2.1.1 tibble()

tibble(a = 2.1, b = "Hello", c = TRUE, d = 9L)

## # A tibble: 1 x 4
##       a b     c         d
##   <dbl> <chr> <lgl> <int>
## 1   2.1 Hello TRUE      9

In our code, we specify the variable names (or the names of columns) as , a, b, and c. Under each variable, we give a different value. Under each variable name, the data type is specified for the data within that column. A tibble can consist of one or more atomic vectors, the most important types of which are double, character, logical, and integer. The tibble above includes a variable of each type. (The “L” in “9L” tells R that we want d to be an integer rather than the default, which would be a double.) When you print out a tibble, the variable type is shown below the variable name.

Variables should not begin with a number (like 54abc) or include spaces (like my var). If you insist on using variable names like that, you must include backticks around each name when you reference it.

tibble(`54abc` = 1, `my var` = 2, c = 3)

## # A tibble: 1 x 3
##   `54abc` `my var`     c
##     <dbl>    <dbl> <dbl>
## 1       1        2     3

If we did not include the backticks, R would give us an error.

tibble(54abc = 1, my var = 2, c = 3)

## Error: <text>:1:10: unexpected symbol
## 1: tibble(54abc
##              ^

It is sometimes easier to use the function tribble() to create tibbles.

tribble(
  ~ var1, ~ `var 2`, ~ myvar,
  1,           3,      5,
  4,           6,      8,
)

## # A tibble: 2 x 3
##    var1 `var 2` myvar
##   <dbl>   <dbl> <dbl>
## 1     1       3     5
## 2     4       6     8

The tildes — as in ~ var1 — specify which row has the column names. The formatting makes it easier, relative to specifying raw vectors, to see which values are from the same observation.

2.2.1 Character vectors

tbl_fruit %>% 
  slice_sample(n = 8)

## # A tibble: 8 x 1
##   fruit     
##   <chr>     
## 1 clementine
## 2 ugli fruit
## 3 peach     
## 4 elderberry
## 5 gooseberry
## 6 mandarine 
## 7 mulberry  
## 8 nectarine

Note that slice_sample() selects a random set of rows from the tibble. If you use the argument n, as here, you get back that many rows. Use the argue prop to return a specific percentage of all the rows in the tibble.

str_detect() determines if a character vector matches a pattern. It returns a logical vector that is the same length as the input. Recall logicals are either TRUE or FALSE.

Which fruit names actually include the letter “c?”

tbl_fruit %>% 
  mutate(fruit_in_name = str_detect(fruit, pattern = "c")) 

## # A tibble: 80 x 2
##    fruit        fruit_in_name
##    <chr>        <lgl>        
##  1 apple        FALSE        
##  2 apricot      TRUE         
##  3 avocado      TRUE         
##  4 banana       FALSE        
##  5 bell pepper  FALSE        
##  6 bilberry     FALSE        
##  7 blackberry   TRUE         
##  8 blackcurrant TRUE         
##  9 blood orange FALSE        
## 10 blueberry    FALSE        
## # … with 70 more rows

str_length() counts characters in your strings. Note this is different from the length() of the character vector itself.

tbl_fruit %>% 
  mutate(name_length = str_length(fruit)) 

## # A tibble: 80 x 2
##    fruit        name_length
##    <chr>              <int>
##  1 apple                  5
##  2 apricot                7
##  3 avocado                7
##  4 banana                 6
##  5 bell pepper           11
##  6 bilberry               8
##  7 blackberry            10
##  8 blackcurrant          12
##  9 blood orange          12
## 10 blueberry              9
## # … with 70 more rows

str_sub() extracts parts of a string. The function takes start and end arguments which are vectorised.

tbl_fruit %>% 
  mutate(first_three_letters = str_sub(fruit, 1, 3)) 

## # A tibble: 80 x 2
##    fruit        first_three_letters
##    <chr>        <chr>              
##  1 apple        app                
##  2 apricot      apr                
##  3 avocado      avo                
##  4 banana       ban                
##  5 bell pepper  bel                
##  6 bilberry     bil                
##  7 blackberry   bla                
##  8 blackcurrant bla                
##  9 blood orange blo                
## 10 blueberry    blu                
## # … with 70 more rows

str_c() combines a character vector of length to a single string. This is similar to the normal c() function for creating a vector.

tbl_fruit %>% 
  mutate(name_with_s = str_c(fruit, "s")) 

## # A tibble: 80 x 2
##    fruit        name_with_s  
##    <chr>        <chr>        
##  1 apple        apples       
##  2 apricot      apricots     
##  3 avocado      avocados     
##  4 banana       bananas      
##  5 bell pepper  bell peppers 
##  6 bilberry     bilberrys    
##  7 blackberry   blackberrys  
##  8 blackcurrant blackcurrants
##  9 blood orange blood oranges
## 10 blueberry    blueberrys   
## # … with 70 more rows

str_replace() replaces a pattern within a string.

tbl_fruit %>% 
  mutate(capital_A = str_replace(fruit, 
                                 pattern = "a", 
                                 replacement = "A")) 

## # A tibble: 80 x 2
##    fruit        capital_A   
##    <chr>        <chr>       
##  1 apple        Apple       
##  2 apricot      Apricot     
##  3 avocado      Avocado     
##  4 banana       bAnana      
##  5 bell pepper  bell pepper 
##  6 bilberry     bilberry    
##  7 blackberry   blAckberry  
##  8 blackcurrant blAckcurrant
##  9 blood orange blood orAnge
## 10 blueberry    blueberry   
## # … with 70 more rows

2.2.2 Regular expressions with stringr

Sometimes, your string tasks cannot be expressed in terms of a fixed string, but can be described in terms of a pattern. Regular expressions, also know as “regexes,” are the standard way to specify these patterns. In regexes, specific characters and constructs take on special meaning in order to match multiple strings.

To explore regular expressions, we will use the str_detect() function, which reports TRUE for any string which matches the pattern, and then filter() to see all the matches. For example, here are all the fruits which include a “w” in their name.

tbl_fruit %>%
  filter(str_detect(fruit, pattern = "w"))

## # A tibble: 4 x 1
##   fruit     
##   <chr>     
## 1 honeydew  
## 2 kiwi fruit
## 3 strawberry
## 4 watermelon

In the code below, the first metacharacter is the period . , which stands for any single character, except a newline (which, by the way, is represented by \n). The regex b.r will match all fruits that have an “b,” followed by any single character, followed by “r.” Regexes are case sensitive.

tbl_fruit %>%
  filter(str_detect(fruit, pattern = "b.r"))

## # A tibble: 15 x 1
##    fruit      
##    <chr>      
##  1 bilberry   
##  2 blackberry 
##  3 blueberry  
##  4 boysenberry
##  5 cloudberry 
##  6 cranberry  
##  7 cucumber   
##  8 elderberry 
##  9 goji berry 
## 10 gooseberry 
## 11 huckleberry
## 12 mulberry   
## 13 raspberry  
## 14 salal berry
## 15 strawberry

Anchors can be included to express where the expression must occur within the string. The ^ indicates the beginning of string and $ indicates the end.

tbl_fruit %>%
  filter(str_detect(fruit, pattern = "^w"))

## # A tibble: 1 x 1
##   fruit     
##   <chr>     
## 1 watermelon

tbl_fruit %>%
  filter(str_detect(fruit, pattern = "o$"))

## # A tibble: 5 x 1
##   fruit    
##   <chr>    
## 1 avocado  
## 2 mango    
## 3 pamelo   
## 4 pomelo   
## 5 tamarillo

2.2.3 Raw strings

There are certain characters with special meaning in regexes, including $ * + . ? [ ] ^ { } | ( ) \. This makes things difficult when we want to use these characters within the strings instead of as regexes. Patterns which seek to match these characters, rather than use them as metacharacters, should make use of r"()", the raw string constant in R. Consider:

r"(Do you use "airquotes" much?)"

## [1] "Do you use \"airquotes\" much?"

The \ preceding each quote is what is known as escaping. It signals to the computer that the quotations are part of the string. Now, what happens if we have a backslash within the string.

r"(\)"

## [1] "\\"

As you can see, a backslash is used to escape the initial backslash.

2.3.1 Dropping unused levels

Removing all the rows corresponding to a specific factor level does not remove the level itself. These unused levels can come back to haunt you later, e.g., in figure legends.

Watch what happens to the levels of country when we filter gapminder to a handful of countries.

nlevels(gapminder$country)

## [1] 142

h_gap <- gapminder %>%
  filter(country %in% c("Egypt", "Haiti", 
                        "Romania", "Thailand", 
                        "Venezuela"))
nlevels(h_gap$country)

## [1] 142

Even though h_gap only has data for a handful of countries, we are still schlepping around all levels from the original gapminder tibble.

How can you get rid of them? The base function droplevels() operates on all the factors in a data frame or on a single factor. The function fct_drop() operates on a single factor variable.

h_gap_dropped <- h_gap %>% 
  droplevels()
nlevels(h_gap_dropped$country)

## [1] 5

# Use fct_drop() on a free-range factor

h_gap$country %>%
  fct_drop() %>%
  levels()

## [1] "Egypt"     "Haiti"     "Romania"   "Thailand"  "Venezuela"

2.3.2 Change the order of the levels

By default, factor levels are ordered alphabetically.

gapminder$continent %>%
  levels()

## [1] "Africa"   "Americas" "Asia"     "Europe"   "Oceania"

We can also order factors by:

1. Frequency: Make the most common level the first and so on.
2. Another variable: Order factor levels according to a summary statistic for another variable.

Let’s order continent by frequency using fct_infreq().

gapminder$continent %>% 
  fct_infreq() %>%
  levels()

## [1] "Africa"   "Asia"     "Europe"   "Americas" "Oceania"

We can also have the frequency print out backwards using fct_rev().

gapminder$continent %>% 
  fct_infreq() %>%
  fct_rev() %>% 
  levels()

## [1] "Oceania"  "Americas" "Europe"   "Asia"     "Africa"

These two barcharts of frequency by continent differ only in the order of the continents. Which do you prefer? We show the code for just the second one.

gapminder %>% 
  mutate(continent = fct_infreq(continent)) %>% 
  mutate(continent = fct_rev(continent)) %>% 
  ggplot(aes(x = continent)) +
    geom_bar() +
    coord_flip()

Let’s now order country by another variable, forwards and backwards. This other variable is usually quantitative and you will order the factor according to a grouped summary. The factor is the grouping variable and the default summarizing function is median() but you can specify something else.

# Order countries by median life expectancy

fct_reorder(gapminder$country, 
            gapminder$lifeExp) %>% 
  levels() %>% 
  head()

## [1] "Sierra Leone"  "Guinea-Bissau" "Afghanistan"   "Angola"       
## [5] "Somalia"       "Guinea"

# Order according to minimum life exp instead of median

fct_reorder(gapminder$country, 
            gapminder$lifeExp, min) %>% 
  levels() %>% 
  head()

## [1] "Rwanda"       "Afghanistan"  "Gambia"       "Angola"       "Sierra Leone"
## [6] "Cambodia"

# Backwards!

fct_reorder(gapminder$country, 
            gapminder$lifeExp, 
            .desc = TRUE) %>% 
  levels() %>% 
  head()

## [1] "Iceland"     "Japan"       "Sweden"      "Switzerland" "Netherlands"
## [6] "Norway"

Why do we reorder factor levels? It often makes plots much better! When plotting a factor against a numeric variable, it is generally a good idea to order the factors by some function of the numeric variable. Alphabetic ordering is rarely best.

Compare the interpretability of these two plots of life expectancy in the Aericas in 2007. The only difference is the order of the country factor. Which is better?

gapminder %>% 
  filter(year == 2007, 
         continent == "Americas") %>% 
  ggplot(aes(x = lifeExp, y = country)) + 
    geom_point()

gapminder %>% 
  filter(year == 2007, 
         continent == "Americas") %>% 
  ggplot(aes(x = lifeExp, 
             y = fct_reorder(country, lifeExp))) + 
    geom_point()

2.3.3 Recode the levels

Use fct_recode() to change the names of the levels for a factor.

i_gap <- gapminder %>% 
  filter(country %in% c("United States", "Sweden", 
                        "Australia")) %>% 
  droplevels()

i_gap$country %>% 
  levels()

## [1] "Australia"     "Sweden"        "United States"

i_gap$country %>%
  fct_recode("USA" = "United States", "Oz" = "Australia") %>% 
  levels()

## [1] "Oz"     "Sweden" "USA"

2.3.4 Grow a factor

Let’s create two tibbles, each with data from two countries and dropped unused factor levels.

df1 <- gapminder %>%
  filter(country %in% c("United States", "Mexico"), year > 2000) %>%
  droplevels()
df2 <- gapminder %>%
  filter(country %in% c("France", "Germany"), year > 2000) %>%
  droplevels()

The country factors in df1 and df2 have different levels.

levels(df1$country)

## [1] "Mexico"        "United States"

levels(df2$country)

## [1] "France"  "Germany"

Can you just combine them?

c(df1$country, df2$country)

## [1] Mexico        Mexico        United States United States France       
## [6] France        Germany       Germany      
## Levels: Mexico United States France Germany

Umm, no. That is wrong on many levels! Use fct_c() to do this.

fct_c(df1$country, df2$country)

## [1] Mexico        Mexico        United States United States France       
## [6] France        Germany       Germany      
## Levels: Mexico United States France Germany

2.4.1 Visualizing lists

To explain more complicated list manipulation functions, it’s helpful to have a visual representation of lists. For example, take these three lists:

x1 <- list(c(1, 2), c(3, 4))
x2 <- list(list(1, 2), list(3, 4))
x3 <- list(1, list(2, list(3)))

I’ll draw them as follows:

There are three principles:

1. Lists have rounded corners. Atomic vectors have square corners.

2. Children are drawn inside their parent, and have a slightly darker background to make it easier to see the hierarchy.

3. The orientation of the children (i.e. rows or columns) isn’t important, so I’ll pick a row or column orientation to either save space or illustrate an important property in the example.

2.4.2 Subsetting

There are three ways to subset a list, which I’ll illustrate with a list named a:

a <- list(a = 1:3, b = "a string", c = pi, d = list(-1, -5))

[ ] extracts a sub-list. The result will always be a list.

str(a[1:2])

## List of 2
##  $ a: int [1:3] 1 2 3
##  $ b: chr "a string"

str(a[4])

## List of 1
##  $ d:List of 2
##   ..$ : num -1
##   ..$ : num -5

Like with vectors, you can subset with a logical, integer, or character vector.

[[ ]] extracts a single component from a list. It removes a level of hierarchy from the list.

str(a[[1]])

##  int [1:3] 1 2 3

str(a[[4]])

## List of 2
##  $ : num -1
##  $ : num -5

$ is a shorthand for extracting named elements of a list. It works similarly to [[ ]] except that you don’t need to use quotes.

a$a

## [1] 1 2 3

a[["a"]]

## [1] 1 2 3

The distinction between [ ] and [[ ]] is really important for lists, because [[ ]] drills down into the list while [ ] returns a new, smaller list. Compare the code and output above with the visual representation.

2.5.1 From strings

Date/time data often comes as strings. The lubridate functions automatically work out the format once you specify the order of the component. To use them, identify the order in which year, month, and day appear in your dates, then arrange “y,” “m,” and “d” in the same order. That gives you the name of the lubridate function that will parse your date. For example:

ymd("2017-01-31")

## [1] "2017-01-31"

mdy("January 31st, 2017")

## [1] "2017-01-31"

dmy("31-Jan-2017")

## [1] "2017-01-31"

These functions also take unquoted numbers. This is the most concise way to create a single date/time object, as you might need when filtering date/time data. ymd() is short and unambiguous:

ymd(20170131)

## [1] "2017-01-31"

ymd() and friends create dates. To create a date-time, add an underscore and one or more of “h,” “m,” and “s” to the name of the parsing function:

ymd_hms("2017-01-31 20:11:59")

## [1] "2017-01-31 20:11:59 UTC"

mdy_hm("01/31/2017 08:01")

## [1] "2017-01-31 08:01:00 UTC"

You can also force the creation of a date-time from a date by supplying a timezone:

ymd(20170131, tz = "UTC")

## [1] "2017-01-31 UTC"

2.5.2 From individual components

Instead of a single string, sometimes you’ll have the individual components of the date-time spread across multiple columns. This is what we have in the flights data:

flights %>% 
  select(year, month, day, hour, minute)

## # A tibble: 336,776 x 5
##     year month   day  hour minute
##    <int> <int> <int> <dbl>  <dbl>
##  1  2013     1     1     5     15
##  2  2013     1     1     5     29
##  3  2013     1     1     5     40
##  4  2013     1     1     5     45
##  5  2013     1     1     6      0
##  6  2013     1     1     5     58
##  7  2013     1     1     6      0
##  8  2013     1     1     6      0
##  9  2013     1     1     6      0
## 10  2013     1     1     6      0
## # … with 336,766 more rows

To create a date/time from this sort of input, use make_date() for dates, or make_datetime() for date-times:

flights %>% 
  select(year, month, day, hour, minute) %>% 
  mutate(departure = make_datetime(year, month, day, hour, minute))

## # A tibble: 336,776 x 6
##     year month   day  hour minute departure          
##    <int> <int> <int> <dbl>  <dbl> <dttm>             
##  1  2013     1     1     5     15 2013-01-01 05:15:00
##  2  2013     1     1     5     29 2013-01-01 05:29:00
##  3  2013     1     1     5     40 2013-01-01 05:40:00
##  4  2013     1     1     5     45 2013-01-01 05:45:00
##  5  2013     1     1     6      0 2013-01-01 06:00:00
##  6  2013     1     1     5     58 2013-01-01 05:58:00
##  7  2013     1     1     6      0 2013-01-01 06:00:00
##  8  2013     1     1     6      0 2013-01-01 06:00:00
##  9  2013     1     1     6      0 2013-01-01 06:00:00
## 10  2013     1     1     6      0 2013-01-01 06:00:00
## # … with 336,766 more rows

2.5.3 From other types

You may want to switch between a date-time and a date. That’s the job of as_datetime() and as_date():

as_datetime(today())

## [1] "2021-06-05 UTC"

as_date(now())

## [1] "2021-06-05"

Sometimes you’ll get date/times as numeric offsets from the “Unix Epoch,” 1970-01-01. If the offset is in seconds, use as_datetime(); if it’s in days, use as_date().

as_datetime(60 * 60 * 10)

## [1] "1970-01-01 10:00:00 UTC"

as_date(365 * 10 + 2)

## [1] "1980-01-01"

2.5.4 Date-time components

Now that you know how to get date-time data into R’s date-time data structures, let’s explore what you can do with them. This section will focus on the accessor functions that let you get and set individual components. The next section will look at how arithmetic works with date-times.

You can pull out individual parts of the date with the accessor functions year(), month(), mday() (day of the month), yday() (day of the year), wday() (day of the week), hour(), minute(), and second().

datetime <- ymd_hms("2016-07-08 12:34:56")
year(datetime)

## [1] 2016

month(datetime)

## [1] 7

mday(datetime)

## [1] 8

yday(datetime)

## [1] 190

wday(datetime)

## [1] 6

For month() and wday() you can set label = TRUE to return the abbreviated name of the month or day of the week. Set abbr = FALSE to return the full name.

month(datetime, label = TRUE)

## [1] Jul
## 12 Levels: Jan < Feb < Mar < Apr < May < Jun < Jul < Aug < Sep < ... < Dec

wday(datetime, label = TRUE, abbr = FALSE)

## [1] Friday
## 7 Levels: Sunday < Monday < Tuesday < Wednesday < Thursday < ... < Saturday

2.5.5 Setting components

You can create a new date-time with update().

update(datetime, year = 2020, month = 2, mday = 2, hour = 2)

## [1] "2020-02-02 02:34:56 UTC"

If values are too big, they will roll-over:

ymd("2015-02-01") %>% 
  update(mday = 30)

## [1] "2015-03-02"

ymd("2015-02-01") %>% 
  update(hour = 400)

## [1] "2015-02-17 16:00:00 UTC"

2.5.6 Time zones

Time zones are an enormously complicated topic because of their interaction with geopolitical entities. Fortunately we don’t need to dig into all the details as they’re not all that important for data analysis. You can see the complete list of possible timezones with the function OlsonNames(). Unless otherwise specified, lubridate always uses UTC (Coordinated Universal Time).

In R, the time zone is an attribute of the date-time that only controls printing. For example, these three objects represent the same instant in time:

(x1 <- ymd_hms("2015-06-01 12:00:00", tz = "America/New_York"))

## [1] "2015-06-01 12:00:00 EDT"

(x2 <- ymd_hms("2015-06-01 18:00:00", tz = "Europe/Copenhagen"))

## [1] "2015-06-01 18:00:00 CEST"

(x3 <- ymd_hms("2015-06-02 04:00:00", tz = "Pacific/Auckland"))

## [1] "2015-06-02 04:00:00 NZST"

2.6.1 Joins

Consider two tibbles: superheroes and publishers.

superheroes <- tibble::tribble(
       ~name,   ~gender,     ~publisher,
   "Magneto",   "male",       "Marvel",
     "Storm",   "female",     "Marvel",
    "Batman",   "male",       "DC",
  "Catwoman",   "female",     "DC",
   "Hellboy",   "male",       "Dark Horse Comics"
  )

publishers <- tibble::tribble(
  ~publisher, ~yr_founded,
        "DC",       1934L,
    "Marvel",       1939L,
     "Image",       1992L
  )

Note how easy it is to use tribble(), from the tibble package (part of the Tidyverse) to create a tibble on the fly using text which organized for easy entry and reading. Recall that a double colon — :: — is how we indicate that a function comes from a specific package.

2.6.1.1 inner_join()

inner_join(x, y): Return all rows from x where there are matching values in y, and all columns from x and y. If there are multiple matches between x and y, all combination of the matches are returned.

FIGURE 2.4: Inner join.

inner_join(superheroes, publishers)

## Joining, by = "publisher"

## # A tibble: 4 x 4
##   name     gender publisher yr_founded
##   <chr>    <chr>  <chr>          <int>
## 1 Magneto  male   Marvel          1939
## 2 Storm    female Marvel          1939
## 3 Batman   male   DC              1934
## 4 Catwoman female DC              1934

We lose Hellboy in the join because, although he appears in x = superheroes, his publisher Dark Horse Comics does not appear in y = publishers. The join result has all variables from x = superheroes plus yr_founded, from y.

Note the message that we are ‘Joining, by = “publisher”.’ Whenever joining, R checks to see if there are variables in common between the two tibbles and, if there are, uses them to join. However, it is concerned that you may not be aware that this is what it is doing, so R tells you. Such messages are both annoying and a signal that we have not made our code as robust as we should. Fortunately, we can specify precisely which variables we want to join by. Always do this.

inner_join(superheroes, publishers, by = "publisher")

by also takes a vector of key variables if you want to merge by multiple variables.

Now compare this result to that of using inner_join() with the two datasets in opposite positions.

inner_join(publishers, superheroes, by = "publisher")

## # A tibble: 4 x 4
##   publisher yr_founded name     gender
##   <chr>          <int> <chr>    <chr> 
## 1 DC              1934 Batman   male  
## 2 DC              1934 Catwoman female
## 3 Marvel          1939 Magneto  male  
## 4 Marvel          1939 Storm    female

In a way, this does illustrate multiple matches, if you think about it from the x = publishers direction. Every publisher that has a match in y = superheroes appears multiple times in the result, once for each match. In fact, we’re getting the same result as with inner_join(superheroes, publishers), up to variable order (which you should also never rely on in an analysis).

full_join(superheroes, publishers, by = "publisher")

## # A tibble: 6 x 4
##   name     gender publisher         yr_founded
##   <chr>    <chr>  <chr>                  <int>
## 1 Magneto  male   Marvel                  1939
## 2 Storm    female Marvel                  1939
## 3 Batman   male   DC                      1934
## 4 Catwoman female DC                      1934
## 5 Hellboy  male   Dark Horse Comics         NA
## 6 <NA>     <NA>   Image                   1992

left_join(superheroes, publishers, by = "publisher")

## # A tibble: 5 x 4
##   name     gender publisher         yr_founded
##   <chr>    <chr>  <chr>                  <int>
## 1 Magneto  male   Marvel                  1939
## 2 Storm    female Marvel                  1939
## 3 Batman   male   DC                      1934
## 4 Catwoman female DC                      1934
## 5 Hellboy  male   Dark Horse Comics         NA

left_join(publishers, superheroes, by = "publisher")

## # A tibble: 5 x 4
##   publisher yr_founded name     gender
##   <chr>          <int> <chr>    <chr> 
## 1 DC              1934 Batman   male  
## 2 DC              1934 Catwoman female
## 3 Marvel          1939 Magneto  male  
## 4 Marvel          1939 Storm    female
## 5 Image           1992 <NA>     <NA>

semi_join(superheroes, publishers, by = "publisher")

## # A tibble: 4 x 3
##   name     gender publisher
##   <chr>    <chr>  <chr>    
## 1 Magneto  male   Marvel   
## 2 Storm    female Marvel   
## 3 Batman   male   DC       
## 4 Catwoman female DC

semi_join(x = publishers, y = superheroes, by = "publisher")

## # A tibble: 2 x 2
##   publisher yr_founded
##   <chr>          <int>
## 1 DC              1934
## 2 Marvel          1939

anti_join(superheroes, publishers, by = "publisher")

## # A tibble: 1 x 3
##   name    gender publisher        
##   <chr>   <chr>  <chr>            
## 1 Hellboy male   Dark Horse Comics

anti_join(publishers, superheroes, by = "publisher")

## # A tibble: 1 x 2
##   publisher yr_founded
##   <chr>          <int>
## 1 Image           1992

2.6.2 Example

Consider the relationships among the tibbles in the nycflights package:

FIGURE 2.5: Relationships among nycflights tables

In both the flights and airlines data frames, the key variable we want to join/merge/match the rows by has the same name: carrier. Let’s use inner_join() to join the two data frames, where the rows will be matched by the variable carrier

flights %>% 
  inner_join(airlines, by = "carrier")

## # A tibble: 336,776 x 20
##     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     1     1      517            515         2      830            819
##  2  2013     1     1      533            529         4      850            830
##  3  2013     1     1      542            540         2      923            850
##  4  2013     1     1      544            545        -1     1004           1022
##  5  2013     1     1      554            600        -6      812            837
##  6  2013     1     1      554            558        -4      740            728
##  7  2013     1     1      555            600        -5      913            854
##  8  2013     1     1      557            600        -3      709            723
##  9  2013     1     1      557            600        -3      838            846
## 10  2013     1     1      558            600        -2      753            745
## # … with 336,766 more rows, and 12 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>,
## #   name <chr>

This is our first example of using a join function with a pipe. flights is fed as the first argument to inner_join(). That is what the pipe does. The above code is equivalent to:

inner_join(flights, airlines, by = "carrier")

The airports data frame contains the airport codes for each airport:

airports

## # A tibble: 1,458 x 8
##    faa   name                       lat    lon   alt    tz dst   tzone          
##    <chr> <chr>                    <dbl>  <dbl> <dbl> <dbl> <chr> <chr>          
##  1 04G   Lansdowne Airport         41.1  -80.6  1044    -5 A     America/New_Yo…
##  2 06A   Moton Field Municipal A…  32.5  -85.7   264    -6 A     America/Chicago
##  3 06C   Schaumburg Regional       42.0  -88.1   801    -6 A     America/Chicago
##  4 06N   Randall Airport           41.4  -74.4   523    -5 A     America/New_Yo…
##  5 09J   Jekyll Island Airport     31.1  -81.4    11    -5 A     America/New_Yo…
##  6 0A9   Elizabethton Municipal …  36.4  -82.2  1593    -5 A     America/New_Yo…
##  7 0G6   Williams County Airport   41.5  -84.5   730    -5 A     America/New_Yo…
##  8 0G7   Finger Lakes Regional A…  42.9  -76.8   492    -5 A     America/New_Yo…
##  9 0P2   Shoestring Aviation Air…  39.8  -76.6  1000    -5 U     America/New_Yo…
## 10 0S9   Jefferson County Intl     48.1 -123.    108    -8 A     America/Los_An…
## # … with 1,448 more rows

However, if you look at both the airports and flights data frames, you’ll find that the airport codes are in variables that have different names. In airports the airport code is in faa, whereas in flights the airport codes are in origin and dest.

In order to join these two data frames by airport code, our inner_join() operation will use by = c("dest" = "faa") thereby allowing us to join two data frames where the key variable has a different name.

flights %>% 
  inner_join(airports, by = c("dest" = "faa"))

## # A tibble: 329,174 x 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     1     1      517            515         2      830            819
##  2  2013     1     1      533            529         4      850            830
##  3  2013     1     1      542            540         2      923            850
##  4  2013     1     1      554            600        -6      812            837
##  5  2013     1     1      554            558        -4      740            728
##  6  2013     1     1      555            600        -5      913            854
##  7  2013     1     1      557            600        -3      709            723
##  8  2013     1     1      557            600        -3      838            846
##  9  2013     1     1      558            600        -2      753            745
## 10  2013     1     1      558            600        -2      849            851
## # … with 329,164 more rows, and 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>,
## #   name <chr>, lat <dbl>, lon <dbl>, alt <dbl>, tz <dbl>, dst <chr>,
## #   tzone <chr>

Let’s construct the chain of pipe operators %>% that computes the number of flights from NYC to each destination, but also includes information about each destination airport:

flights %>%
  group_by(dest) %>%
  summarize(num_flights = n(),
            .groups = "drop") %>%
  arrange(desc(num_flights)) %>%
  inner_join(airports, by = c("dest" = "faa")) %>%
  rename(airport_name = name)

## # A tibble: 101 x 9
##    dest  num_flights airport_name          lat    lon   alt    tz dst   tzone   
##    <chr>       <int> <chr>               <dbl>  <dbl> <dbl> <dbl> <chr> <chr>   
##  1 ORD         17283 Chicago Ohare Intl   42.0  -87.9   668    -6 A     America…
##  2 ATL         17215 Hartsfield Jackson…  33.6  -84.4  1026    -5 A     America…
##  3 LAX         16174 Los Angeles Intl     33.9 -118.    126    -8 A     America…
##  4 BOS         15508 General Edward Law…  42.4  -71.0    19    -5 A     America…
##  5 MCO         14082 Orlando Intl         28.4  -81.3    96    -5 A     America…
##  6 CLT         14064 Charlotte Douglas …  35.2  -80.9   748    -5 A     America…
##  7 SFO         13331 San Francisco Intl   37.6 -122.     13    -8 A     America…
##  8 FLL         12055 Fort Lauderdale Ho…  26.1  -80.2     9    -5 A     America…
##  9 MIA         11728 Miami Intl           25.8  -80.3     8    -5 A     America…
## 10 DCA          9705 Ronald Reagan Wash…  38.9  -77.0    15    -5 A     America…
## # … with 91 more rows

"ORD" is the airport code of Chicago O’Hare airport and "FLL" is the code for the main airport in Fort Lauderdale, Florida, which can be seen in the airport_name variable.

2.7.1 Definition of “tidy” data

What does it mean for your data to be “tidy?” While “tidy” has a clear English meaning of “organized,” the word “tidy” in data science using R means that your data follows a standardized format.

“Tidy” data is a standard way of mapping the meaning of a dataset to its structure. A dataset is messy or tidy depending on how rows, columns and tables are matched up with observations, variables and types. In tidy data:

1. Each variable forms a column.
2. Each observation forms a row.
3. Each type of observational unit forms a table.

2.7.2 Converting to “tidy” data

In this book so far, you’ve only seen data frames that were already in “tidy” format. Furthermore, for the rest of this book, you’ll mostly only see data frames that are already in “tidy” format as well. This is not always the case however with all datasets in the world. If your original data frame is in wide (non-“tidy”) format and you would like to use the ggplot2 or dplyr packages, you will first have to convert it to “tidy” format. To do so, we recommend using the pivot_longer() function in the tidyr package (Wickham 2021d).

Going back to our drinks_smaller data frame from earlier:

drinks_smaller

## # A tibble: 4 x 4
##   country       beer spirit  wine
##   <chr>        <int>  <int> <int>
## 1 China           79    192     8
## 2 Italy           85     42   237
## 3 Saudi Arabia     0      5     0
## 4 USA            249    158    84

We convert it to “tidy” format by using the pivot_longer() function from the tidyr package as follows:

drinks_smaller_tidy <- drinks_smaller %>% 
  pivot_longer(names_to = "type", 
               values_to = "servings", 
               cols = -country)
drinks_smaller_tidy

## # A tibble: 12 x 3
##    country      type   servings
##    <chr>        <chr>     <int>
##  1 China        beer         79
##  2 China        spirit      192
##  3 China        wine          8
##  4 Italy        beer         85
##  5 Italy        spirit       42
##  6 Italy        wine        237
##  7 Saudi Arabia beer          0
##  8 Saudi Arabia spirit        5
##  9 Saudi Arabia wine          0
## 10 USA          beer        249
## 11 USA          spirit      158
## 12 USA          wine         84

Let’s dissect the arguments to pivot_longer().

1. the first argumentnames_to corresponds to the name of the variable in the new “tidy”/long data frame that will contain the column names of the original data. Observe how we set names_to = "type". In the resulting drinks_smaller_tidy, the column type contains the three types of alcohol beer, spirit, and wine. Since type is a variable name that doesn’t appear in drinks_smaller, we use quotation marks around it. You’ll receive an error if you just use names_to = type here.

2. The second argument values_to corresponds to the name of the variable in the new “tidy” data frame that will contain the values of the original data. Observe how we set values_to = "servings" since each of the numeric values in each of the beer, wine, and spirit columns of the drinks_smaller data corresponds to a value of servings. In the resulting drinks_smaller_tidy, the column servings contains the 4 \(\times\) 3 = 12 numerical values. Note again that servings doesn’t appear as a variable in drinks_smaller so it again needs quotation marks around it for the values_to argument.

3. The third argument cols is the columns in the drinks_smaller data frame you either want to or don’t want to “tidy.” Observe how we set this to -country indicating that we don’t want to “tidy” the country variable in drinks_smaller and rather only beer, spirit, and wine. Since country is a column that appears in drinks_smaller we don’t put quotation marks around it.

The third argument here of cols is a little nuanced, so let’s consider code that’s written slightly differently but that produces the same output:

drinks_smaller %>% 
  pivot_longer(names_to = "type", 
               values_to = "servings", 
               cols = c(beer, spirit, wine))

Note that the third argument now specifies which columns we want to “tidy” with c(beer, spirit, wine), instead of the columns we don’t want to “tidy” using -country. We use the c() function to create a vector of the columns in drinks_smaller that we’d like to “tidy.” Note that since these three columns appear one after another in the drinks_smaller data frame, we could also do the following for the cols argument:

drinks_smaller %>% 
  pivot_longer(names_to = "type", 
               values_to = "servings", 
               cols = beer:wine)

Converting “wide” format data to “tidy” format often confuses new R users. The only way to learn to get comfortable with the pivot_longer() function is with practice, practice, and more practice using different datasets. For example, run ?pivot_longer and look at the examples in the bottom of the help file.

If however you want to convert a “tidy” data frame to “wide” format, you will need to use the pivot_wider() function instead. Run ?pivot_wider and look at the examples in the bottom of the help file for examples.

You can also view examples of both pivot_longer() and pivot_wider() on the tidyverse.org webpage. There’s a nice example to check out the different functions available for data tidying and a case study using data from the World Health Organization on that webpage. Furthermore, each week the R4DS Online Learning Community posts a dataset in the weekly #TidyTuesday event that might serve as a nice place for you to find other data to explore and transform.

2.8.1 Scaling a distribution

Consider the vector which is the result of rolling one die 10 times.

rolls <- c(5, 5, 1, 5, 4, 2, 6, 2, 1, 5)

There are other ways of storing the data in this vector. Instead of recording every draw, we could just record the number of times each value appears.

table(rolls)

## rolls
## 1 2 4 5 6 
## 2 2 1 4 1

In this case, with only 10 vlaues, it is actually less efficient to store the data like this. But what happens when we have 10,000 rolls.

more_rolls <- rep(rolls, 1000)
table(more_rolls)

## more_rolls
##    1    2    4    5    6 
## 2000 2000 1000 4000 1000

Instead of keeping around a vector of length 10,000, we can just keep 10 values, without losing any information.

This example also highlights the fact that, graphically, two distributions can be identical even if they are of very different lengths.

The two vectors — rolls and more_rolls — have the exact same shape because, even though their lengths differ, they are the same thing. The total count for each value does not matter. What matters is the relative proportions.

Since shape is what matters, we will often “normalize” distributions so that the sum of the counts equals one, meaning that the y-axis is a percentage of the total. Example:

2.8.2 sample()

The most common distributions you will work with are empirical or frequency distributions, the values of age in the trains tibble, the values of poverty in the kenya tibble, and so on. But we can also create our own data by making “draws” from a distribution which we have concocted.

Consider the distribution of the possible values from rolling a fair die. We can use the sample() function to create draws from this distribution, meaning it will change (or sometimes stay the same) for every subsequent draw.

die <- c(1, 2, 3, 4, 5, 6)

sample(x = die, size = 1)

## [1] 3

This produces one “draw” from the distribution of the possible values of one roll of fair six-sided die.

Now, suppose we wanted to roll this die 10 times. One of the arguments of the sample() function is replace. We must specify it as TRUE if values can appear more than once. Since, when rolling a die 10 times, we expect that a value like 3 can appear more than once, we need to set replace = TRUE.

sample(x = die, size = 10, replace = TRUE)

##  [1] 5 6 3 3 3 4 3 6 4 5

In other words, rolling a 1 on the first roll should not preclude you from rolling a 1 on a later roll.

What if the die is not “fair,” meaning that some sides are more likely to appear than others? The final argument of the sample() function is the prob argument. It takes a vector (of the same length as the initial vector x) that contains all of the probabilities of selecting each one of the elements of x. Suppose that the probability of rolling a 1 was 0.5, and the probability of rolling any other value is 0.1. (These probabilities should sum to 1. If they don’t sample() will automatically re-scale them.)

sample(x = die, 
       size = 10, 
       replace = TRUE, 
       prob = c(0.5, 0.1, 0.1, 0.1, 0.1, 0.1))

##  [1] 1 1 3 1 2 1 1 6 1 1

Remember: There is no real data here. We have not actually rolled a die. We have just made some assumptions about what would happen if we were to roll a die. With those assumptions we have built an urn — a data generating mechanism — from which we can draw as many values as we like. Let’s roll the unfair die 10,000 times.

tibble(result = sample(x = die, 
                       size = 10000, 
                       replace = TRUE, 
                       prob = c(0.5, rep(0.1, 5)))) %>% 
  ggplot(aes(x = result)) +
    geom_bar() +
    labs(title = "Distribution of Results of an Unfair Die",
         x = "Result of One Roll",
         y = "Count") +
    scale_x_continuous(breaks = 1:6,
                       labels = as.character(1:6)) +
    scale_y_continuous(labels = scales::comma_format())

This makes the dual nature of distributions more clear. A distribution is the “thing” you see in this plot. A mathematical object with several different parts, including a set of possible values (1 through 6, in this case) and a record of the number of times each value appears. A distribution is also the simple vector of numbers we used to create this plot.

In general, we travel back-and-forth between distribution as a thing and distribution as a vector of draws from the thing, depending on what we are trying to accomplish.

sample() is just one of many functions for creating draws — or, more colloquially, “drawing” — from a distribution. Three of the most important functions are: runif(), rbinom(), and rnorm().

2.8.3 runif()

Consider a “uniform” distribution. This is the case in which every outcome in the range of possible outcomes has the same chance of occurring. The function runif() (spoken as “r-unif”) enables us to draw from a uniform contribution. runif() has three arguments: n, min, and max. runif() will produce n draws from between min and max, with each value having an equal chance of occurring.

runif(n = 10, min = 4, max = 6)

##  [1] 5.6 5.0 4.6 5.7 4.9 5.2 4.3 5.6 4.8 5.6

Mathematically, we would write:

\[y_i \sim U(4, 6)\],

This means that the each value for \(y\) is drawn from a uniform distribution between four and six.

2.8.4 rbinom()

Consider binomial distribution, the case in which the probability of some Boolean variable (for instance success or failure) is calculated for repeated, independent trials. One common example would be the probability of flipping a coin and landing on heads. The function rbinom() allows us to draw from a binomial distribution. This function takes three arguments, n, size, and prob.

n is the number of values we seek to draw. size is the number of trials for each n. *prob is the probability of success on each trial.

Suppose we wanted to flip a fair coin one time, and let landing on heads represent success.

rbinom(n = 1 , size = 1, prob = 0.5)

## [1] 0

Do the same thing 100 times:

tibble(heads = rbinom(n = 100, size = 1, prob = 0.5)) %>% 
  ggplot(aes(x = heads)) +
    geom_bar() +
    labs(title = "Flipping a Fair Coin 100 Times",
         x = "Result",
         y = "Count") +
    scale_x_continuous(breaks = c(0, 1),
                       labels = c("Tails", "Heads"))

In our graph above, we use the function scale_x_continuous() because our x-axis variable is continuous, meaning it can take on any real values. The breaks argument to scale_x_continuous() converts our x-axis to having two different “tick marks.” There is a fairly even distribution of Tails and Heads. More draws would typically result in an even more equal split.

Randomness creates (inevitable) tension between distribution as a “thing” and distribution as a vector of draws from that thing. In this case, the vector of draws is not balanced between Tails and Heads. Yet, we “know” that it should be since the coin is, by definition, fair. In a sense, the mathematics require an even split. Yet, randomness means that the vector of draws will rarely match the mathematically “true” result. And that is OK! First, randomness is an intrinsic property of the real world. Second, we can make the effect of randomness be as small as we want by increasing the number of draws.

Suppose instead we wanted to simulate an unfair coin, where the probability of landing on Heads was 0.75 instead of 0.25.

tibble(heads = rbinom(n = 100, size = 1, prob = 0.75)) %>% 
  ggplot(aes(x = heads)) +
    geom_bar() +
    labs(title = "Flipping a Fair Coin 100 Times",
         x = "Result",
         y = "Count") +
    scale_x_continuous(breaks = c(0, 1),
                       labels = c("Tails", "Heads"))

The distribution — the imaginary urn — from which we draw the results of a coin flip for a fair coin is a different distribution — a different imaginary urn — from the distribution for a biased coin. In fact, there are an infinite number of distributions. Yet as long as we can draw values from a distribution, we can work with it. Mathematics:

\[y_i \sim B(n, p)\].

Each value for \(y\) is drawn from a binomial distribution with parameters \(n\) for the number of trials and \(p\) for the probability of success.

Instead of each n consisting of a single trial, we could have situation in which we are, 10,000 times, flipping a coin 10 times and summing up, for each experiment, the number of heads. In that case:

set.seed(9)
tibble(heads = rbinom(n = 10000, size = 10, prob = 0.5)) %>% 
  ggplot(aes(x = heads)) +
    geom_bar() +
    labs(title = "Flipping a Fair Coin 10 Times",
         subtitle = "Extreme results are possible with enough experiments",
         x = "Total Number of Heads in Ten Flips",
         y = "Count") +
    scale_x_continuous(breaks = 0:10)

2.8.5 rnorm()

The most important distribution is the normal distribution. Mathematics:

\[y_i \sim N(\mu, \sigma^2)\].

Each value \(y_i\) is drawn from a normal distribution with parameters \(\mu\) for the mean and \(\sigma\) for the standard deviation.

This bell-shaped distribution is defined by two parameters: (1) the mean \(\mu\) (spoken as “mu”) which locates the center of the distribution and (2) the standard deviation \(\sigma\) (spoken as “sigma”) which determines the variation of values around that center. In the figure below, we plot three normal distributions where:

1. The solid normal curve has mean \(\mu = 5\) & standard deviation \(\sigma = 2\).
2. The dotted normal curve has mean \(\mu = 5\) & standard deviation \(\sigma = 5\).
3. The dashed normal curve has mean \(\mu = 15\) & standard deviation \(\sigma = 2\).

FIGURE 2.6: Three normal distributions.

Notice how the solid and dotted line normal curves have the same center due to their common mean \(\mu\) = 5. However, the dotted line normal curve is wider due to its larger standard deviation of \(\sigma = 5\). On the other hand, the solid and dashed line normal curves have the same variation due to their common standard deviation \(\sigma = 2\). However, they are centered at different locations.

When the mean \(\mu = 0\) and the standard deviation \(\sigma = 1\), the normal distribution has a special name. It’s called the standard normal distribution or the \(z\)-curve.

Furthermore, if a variable follows a normal curve, there are three rules of thumb we can use:

1. 68% of values will lie within \(\pm\) 1 standard deviation of the mean.
2. 95% of values will lie within \(\pm\) 1.96 \(\approx\) 2 standard deviations of the mean.
3. 99.7% of values will lie within \(\pm\) 3 standard deviations of the mean.

Let’s illustrate this on a standard normal curve. The dashed lines are at -3, -1.96, -1, 0, 1, 1.96, and 3. These 7 lines cut up the x-axis into 8 segments. The areas under the normal curve for each of the 8 segments are marked and add up to 100%. For example:

1. The middle two segments represent the interval -1 to 1. The shaded area above this interval represents 34% + 34% = 68% of the area under the curve. In other words, 68% of values.
2. The middle four segments represent the interval -1.96 to 1.96. The shaded area above this interval represents 13.5% + 34% + 34% + 13.5% = 95% of the area under the curve. In other words, 95% of values.
3. The middle six segments represent the interval -3 to 3. The shaded area above this interval represents 2.35% + 13.5% + 34% + 34% + 13.5% + 2.35% = 99.7% of the area under the curve. In other words, 99.7% of values.

FIGURE 2.7: Rules of thumb about areas under normal curves.

The function rnorm() (spoken as “r-norm”) returns draws from a normal distribution. rnorm() has three arguments: n, mean, and sd. n corresponds to the number of draws, mean and sd are the \(\mu\) and \(\sigma\) of the distribution from which we want to draw. Again, imagine an urn filled with beads. Each bead has a number written on it. If the distribution is standard normal, then we can draw 10 beads from the urn by running:

rnorm(10)

##  [1] -0.84  1.38 -1.26  0.07  1.71 -0.60 -0.47 -0.64 -0.29  0.14

These 10 draws come from a distribution with the default mean of 0 and the default sd of 1. What if we create a histogram of the values?

tibble(value = rnorm(10)) %>% 
  ggplot(aes(x = value)) + 
    geom_histogram(bins = 10)

As you can see, it is not as symmetrical as the one displayed above. This is not surprising! If you just draw 10 beads from the urn, you can not possibly have a very good sense of what all the numbers on all the beads in the urn look like. What if we draw 100 values? What about 100,000?

tibble(value = rnorm(100)) %>% 
  ggplot(aes(x = value)) +
    geom_histogram(bins = 10)

tibble(value = rnorm(100000)) %>% 
  ggplot(aes(x = value)) +
    geom_histogram(bins = 1000)

Now it’s looking a lot more similar to the “truth,” although still imperfect.

Now, let’s compare normal distributions with varying means and standard deviations, which can be set using the mean and sd arguments included with the function.

tibble(rnorm_5_1 = rnorm(n = 1000, mean = 5, sd = 1), 
       rnorm_0_3 = rnorm(n = 1000, mean = 0, sd = 3),
       rnorm_0_1 = rnorm(n = 1000, mean = 0, sd = 1)) %>%
  pivot_longer(cols = everything(), 
               names_to = "distribution", 
               values_to = "value") %>% 
  ggplot(aes(x = value, fill = distribution)) +
    geom_density(alpha = 0.5) +
    labs(title = "Comparison of Normal Distributions with Differing Mean and Standard Deviation Values", 
         fill = "Distribution",
         x = "Value",
         y = "Density")

2.8.6 Working with draws

Once we have a vector of draws, we can examine various aspects of the distribution. Examples:

draws <- rnorm(100, mean = 2, sd = 1)

This is a case in which there is no distinction between the true distribution and the estimated distribution. We know, by assumption, what the truth is.

Even though we know, because we wrote the code, that the draws come from a normal distribution with a mean of 2 and a standard deviation of 1, the calculated results will not match those values exactly because the draws themselves are random.

mean(draws)

## [1] 1.9

sd(draws)

## [1] 1

Note that we are more likely to use the median and the mad to summarize a distribution. In this case, they are very similar to the mean and standard deviation.

median(draws)

## [1] 1.9

mad(draws)

## [1] 1.1

In practice, we will not know the exact distribution which generates our data. (If we did know, then estimation would not be necessary.) The inherent randomness of the world means that calculated statistics will not match the underlying truth perfectly. But the more data that we collect, the closer the match will be.

In addition to the mean and standard deviation of the draws, we will often be interested in various quantiles of the distribution, most commonly because we want to create intervals which cover a specified portion of the draws. Examples:

quantile(draws, probs = c(0.25, 0.75))

## 25% 75% 
## 1.2 2.7

quantile(draws, probs = c(0.05, 0.95))

##  5% 95% 
## 0.4 3.5

quantile(draws, probs = c(0.025, 0.975))

##  2.5% 97.5% 
##  0.15  3.80

Note that these draws come from a distribution which is centered around 2 rather than 0. There is nothing intrinsically special about any of these ranges. They are mere convention, especially the 95% interval.

Note how cavalier we are in sometimes using the word “distribution” and sometimes the word “draws.” These are two different things! The distribution is the underlying reality, which we will only know for certain when we create it ourselves, as in this example. The draws are a vector of numbers which, we assume, are “drawn” from some underlying distribution which, in general, we do not know.

By assumption, we can analyze the draws to make inferences about the distribution.

Although distributions (and the draws therefrom) are complex, we can often treat them in the same way that we treat simple numbers. For example, we can add two distributions together.

n <- 100000
tibble(Normal = rnorm(n, mean = 1),
       Uniform = runif(n, min = 2, max = 3),
       Combined = Normal + Uniform) %>% 
  pivot_longer(cols = everything(),
               names_to = "Distribution",
               values_to = "draw") %>% 
  ggplot(aes(x = draw, fill = Distribution)) +
    geom_histogram(aes(y = after_stat(count/sum(count))),
                   alpha = 0.5, 
                   bins = 100, 
                   position = "identity") +
    labs(title = "Two Distributions and Their Sum",
         subtitle = "You can sum distributions just like you sum numbers",
         x = "Value",
         y = "Probability")

Drawing from a distribution also allows us to answer questions via simulation. For example, imagine that A and B are both flipping fair coins. A flips the coin 3 times. B flips the coin 6 times. What is the probability that A flips more heads than B?

It is obvious that B will win this game more often than A. It is also obvious that A will win some of the time. But in order to estimate the chances of A winning, we can simply simulate playing the game 1,000 times.

set.seed(56)
games <- 1000 

tibble(A_heads = rbinom(n = games, size = 3, prob = 0.5),
       B_heads = rbinom(n = games, size = 6, prob = 0.5)) %>% 
  mutate(A_wins = ifelse(A_heads > B_heads, 1, 0)) %>% 
  summarize(A_chances = mean(A_wins))

## # A tibble: 1 x 1
##   A_chances
##       <dbl>
## 1     0.091

A has about a 9% chance of winning the game.

In data science, the most important kind of distribution is a probability distribution, a concept which we will introduce in Chapter 5.

2.9.1 Matrices

Recall that a “matrix” in R is a rectangular array of data, shaped like a data frame or tibble, but containing only one type of data, e.g., numeric. Large matrices also print out ugly. (There are other differences, none of which we care about here.) Example:

m <- matrix(c(3, 4, 8, 9, 12, 13, 0, 15, -1), ncol = 3)
m

##      [,1] [,2] [,3]
## [1,]    3    9    0
## [2,]    4   12   15
## [3,]    8   13   -1

The easiest way to pull information from a matrix is to use [ ], the subset operator. Here is how we grab the second and third columns of m:

m[, 2:3]

##      [,1] [,2]
## [1,]    9    0
## [2,]   12   15
## [3,]   13   -1

Note, however, that matrices with just one dimension “collapse” into single vectors:

m[, 2]

## [1]  9 12 13

Tibbles, on the other hand, always maintain their rectangular shapes, even with only one column or row.

We can turn matrices into tibbles with as_tibble().

m %>% 
  as_tibble()

## Warning: The `x` argument of `as_tibble.matrix()` must have unique column names if `.name_repair` is omitted as of tibble 2.0.0.
## Using compatibility `.name_repair`.

## # A tibble: 3 x 3
##      V1    V2    V3
##   <dbl> <dbl> <dbl>
## 1     3     9     0
## 2     4    12    15
## 3     8    13    -1

Because m does not have column names, as_tibble() creates its won variables names, using “V” for variable.

2.9.2 Missing Values

Some observations in a tibble are blank. These are called missing values, and they are often marked as NA. We can create such a tibble as follows:

tbl <- tribble(
  ~ a, ~ b, ~ c,
    2,   3,   5,
    4,  NA,   8,
   NA,   7,   9,
)

tbl

## # A tibble: 3 x 3
##       a     b     c
##   <dbl> <dbl> <dbl>
## 1     2     3     5
## 2     4    NA     8
## 3    NA     7     9

The presence of NA values can be problematic.

tbl %>% 
  summarize(avg_a = mean(a))

## # A tibble: 1 x 1
##   avg_a
##   <dbl>
## 1    NA

Fortunately, most R functions take an argument, na.rm, which, when set to TRUE, removes NA values from any calculations.

tbl %>% 
  summarize(avg_a = mean(a, na.rm = TRUE))

## # A tibble: 1 x 1
##   avg_a
##   <dbl>
## 1     3

Another approach is to use drop_na().

tbl %>% 
  drop_na(a)

## # A tibble: 2 x 3
##       a     b     c
##   <dbl> <dbl> <dbl>
## 1     2     3     5
## 2     4    NA     8

Be careful, however, if you use drop_na() without a specific variable provided. In that case, you will remove all rows with a missing value for any variable in the tibble.

tbl %>% 
  drop_na()

## # A tibble: 1 x 3
##       a     b     c
##   <dbl> <dbl> <dbl>
## 1     2     3     5

A final approach is to use is.na() to explicitly determine if a value is missing.

tbl %>% 
  mutate(a_missing = is.na(a))

## # A tibble: 3 x 4
##       a     b     c a_missing
##   <dbl> <dbl> <dbl> <lgl>    
## 1     2     3     5 FALSE    
## 2     4    NA     8 FALSE    
## 3    NA     7     9 TRUE

2.9.3 Working by rows

Tibbles and the main Tidyverse functions are designed to work by columns. You do something to all the values for variable a. But, sometimes, we want to work across the tibble, comparing the value for a in the first row to the value of b in the first row, and so on. To do that, we need two tricks. First, we use rowwise() to inform R that the next set of commands should be executed across the rows.

tbl %>% 
  rowwise()

## # A tibble: 3 x 3
## # Rowwise: 
##       a     b     c
##   <dbl> <dbl> <dbl>
## 1     2     3     5
## 2     4    NA     8
## 3    NA     7     9

Note how “# Rowwise:” is printed out. Having set up the pipe to work across rows, we need to pass c_across() to whichever function we are using, generally specifying the variables we want to use. If we don’t provide any arguments to c_across(), it will use all the columns in the tibble.

tbl %>% 
  rowwise() %>% 
  mutate(sum_a_c = sum(c_across(c(a, c)))) %>% 
  mutate(largest = max(c_across())) %>% 
  mutate(largest_na = max(c_across(), na.rm = TRUE))

## # A tibble: 3 x 6
## # Rowwise: 
##       a     b     c sum_a_c largest largest_na
##   <dbl> <dbl> <dbl>   <dbl>   <dbl>      <dbl>
## 1     2     3     5       7       7          7
## 2     4    NA     8      12      NA         12
## 3    NA     7     9      NA      NA          9