logicals.qmd

# Logical vectors {#sec-logicals}

```{r}
#| echo: false

source("_common.R")
```

## Introduction

In this chapter, you'll learn tools for working with logical vectors.
Logical vectors are the simplest type of vector because each element can only be one of three possible values: `TRUE`, `FALSE`, and `NA`.
It's relatively rare to find logical vectors in your raw data, but you'll create and manipulate them in the course of almost every analysis.

We'll begin by discussing the most common way of creating logical vectors: with numeric comparisons.
Then you'll learn about how you can use Boolean algebra to combine different logical vectors, as well as some useful summaries.
We'll finish off with `if_else()` and `case_when()`, two useful functions for making conditional changes powered by logical vectors.

### Prerequisites

Most of the functions you'll learn about in this chapter are provided by base R, so we don't need the tidyverse, but we'll still load it so we can use `mutate()`, `filter()`, and friends to work with data frames.
We'll also continue to draw examples from the `nycflights13::flights` dataset.

```{r}
#| label: setup
#| message: false

library(tidyverse)
library(nycflights13)
```

However, as we start to cover more tools, there won't always be a perfect real example.
So we'll start making up some dummy data with `c()`:

```{r}
x <- c(1, 2, 3, 5, 7, 11, 13)
x * 2
```

This makes it easier to explain individual functions at the cost of making it harder to see how it might apply to your data problems.
Just remember that any manipulation we do to a free-floating vector, you can do to a variable inside a data frame with `mutate()` and friends.

```{r}
df <- tibble(x)
df |> 
  mutate(y = x * 2)
```

## Comparisons

A very common way to create a logical vector is via a numeric comparison with `<`, `<=`, `>`, `>=`, `!=`, and `==`.
So far, we've mostly created logical variables transiently within `filter()` --- they are computed, used, and then thrown away.
For example, the following filter finds all daytime departures that arrive roughly on time:

```{r}
flights |> 
  filter(dep_time > 600 & dep_time < 2000 & abs(arr_delay) < 20)
```

It's useful to know that this is a shortcut and you can explicitly create the underlying logical variables with `mutate()`:

```{r}
flights |> 
  mutate(
    daytime = dep_time > 600 & dep_time < 2000,
    approx_ontime = abs(arr_delay) < 20,
    .keep = "used"
  )
```

This is particularly useful for more complicated logic because naming the intermediate steps makes it easier to both read your code and check that each step has been computed correctly.

All up, the initial filter is equivalent to:

```{r}
#| results: false

flights |> 
  mutate(
    daytime = dep_time > 600 & dep_time < 2000,
    approx_ontime = abs(arr_delay) < 20,
  ) |> 
  filter(daytime & approx_ontime)
```

### Floating point comparison {#sec-fp-comparison}

Beware of using `==` with numbers.
For example, it looks like this vector contains the numbers 1 and 2:

```{r}
x <- c(1 / 49 * 49, sqrt(2) ^ 2)
x
```

But if you test them for equality, you get `FALSE`:

```{r}
x == c(1, 2)
```

What's going on?
Computers store numbers with a fixed number of decimal places so there's no way to exactly represent 1/49 or `sqrt(2)` and subsequent computations will be very slightly off.
We can see the exact values by calling `print()` with the `digits`[^logicals-1] argument:

[^logicals-1]: R normally calls print for you (i.e. `x` is a shortcut for `print(x)`), but calling it explicitly is useful if you want to provide other arguments.

```{r}
print(x, digits = 16)
```

You can see why R defaults to rounding these numbers; they really are very close to what you expect.

Now that you've seen why `==` is failing, what can you do about it?
One option is to use `dplyr::near()` which ignores small differences:

```{r}
near(x, c(1, 2))
```

### Missing values {#sec-na-comparison}

Missing values represent the unknown so they are "contagious": almost any operation involving an unknown value will also be unknown:

```{r}
NA > 5
10 == NA
```

The most confusing result is this one:

```{r}
NA == NA
```

It's easiest to understand why this is true if we artificially supply a little more context:

```{r}
# We don't know how old Mary is
age_mary <- NA

# We don't know how old John is
age_john <- NA

# Are Mary and John the same age?
age_mary == age_john
# We don't know!
```

So if you want to find all flights where `dep_time` is missing, the following code doesn't work because `dep_time == NA` will yield `NA` for every single row, and `filter()` automatically drops missing values:

```{r}
flights |> 
  filter(dep_time == NA)
```

Instead we'll need a new tool: `is.na()`.

### `is.na()`

`is.na(x)` works with any type of vector and returns `TRUE` for missing values and `FALSE` for everything else:

```{r}
is.na(c(TRUE, NA, FALSE))
is.na(c(1, NA, 3))
is.na(c("a", NA, "b"))
```

We can use `is.na()` to find all the rows with a missing `dep_time`:

```{r}
flights |> 
  filter(is.na(dep_time))
```

`is.na()` can also be useful in `arrange()`.
`arrange()` usually puts all the missing values at the end but you can override this default by first sorting by `is.na()`:

```{r}
flights |> 
  filter(month == 1, day == 1) |> 
  arrange(dep_time)

flights |> 
  filter(month == 1, day == 1) |> 
  arrange(desc(is.na(dep_time)), dep_time)
```

We'll come back to cover missing values in more depth in @sec-missing-values.

### Exercises

1.  How does `dplyr::near()` work? Type `near` to see the source code. Is `sqrt(2)^2` near 2?
2.  Use `mutate()`, `is.na()`, and `count()` together to describe how the missing values in `dep_time`, `sched_dep_time` and `dep_delay` are connected.

## Boolean algebra

Once you have multiple logical vectors, you can combine them together using Boolean algebra.
In R, `&` is "and", `|` is "or", `!` is "not", and `xor()` is exclusive or[^logicals-2].
For example, `df |> filter(!is.na(x))` finds all rows where `x` is not missing and `df |> filter(x < -10 | x > 0)` finds all rows where `x` is smaller than -10 or bigger than 0.
@fig-bool-ops shows the complete set of Boolean operations and how they work.

[^logicals-2]: That is, `xor(x, y)` is true if x is true, or y is true, but not both.
    This is how we usually use "or" In English.
    "Both" is not usually an acceptable answer to the question "would you like ice cream or cake?".

```{r}
#| label: fig-bool-ops
#| echo: false
#| out-width: NULL
#| fig-cap: | 
#|    The complete set of Boolean operations. `x` is the left-hand
#|    circle, `y` is the right-hand circle, and the shaded regions show 
#|    which parts each operator selects.
#| fig-alt: |
#|    Seven Venn diagrams, each explaining a given logical operator. The
#|    circles (sets) in each of the Venn diagrams represent x and y. x &
#|    !y is x but none of y; x & y is the intersection of x and y; !x & y is
#|    y but none of x; x is all of x; xor(x, y) is everything
#|    except the intersection of x and y; y is all of y; and 
#|    x | y is everything.
knitr::include_graphics("diagrams/transform.png", dpi = 270)
```

As well as `&` and `|`, R also has `&&` and `||`.
Don't use them in dplyr functions!
These are called short-circuiting operators and only ever return a single `TRUE` or `FALSE`.
They're important for programming, not data science.

### Missing values {#sec-na-boolean}

The rules for missing values in Boolean algebra are a little tricky to explain because they seem inconsistent at first glance:

```{r}
df <- tibble(x = c(TRUE, FALSE, NA))

df |> 
  mutate(
    and = x & NA,
    or = x | NA
  )
```

To understand what's going on, think about `NA | TRUE` (`NA` or `TRUE`).
A missing value in a logical vector means that the value could either be `TRUE` or `FALSE`.
`TRUE | TRUE` and `FALSE | TRUE` are both `TRUE` because at least one of them is `TRUE`.
`NA | TRUE` must also be `TRUE` because `NA` can either be `TRUE` or `FALSE`.
However, `NA | FALSE` is `NA` because we don't know if `NA` is `TRUE` or `FALSE`.
Similar reasoning applies with `NA & FALSE`.

### Order of operations {#sec-order-operations-boolean}

Note that the order of operations doesn't work like English.
Take the following code that finds all flights that departed in November or December:

```{r}
#| eval: false

flights |> 
   filter(month == 11 | month == 12)
```

You might be tempted to write it like you'd say in English: "Find all flights that departed in November or December.":

```{r}
flights |> 
   filter(month == 11 | 12)
```

This code doesn't error but it also doesn't seem to have worked.
What's going on?
Here, R first evaluates `month == 11` creating a logical vector, which we call `nov`.
It computes `nov | 12`.
When you use a number with a logical operator it converts everything apart from 0 to `TRUE`, so this is equivalent to `nov | TRUE` which will always be `TRUE`, so every row will be selected:

```{r}
flights |> 
  mutate(
    nov = month == 11,
    final = nov | 12,
    .keep = "used"
  )
```

### `%in%`

An easy way to avoid the problem of getting your `==`s and `|`s in the right order is to use `%in%`.
`x %in% y` returns a logical vector the same length as `x` that is `TRUE` whenever a value in `x` is anywhere in `y` .

```{r}
1:12 %in% c(1, 5, 11)
letters[1:10] %in% c("a", "e", "i", "o", "u")
```

So to find all flights in November and December we could write:

```{r}
#| eval: false

flights |> 
  filter(month %in% c(11, 12))
```

Note that `%in%` obeys different rules for `NA` to `==`, as `NA %in% NA` is `TRUE`.

```{r}
c(1, 2, NA) == NA
c(1, 2, NA) %in% NA
```

This can make for a useful shortcut:

```{r}
flights |> 
  filter(dep_time %in% c(NA, 0800))
```

### Exercises

1.  Find all flights where `arr_delay` is missing but `dep_delay` is not. Find all flights where neither `arr_time` nor `sched_arr_time` are missing, but `arr_delay` is.
2.  How many flights have a missing `dep_time`? What other variables are missing in these rows? What might these rows represent?
3.  Assuming that a missing `dep_time` implies that a flight is cancelled, look at the number of cancelled flights per day. Is there a pattern? Is there a connection between the proportion of cancelled flights and the average delay of non-cancelled flights?

## Summaries {#sec-logical-summaries}

The following sections describe some useful techniques for summarizing logical vectors.
As well as functions that only work specifically with logical vectors, you can also use functions that work with numeric vectors.

### Logical summaries

There are two main logical summaries: `any()` and `all()`.
`any(x)` is the equivalent of `|`; it'll return `TRUE` if there are any `TRUE`'s in `x`.
`all(x)` is equivalent of `&`; it'll return `TRUE` only if all values of `x` are `TRUE`'s.
Like all summary functions, they'll return `NA` if there are any missing values present, and as usual you can make the missing values go away with `na.rm = TRUE`.

For example, we could use `all()` and `any()` to find out if every flight was delayed on departure by at most an hour or if any flights were delayed on arrival by five hours or more.
And using `group_by()` allows us to do that by day:

```{r}
flights |> 
  group_by(year, month, day) |> 
  summarize(
    all_delayed = all(dep_delay <= 60, na.rm = TRUE),
    any_long_delay = any(arr_delay >= 300, na.rm = TRUE),
    .groups = "drop"
  )
```

In most cases, however, `any()` and `all()` are a little too crude, and it would be nice to be able to get a little more detail about how many values are `TRUE` or `FALSE`.
That leads us to the numeric summaries.

### Numeric summaries of logical vectors {#sec-numeric-summaries-of-logicals}

When you use a logical vector in a numeric context, `TRUE` becomes 1 and `FALSE` becomes 0.
This makes `sum()` and `mean()` very useful with logical vectors because `sum(x)` gives the number of `TRUE`s and `mean(x)` gives the proportion of `TRUE`s (because `mean()` is just `sum()` divided by `length()`).

That, for example, allows us to see the proportion of flights that were delayed on departure by at most an hour and the number of flights that were delayed on arrival by five hours or more:

```{r}
flights |> 
  group_by(year, month, day) |> 
  summarize(
    proportion_delayed = mean(dep_delay <= 60, na.rm = TRUE),
    count_long_delay = sum(arr_delay >= 300, na.rm = TRUE),
    .groups = "drop"
  )
```

### Logical subsetting

There's one final use for logical vectors in summaries: you can use a logical vector to filter a single variable to a subset of interest.
This makes use of the base `[` (pronounced subset) operator, which you'll learn more about in @sec-subset-many.

Imagine we wanted to look at the average delay just for flights that were actually delayed.
One way to do so would be to first filter the flights and then calculate the average delay:

```{r}
flights |> 
  filter(arr_delay > 0) |> 
  group_by(year, month, day) |> 
  summarize(
    behind = mean(arr_delay),
    n = n(),
    .groups = "drop"
  )
```

This works, but what if we wanted to also compute the average delay for flights that arrived early?
We'd need to perform a separate filter step, and then figure out how to combine the two data frames together[^logicals-3].
Instead you could use `[` to perform an inline filtering: `arr_delay[arr_delay > 0]` will yield only the positive arrival delays.

[^logicals-3]: We'll cover this in @sec-joins.

This leads to:

```{r}
flights |> 
  group_by(year, month, day) |> 
  summarize(
    behind = mean(arr_delay[arr_delay > 0], na.rm = TRUE),
    ahead = mean(arr_delay[arr_delay < 0], na.rm = TRUE),
    n = n(),
    .groups = "drop"
  )
```

Also note the difference in the group size: in the first chunk `n()` gives the number of delayed flights per day; in the second, `n()` gives the total number of flights.

### Exercises

1.  What will `sum(is.na(x))` tell you? How about `mean(is.na(x))`?
2.  What does `prod()` return when applied to a logical vector? What logical summary function is it equivalent to? What does `min()` return when applied to a logical vector? What logical summary function is it equivalent to? Read the documentation and perform a few experiments.

## Conditional transformations

One of the most powerful features of logical vectors are their use for conditional transformations, i.e. doing one thing for condition x, and something different for condition y.
There are two important tools for this: `if_else()` and `case_when()`.

### `if_else()`

If you want to use one value when a condition is `TRUE` and another value when it's `FALSE`, you can use `dplyr::if_else()`[^logicals-4].
You'll always use the first three argument of `if_else()`. The first argument, `condition`, is a logical vector, the second, `true`, gives the output when the condition is true, and the third, `false`, gives the output if the condition is false.

[^logicals-4]: dplyr's `if_else()` is very similar to base R's `ifelse()`.
    There are two main advantages of `if_else()`over `ifelse()`: you can choose what should happen to missing values, and `if_else()` is much more likely to give you a meaningful error if your variables have incompatible types.

Let's begin with a simple example of labeling a numeric vector as either "+ve" (positive) or "-ve" (negative):

```{r}
x <- c(-3:3, NA)
if_else(x > 0, "+ve", "-ve")
```

There's an optional fourth argument, `missing` which will be used if the input is `NA`:

```{r}
if_else(x > 0, "+ve", "-ve", "???")
```

You can also use vectors for the `true` and `false` arguments.
For example, this allows us to create a minimal implementation of `abs()`:

```{r}
if_else(x < 0, -x, x)
```

So far all the arguments have used the same vectors, but you can of course mix and match.
For example, you could implement a simple version of `coalesce()` like this:

```{r}
x1 <- c(NA, 1, 2, NA)
y1 <- c(3, NA, 4, 6)
if_else(is.na(x1), y1, x1)
```

You might have noticed a small infelicity in our labeling example above: zero is neither positive nor negative.
We could resolve this by adding an additional `if_else()`:

```{r}
if_else(x == 0, "0", if_else(x < 0, "-ve", "+ve"), "???")
```

This is already a little hard to read, and you can imagine it would only get harder if you have more conditions.
Instead, you can switch to `dplyr::case_when()`.

### `case_when()`

dplyr's `case_when()` is inspired by SQL's `CASE` statement and provides a flexible way of performing different computations for different conditions.
It has a special syntax that unfortunately looks like nothing else you'll use in the tidyverse.
It takes pairs that look like `condition ~ output`.
`condition` must be a logical vector; when it's `TRUE`, `output` will be used.

This means we could recreate our previous nested `if_else()` as follows:

```{r}
x <- c(-3:3, NA)
case_when(
  x == 0   ~ "0",
  x < 0    ~ "-ve", 
  x > 0    ~ "+ve",
  is.na(x) ~ "???"
)
```

This is more code, but it's also more explicit.

To explain how `case_when()` works, let's explore some simpler cases.
If none of the cases match, the output gets an `NA`:

```{r}
case_when(
  x < 0 ~ "-ve",
  x > 0 ~ "+ve"
)
```

Use `.default` if you want to create a "default"/catch all value:

```{r}
case_when(
  x < 0 ~ "-ve",
  x > 0 ~ "+ve",
  .default = "???"
)
```

And note that if multiple conditions match, only the first will be used:

```{r}
case_when(
  x > 0 ~ "+ve",
  x > 2 ~ "big"
)
```

Just like with `if_else()` you can use variables on both sides of the `~` and you can mix and match variables as needed for your problem.
For example, we could use `case_when()` to provide some human readable labels for the arrival delay:

```{r}
flights |> 
  mutate(
    status = case_when(
      is.na(arr_delay)      ~ "cancelled",
      arr_delay < -30       ~ "very early",
      arr_delay < -15       ~ "early",
      abs(arr_delay) <= 15  ~ "on time",
      arr_delay < 60        ~ "late",
      arr_delay < Inf       ~ "very late",
    ),
    .keep = "used"
  )
```

Be wary when writing this sort of complex `case_when()` statement; my first two attempts used a mix of `<` and `>` and I kept accidentally creating overlapping conditions.

### Compatible types

Note that both `if_else()` and `case_when()` require **compatible** types in the output.
If they're not compatible, you'll see errors like this:

```{r}
#| error: true

if_else(TRUE, "a", 1)

case_when(
  x < -1 ~ TRUE,  
  x > 0  ~ now()
)
```

Overall, relatively few types are compatible, because automatically converting one type of vector to another is a common source of errors.
Here are the most important cases that are compatible:

-   Numeric and logical vectors are compatible, as we discussed in @sec-numeric-summaries-of-logicals.
-   Strings and factors (@sec-factors) are compatible, because you can think of a factor as a string with a restricted set of values.
-   Dates and date-times, which we'll discuss in @sec-dates-and-times, are compatible because you can think of a date as a special case of date-time.
-   `NA`, which is technically a logical vector, is compatible with everything because every vector has some way of representing a missing value.

We don't expect you to memorize these rules, but they should become second nature over time because they are applied consistently throughout the tidyverse.

### Exercises

1.  A number is even if it's divisible by two, which in R you can find out with `x %% 2 == 0`.
    Use this fact and `if_else()` to determine whether each number between 0 and 20 is even or odd.

2.  Given a vector of days like `x <- c("Monday", "Saturday", "Wednesday")`, use an `if_else()` statement to label them as weekends or weekdays.

3.  Use `if_else()` to compute the absolute value of a numeric vector called `x`.

4.  Write a `case_when()` statement that uses the `month` and `day` columns from `flights` to label a selection of important US holidays (e.g., New Years Day, 4th of July, Thanksgiving, and Christmas).
    First create a logical column that is either `TRUE` or `FALSE`, and then create a character column that either gives the name of the holiday or is `NA`.

## Summary

The definition of a logical vector is simple because each value must be either `TRUE`, `FALSE`, or `NA`.
But logical vectors provide a huge amount of power.
In this chapter, you learned how to create logical vectors with `>`, `<`, `<=`, `>=`, `==`, `!=`, and `is.na()`, how to combine them with `!`, `&`, and `|`, and how to summarize them with `any()`, `all()`, `sum()`, and `mean()`.
You also learned the powerful `if_else()` and `case_when()` functions that allow you to return values depending on the value of a logical vector.

We'll see logical vectors again and again in the following chapters.
For example in @sec-strings you'll learn about `str_detect(x, pattern)` which returns a logical vector that's `TRUE` for the elements of `x` that match the `pattern`, and in @sec-dates-and-times you'll create logical vectors from the comparison of dates and times.
But for now, we're going to move onto the next most important type of vector: numeric vectors.