Objects

• In R, everything is an object.

• Objects have a name that is assigned with <- (recommended) or =.

• Names have to start with a letter and include only letters, numbers, and characters such as “.” and “_”.

• R is case sensitive: \(\Rightarrow Name\neq name\)!

• Objects can store vectors, matrices, lists, data frames, functions…

# generate object x (no output):
x <- 5
# display log(x)
log(x)

[1] 1.609438

# object X is not defined => error message 
X

Error: object 'X' not found

Vectors

• Vectors can store multiple types of information (e.g., numbers or “characters”).
• To define a 3-dimensional vector named “vec”, use vec <- c(value1, value2, value3).
• Operators and functions can be applied to vectors, which means they are applied to each of the elements individually.

# define vector named 'vec'
vec <- c(1, 2, 3)
# take the square root of 'vec' and store the result in 'sqrt_vec'
sqrt_vec <- sqrt(vec)
# display sqrt_vec
print(sqrt_vec)

[1] 1.000000 1.414214 1.732051

Vectors - some helpful shortcuts

• R has built-in functions that generate sequences (useful for loops or plots, among other things).
• We can also repeat elements using rep().

# generate sequence 5,6,...,10
5:10

[1]  5  6  7  8  9 10

# generate sequence from 1 to 10 in steps of 0.5
seq(from = 1, to = 5, by = 0.5)

[1] 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0

#generate 4-dimensional vector of ones
rep(1, 4)

[1] 1 1 1 1

Order matters!

• Be aware of the order of operations!
• compare the following:

1+2:3^2 # '^2' evaluated before ':', only then '+1' is evaluated

[1]  3  4  5  6  7  8  9 10

1+2:3*4 # first ':', then '*4', then '+1'

[1]  9 13

# use brackets to avoid confusion or mistakes
(1+2):(3*4)

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

Summarizing vectors

• R has built-in functions to summarize the information stored in vectors.
• Remark: R is very good at generating random numbers! (Such functions are studied in more detail in the next session.)

# Example: generate 100 random draws from a normal distribution with mean 1
# and standard deviation 2
norm.vec <- rnorm(n = 100, mean = 1, sd = 2)
# get mean 
mean(norm.vec)

[1] 1.017192

# get standard deviation 
sd(norm.vec)

[1] 2.16042

# get maximum
max(norm.vec)

[1] 6.14331

Exercise 2.1: generating and summarizing vectors

• draw 50 random numbers from a normal distribution with mean 0 and variance 1. Store your results in the object norm.vec.
• calculate the mean and standard deviation of norm.vec.
• use rep() to repeat each element of norm.vec 3 times. Store the result in the object norm.vec.rep.
• Is mean(norm.vec.rep^2) equal to mean(norm.vec.rep)^2?

logical operators

• logical operators can be either TRUE or FALSE.
• Extremely useful for conditional statements, e.g. if(condition is TRUE){do this}else{do that}.
• We can check if two objects are equal by ==, different by != or compare them with < and >.
• We can combine logical statements with “AND” & and “OR” |

# define  objects 
obj1 <- 1
obj2 <- 2
obj3 <- 1 # same value as obj1
obj1 == obj2 # false statement

[1] FALSE

obj1 != obj2 # true statement

[1] TRUE

obj1 == obj2 & obj1 == obj3 # FALSE AND TRUE => FALSE

[1] FALSE

obj1 == obj2 | obj1 == obj3 # FALSE OR TRUE => TRUE

[1] TRUE

• We can also use logical operators in vectors.
• the AND and OR operators & and | are then applied element-wise.

vec2 <- 1:5 # defines vector vec2=(1,2,3,4,5)
vec2 == 3  # =FALSE if element is not 3, =TRUE if element is 3

[1] FALSE FALSE  TRUE FALSE FALSE

vec2 >= 2 & vec2 < 5 # Only TRUE for elements >=2 and <5

[1] FALSE  TRUE  TRUE  TRUE FALSE

vec2 >= 2 | vec2 < 5 # TRUE for all elements since either >=2 or <5

[1] TRUE TRUE TRUE TRUE TRUE

Characters

• Vectors can also store characters.
• characters are enclosed in ""or ''.

# define a vector of 2 cities
cities <- c('Maastricht', "Amsterdam",'Rotterdam')
print(cities)

[1] "Maastricht" "Amsterdam"  "Rotterdam" 

Exercise 2.2: type coersion

• R tries to make objects comparable by coercing one object into the type of another.
• This can sometimes be handy, but sometimes it leads to unforeseen errors (e.g., when loading new data). To illustrate this, do the following:
  • compare the character "1" to the numeric 1.
  • try computing the sum of "1" and "2".
  • try computing the sum of as.numeric("1") and as.numeric("2"). What happened?
  • create a mixed vector containing the numeric 1 and the character "2". Of which type are the elements of the vector?

factors

• Many variables are qualitative rather than quantitative.
• While they are often coded using numbers, they don’t have a numerical meaning.
• Examples: gender, nationality…
• Can also be ordinal, i.e., the outcomes can be ranked (e.g., “bad”, “meh”, “great”).

x <- c(1, 3, 3, 2, 1, 3)
xf <- factor(x, labels = c("bad", "ok", "good"))# no ranking
xf

[1] bad  good good ok   bad  good
Levels: bad ok good

# now with ranking
xf.ordered <- factor(x, labels = c("bad", "ok", "good"), ordered = TRUE)
xf.ordered

[1] bad  good good ok   bad  good
Levels: bad < ok < good

Names

• You can give the elements of your vector names either directly or using the names() command.
• This is very useful for accessing elements (see next slide)

avg_temp <- c(Maastricht = 14.2, Amsterdam = 13.4, Rotterdam = 13.7)
print(avg_temp) # names appear on top of elements

Maastricht  Amsterdam  Rotterdam 
      14.2       13.4       13.7 

names(avg_temp) # returns names of elements

[1] "Maastricht" "Amsterdam"  "Rotterdam" 

# Alternatively, we can define data and names separately
temp <- c(14.2, 13.4, 13.7)
names(temp) <- cities # recall that we have defined "cities" earlier!
print(temp)

Maastricht  Amsterdam  Rotterdam 
      14.2       13.4       13.7 

Accessing elements

• One can access the elements of a vector either by name or position.

# return the second element of "avg_temp" defined before
avg_temp[2] 

Amsterdam 
     13.4 

# return the element corresponding to "Maastricht"
avg_temp["Maastricht"]

Maastricht 
      14.2 

# trying to access a non-existing element yields "NA"
# ( for "not available"), i.e., a missing value
avg_temp[4]

<NA> 
  NA 

• By using the minus sign [-k], we can get the vector except for the \(k\)-th element.
• We can also add elements to an existing vector.

# get the vector except for the third element
avg_temp[-3]

Maastricht  Amsterdam 
      14.2       13.4 

# now add another city to avg_temp
avg_temp["Tilburg"] <- 14.7
# now the fourth element is defined!
avg_temp[4]

Tilburg 
   14.7 

More on NA, NaN, Inf

• NA (“not available”) indicates missing values.
• Anything combined with NA yields NA.
• NaN(“not a number”) indicates the result of a mathematically undefined operation.

#define another vector
vec3 <- c(-1.2, NA, 0)
# combine avg_temp and vec3
vec4 <- c(avg_temp, vec3) 
# divide elements by 0; notice the different outcomes
vec4 / 0 

Maastricht  Amsterdam  Rotterdam    Tilburg                                  
       Inf        Inf        Inf        Inf       -Inf         NA        NaN 

Matrices

• We can create a matrix with m rows directly using matrix(vector,nrow=m).

# create matrix with 3 rows; fill numbers by row
mat1 <- matrix(1:12, nrow = 3, byrow = TRUE) # by default, R fills matrices by column
mat1

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

• We can also combine vectors of the same length by row with rbind(v1,v2,...) or by column by cbind(v1,v2,...).

# create vectors v1, v2 and v3 and combine them for same result
v1 <- 1:4
v2 <- 5:8
v3 <- 9:12
mat2 <- rbind(v1, v2, v3)
mat2

   [,1] [,2] [,3] [,4]
v1    1    2    3    4
v2    5    6    7    8
v3    9   10   11   12

Matrix indexing

• We can give the rows and columns names using rownames() and colnames().

# assign names to columns
colnames(mat1) <- c("col1", "col2", "col3", "col4")
mat1

     col1 col2 col3 col4
[1,]    1    2    3    4
[2,]    5    6    7    8
[3,]    9   10   11   12

# assign names to rows
rownames(mat1) <- c("row1","row2","row3")
mat1

     col1 col2 col3 col4
row1    1    2    3    4
row2    5    6    7    8
row3    9   10   11   12

Accessing elements

• We can access single elements by [rownumber,colnumber], the k-th row by [k,] and the k-th column by [,k].

# get element in second row in third column
mat1[2,3]

[1] 7

# get second row
mat1[2,]

col1 col2 col3 col4 
   5    6    7    8 

# get third column
mat1[,3]

row1 row2 row3 
   3    7   11 

• If rows/columns have names, we can also use those.
• Using vectors, we can also create more complicated subsets of matrices.

# get sub-matrix using vectors
mat1[c(2,3),c(1,3)]

     col1 col3
row2    5    7
row3    9   11

# get second row using names (recall definition of mat2)
mat2["v2",]

[1] 5 6 7 8

Exercise 2.3: creating matrices, accessing elements

1. Create the 3x3 identity matrix “by hand”. To do so:
   1. create 3 vectors with zeros and ones in the appropriate spots.
   2. use rbind() or cbind() to combine them into the identity matrix.
   3. store the identity matrix as the object “I_mat”.
   4. R makes your life easy: type diag(3) in your console.
2. Replicate the following Excel-matrix:

3. Get the data for April and May by - including only the first and second row - excluding the third row - using the names

Bonus: Matrix algebra

• R can do matrix “regular” algebra, and even lets you do operations that are not well-defined mathematically.

• t(A) is the transpose of the matrix A.

# define matrix containing normal data
data.vec <- rnorm(9, mean = 0, sd = 1)
A <- matrix(data.vec, nrow = 3)
A # return A

           [,1]        [,2]      [,3]
[1,] -0.7569037 -0.02340064  1.124970
[2,] -0.6589236  0.22297959  1.963878
[3,]  0.6227039 -1.31762563 -1.514380

t(A) # return the transpose

            [,1]       [,2]       [,3]
[1,] -0.75690372 -0.6589236  0.6227039
[2,] -0.02340064  0.2229796 -1.3176256
[3,]  1.12496977  1.9638780 -1.5143799

• solve(A) returns the inverse of an invertible matrix.

solve(A) # return the inverse of A

           [,1]       [,2]       [,3]
[1,] -2.5344123  1.7095904  0.3343216
[2,] -0.2535042 -0.5020622 -0.8394019
[3,] -0.8215672  1.1398055  0.2074782

• *does element-wise multiplication.
• %*% does matrix multiplication .

# element-wise multiplication
A * solve(A) # NOT the identity matrix

           [,1]        [,2]       [,3]
[1,]  1.9183061 -0.04000552  0.3761016
[2,]  0.1670399 -0.11194962 -1.6484830
[3,] -0.5115931 -1.50183698 -0.3142008

# matrix multiplication
A %*% solve(A) 

     [,1]          [,2] [,3]
[1,]    1  0.000000e+00    0
[2,]    0  1.000000e+00    0
[3,]    0 -2.220446e-16    1

# yields the identity (up to a small error due to the
# numerical computation of the inverse)

Lists

• A list is a generic collection of objects.
• Unlike vectors, the components can have different types (e.g., numeric and character).
• Many functions output lists, so knowing how to access elements is very useful.
• Generate a list with mylist<- list(name1=component1, name2=component2,...).

mylist <- list(num.vec = 1:3, city = "Maastricht") 
print(mylist)

$num.vec
[1] 1 2 3

$city
[1] "Maastricht"

• Get names of the components with names(mylist).
• You can access components with the $ (dollar sign) operator, e.g., mylist$name1, or by position with [[]].

mylist$city

[1] "Maastricht"

mylist[[2]] # same result

[1] "Maastricht"

Data frames

• data frames are simply data sets in R terminology.
• So-called data files can contain multiple data sets.
• We can create a data frame by data.frame() or transform a matrix mat into a data frame by as.data.frame(mat).
• Many functions (e.g. lm() for regressions) need a data frame as input (see later sessions).

# generate a data frame
ID <- 1:4
hourly_wage <- rnorm(n = 4, mean = 20, sd = 1) # create 4 draws from N(20,1)
city <- c("Maastricht", "Eindhoven", "Amsterdam", NA)
dats <- data.frame(ID, hourly_wage, city) # add new variable
dats

  ID hourly_wage       city
1  1    20.60223 Maastricht
2  2    19.20833  Eindhoven
3  3    19.56172  Amsterdam
4  4    19.21373       <NA>

• As with lists, we can access variables using the $ operator.
• We can also add new variables using the $ operator.
• View() opens a data-viewer. Very useful (but difficult to demonstrate on these slides).

dats$city # "city" is NA for ID 4.

[1] "Maastricht" "Eindhoven"  "Amsterdam"  NA          

dats$city[4] <- 'Tilburg' # assign city to ID 4
dats$educ <- c(12, 21, 9, 10)
dats

  ID hourly_wage       city educ
1  1    20.60223 Maastricht   12
2  2    19.20833  Eindhoven   21
3  3    19.56172  Amsterdam    9
4  4    19.21373    Tilburg   10

• Using subset(data_frame,condition), we can easily get a subset of the original data frame where condition is TRUE.

# only keep individuals with at least 10 years of education
sub_dats <- subset(dats, educ > 10)
sub_dats

  ID hourly_wage       city educ
1  1    20.60223 Maastricht   12
2  2    19.20833  Eindhoven   21

Exercise 2.4

• Create your own data frame:
  • create a vector ID that contains the sequence 1,2,…,100.
  • create a vector income that contains 100 random draws from N(10,1).
  • create a dummy female that is 1 for ID=1,...,50 and 0 otherwise. (hint: you can achieve this by using rep() twice and combining two vectors with c())
  • collect the variables in a data frame my_df.
  • inspect your data with View(my_df)
  • bonus: create a subset sub_my_df that contains only individuals with income larger than 10.

Bonus: teaching regression with R

• To give students a feeling for the behavior of the least squares estimator, it can be very useful to use simulated data.
• This allows teachers to visualize the effects of various quantities of interest, e.g., sample size, variation in the observed and unobserved variables, or omitted variables.

n <- 100 # set the sample size
X <- rnorm(n, mean = 1, sd = 2)# define the observed covariate X
epsilon <- rnorm(n, mean = 0, sd = 1) # define the model error
beta0 <- 1 # define true intercept
beta1 <- 2 # define true slope
Y <- beta0 + beta1 * X + epsilon # generate Y according to a linear model
# recall the formula in a bivariate model
beta1.hat <- cov(X,Y) / var(X)
beta0.hat <- mean(Y) - beta1.hat * mean(X)
# print estimators
beta0.hat

[1] 1.166459

beta1.hat

[1] 1.924542