Lecture 5: An Introduction to R

• It originated from Bell Laboratories in the 1970s, as the S language, from which the commercial version S-Plus was developed in 1987.
• R was initially developed by Robert Gentleman and Ross Ihaka at the University of Auckland, New Zealand in 1996.
• Since its first release in 2000, the development of the R project has been tremendous in the last two decades.

• It is open source, free to download.
• It has a huge number of packages that can implement almost all state-of-art statistical techniques.
• It has a powerful and flexible capabilities of making graphs.
• It is a programming language designed specifically for statistics, enabling you to accomplish almost anything a programming language can do for statistics.

The installation files can be downloaded from https://mirrors.tuna.tsinghua.edu.cn/CRAN/. You can download the installation files for Windows, OS X, and Linux(ubuntu).

The base R comes with a simple Graphic User Interface (GUI). RStudio supplies with a more user-friendly GUI and provides other powerful functionalities, such as writing dynamic documents with nitro and rmarkdown.

• RStudio can be downloaded from https://www.rstudio.com/products/rstudio/download/

• The window of RStudio looks like Figure 1

  

  Figure 1: The Window of RStudio

R supports the following arithmetic operators

+, -, *, /, ^, %%, %/%

Hence,

## R as a calculator ------------------------------

#+ Binary operations
1 + 2; 2*3; 2^3; 5/2;
5 %% 2  # get x mod y
5 %/% 2 # get the integer division

[1] 3
[1] 6
[1] 8
[1] 2.5
[1] 1
[1] 2

R also have many built-in mathematical functions, such as, log(), exp(), sin(), sqrt(), min(), etc.

# Use built-in functions
log(exp(sin(pi/2)^2) * exp(cos(pi/3)^2))

[1] 1.25

A vector can be generated by the function c(), which can also be used to concatenate two vectors

## Vector operations ------------------------------

# Create a vector with c()
x <- c(0.3, 1.5, 7.3, 2)
y <- c(3, 2, 1)
z <- c(x, y)
z

[1] 0.3 1.5 7.3 2.0 3.0 2.0 1.0

The symbol <- is to assign a value to a variable. You can also use = to assign values, but <- is more commonly used by convention and = is used within a function calling for assigning values to the arguments of the function.

Note that by concatenating x and y, integers are converted to floating point numbers. That means the elements in a vector must have the same mode (data types), including numeric, character, and logical.

# Vectors with different data types
student.names <- c("John", "Mary", "Bob", "Ann")
student.male <- c(TRUE, FALSE, TRUE, FALSE)
student.age <- c(20, 19, 21, 20)

class(student.names)
class(student.male)
class(student.age)

students <- c(student.names, student.male, student.age)
students

[1] "character"
[1] "logical"
[1] "numeric"
 [1] "John"  "Mary"  "Bob"   "Ann"   "TRUE"  "FALSE" "TRUE"  "FALSE" "20"
[10] "19"    "21"    "20"

A vector can also be generated by the functions, like rep(), seq(), and :.

• seq() generates a vector by some patterns and a:b is a shorthand for seq(from=a, to=b, by=1).

  # Create a sequence
  even <- seq(from = 2, to = 20, by = 2)
  even
  years <- 1995:2005
  years
  

  [1]  2  4  6  8 10 12 14 16 18 20
  [1] 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005
  

• rep() generates a vector by repeating some values

  # Create repetition
  ones <- rep(1, times = 10)
  ones
  rep13 <- rep(1:3, times = 3, each = 2)
  rep13
  

  [1] 1 1 1 1 1 1 1 1 1 1
  [1] 1 1 2 2 3 3 1 1 2 2 3 3 1 1 2 2 3 3
  

Arithmetic operators and mathematical functions can be applied to vector in an element-by-element way in R.

Let's first draw random numbers for the uniform distribution \(x \sim Uniform(0, 1)\). The length of \(x\) is 10. We can use the length() function to check the length of a vector.

# Draw a random vector
x <- runif(10); x
length(x)

 [1] 0.51265670 0.86873322 0.01954781 0.84745505 0.50474095 0.28941441
 [7] 0.50314069 0.80165746 0.12290754 0.52574878
[1] 10

The arithmetic operations and built-in math functions are all applied for each element of a vector.

2 * x + 3
log(x)

[1] 4.025313 4.737466 3.039096 4.694910 4.009482 3.578829 4.006281 4.603315
[9] 3.245815 4.051498
[1] -0.6681489 -0.1407192 -3.9348922 -0.1655175 -0.6837099 -1.2398957
[7] -0.6868855 -0.2210739 -2.0963229 -0.6429318

If two vectors with different lengths are computed within one operation, the elements of the vector with a shorter length will be used in an iterated way. We must keep in mind this feature of R, which in some cases may give rise to unintended results.

y <- runif(5)
x + y

[1] 0.8453125 0.9259139 0.6906356 1.7553697 0.6985153 0.6220702 0.5603213
[8] 1.4727453 1.0308222 0.7195231

Element(s) in a vector can be selected by [position], in which position can be a vector indicating the position of each element in a vector, a negative value to exclude an element with the corresponding position, and a condition to select elements satisfying the condition.

# Selecting elements in a vector
x[1:5]
x[c(1, length(x))]
x[-4]
x[x > 0.5]

[1] 0.51265670 0.86873322 0.01954781 0.84745505 0.50474095
[1] 0.5126567 0.5257488
[1] 0.51265670 0.86873322 0.01954781 0.50474095 0.28941441 0.50314069 0.80165746
[8] 0.12290754 0.52574878
[1] 0.5126567 0.8687332 0.8474551 0.5047410 0.5031407 0.8016575 0.5257488

Instead of selecting elements in a vector by their positions, we can also give each element a particular name so that we can use their names to choose elements.

student.names
student.age
# Give elements names
names(student.age) <- student.names
student.age
student.age[c("John", "Bob")]

[1] "John" "Mary" "Bob"  "Ann"
[1] 20 19 21 20
John Mary  Bob  Ann
  20   19   21   20
John  Bob
  20   21

We can create a matrix with the matrix() function, in which the first argument is a vector. We specify the two dimensions by the arguments of nrow and ncol. By default, matrix() arranges all the elements of the vector in its first argument into a matrix by column. We can change it by adding byrow=TRUE.

# Create a matrix
A <- matrix(1:12, nrow = 3, ncol = 4); A
matrix(1:12, nrow = 3, ncol = 4, byrow = TRUE)

     [,1] [,2] [,3] [,4]
[1,]    1    4    7   10
[2,]    2    5    8   11
[3,]    3    6    9   12
     [,1] [,2] [,3] [,4]
[1,]    1    2    3    4
[2,]    5    6    7    8
[3,]    9   10   11   12

We can also juxtapose vectors of the same length to create a matrix by cbind(), or stack over vectors by rbind().

# Create a matrix by combining vectors
a <- 1:4; b <- 2:5; c <- 3:6
cbind(a, b, c)
rbind(a, b, c)

     a b c
[1,] 1 2 3
[2,] 2 3 4
[3,] 3 4 5
[4,] 4 5 6
  [,1] [,2] [,3] [,4]
a    1    2    3    4
b    2    3    4    5
c    3    4    5    6

Like vectors, we can also give each row and each column in a matrix their specific names. Here we use the function of paste() to combine two (character) vectors together to generate a new character vector.

# Give names to rows and columns
rownames(A) <- paste("X", 1:3, sep = "")
colnames(A) <- paste("Y", 1:4, sep = "")
A

   Y1 Y2 Y3 Y4
X1  1  4  7 10
X2  2  5  8 11
X3  3  6  9 12

We select elements from a matrix using [rows, cols]. rows and cols are two vectors to set the rows and columns of elements to be selected.

# Selecting elements in a matrix
A[1, 3]
A["X1", "Y3"]
A[1:3, c(2, 4)]
A[, 2]
A[3, ]

[1] 7
[1] 7
   Y2 Y4
X1  4 10
X2  5 11
X3  6 12
X1 X2 X3
 4  5  6
Y1 Y2 Y3 Y4
 3  6  9 12

We can do all matrix operations that we have reviewed in Lecture 4.

The function diag() can create a diagonal matrix.

diag(1:3)

     [,1] [,2] [,3]
[1,]    1    0    0
[2,]    0    2    0
[3,]    0    0    3

An identity matrix is a special case of a diagonal matrix.

diag(3)

     [,1] [,2] [,3]
[1,]    1    0    0
[2,]    0    1    0
[3,]    0    0    1

Vectors and matrices are special cases of arrays. The former is one-dimensional array, and the latter is two-dimensional. We can also create higher-dimensional arrays by array().

array(1:18, dim = c(3, 3, 2))

, , 1

     [,1] [,2] [,3]
[1,]    1    4    7
[2,]    2    5    8
[3,]    3    6    9

, , 2

     [,1] [,2] [,3]
[1,]   10   13   16
[2,]   11   14   17
[3,]   12   15   18

mydata <- data.frame(X = 1:5, Y = letters[1:5], Z = rep(c(TRUE, FALSE), length = 5)); mydata

  X Y     Z
1 1 a  TRUE
2 2 b FALSE
3 3 c  TRUE
4 4 d FALSE
5 5 e  TRUE

We use as.data.frame() to convert a matrix to a data frame. In creating the matrix, we use sample.int() that is a special case of the function sample() to draw random samples from a vector.

A <- matrix(sample.int(100, size = 20), nrow = 5)
A.df <- as.data.frame(A); A.df

  V1 V2 V3 V4
1 88 37 26 81
2 95 24 14 69
3 56 80 84 34
4 57  1  9 50
5 47 23 43 83

We can assign each variable (column) a name. Here we use the function paste() to combine a string VAR with each element of the vector 1:4, joined with _.

names(A.df) <- paste("VAR", 1:4, sep = "_"); A.df

  VAR_1 VAR_2 VAR_3 VAR_4
1    88    37    26    81
2    95    24    14    69
3    56    80    84    34
4    57     1     9    50
5    47    23    43    83

Compute the sample mean for average hourly earnings (ahe) in 1992 and
in 2008. Construct a 95% confidence interval for the population means
for ahe in 1992 and 2008 and the change between 1992 and 2008

Now we need to adjust the average hourly earnings in the 1992 dollars to the 2008 dollars with the inflation rate, computed as CPI2008/CPI1992.

# CPI in 1992 and 2008
cpi.92 <- 140.3
cpi.08 <- 215.2
# Inflation adjustment
inflator <- cpi.08 / cpi.92
cpsdat$ahe.adj <- with(cpsdat, ifelse(year == 1992, ahe * inflator, ahe))

In the code block above, we first use the function with() to attach the data frame cpsdat within its own environment so that when we refer to variables in cpsdat, such as ahe and year, we do not need to write cpsdat$ and every time we use its variables.

The function ifesle() set the values of ahe based on the condition year == 1992. If the condition is true, we do ahe * inflator; if not, leave ahe as it is.

Then we repeat what we've done in Question (a) with the inflation-adjusted earnings in 1992.

ahe.92.adj <- with(cpsdat, ahe.adj[year == 1992])
mean.ahe.92.adj <- mean(ahe.92.adj)
t.ahe.92.adj <- t.test(ahe.92.adj)
t.ahe.diff.adj <- t.test(ahe.08, ahe.92.adj)

• The sample average of the inflation-adjusted earnings in 1992 is 17.83 in the 2008 dollars.
• The confidence interval for the inflation-adjusted average hourly earnings in 1992 is (17.64, 18.03).
• The confidence interval for the change between 1992 and 2008 is (0.85, 1.44).

If we are interested in the change in workers' purchasing power, the results with the inflation-adjusted earnings should be used in comparison.

Now let's compute the average earnings for high school graduates and college graduates with the 2008 data. First thing to do is to select the 2008 data from cpsdat using the function subset()

# select data in 2008
cps08 <- subset(cpsdat, year == 2008, select = c(year, ahe, bachelor))

# calculate means
ahe.educ.08 <- aggregate(ahe ~ bachelor, FUN = mean, data = cps08)

# select ahe and filter by bachelor
ahe.high.08 <- with(cps08, ahe[bachelor == 0])
ahe.bach.08 <- with(cps08, ahe[bachelor == 1])

# construct confindence interval
t.ahe.high.08 <- t.test(ahe.high.08)
t.ahe.bach.08 <- t.test(ahe.bach.08)
t.ahe.gap.08  <- t.test(ahe.bach.08, ahe.high.08)

• The mean of the average hourly earnings of high school graduates in 2008 is 15.33 dollars with the 95% confidence interval (15.09, 15.57)
• The mean of the average hourly earnings of college graduates is 22.91 dollars with the 95% confidence interval (22.56, 23.26)
• The 95% confidence interval of the gap in earnings between the two groups is (7.15, 8)

We can create a boxplot to compare the means and confidence intervals of average hourly earnings between high school graduates and college graduates.

boxplot(ahe ~ bachelor, data = cps08,
        main = "Average Hourly Earnings by Education",
        col = c("red", "orange"),
        xlab = "Bachelor degres = 1, high school = 0",
        ylab = "US$ 2008")

We leave Question (e)-(g) to students as exercises.

I include all the files to generate a complete answer to Empirical exercise 3.1 in the following package, rfiles.zip, including the R code file, Rmarkdown file, the data file, and the html and pdf files containing the answers.