3 -4.9        0.00000244             0.000000479
 4 -4.85       0.00000311             0.000000617
 5 -4.8        0.00000396             0.000000793
 6 -4.75       0.00000503             0.00000102 
 7 -4.7        0.00000637             0.00000130 
 8 -4.65       0.00000805             0.00000166 
 9 -4.6        0.0000101              0.00000211 
10 -4.55       0.0000127              0.00000268 
# … with 191 more rows
```
]

---
count: false
 
## What's pnorm()
.panel1-dnorm_pnorm-auto[

```r
((-100:100)/20) %>%
  tibble(sd = .) %>%
  mutate(prob_from_normal =
           dnorm(sd)) %>%
  mutate(cummulative_from_normal =
           pnorm(sd)) %>%
* ggplot()
```
]
 
.panel2-dnorm_pnorm-auto[
![](lesson_07_generalization_validity_files/figure-html/dnorm_pnorm_auto_05_output-1.png)
]

---
count: false
 
## What's pnorm()
.panel1-dnorm_pnorm-auto[

```r
((-100:100)/20) %>%
  tibble(sd = .) %>%
  mutate(prob_from_normal =
           dnorm(sd)) %>%
  mutate(cummulative_from_normal =
           pnorm(sd)) %>%
  ggplot() +
* aes(x = sd)
```
]
 
.panel2-dnorm_pnorm-auto[
![](lesson_07_generalization_validity_files/figure-html/dnorm_pnorm_auto_06_output-1.png)
]

---
count: false
 
## What's pnorm()
.panel1-dnorm_pnorm-auto[

```r
((-100:100)/20) %>%
  tibble(sd = .) %>%
  mutate(prob_from_normal =
           dnorm(sd)) %>%
  mutate(cummulative_from_normal =
           pnorm(sd)) %>%
  ggplot() +
  aes(x = sd) +
* aes(y = prob_from_normal)
```
]
 
.panel2-dnorm_pnorm-auto[
![](lesson_07_generalization_validity_files/figure-html/dnorm_pnorm_auto_07_output-1.png)
]

---
count: false
 
## What's pnorm()
.panel1-dnorm_pnorm-auto[

```r
((-100:100)/20) %>%
  tibble(sd = .) %>%
  mutate(prob_from_normal =
           dnorm(sd)) %>%
  mutate(cummulative_from_normal =
           pnorm(sd)) %>%
  ggplot() +
  aes(x = sd) +
  aes(y = prob_from_normal) +
* labs(title = "Probability Distribution for Normal Distribution")
```
]
 
.panel2-dnorm_pnorm-auto[
![](lesson_07_generalization_validity_files/figure-html/dnorm_pnorm_auto_08_output-1.png)
]

---
count: false
 
## What's pnorm()
.panel1-dnorm_pnorm-auto[

```r
((-100:100)/20) %>%
  tibble(sd = .) %>%
  mutate(prob_from_normal =
           dnorm(sd)) %>%
  mutate(cummulative_from_normal =
           pnorm(sd)) %>%
  ggplot() +
  aes(x = sd) +
  aes(y = prob_from_normal) +
  labs(title = "Probability Distribution for Normal Distribution") +
* geom_line()
```
]
 
.panel2-dnorm_pnorm-auto[
![](lesson_07_generalization_validity_files/figure-html/dnorm_pnorm_auto_09_output-1.png)
]

---
count: false
 
## What's pnorm()
.panel1-dnorm_pnorm-auto[

```r
((-100:100)/20) %>%
  tibble(sd = .) %>%
  mutate(prob_from_normal =
           dnorm(sd)) %>%
  mutate(cummulative_from_normal =
           pnorm(sd)) %>%
  ggplot() +
  aes(x = sd) +
  aes(y = prob_from_normal) +
  labs(title = "Probability Distribution for Normal Distribution") +
  geom_line() +
* aes(y = cummulative_from_normal)
```
]
 
.panel2-dnorm_pnorm-auto[
![](lesson_07_generalization_validity_files/figure-html/dnorm_pnorm_auto_10_output-1.png)
]

---
count: false
 
## What's pnorm()
.panel1-dnorm_pnorm-auto[

```r
((-100:100)/20) %>%
  tibble(sd = .) %>%
  mutate(prob_from_normal =
           dnorm(sd)) %>%
  mutate(cummulative_from_normal =
           pnorm(sd)) %>%
  ggplot() +
  aes(x = sd) +
  aes(y = prob_from_normal) +
  labs(title = "Probability Distribution for Normal Distribution") +
  geom_line() +
  aes(y = cummulative_from_normal) +
* labs(title = "Cumulative Distribution Function (CDF) for Normal Distribution")
```
]
 
.panel2-dnorm_pnorm-auto[
![](lesson_07_generalization_validity_files/figure-html/dnorm_pnorm_auto_11_output-1.png)
]

---

# Using `prop.test()` function

### Is the proportion of Yuks is .25?

```r
prop.test(x = 13, # number "successes" observed 
          n = 15, # total number
          p = .25, # proportion expected under null
          alternative = # which extremes count
          "two.sided", # other options are greater and less
          correct = F) # use classic prop test
```

```

1-sample proportions test without continuity correction

data:  13 out of 15, null probability 0.25
X-squared = 30.422, df = 1, p-value = 0.00000003475
alternative hypothesis: true p is not equal to 0.25
95 percent confidence interval:
 0.6211802 0.9626387
sample estimates:
        p 
0.8666667 
```

---

# Boxing

```r
prop.test(x = 248, # number observed "successes"
          n = 457, # total number
          p = .5, # proportion expected under null
          alternative = # which extremes count
          "two.sided", # other options are greater and less
          correct = F) # use classic prop test
```

```

1-sample proportions test without continuity correction

data:  248 out of 457, null probability 0.5
X-squared = 3.3282, df = 1, p-value = 0.0681
alternative hypothesis: true p is not equal to 0.5
95 percent confidence interval:
 0.4968289 0.5877989
sample estimates:
        p 
0.5426696 
```