Lecture 2: PY 0794 - Advanced Quantitative Research Methods

class: center, middle, inverse, title-slide

.title[
# Lecture 2: PY 0794 - Advanced Quantitative Research Methods
]
.author[
### Dr. Thomas Pollet, Northumbria University (<a href="mailto:thomas.pollet@northumbria.ac.uk" class="email">thomas.pollet@northumbria.ac.uk</a>)
]
.date[
### 2024-01-30 | <a href="http://tvpollet.github.io/disclaimer">disclaimer</a>
]

---

## PY0794: Advanced Quantitative research methods.
* Last week: Introduction.
* Today all about visualization.

---
## Goals (today)

Data manipulation.

Data visualization.

Basic tests (_t_-tests, and a non-parametric alternative.)

---
## Assignment

After today you should be able to complete the following sections:

Descriptive statistics and data wrangling.

Histogram

Violin plot

_t_-test and non-parametric alternative

---
## Flights dataset

Again this is based on [Whickham & Grolemund (2017)](http://r4ds.had.co.nz/) and on work by [Gert Stulp](http://www.gertstulp.com/)

```r
setwd("~/Dropbox/Teaching_MRes_Northumbria/Lecture2")
library(nycflights13)
flights<-nycflights13::flights
```

---
## 'dplyr' 
 **Last week:** Pick observations by their values: **filter()**.

Reorder the rows: **arrange()**. (read about this [here](http://r4ds.had.co.nz/transform.html#arrange-rows-with-arrange))

_Pick variables by their names **select()**._

_Create new variables with functions of existing variables **mutate()**._

_Collapse many values down to a single summary **summarise()**._

---
## Load dplyr

Load dplyr:

```r
library(dplyr)
```

```
## 
## Attaching package: 'dplyr'
```

```
## The following objects are masked from 'package:stats':
## 
##     filter, lag
```

```
## The following objects are masked from 'package:base':
## 
##     intersect, setdiff, setequal, union
```

---
## select()

Sometimes you have dozens, even >100 or >1000 variables.

You can use select() to make a reduced data frame.

```r
flights_red<-select(flights, year, month, day)
### This overrides our earlier attempt
flights_red<-select(flights, year:day)
### all columns excluding those from year:day (inclusive)
flights_red<-select(flights, -(year:day))
```

---
## Try it yourself. 
 Team up with a partner.
 
 Load the flights data.
 
 Make a new dataframe with all variables starting from dep_time to arr_delay (inclusive).

---
## helper functions.
For more complex cases:

starts_with("abc"): matches names that begin with “abc”.
 
--

ends_with("xyz"): matches names that end with “xyz”.

contains("ijk"): matches names that contain “ijk”.
 
--

matches("(.)\\1"): selects variables that match a "regular expression". This one matches any variables that contain repeated characters. ??regex

--
 
num_range("x", 1:3): matches x1, x2 and x3.

---
## rename()

* select() can be used to rename variables, **but** it drops all of the variables not explicitly mentioned. 
* Instead, use rename(), which is a variant of select() which keeps all the variables that are not explicitly mentioned. 
* However, I found this to be [buggy](https://stackoverflow.com/questions/34275576/avoiding-error-when-using-rename-in-dplyr-and-column-doesnt-exist) so back to base R but feel free to use rename().

---
## names

```r
# Note the square brackets for indexing the columns.
names(flights_red)[1:3]=c("Year", "Month", "Day")
head(flights_red)
```

```
## # A tibble: 6 × 16
##    Year Month   Day arr_time sched_arr_time arr_delay carrier flight tailnum
##   <int> <int> <dbl>    <int>          <int>     <dbl> <chr>    <int> <chr>  
## 1   517   515     2      830            819        11 UA        1545 N14228 
## 2   533   529     4      850            830        20 UA        1714 N24211 
## 3   542   540     2      923            850        33 AA        1141 N619AA 
## 4   544   545    -1     1004           1022       -18 B6         725 N804JB 
## 5   554   600    -6      812            837       -25 DL         461 N668DN 
## 6   554   558    -4      740            728        12 UA        1696 N39463 
## # ℹ 7 more variables: origin <chr>, dest <chr>, air_time <dbl>, distance <dbl>,
## #   hour <dbl>, minute <dbl>, time_hour <dttm>
```

---
## mutate()

%>% : the pipe. Basically means apply function to.

mutate() allows you to make new variables

```r
flights <- flights %>% mutate(speed = distance / (air_time * 60), log_speed=log(speed))
```

--
 transmute() if you want **only** the things you have calculated.

---
## helper functions() for mutate()

Again a whole load of helper functions exist. You can read more about them [here](http://r4ds.had.co.nz/transform.html#add-new-variables-with-mutate)

the key thing to remember is that you can apply all the functions we covered last week.

---
## Centering and standardizing.

**centering**: subtracting the mean. (Often used to make scores more interpretable and to avoid collinearity, which is a big [deal](https://link.springer.com/article/10.1007/s00265-010-1045-6)). More about this when we discuss OLS regression.

**standardizing**: Z-scores. We use the scale() function.

```r
flights<- flights %>% mutate(speed_cent = speed - mean(speed, na.rm = TRUE))
flights<- flights %>% mutate(speed_stand = scale(speed)) 
```

* now about that 'na.rm = TRUE' --> this removes any missings.

* Reminder about Z-scores.
<img src="http://deafstudeerconsultant.nl/wp-content/uploads/2015/01/z-score-normale-verdeling.jpg" width="350px" />

---
## summarise()

Get one row with data for the dataframe.

```r
summarise(flights, mean_speed = mean(speed, na.rm = TRUE))
```

```
## # A tibble: 1 × 1
##   mean_speed
##        <dbl>
## 1      0.110
```

---
## group_by
 more useful if combined with group_by , let's look at delays by [carrier](http://aspmhelp.faa.gov/index.php/ASQP:_Carrier_Codes_and_Names)

```r
by_carrier <- group_by(flights, carrier)
summarise(by_carrier, delay = mean(dep_delay, na.rm = TRUE))
```

```
## # A tibble: 16 × 2
##    carrier delay
##    <chr>   <dbl>
##  1 9E      16.7 
##  2 AA       8.59
##  3 AS       5.80
##  4 B6      13.0 
##  5 DL       9.26
##  6 EV      20.0 
##  7 F9      20.2 
##  8 FL      18.7 
##  9 HA       4.90
## 10 MQ      10.6 
## 11 OO      12.6 
## 12 UA      12.1 
## 13 US       3.78
## 14 VX      12.9 
## 15 WN      17.7 
## 16 YV      19.0
```
---
## Try it yourself.

* calculate 'gain' which is arr_delay - dep_delay.
* calculate 'airtime_hours' which is airtime (in mins.) divided by the number of minutes in an hour. 
* now calculate 'gain per hour'. (gain/airtime_hours)
* standardize 'gain per hour'.
* use group_by to calculate the means of gain per hour by carrier.

---
## Replacing missings.

Often we want to replace missings. For example, we want to replace the missings with the means or median.

```r
flights <- flights %>% mutate(dep_delay_no_miss= ifelse(is.na(dep_delay), 
           mean(dep_delay, na.rm = T), dep_delay))
mean(flights$dep_delay_no_miss)
```

```
## [1] 12.63907
```

Here we use an 'ifelse' function to replace the missings.

True can be abbreviated to 'T'.

---
## So far, so good.

Now that we know the basics of manipulating dataframes let's move to graphs.

Why is plotting your data important?

[Anscombe's quartet (1973)](http://www.jstor.org/stable/pdf/2682899.pdf?refreqid=excelsior%3A19d1f2fcfb1dacfb0ec692faa1512a37). _r_= .816

Often: testing important assumptions.

---
## Datasaurus.

* An extension of Anscombe's quartet. Read more [here](https://www.autodeskresearch.com/publications/samestats)

---
## Now let's make some graphs.

ggplot2

some 'helper' packages, 'scales' helps with axes, 'Rcolorbrewer' use, 'ggthemes' allows standardized templates, 'cow plot' allows panelling graphs.

```r
library(ggplot2)
library(scales) ### no longer needed in newer versions.
library(RColorBrewer)
library(ggthemes)
```

---
## Scatter plot

* Our data file is truly massive, so for this exercise we are just going to select all flights from JFK airport on christmas day.

```r
flights_jfk_new<-filter(flights, month == 12, day == 25, origin == 'JFK')
```

* Scatter plot

```r
ggplot(flights_jfk_new, aes(x=dep_delay, y=arr_delay)) + geom_point() 
```

```
## Warning: Removed 2 rows containing missing values (`geom_point()`).
```

---
## What is that warning about?

* 2 missing cases.
* Let us replace those, in fact we already did that for dep_delay, but that was based on the **whole** set. For this exercise, we'll replace it again (so now it is replaced with the mean on X-mas day)

```r
flights_jfk_new <- flights_jfk_new %>% 
                   mutate(dep_delay_no_miss= ifelse(is.na(dep_delay), 
                   mean(dep_delay, na.rm = T), dep_delay))
flights_jfk_new <- flights_jfk_new %>% 
                   mutate(arr_delay_no_miss= ifelse(is.na(arr_delay), 
                   mean(arr_delay, na.rm = T), arr_delay))
```

---
## Plot again

```r
graph_delays<-ggplot(flights_jfk_new, aes(x=dep_delay_no_miss, y=arr_delay_no_miss)) 
graph_delays + geom_point() 
```

---
## Colour by airline

Recap: ggplot(your_data, aes(x=x_variable, y=y_variable)) + geom_point()

Now let's colour it by carrier.

```r
graph_colour <- ggplot(flights_jfk_new, aes(x=dep_delay_no_miss, y=arr_delay_no_miss, 
                                             colour=carrier)) 
graph_colour + geom_point()
```

---
## Flexibility.

* The beauty of ggplot is that using a different “geom”-function will create a different graph:

```r
graph_delays + geom_count() 
```

---
## Let's try another one.

```r
graph_delays +  geom_count() + geom_smooth()
```

---
## Try it yourself.
* Make a similar set as we did but now: flights leaving LaGuardia (LGA) airport on X-mas day.
* Make a plot with departure delay on the X-axis and speed on the Y-axis.
* Colour the points by carrier.

---
## You can tweak everything!

Example with Delta and United airlines.

```r
flights_jfk_new_2_airlines<-filter(flights_jfk_new, carrier== "DL" | carrier== "UA" )
graph_beauty<-ggplot(flights_jfk_new_2_airlines, 
                     aes(x=dep_delay_no_miss, y=arr_delay_no_miss, 
                                             colour=carrier))  + 
  geom_point(size=3, shape=13, alpha=0.5) + ### size/shape of points + transparency
  geom_smooth(method="lm") + ### Add linear regression lines
  scale_colour_brewer(palette = "Set1") + # Change the colours
  labs(x="Delay departure", y="Delay arrival",
       title="United vs. Delta - Showdown") +  
  # Change labels on axes and add title
  theme_bw() # Remove the gray background
```

---
## Look at the pretty, pretty!

```r
graph_beauty
```

---
## Many types of graphs.
| #  | geom      | graph            |
|----|:----------|:-----------------|
| 1  | [geom_density()](#anchor1) | Density plot (distribution) |
| 2  | [geom_histogram()](#anchor2) | Histogram (distribution) |
| 3  | [geom_violin()](#anchor3) | Violin pot (multiple distributions) |
| 4  | [geom_boxplot()](#anchor4) | Boxplot (multiple distributions) |
| 5  | [geom_bar()](#anchor5) | Bar chart (categorical + continuous/categorical) |
| 6  | [geom_point()](#anchor6) | Scatter (continuous + continuous) |
| 7  | [geom_count()](#anchor7) | Bubble plot (continuous + continuous/categorical) |
| 8  | [geom_jitter()](#anchor8) | Scatter (continuous + continuous/categorical) |

---
## Many types of graphs, continued
| #  | geom      | graph            |
|----|:----------|:-----------------|
| 9  | [geom_smooth()](#anchor9) | Prediction line (continuous + continuous) |
| 10 | [geom_line()](#anchor10) | Line (continuous + continuous/categorical) |
| 11 | [geom_errorbar()](#anchor11) | Errorbar (continuous + continuous/categorical) |

We will not cover all of these today.

Need any help / reference? All the codes can be found [here](http://ggplot2.tidyverse.org/reference/)

---
## Histogram.

Let's make a panelled histogram comparing the delays in departure of two major airlines Jetblue (code= B6) and Delta airlines on X-mas day.

To help ourselves we'll make to separate sets. We'll also save them.

```r
### Note that I now went for new, short names. 
### Ideally you want your naming to be consistent.
jfk_delta<-filter(flights_jfk_new, carrier== "DL")
jfk_jetblue<-filter(flights_jfk_new, carrier== "B6")
write.csv(jfk_delta,row.names = F, file= 'jfk_delta.csv')
write.csv(jfk_jetblue,row.names = F, file= 'jfk_jetblue.csv')
```

---
## Basics.

```r
plot_dl<-ggplot(jfk_delta, aes(dep_delay))
plot_jetblue<-ggplot(jfk_jetblue, aes(dep_delay))
### Plot Delta.
plot_dl+geom_histogram(colour = "black", fill = "white")
```

```
## `stat_bin()` using `bins = 30`. Pick better value with `binwidth`.
```

---
## Binwidth?

* Defaults used. Binwidth is [important](http://www.stat.cmu.edu/~rnugent/PCMI2016/papers/WandBinWidth.pdf). Not too large/small.

```r
plot_dl+geom_histogram(binwidth=1)
```

```r
plot_dl+geom_histogram(binwidth=40)
```

---
## Settle on 5?

```r
plot_dl<-plot_dl+geom_histogram(binwidth=5, colour = "black", fill = "white")
plot_jetblue<-plot_jetblue + geom_histogram(binwidth=5, colour = "black", fill = "white")
plot_dl
```

```r
plot_jetblue
```

---
## Change labels.

```r
plot_dl+ylab("Count") + xlab("Departure Delay - Delta")
```

```r
plot_jetblue + ylab("Count") + xlab("Departure Delay - Jet Blue")
```

---
## Change the ticks / axes limits.

```r
plot_dl + scale_y_continuous(limits=c(0,50),breaks = pretty_breaks(6))
```

```r
plot_dl + scale_x_continuous(limits=c(-20,300),breaks = pretty_breaks(6)) 
```

---
## Intermediate formatting lost.

* In the end we will need ***all*** the commands in one go! Or we need to store intermediate graphs and update those!

---
## Themes.

* ggthemes. ??ggthemes for all options.
* google doc theme. Base size = fonts.

```r
plot_dl+theme_gdocs(base_size = 14)
```

---
## All in one!

```r
plot_dl <- plot_dl +
    geom_histogram(binwidth = 5,
        colour = "black",
        fill = "white") +
    ylab("Count") + xlab("Departure Delay - Delta") +
    scale_y_continuous(limits = c(0,
        50), breaks = pretty_breaks(6)) +
    scale_x_continuous(limits = c(-15,
        250), breaks = pretty_breaks(6)) +
    theme_gdocs(base_size = 14)
plot_dl
```

---
## Try it yourself.

* Make a plot for jetblue, similar to the one for Delta Airlines

* Solution

```r
plot_jetblue <- plot_jetblue + geom_histogram(binwidth = 5,
    colour = "black", fill = "white") +
    ylab("Count") + xlab("Departure Delay - Jet Blue") +
    scale_y_continuous(limits = c(0,
        50), breaks = pretty_breaks(6)) +
    scale_x_continuous(limits = c(-15,
        250), breaks = pretty_breaks(6)) +
    theme_gdocs(base_size = 14)
plot_jetblue
```

---
## Saving it. ggsave??

```r
ggsave("plot_jetblue.pdf", width = 250, height = 250, units = "mm")
```

default is to save last plot displayed but you can explicitly name it (plot = )

You can change the output size to different units ("in")

dpi for resolution.

different formats (e.g., .png, .tiff)

It will be saved in your working directory. (change that with path = ).

---
## Combing multiple charts. plot_grid()

```r
library(cowplot)
plot_grid(plot_dl, plot_jetblue, labels = c("A", "B"), label_size = 10,
    align = "h")
```

```r
ggsave("plot_panelled.pdf", width = 250, height = 250, units = "mm")
```

---
## Interpretation.

---
## What is good/bad about this chart?

* Quite clear.

* Only 2 groups? Overlay?

* Varying N. Sample size.

* White space.

* Units? (in minutes).

* (Suppressed some 'missing' values? - binning can obscure things - not always major issue)

---
## Violin plot (with box plot)

* Now let's approach this in a different way.

* Violin plots combined with a box plot (remember last week) give us good overview.

---
## Basic violin plot.

```r
### combine them into a single dataframe.
violinplot_data <- rbind.data.frame(jfk_delta, jfk_jetblue)
library(dplyr)
violinplot_data <- violinplot_data %>%
    mutate(carrier = recode(carrier, B6 = "Jet Blue",
        DL = "Delta Airlines"))
### make plot
violinplot <- ggplot(violinplot_data, aes(x = carrier,
    y = dep_delay)) + geom_violin() + geom_boxplot(width = 0.2)
violinplot
```

---
## Add some 'oomph'.

```r
violinplot + xlab("Carrier") + ylab("Delay Departure (mins.)") +
    theme_stata(14)
```

---
## Can you describe the graph?

* What are the key points?

---
## Changing the order.

* Suppose we want Jet Blue on the left for the X-axis.

```r
### recode.
library(haven)  # needed for as_factor
violinplot_data$carrier_rec <- as_factor(violinplot_data$carrier)
ordered_levels <- c("Jet Blue", "Delta Airlines")
violinplot_data <- mutate(violinplot_data, carrier_rec = factor(carrier_rec,
    levels = ordered_levels))
### violinplot_2
violinplot_2 <- ggplot(violinplot_data, aes(x = carrier_rec,
    y = dep_delay)) + geom_violin() + geom_boxplot(width = 0.2) +
    xlab("Carrier") + ylab("Delay Departure (mins.)") +
    theme_stata(14)
violinplot_2
```

---
## _t_-test: Are there more delays with Delta than with Jet Blue?

* Let's run an independent samples _t_-test

```r
t.test(dep_delay~carrier, data=violinplot_data)
```

```
## 
## 	Welch Two Sample t-test
## 
## data:  dep_delay by carrier
## t = 1.4522, df = 52.832, p-value = 0.1524
## alternative hypothesis: true difference in means between group Delta Airlines and group Jet Blue is not equal to 0
## 95 percent confidence interval:
##  -4.351906 27.177268
## sample estimates:
## mean in group Delta Airlines       mean in group Jet Blue 
##                    17.304348                     5.891667
```

---
## Let's break it down.

* dependent variable~independent variable

* Welch correction, this corrects for assumption of equality of variances. And should be the [default](https://www.rips-irsp.com/articles/10.5334/irsp.82/galley/42/download/), also check [this](https://academic.oup.com/beheco/article/17/4/688/215960/The-unequal-variance-t-test-is-an-underused).

---
## No SD?

* We need standard deviations for our report!

```r
sd(jfk_delta$dep_delay_no_miss)
```

```
## [1] 51.17242
```

```r
sd(jfk_jetblue$dep_delay_no_miss)
```

```
## [1] 24.09847
```

---
## APA style write up.

You have done this before!

An independent-samples _t_-test was used to compare the delays in departure between Delta Airlines and Jet Blue in minutes. There was no significant difference between Delta airlines (_M_ = 17.30, _SD_ = 51.17) and Jet Blue (_M_ = 5.89, _SD_= 24.10) in average departure delay time, _t_(52.83)= 1.45, _p_=.15.

---
## Non-parametric tests.

* Do not make typical assumptions associated with parametric statistics (e.g., (roughly) normal distribution).
* Useful in our case, given the clear extreme values!
* Downside: information lost, often less powerful.

---
## Wilcoxon-Mann-Whitney U.

*  Other alternatives exist (e.g., Mood's median test)
*  Wilcoxon-Mann-Whitney U is common [alternative](http://vassarstats.net/textbook/ch11a.html) to _t_-test. (Sometimes, just called Mann-Whitney U).

```r
wilcox.test(dep_delay~carrier, data=violinplot_data)
```

```
## 
## 	Wilcoxon rank sum test with continuity correction
## 
## data:  dep_delay by carrier
## W = 3375.5, p-value = 0.02619
## alternative hypothesis: true location shift is not equal to 0
```

---
## Kruskal-Wallis.

* We need as_factor. Read more [here](http://vassarstats.net/textbook/ch14a.html)

```r
kruskal.test(dep_delay~as_factor(carrier), data=violinplot_data)
```

```
## 
## 	Kruskal-Wallis rank sum test
## 
## data:  dep_delay by as_factor(carrier)
## Kruskal-Wallis chi-squared = 4.9518, df = 1, p-value = 0.02606
```

---
## Medians?

* We need the medians for our report!

```r
median(jfk_delta$dep_delay_no_miss)
```

```
## [1] 1
```

```r
median(jfk_jetblue$dep_delay_no_miss)
```

```
## [1] -1
```

---
## APA style write up.

* A Wilcoxon-Mann-Whitney test showed that there was a significant difference between Delta airlines (_Mdn_ = 1) and Jet Blue (_Mdn_= -1) in delays, _W_= 3375.5, _p_= .026.
* So: Delta airlines significantly later than Jet Blue.

---
## Bootstrapping!

Bootstrapping is sampling ***with replacement***

It is a non-parametric technique.

Basically repeatedly drawing a sample and calculating a statistic.

Can be applied to nearly any technique.

---
## How to do it.

boot package. Find out more [here](http://people.tamu.edu/~alawing/materials/ESSM689/Btutorial.pdf).

We make a function with two arguments (data and indices). We will then feed our data to the function!

```r
### Bootstrap 95% CI for t-test
library(boot)
### function to obtain p.values from t.test
### from the data
p_value <- function(data, indices) {
    data_boot <- data[indices, ]  # allows boot to select sample 
    T_test_boot <- t.test(dep_delay ~ carrier,
        data = data_boot)
    return(T_test_boot[[3]])  ### [[]] denotes the position in output - 3rd element
}
```

---
## Now let's run it.

```r
### Bootstrapping is truly random.  We need a 'seed'
### to make it replicable.
set.seed(2019)
### R is number of samples we want to draw?  we store
### it in results.
results <- boot(data = violinplot_data, statistic = p_value,
    R = 10000)
```

---
## Results.

```r
## view results
head(results) 
```

```
## $t0
## [1] 0.1523715
## 
## $t
##                  [,1]
##     [1,] 0.0941157274
##     [2,] 0.0957792261
##     [3,] 0.4402625512
##     [4,] 0.1290377603
##     [5,] 0.0243861790
##     [6,] 0.1246678194
##     [7,] 0.7349298519
##     [8,] 0.0175446348
##     [9,] 0.0130285588
##    [10,] 0.3148152847
##    [11,] 0.0522036278
##    [12,] 0.3243698718
##    [13,] 0.2937255761
##    [14,] 0.2283020512
##    [15,] 0.3008869621
##    [16,] 0.2972206962
##    [17,] 0.0897802446
##    [18,] 0.6525037309
##    [19,] 0.0436531340
##    [20,] 0.0423311931
##    [21,] 0.0338700753
##    [22,] 0.1700023727
##    [23,] 0.2523342823
##    [24,] 0.2917196961
##    [25,] 0.1847427890
##    [26,] 0.1869619673
##    [27,] 0.4314130177
##    [28,] 0.3857868275
##    [29,] 0.0242086336
##    [30,] 0.4039713337
##    [31,] 0.4185225300
##    [32,] 0.6545329116
##    [33,] 0.1283350654
##    [34,] 0.0364932851
##    [35,] 0.4457353051
##    [36,] 0.3450742123
##    [37,] 0.0398514541
##    [38,] 0.1644253518
##    [39,] 0.0515198861
##    [40,] 0.1271995109
##    [41,] 0.1192853986
##    [42,] 0.1811656317
##    [43,] 0.0709811106
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##  [9735,] 0.1000235959
##  [9736,] 0.2467697775
##  [9737,] 0.0697436357
##  [9738,] 0.1390773541
##  [9739,] 0.0229306067
##  [9740,] 0.0754447704
##  [9741,] 0.6017317045
##  [9742,] 0.0093645964
##  [9743,] 0.1341204780
##  [9744,] 0.0295317894
##  [9745,] 0.1984925652
##  [9746,] 0.2289565036
##  [9747,] 0.2451096642
##  [9748,] 0.1772053339
##  [9749,] 0.4107080259
##  [9750,] 0.0246352696
##  [9751,] 0.6806590404
##  [9752,] 0.1742797134
##  [9753,] 0.2910188968
##  [9754,] 0.1363399100
##  [9755,] 0.1488393942
##  [9756,] 0.6427799139
##  [9757,] 0.8152190423
##  [9758,] 0.4176851263
##  [9759,] 0.0567546777
##  [9760,] 0.1521638493
##  [9761,] 0.1881798579
##  [9762,] 0.9418002410
##  [9763,] 0.4338114228
##  [9764,] 0.0673071745
##  [9765,] 0.7465064643
##  [9766,] 0.0941707885
##  [9767,] 0.8627975300
##  [9768,] 0.0912004006
##  [9769,] 0.1373006361
##  [9770,] 0.1163695457
##  [9771,] 0.3142161822
##  [9772,] 0.3524750819
##  [9773,] 0.0231002778
##  [9774,] 0.4131131487
##  [9775,] 0.1519836863
##  [9776,] 0.5447998537
##  [9777,] 0.8021341036
##  [9778,] 0.8463516596
##  [9779,] 0.8834826364
##  [9780,] 0.0823900268
##  [9781,] 0.2756251651
##  [9782,] 0.9871789036
##  [9783,] 0.3351757794
##  [9784,] 0.8873976989
##  [9785,] 0.2352501627
##  [9786,] 0.0341940444
##  [9787,] 0.1277592927
##  [9788,] 0.1048561746
##  [9789,] 0.9739846147
##  [9790,] 0.1267158509
##  [9791,] 0.0025372104
##  [9792,] 0.0581959322
##  [9793,] 0.0504691214
##  [9794,] 0.0592068788
##  [9795,] 0.8838720884
##  [9796,] 0.3344654131
##  [9797,] 0.1751640056
##  [9798,] 0.0073694351
##  [9799,] 0.0256980200
##  [9800,] 0.1522773248
##  [9801,] 0.0586782798
##  [9802,] 0.0007435812
##  [9803,] 0.0815421151
##  [9804,] 0.8565312594
##  [9805,] 0.0544691464
##  [9806,] 0.0103897344
##  [9807,] 0.9727486544
##  [9808,] 0.2132319425
##  [9809,] 0.3935079494
##  [9810,] 0.1532382366
##  [9811,] 0.7416409605
##  [9812,] 0.2222039472
##  [9813,] 0.3644548421
##  [9814,] 0.0169749776
##  [9815,] 0.0559174987
##  [9816,] 0.0793847924
##  [9817,] 0.0270461226
##  [9818,] 0.0565482620
##  [9819,] 0.2224406622
##  [9820,] 0.2375837598
##  [9821,] 0.2690978238
##  [9822,] 0.1891143804
##  [9823,] 0.2892190836
##  [9824,] 0.0429249538
##  [9825,] 0.2972750139
##  [9826,] 0.0857017034
##  [9827,] 0.8652219306
##  [9828,] 0.1450442717
##  [9829,] 0.1774338034
##  [9830,] 0.1115569966
##  [9831,] 0.0869501606
##  [9832,] 0.0943922415
##  [9833,] 0.0940921972
##  [9834,] 0.0248823359
##  [9835,] 0.3554360963
##  [9836,] 0.2006679880
##  [9837,] 0.5287102276
##  [9838,] 0.3404906709
##  [9839,] 0.0677206398
##  [9840,] 0.4227576676
##  [9841,] 0.0620875852
##  [9842,] 0.0071410724
##  [9843,] 0.1505584017
##  [9844,] 0.0180803823
##  [9845,] 0.5481394440
##  [9846,] 0.7915810547
##  [9847,] 0.1039887006
##  [9848,] 0.3694767465
##  [9849,] 0.8770239689
##  [9850,] 0.3512208334
##  [9851,] 0.0605453285
##  [9852,] 0.4967526726
##  [9853,] 0.1642845514
##  [9854,] 0.1326887568
##  [9855,] 0.2312978848
##  [9856,] 0.0134167552
##  [9857,] 0.3315157180
##  [9858,] 0.0132995960
##  [9859,] 0.1502212315
##  [9860,] 0.8316323284
##  [9861,] 0.2400289785
##  [9862,] 0.5146556504
##  [9863,] 0.0388681901
##  [9864,] 0.4220728243
##  [9865,] 0.2581036599
##  [9866,] 0.1594774361
##  [9867,] 0.3884764176
##  [9868,] 0.3370461240
##  [9869,] 0.0502795562
##  [9870,] 0.0655142003
##  [9871,] 0.6263535604
##  [9872,] 0.5434592391
##  [9873,] 0.9775513410
##  [9874,] 0.0450823253
##  [9875,] 0.0796929904
##  [9876,] 0.3404733707
##  [9877,] 0.0682291088
##  [9878,] 0.4482152487
##  [9879,] 0.0289662463
##  [9880,] 0.1054554151
##  [9881,] 0.3108189104
##  [9882,] 0.0431118599
##  [9883,] 0.9221561982
##  [9884,] 0.1254377549
##  [9885,] 0.0950325638
##  [9886,] 0.2957223663
##  [9887,] 0.6563690783
##  [9888,] 0.2419745865
##  [9889,] 0.0891662996
##  [9890,] 0.0956944718
##  [9891,] 0.0771475702
##  [9892,] 0.2028055970
##  [9893,] 0.4243036311
##  [9894,] 0.6608789984
##  [9895,] 0.2732572806
##  [9896,] 0.2259104355
##  [9897,] 0.1508971050
##  [9898,] 0.2191645317
##  [9899,] 0.0527897418
##  [9900,] 0.6867223430
##  [9901,] 0.0925535091
##  [9902,] 0.0257593424
##  [9903,] 0.1099942362
##  [9904,] 0.6278250222
##  [9905,] 0.0524130157
##  [9906,] 0.0184137269
##  [9907,] 0.0428720975
##  [9908,] 0.0845822038
##  [9909,] 0.0058976631
##  [9910,] 0.0958739487
##  [9911,] 0.2241054559
##  [9912,] 0.2425918956
##  [9913,] 0.1875624707
##  [9914,] 0.1766165743
##  [9915,] 0.0403801607
##  [9916,] 0.2623550332
##  [9917,] 0.7282149803
##  [9918,] 0.2566498727
##  [9919,] 0.7781165667
##  [9920,] 0.0605113627
##  [9921,] 0.0282669253
##  [9922,] 0.0502939471
##  [9923,] 0.3031239554
##  [9924,] 0.5019015357
##  [9925,] 0.9483457905
##  [9926,] 0.7453124246
##  [9927,] 0.1094312783
##  [9928,] 0.0677521062
##  [9929,] 0.1088959764
##  [9930,] 0.7924774559
##  [9931,] 0.6298090186
##  [9932,] 0.0446101999
##  [9933,] 0.6427649612
##  [9934,] 0.0414592095
##  [9935,] 0.2110800176
##  [9936,] 0.3389116756
##  [9937,] 0.9475137517
##  [9938,] 0.4974641732
##  [9939,] 0.0844153320
##  [9940,] 0.2042909844
##  [9941,] 0.6503734917
##  [9942,] 0.0855396824
##  [9943,] 0.5376314928
##  [9944,] 0.2835940844
##  [9945,] 0.1582428118
##  [9946,] 0.2749569404
##  [9947,] 0.4275183824
##  [9948,] 0.0249563222
##  [9949,] 0.2867268378
##  [9950,] 0.0785889601
##  [9951,] 0.0508177025
##  [9952,] 0.2882079598
##  [9953,] 0.0407645334
##  [9954,] 0.0572991935
##  [9955,] 0.0165693333
##  [9956,] 0.0570326550
##  [9957,] 0.2866314195
##  [9958,] 0.0387987448
##  [9959,] 0.0504958610
##  [9960,] 0.0158492929
##  [9961,] 0.4637434529
##  [9962,] 0.0858544245
##  [9963,] 0.2653480451
##  [9964,] 0.1365439304
##  [9965,] 0.0851960364
##  [9966,] 0.1764546893
##  [9967,] 0.2847609816
##  [9968,] 0.3212128320
##  [9969,] 0.2548460073
##  [9970,] 0.6771864589
##  [9971,] 0.0704167075
##  [9972,] 0.2265359831
##  [9973,] 0.4508273689
##  [9974,] 0.5368006379
##  [9975,] 0.1154162922
##  [9976,] 0.0326590147
##  [9977,] 0.4392281650
##  [9978,] 0.0721471760
##  [9979,] 0.3863676749
##  [9980,] 0.0904626769
##  [9981,] 0.1656002504
##  [9982,] 0.7783497575
##  [9983,] 0.7114520289
##  [9984,] 0.8609910692
##  [9985,] 0.1609037502
##  [9986,] 0.2328274337
##  [9987,] 0.4264028151
##  [9988,] 0.1402483860
##  [9989,] 0.2417605880
##  [9990,] 0.0098812368
##  [9991,] 0.0005593404
##  [9992,] 0.0148372864
##  [9993,] 0.7737216068
##  [9994,] 0.1256840094
##  [9995,] 0.1526719521
##  [9996,] 0.0394360224
##  [9997,] 0.1439194982
##  [9998,] 0.0604766426
##  [9999,] 0.2833049659
## [10000,] 0.7071737764
## 
## $R
## [1] 10000
## 
## $data
## # A tibble: 166 × 26
##     year month   day dep_time sched_dep_time dep_delay arr_time sched_arr_time
##    <int> <int> <int>    <int>          <int>     <dbl>    <int>          <int>
##  1  2013    12    25      612            615        -3      859            852
##  2  2013    12    25      654            656        -2     1118           1139
##  3  2013    12    25      656            700        -4     1026           1039
##  4  2013    12    25      700            700         0     1017           1025
##  5  2013    12    25      708            700         8     1117           1025
##  6  2013    12    25      811            810         1     1027           1039
##  7  2013    12    25      820            820         0     1301           1340
##  8  2013    12    25      903            900         3     1215           1226
##  9  2013    12    25     1006            740       146     1253           1053
## 10  2013    12    25     1035           1030         5     1308           1310
## # ℹ 156 more rows
## # ℹ 18 more variables: arr_delay <dbl>, carrier <chr>, flight <int>,
## #   tailnum <chr>, origin <chr>, dest <chr>, air_time <dbl>, distance <dbl>,
## #   hour <dbl>, minute <dbl>, time_hour <dttm>, speed <dbl>, log_speed <dbl>,
## #   speed_cent <dbl>, speed_stand <dbl[,1]>, dep_delay_no_miss <dbl>,
## #   arr_delay_no_miss <dbl>, carrier_rec <fct>
## 
## $seed
##   [1]       10403         624 -1529921617 -1075781084  -189469995   262266258
##   [7] -1703880725  1061082672 -1402866575  -129270914   624249319  -900470532
##  [13]   822238669 -1149925846 -1693019741  -400890232   566723305 -1319982698
##  [19]  -614175969   871979732 -1558282299 -1414639166 -1501539749   726372832
##  [25]   332148833  1767418542 -2024176809  1907398572 -1714868035 -1907187622
##  [31]  -894895597  -669632456  1658707161  1287249606 -1074957681   847576964
##  [37]   889082037 -1435184654  1076324555  -928237168 -1436129199   262055390
##  [43]   911546567  -315755940   896909229 -1984739702 -1515291005   128644584
##  [49]  -934563127  -277127178  1772180991   495898676 -1138532955  -776446430
##  [55] -1416815813 -1516516032  1416600641  -460296946  -901838281   842463500
##  [61]  -105444707  1304066234   791551731  1047797656   137364665    57290022
##  [67]  1299649903   702663908  -836983147   670824018  -963090005   789925104
##  [73]   367447089   241238078  1894668199  -520705092  1476137357  1397438186
##  [79] -1141974173  2060236104  1885983401   924403286 -1373261601   252245396
##  [85]  1949903749   612539010  2029016091  1624488096    42661921   268656494
##  [91]  1576665367  -195556756   762565757   470319386  1652011987 -1580222728
##  [97]  -694674279 -1378112634   203696207 -1219540412   461892725  -548451662
## [103]   603708043  1968312400   769335313  -114652514   985204359  1913001244
## [109] -1290972307  1649760074  1796188227   794305704 -2011569527   801397942
## [115] -1784054849  -272643340 -2066224795  1092945634   316800251 -1781836800
## [121] -1693039615 -1773567538 -1774032905   443272140  1836551773  1230289274
## [127] -1078085453  -436142504   974005369  1504074214 -1767068881   405108644
## [133] -1263001003   864472850  -360072341 -1975876688   758866929 -1600961282
## [139]  1433216359   414461564   488736077 -1930844246  1548281123  2133148680
## [145]  -111943063  -854285546  -387664225  -754233260  -478701755  -793271486
## [151]   429528539  1767546720 -1760985119  -302328786   584072919 -1295368916
## [157] -1371916227 -1313398310 -1103614573  1524626872   445258841  1617049670
## [163]  1674092047 -1245948668  1831159861 -1816493198  1215958091  1218881296
## [169]  1258264529 -1590481058  -557665209  -111609892   682665773   891299850
## [175]  1423086083   806097768   705679945  1319246198  1284465023  -129792588
## [181] -1047513819 -2007863390  -917463365  -329672000  1801235393  1504663182
## [187]   257656247  2009816716 -1607216611 -1119372742  -249621901 -1136354024
## [193]  -603716551  1739015846  -969932561   627827300  1413963285  2063755218
## [199]   134532395  2003729008 -1172325455  1671580094  1349628711  -464752324
## [205]   607303949  1225840746   904179427  1944359624   -62541271  1074053078
## [211]  1096907871  -893359340 -1936080123   688745474   -96626789   463488544
## [217] -1995978847  -402716434 -1013171049  -809029652 -1344513027  1612612250
## [223]   128608083   839316600  1584651289  1628275974  -325397553   680271812
## [229] -1248435211  1886813234 -1889403381  -915800624 -1449930863   477664286
## [235] -2144198137  1441176220   438287085  1121171658   -55100477  -393821656
## [241]  -840072695  2027198006   605226815  -505806732  -301186843 -2147444638
## [247]   546898043  -536436352  1560466305  1937894222    55206775  -854216372
## [253]    95147485   441678586  -854447053  1342110680   -36592647 -1976765594
## [259]  -507439441 -1401615068   411720149   120504466  -523657493  -579784400
## [265]  1146344305  -867299714 -1615068953  1988064252  -789599027   679528746
## [271] -1031653213 -1748328056  2029994473   480877718   761004575    15222228
## [277]  -882921787  1715729602  1285216603   241478880  1346752353 -1563427410
## [283]  -300025257   772727468 -2045100099  -134306982   680419603   369406776
## [289] -1844093991  1822265798 -1940207217  -897668476  1083916213  -566020878
## [295] -1703694389  1046221968 -1319647407  1069201630  1019695047   443481436
## [301]  -887451987 -1943008886 -1132616317  -142069528  1400041929 -1692675338
## [307]  1911844095   390282036  1175194789 -1058045662   712712763  1737648192
## [313]  -943146175  -394182642    -3210953  1560986636  -778069603 -1858602054
## [319]   492242419  -269477224  1865876409  -570990554 -1356862353  -939460636
## [325]   759240085 -1597253294  -157798229  1661118448   328706865   257332030
## [331]  1091309223 -1039321412  1192779917 -1946584598  1004634723  -375951288
## [337]   792760745 -1346160298  -493598753  1078124692 -1148626299 -2128921214
## [343]    41016091 -1738026080   612603681 -2094157202   269842455  1885427052
## [349]  1152639869   169313306  -895214893 -1248251400  1697553305    -2990458
## [355]  -389513393   374601028   435411829    98610610  -881909365 -1371739312
## [361] -1978958063 -1615225442  -230539897 -1716379620  1063330413  -772466102
## [367] -1377448125 -1143971928  1516249481  1647824822  1274254015 -1019267596
## [373]  -984639387 -1545655838 -1195965445   968683264 -1016175871  2104317134
## [379]  1386831607   711628492  -137421475   313867386  1806540723 -1428686504
## [385] -1074519175  1069976806 -1274248657  1389339300  2116784469  -795233774
## [391] -1959755157  1860361904   799752945   561764350  -232661913  2041952636
## [397]  1785517133 -1808078166 -1681748957   671809288 -1630953111   101815830
## [403]  1456098719   328219476   953599557   844318274  -847078181  -798377376
## [409]   -41869599 -1378343122  1463989719  -109148116 -1083619523  -416733990
## [415]  1470862483  2139386040  1199545177  1466690374  1738798351  1187774468
## [421]   224409397  -804329870  1142182731  -540245488   460256977 -1858780578
## [427]  1046670151  -428866852   984839725 -1797068022 -1431324413  1411337832
## [433]  1114968393   988318838 -2016380801  -796004172   664200229  1099928226
## [439]  1561109947  -851036736   666094273 -1224332914   161053879  -134927988
## [445]   737829149  1334525242    30800243  1333172248  1070138169  1280997798
## [451]   910612463  -408875676 -1224094443  -547861806 -1605200853   668063088
## [457]  1615764145 -1216488258   817719847   272170044  -559085555   570760042
## [463] -1679469085  -773501496   123367721  -329993514  1069421407  -980396524
## [469]  -553127419  -223604990   529342107 -1926602464 -1828821343   126835694
## [475] -1318734953  -333022484 -1980077315 -1611430502  -110834093 -1568247944
## [481]  1757960985  1881815046   783100623  1453771460 -1499442443  -443687118
## [487]  -433896181  1506466000   222441105   686663454  2129221895  -748116580
## [493]  1087474157   364477386  1285251779 -1623917272   728875273  1374554422
## [499] -1152704961  -370185356  -392040475  1925233506  1765541755  1492008064
## [505] -1940327807  -636764594  -228631945  1214938188  -505322275 -1199964678
## [511]  -376433869  1821112024   -25844999  1618159206   999222703  -516605916
## [517]  1134282965  -708336750  -138517013  1955564592   759280241  1176281470
## [523]   983480295 -1202258180   127466445  -703067094 -1240866909   635853960
## [529]  1756561641  -280195178   321886495  1627599060   165469637   -85616702
## [535]   709976155  1701431264  1555787361   890458286  -868322985   723070380
## [541]  -131641667    87467610 -1718630381   224341560 -1194796327   114115782
## [547]   597958799    10769796   829697717 -1357015054  1398535883  1660415888
## [553]  -951769519  1000524766  -778729785  -212072356 -1787077203  1429785738
## [559]  -311899005  1003580392  -183095095 -1811397130  1008960511  2049156660
## [565]  1144044453  -801987550  -284874437  -747306176  1226693185  -348253426
## [571] -1699022793  1586138892  1296312477  2118221498  -325325581  1356338584
## [577]  -877002055 -1731161306 -1985690769  1514939108  1335985301  2091367506
## [583]   322172843   -69467408  -565933519    35137086   226870695  1691337148
## [589]   149991309   288603370   602941795   583936840 -2104274775  1538862166
## [595]   115269343 -1331115116  -720008827  1136648322  -544251365 -1343754592
## [601] -1837615583 -1692138130   361557783  1494655084   498070141 -1481472230
## [607]  -504593965  1859038456  -415163751 -1743449722  -330770865 -1081827260
## [613] -1180972427  1434325170  -310482805    34850384  1899486737  1578414238
## [619]   338130055 -1727386852  1014034797   337698122 -1495796157 -2001429848
## [625]  1059217545 -1376304458
## 
## $statistic
## function(data, indices) {
##     data_boot <- data[indices, ]  # allows boot to select sample 
##     T_test_boot <- t.test(dep_delay ~ carrier,
##         data = data_boot)
##     return(T_test_boot[[3]])  ### [[]] denotes the position in output - 3rd element
## }
## <bytecode: 0x1294190b0>
```

---
## Plot Results.

```r
plot(results)
```

---
## 95%CI for _p_.

* Various types. Read all about [here](http://www.tau.ac.il/~saharon/Boot/10.1.1.133.8405.pdf)

* One of the ['best'](https://faculty.washington.edu/heagerty/Courses/b572/public/GregImholte-1.pdf) methods is Bias-Corrected Accelerated.

```r
boot.ci(results, type="bca")
```

```
## BOOTSTRAP CONFIDENCE INTERVAL CALCULATIONS
## Based on 10000 bootstrap replicates
## 
## CALL : 
## boot.ci(boot.out = results, type = "bca")
## 
## Intervals : 
## Level       BCa          
## 95%   ( 0.0059,  0.8570 )  
## Calculations and Intervals on Original Scale
```

---
## What would you write in your report?

* The 95% Bias-Corrected Accelerated interval (using 10,000 bootstrap samples) for _p_ was [.006, .857].

---
## Test it yourself.

* Do the same as shown but now do it for delay in arrival.

---
## Exercise. 
- Install and load the 'carData' package. load the professorial 'Salaries' dataset.
 
- This dataset has 397 cases and 6 variables. ??Salaries. Have a look (perhaps use ??skimr)
 
- Calculate the monthly salary (note that these are 9 month salaries)
 
- Make a 'beautiful' scatter plot with yrs.since.phd on X-axis and monthly salary and save it.
 
- Colour the scatter plot by gender.
 
- Make a panelled 'beautiful' histogram with the distribution of salaries by rank and save it.

---
## Exercise (cont'd).

- Make a 'beautiful' violin plot with gender on the X-axis and monthly salary on Y-axis.

- Test whether men earn more than women do. Use a parametric test and bootstrap it. Also use a non-parametric  test. Report it as you would in an academic paper. Reflect and discuss: Why is this a good/bad testing approach?
 
-  Bonus: calculate the mode of monthly income for each gender. calculate the (parametric) 95%CI for the mean of monthly income for each gender. 
 
---
## References (and further reading.)

* Crawley, M. J. (2013). _The R book: Second edition._ New York, NY: John Wiley & Sons.
* Davison, A. C., & Hinkley, D. V. (1997). _Bootstrap methods and their application_. Cambridge, UK: Cambridge University Press.
* Pruim, R., Kaplan, D., & Horton, N. (2012). [mosaic: project MOSAIC statistics and mathematics teaching utilities](https://cran.r-project.org/doc/contrib/Horton+Pruim+Kaplan_MOSAIC-StudentGuide.pdf). R package version 0.6-2. 
* Siegel, S. & Castellan, N.J. Jr. (1988). _Nonparametric statistics for the behavioral sciences. 2nd Edn._ McGraw-Hill, New York.
* Stulp, G. (2017). _[ggplotgui](https://github.com/gertstulp/ggplotgui)_
* Wickham, H., & Grolemund, G. (2017). _[R for data science](http://r4ds.had.co.nz/)_. Sebastopol, CA: O’Reilly.
* Willems, K. (2015) _[How to Make a Histogram with ggplot2](https://www.datacamp.com/community/tutorials/make-histogram-ggplot2#gs.MSjGyL8)_

---
## Acknowledgments.

[Gert Stulp](http://gertstulp.github.io) provided code for the ggplot2 section of this lecture.

These slides were built with [xaringan](https://github.com/yihui/xaringan).