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1. `lm()`

Run the code below to get started. It uses the assignment operator <- a couple of times. For instance, x <- 0:2 should be read “x gets 0:2”. When we run that piece of code, we’re taking the vector 0:2 and giving it the name x. So after running x <- 0:2, x + 1 will be equal to 1:3. Also notice that after you run the code below, if you check the top right pane in RStudio, the Environment tab should let you know that x now refers to 0:2.

library(tidyverse)
x <- 0:2
u <- c(2, -1, 2)
y <- -3 + 3 * x + u

You now have access to 3 vectors of data: x, u, and y.

u is random noise that’s drawn from a distribution with mean = 0, but in any given sample may not necessarily have mean = 0.
y is generated from x and u according to the equation in the line that starts with y <-. This is called the true model or the data generating process: it’s how y was truly generated.

1.1 What’s the true effect of x on y (slope of the true model)?

Take a look at the contents of x, u, and y:

## [1] 0 1 2

## [1]  2 -1  2

## [1] -1 -1  5

Notice that you can apply functions to your vectors like this:

x^2

## [1] 0 1 4

x/y

## [1]  0.0 -1.0  0.4

mean(x)

## [1] 1

1.2 Consider the model: \(y_i = \beta_0 + \beta_1 x_i + u_i\).

Calculate OLS estimates \(\hat{\beta_0}\) and \(\hat{\beta_1}\) using only these functions:

Addition and subtraction: +, -
Multiplication and division: *, /
Square: ^2
and three others: mean(), sum(), and sqrt() for square root.

# b1 <- __
#
# print(b1)
#
# b0 <- __
#
# print(b0)

After doing those calculations, check that they match the lm() results:

tibble(x, y) %>%
  lm(y ~ x, data = .) %>%
  broom::tidy()

## # A tibble: 2 × 5
##   term        estimate std.error statistic p.value
##   <chr>          <dbl>     <dbl>     <dbl>   <dbl>
## 1 (Intercept)       -2      2.24    -0.894   0.535
## 2 x                  3      1.73     1.73    0.333

1.3 Calculate the fitted values \(\hat{y_i}\) and the residuals \(e_i\) using b0 and b1.

Check that you have the correct answer:

tibble(x, y) %>%
  lm(y ~ x, data = .) %>%
  fitted.values()

##  1  2  3 
## -2  1  4

tibble(x, y) %>%
  lm(y ~ x, data = .) %>%
  residuals()

##  1  2  3 
##  1 -2  1

2. `plot()`

This problem is open-ended. You’ll create 3 visualizations of a small dataset. You’ll use plot() for drawing the graphs (or if you’re already comfortable with ggplot, you’re welcome to use that instead).

Run this code to get started:

world_data <- tibble(
  country   = c("Brazil", "Brazil", "Brazil", "Brazil",
                "Haiti", "Haiti", "Haiti", "Haiti",
                "Puerto Rico", "Puerto Rico", "Puerto Rico", "Puerto Rico",
                "Colombia", "Colombia", "Colombia", "Colombia"),
  year      = c(1952, 1972, 1992, 2007, 1952, 1972, 1992, 2007,
                1952, 1972, 1992, 2007, 1952, 1972, 1992, 2007),
  gdpPercap = c(2108.944,  4985.711,  6950.283,  9065.801,
                1840.367,  1654.457,  1456.310,  1201.637,
                3081.960,  9123.042, 14641.587, 19328.709,
                2144.115,  3264.660,  5444.649,  7006.580),
  lifeExp   = c(50.9, 59.5, 67.1, 72.4, 37.6, 48, 55.1, 60.9,
                64.3, 72.2, 73.9, 78.7, 50.6, 61.6, 68.4, 72.9)
)

Your data is called world_data. It tells you the GDP per capita and life expectancy for four different countries during four different years.

print(world_data)

## # A tibble: 16 × 4
##    country      year gdpPercap lifeExp
##    <chr>       <dbl>     <dbl>   <dbl>
##  1 Brazil       1952     2109.    50.9
##  2 Brazil       1972     4986.    59.5
##  3 Brazil       1992     6950.    67.1
##  4 Brazil       2007     9066.    72.4
##  5 Haiti        1952     1840.    37.6
##  6 Haiti        1972     1654.    48  
##  7 Haiti        1992     1456.    55.1
##  8 Haiti        2007     1202.    60.9
##  9 Puerto Rico  1952     3082.    64.3
## 10 Puerto Rico  1972     9123.    72.2
## 11 Puerto Rico  1992    14642.    73.9
## 12 Puerto Rico  2007    19329.    78.7
## 13 Colombia     1952     2144.    50.6
## 14 Colombia     1972     3265.    61.6
## 15 Colombia     1992     5445.    68.4
## 16 Colombia     2007     7007.    72.9

Read the qelp docs for qplot() to find out how to use it:

?qelp::qplot

Rubric:

Make 3 nice visualizations of different aspects of the data.
You should use color or fill at least one time.
Every plot should showcase at least one unique geom so that over the course of your 3 plots, you show at least 3 different geoms.
Each plot should have a title.

Write your code for your 3 plots here:

Classwork 2

Your Name Here

2024-03-17

1. `lm()`

1.1 What’s the true effect of x on y (slope of the true model)?

1.2 Consider the model: \(y_i = \beta_0 + \beta_1 x_i + u_i\).

1.3 Calculate the fitted values \(\hat{y_i}\) and the residuals \(e_i\) using b0 and b1.

2. `plot()`

4.1 Plot 1:

4.2 Plot 2:

4.3 Plot 3:

Classwork 2

Your Name Here

2024-03-17

1. lm()

1.1 What’s the true effect of x on y (slope of the true model)?

1.2 Consider the model: \(y_i = \beta_0 + \beta_1 x_i + u_i\).

1.3 Calculate the fitted values \(\hat{y_i}\) and the residuals \(e_i\) using b0 and b1.

2. plot()

4.1 Plot 1:

4.2 Plot 2:

4.3 Plot 3:

1. `lm()`

2. `plot()`