# 6 Logical

In R, logical values are stored as a distinct data type. Logical values are used both in statistical modeling and in data management. In modeling, logical vectors are often called indicator or dummy variables. In data management, logical values also serve as conditional indicators.

## 6.1 Logical Values

There are three logical values:

• TRUE
• FALSE
• NA (“missing”, or “unknown value”)

The names T and F are used by default as aliases for TRUE and FALSE, but be aware that you can redefine T and F. Do not do this accidentally! The names T and F print the values TRUE and FALSE.

T
 TRUE
F
 FALSE

See help(logical) and help(NA).

## 6.2 Logical Operators

R has the typical binary comparison operators (see help(Comparison)). These take data of arbitrary type as inputs (x and y) and return logical values.

• Greater than, x > y, or equal to, x >= y
• Less than, x < y, or equal to, x <= y
• Equal to, x == y
• Not equal, x != y

R also has the typical Boolean operators for creating compound logical expressions. These take logical values as inputs (x and y) and return logical values.

• And, x & y
• Or, x | y
• Not, !x

### 6.2.1 Making Comparisons

As with the mathematical operators, logical operators work pairwise with the elements of two vectors, returning a vector of comparisons. Where one vector is shorter than the other, recycling occurs.

In this first example, we set up an arbitrary numeric vector, $$a$$, and ask if each element of $$a$$ is a 3.

a <- c(1.1, 3, 5.3, 2)  # a numeric vector

f <- (a == 3)           # a vector of comparisons
f
 FALSE  TRUE FALSE FALSE

We can make comparisons with character values, too, but be aware that the result can depend on what language R thinks you are working in!

A <- c("a", "b", "e")
A > "d"
 FALSE FALSE  TRUE

Comparison of two vectors is done element by element. (The term “vectorized” is sometimes used to mean this sort of operation). In this example, each element of $$a$$ is compared to one corresponding integer in 1 to 4.

a > 1:4
  TRUE  TRUE  TRUE FALSE

If the two vectors being compared are of different lengths, recycling occurs just as we saw with numeric operators.

b <- c(1.1, 2)
a != b                  # silent recycling
 FALSE  TRUE  TRUE FALSE
a > 2:4                 # noisy recycling
Warning in a > 2:4: longer object length is not a multiple of shorter object length
 FALSE FALSE  TRUE FALSE

A somewhat different kind of comparison is the value match. Here we ask if values in the left-hand vector are elements of the set represented by the right-hand vector. Despite the use of two vectors, these are no longer pairwise comparisons and there is no recycling. The length of the output is equal to the length of the vector on the left-hand side. In the first example, 2 %in% a, 2 has length one, so the output has one element. In the second example, 1:4 %in% a, 1:4 has length four, so the output has four elements.

2 %in% a
 TRUE
1:4 %in% a
 FALSE  TRUE  TRUE FALSE

These two operators can be used for different purposes. To illustrate, first create a hypothetical dataset where two columns, state2020 and state2021, contain information about the states where an individual lived in 2020 and 2021, respectively.

n <- 10
set.seed(455)
dat <-
data.frame(id = 1:n,
state2020 = sample(state.name[1:5], n, replace = T),
state2021 = sample(state.name[1:5], n, replace = T))

Use == for elementwise comparisons. The variable same_state represents, “Did this individual live in the same state for both years?”

dat$same_state <- dat$state2020 == dat$state2021 Use %in% if you want to find cases that take a value found in some set. The variable AL_AK_2021 represents, “Did somebody live in either Arizona or Alaska in 2021?” dat$AL_AK_2021 <- dat$state2021 %in% c("Arizona", "Alaska") For illustration purposes, we will also create a variable that uses == to compare a column to a vector with more than one value (a set). This code will recycle our vector c("Arizona", "Alaska"), comparing observation one to “Arizona”, observation two to “Alaska”, observation three to “Arizona”, and so on. This is decidedly a bad idea; results will vary if we sort, add, or remove observations. # DO NOT DO THIS - USE %in% INSTEAD dat$AL_AK_2021_v2 <- dat$state2021 == c("Arizona", "Alaska") Now, print the dataframe to see the TRUEs and FALSEs in our new variables. dat  id state2020 state2021 same_state AL_AK_2021 AL_AK_2021_v2 1 1 Arkansas Alabama FALSE FALSE FALSE 2 2 California Alaska FALSE TRUE TRUE 3 3 Alabama California FALSE FALSE FALSE 4 4 California Arizona FALSE TRUE FALSE 5 5 Arkansas California FALSE FALSE FALSE 6 6 Alaska Alaska TRUE TRUE TRUE 7 7 Arizona Arkansas FALSE FALSE FALSE 8 8 Alabama Alabama TRUE FALSE FALSE 9 9 Alabama California FALSE FALSE FALSE 10 10 Alaska Alabama FALSE FALSE FALSE Different operators are useful for different research questions. ### 6.2.2 Boolean Algebra We also have the usual operators (“and”, “or”, “not”) for combining logical inputs to produce a logical outcome. # &, "and" - satisfy both conditions (a == 2) & (a < 5)   FALSE FALSE FALSE TRUE # |, "or" - satisfy at least one condition (a == 2) | (a < 5)   TRUE TRUE FALSE TRUE ### 6.2.3 Missing Values The logical status of missing values is treated somewhat differently in R than in some other statistical software (Stata, SAS, SPSS). Where in some languages the result of a comparison is either true or false, in R a comparison may produce a “missing” or “unknown” result. b <- c(1:4, NA) b > 3 # in Stata the final value is "true"  FALSE FALSE FALSE TRUE NA b == 3 # in Stata and SAS the final value is "false"  FALSE FALSE TRUE FALSE NA b < 3 # in SAS the final value is "true"  TRUE TRUE FALSE FALSE NA Likewise, Boolean operations on missing values produce missing results. When checking for missing values a common mistake is to use a comparison. However, in R we use a testing function. b == NA # not useful, but doesn't produce an error!  NA NA NA NA NA is.na(b) # the proper way to check  FALSE FALSE FALSE FALSE TRUE ## 6.3 Functions with Logical Vectors ifelse() is a function we will use a lot to recode and modify values in First Steps with Dataframes. To use ifelse(), give it three arguments: a logical test, a value to return if the test evaluates to TRUE, and a value to return if the test evaluates to FALSE. Make a vector of the numbers one to five. x <- 1:5 The first way to use ifelse() is to recode all values to make a binary variable: ifelse(x >= 3, "three_or_more", "less_than_three")  "less_than_three" "less_than_three" "three_or_more" "three_or_more" "three_or_more"  We can also use ifelse() to change some values, and leave others as is. To do so, supply a value in either the second or third argument of ifelse(), and put the name of the vector in the other. For example, we could change one specific value. Wherever x is 3, make it missing, and return its original value otherwise: ifelse(x == 3, NA, x)  1 2 NA 4 5 We could return the corresponding element from another vector. Here, the test evaluates to TRUE in the third position, so the third element of 11:15 is returned (13): ifelse(x == 3, 11:15, x)  1 2 13 4 5 Other applications include top-coding or bottom-coding data: ifelse(x > 4, 4, x) # top-code at 4  1 2 3 4 4 ifelse(x < 2, 2, x) # bottom-code at 2  2 2 3 4 5 We can compare a vector to a set with %in%: ifelse(x %in% c(1, 3, 5), "odd", x)  "odd" "2" "odd" "4" "odd" And of course, complex expressions with Boolean operators are allowed: ifelse(x < 3 | x > 3, "not_three", x)  "not_three" "not_three" "3" "not_three" "not_three" ### 6.3.1 Coercion A generic function is a function which uses different methods (implements different algorithms) depending on the class and type of the input data. (Recall the discussion in the chapter on Data Class.) A very few generic functions have specific methods for logical vectors, while most functions will coerce logical vectors to either a numeric vector or a factor. summary(f) # produces counts, but also notes mode  Mode FALSE TRUE logical 3 1  If you have worked with other statistical software, you won’t be surprised that very often logical values are automatically coerced to the integers 0 and 1. mean(f) # coerced to numeric, a proportion  0.25 f + 1 # coercion in binary operators, too  1 2 1 1 You may also be aware that where numeric values are coerced into logical values, 0 is FALSE and anything else is TRUE (unless it is missing). (Recall Exercise 3 from Data Types.) as.logical(-1:2)  TRUE FALSE TRUE TRUE The fact that we can coerce logical values into zeroes and ones is extremely useful in data wrangling when it is used in combination with sum() or mean(). The cyl column of mtcars takes on several values: mtcars$cyl
  6 6 4 6 8 6 8 4 4 6 6 8 8 8 8 8 8 4 4 4 4 8 8 8 8 4 4 4 8 6 8 4

A logical comparison can tell us which values are equal to 4:

mtcars$cyl == 4   FALSE FALSE TRUE FALSE FALSE FALSE FALSE TRUE TRUE FALSE FALSE FALSE FALSE FALSE  FALSE FALSE FALSE TRUE TRUE TRUE TRUE FALSE FALSE FALSE FALSE TRUE TRUE TRUE  FALSE FALSE FALSE TRUE Taking the sum of this comparison will coerce the TRUEs to 1 and FALSEs to 0, so that the sum is equal to the number of 4s in mtcars$cyl. The mean, then, is the proportion of values equal to 4.

sum(mtcars$cyl == 4)  11 mean(mtcars$cyl == 4)
 0.34375

We can also check the number of cars that have 1 in both vs and am. Recall that comparisons will be made elementwise. Using the & operator allows us to check, whether the first pair of vs and am is all TRUE, whether the second pair is all TRUE, and so on. Summing this will give us the number of cases in mtcars that have a 1 for both. Looking through the numeric values for these two variables, it looks like they are both 1 in seven positions.

mtcars$vs   0 0 1 1 0 1 0 1 1 1 1 0 0 0 0 0 0 1 1 1 1 0 0 0 0 1 0 1 0 0 0 1 mtcars$am
  1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 1 1 1 1 1 1 1
mtcars$vs == 1   FALSE FALSE TRUE TRUE FALSE TRUE FALSE TRUE TRUE TRUE TRUE FALSE FALSE FALSE  FALSE FALSE FALSE TRUE TRUE TRUE TRUE FALSE FALSE FALSE FALSE TRUE FALSE TRUE  FALSE FALSE FALSE TRUE mtcars$am == 1
   TRUE  TRUE  TRUE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
 FALSE FALSE FALSE  TRUE  TRUE  TRUE FALSE FALSE FALSE FALSE FALSE  TRUE  TRUE  TRUE
  TRUE  TRUE  TRUE  TRUE
mtcars$vs == 1 & mtcars$am == 1
  FALSE FALSE  TRUE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
 FALSE FALSE FALSE  TRUE  TRUE  TRUE FALSE FALSE FALSE FALSE FALSE  TRUE FALSE  TRUE
 FALSE FALSE FALSE  TRUE
sum(mtcars$vs == 1 & mtcars$am == 1)
 7

Later, we will use this property of logical values to Quantify Missing Data.

## 6.4 Testing Equality

There is one logical comparison that is particularly problematic when made by a computer: equality. Checking the equality of logical values, character values, and integer values is straightforward, but numeric values with decimal precision (stored as “doubles”) are often imprecise. Think of the decimal representation of $$1/3$$, or 0.3333333, which must be truncated at some point: 0.3333… cannot continue forever.
(A computer works with binary representations, but the problem is conceptually the same.)

Even simple mathematical operations can introduce numerical deviations.

a <- 0.5 - 0.2  # 0.3
b <- 0.4 - 0.1  # 0.3

a == b          # Probably not what you expected!
 FALSE
a - b           # a small difference, but not exactly zero
 -5.551115e-17

We have two general approaches for handling this imprecision with comparisons of numeric vectors. In the special case where we want to know of all elements of two vectors are equivalent, we have a summary function all.equal. In the more general case, we test that the differences between two vectors are less than a numerical tolerance.

# An example with vectors
x <- seq(0, 0.5, by = 0.1)
y <- seq(0.1, 0.6, by=0.1)-0.1

x == y          # Not what you hoped for?  (the third element ....)
  TRUE  TRUE FALSE  TRUE  TRUE  TRUE
x - y
  0.000000e+00  0.000000e+00 -2.775558e-17  0.000000e+00  0.000000e+00  0.000000e+00
all.equal(x,y)  # checking all are equal
 TRUE

The smallest precision available on your computer is given by

.Machine$double.eps  2.220446e-16 We commonly take our maximum imprecision to be the square root of that value. So if we check for numerical equivalence, we get the result we expected earlier with ==. tol <- sqrt(.Machine$double.eps)

x-y < tol
 TRUE TRUE TRUE TRUE TRUE TRUE

## 6.5 Exercises

1. A typical use of logical comparison is to create an indicator variable. Create an object called high_mpg that indicates whether a given value of mtcars$mpg has a value greater than the mean of mtcars$mpg.

2. What proportion of values in mtcars$mpg are greater than the mean? ## 6.6 Advanced Exercises 1. Another use of logical vectors is to select observations from another vector. Use the high_mpg object created above to select values of mtcars$disp associated with high gas mileage. Calculate the mean of this vector.

2. We have seen how logical values are often coerced to numeric, and numeric to logical. However, important differences remain between the two types. Consider this example, which seems like it should produce the same results in two different ways. Why are two different vectors returned?

v <- 1:4
v[c(T,F,T,F)]
 1 3
v[c(1,0,1,0)]
 1 1