Description
Exercise 1 (Using lm)
For this exercise we will use the cats dataset from the MASS package. You should use ?cats to learn about
the background of this dataset.
(a) Suppose we would like to understand the size of a cat’s heart based on the body weight of a cat. Fit
a simple linear model in R that accomplishes this task. Store the results in a variable called cat_model.
Output the result of calling summary() on cat_model.
(b) Output only the estimated regression coefficients. Interpret βˆ
0 and β1 in the context of the problem. Be
aware that only one of those is an estimate.
(c) Use your model to predict the heart weight of a cat that weights 3.1 kg. Do you feel confident in this
prediction? Briefly explain.
(d) Use your model to predict the heart weight of a cat that weights 1.5 kg. Do you feel confident in this
prediction? Briefly explain.
(e) Create a scatterplot of the data and add the fitted regression line. Make sure your plot is well labeled
and is somewhat visually appealing.
(f) Report the value of R2
for the model. Do so directly. Do not simply copy and paste the value from the
full output in the console after running summary() in part (a).
Exercise 2 (Writing Functions)
This exercise is a continuation of Exercise 1.
(a) Write a function called get_sd_est that calculates an estimate of σ in one of two ways depending on
input to the function. The function should take three arguments as input:
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• fitted_vals – A vector of fitted values from a model
• actual_vals – A vector of the true values of the response
• mle – A logical (TRUE / FALSE) variable which defaults to FALSE
The function should return a single value:
• se if mle is set to FALSE.
• σˆ if mle is set to TRUE.
(b) Run the function get_sd_est on the residuals from the model in Exercise 1, with mle set to FALSE.
Explain the resulting estimate in the context of the model.
(c) Run the function get_sd_est on the residuals from the model in Exercise 1, with mle set to TRUE.
Explain the resulting estimate in the context of the model. Note that we are trying to estimate the same
parameter as in part (b).
(d) To check your work, output summary(cat_model)$sigma. It should match at least one of (b) or (c).
Exercise 3 (Simulating SLR)
Consider the model
Yi = 5 + −3xi + i
with
i ∼ N(µ = 0, σ2 = 10.24)
where β0 = 5 and β1 = −3.
This exercise relies heavily on generating random observations. To make this reproducible we will set a
seed for the randomization. Alter the following code to make birthday store your birthday in the format:
yyyymmdd. For example, William Gosset, better known as Student, was born on June 13, 1876, so he would
use:
birthday = 18760613
set.seed(birthday)
(a) Use R to simulate n = 25 observations from the above model. For the remainder of this exercise, use
the following “known” values of x.
x = runif(n = 25, 0, 10)
You may use the sim_slr function provided in the text. Store the data frame this function returns in a
variable of your choice. Note that this function calls y response and x predictor.
(b) Fit a model to your simulated data. Report the estimated coefficients. Are they close to what you would
expect? Briefly explain.
(c) Plot the data you simulated in part (a). Add the regression line from part (b) as well as the line for
the true model. Hint: Keep all plotting commands in the same chunk.
(d) Use R to repeat the process of simulating n = 25 observations from the above model 1500 times. Each
time fit a SLR model to the data and store the value of βˆ
1 in a variable called beta_hat_1. Some hints:
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• Consider a for loop.
• Create beta_hat_1 before writing the for loop. Make it a vector of length 1500 where each element
is 0.
• Inside the body of the for loop, simulate new y data each time. Use a variable to temporarily store
this data together with the known x data as a data frame.
• After simulating the data, use lm() to fit a regression. Use a variable to temporarily store this output.
• Use the coef() function and [] to extract the correct estimated coefficient.
• Use beta_hat_1[i] to store in elements of beta_hat_1.
• See the notes on Distribution of a Sample Mean for some inspiration.
You can do this differently if you like. Use of these hints is not required.
(e) Report the mean and standard deviation of beta_hat_1. Do either of these look familiar?
(f) Plot a histogram of beta_hat_1. Comment on the shape of this histogram.
Exercise 4 (Be a Skeptic)
Consider the model
Yi = 3 + 0 · xi + i
with
i ∼ N(µ = 0, σ2 = 4)
where β0 = 3 and β1 = 0.
Before answering the following parts, set a seed value equal to your birthday, as was done in the previous
exercise.
birthday = 18760613
set.seed(birthday)
(a) Use R to repeat the process of simulating n = 75 observations from the above model 2500 times. For
the remainder of this exercise, use the following “known” values of x.
x = runif(n = 75, 0, 10)
Each time fit a SLR model to the data and store the value of βˆ
1 in a variable called beta_hat_1. You may
use the sim_slr function provided in the text. Hint: Yes β1 = 0.
(b) Plot a histogram of beta_hat_1. Comment on the shape of this histogram.
(c) Import the data in skeptic.csv and fit a SLR model. The variable names in skeptic.csv follow the
same convention as those returned by sim_slr(). Extract the fitted coefficient for β1.
(d) Re-plot the histogram from (b). Now add a vertical red line at the value of βˆ
1 in part (c). To do so,
you’ll need to use abline(v = c, col = “red”) where c is your value.
(e) Your value of βˆ
1 in (c) should be negative. What proportion of the beta_hat_1 values is smaller than
your βˆ
1? Return this proportion, as well as this proportion multiplied by 2.
(f) Based on your histogram and part (e), do you think the skeptic.csv data could have been generated
by the model given above? Briefly explain.
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Exercise 5 (Comparing Models)
For this exercise we will use the Ozone dataset from the mlbench package. You should use ?Ozone to learn
about the background of this dataset. You may need to install the mlbench package. If you do so, do not
include code to install the package in your R Markdown document.
For simplicity, we will perform some data cleaning before proceeding.
data(Ozone, package = “mlbench”)
Ozone = Ozone[, c(4, 6, 7, 8)]
colnames(Ozone) = c(“ozone”, “wind”, “humidity”, “temp”)
Ozone = Ozone[complete.cases(Ozone), ]
We have:
• Loaded the data from the package
• Subset the data to relevant variables
– This is not really necessary (or perhaps a good idea) but it makes the next step easier
• Given variables useful names
• Removed any observation with missing values
– This should be given much more thought in practice
For this exercise we will define the “Root Mean Square Error” of a model as
RMSE =
vuut
1
n
Xn
i=1
(yi − yˆi)
2.
(a) Fit three SLR models, each with “ozone” as the response. For the predictor, use “wind speed,” “humidity
percentage,” and “temperature” respectively. For each, calculate RMSE and R2
. Arrange the results in a
markdown table, with a row for each model. Suggestion: Create a data frame that stores the results, then
investigate the kable() function from the knitr package.
(b) Based on the results, which of the three predictors used is most helpful for predicting ozone readings?
Briefly explain.
Exercise 00 (SLR without Intercept)
This exercise will not be graded and is simply provided for your information. No credit will be
given for the completion of this exercise. Give it a try now, and be sure to read the solutions
later.
Sometimes it can be reasonable to assume that β0 should be 0. That is, the line should pass through the
point (0, 0). For example, if a car is traveling 0 miles per hour, its stopping distance should be 0! (Unlike
what we saw in the book.)
We can simply define a model without an intercept,
Yi = βxi + i
.
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(a) In the Least Squares Approach section of the text you saw the calculus behind the derivation of the
regression estimates, and then we performed the calculation for the cars dataset using R. Here you need to
do, but not show, the derivation for the slope only model. You should then use that derivation of βˆ to write
a function that performs the calculation for the estimate you derived.
In summary, use the method of least squares to derive an estimate for β using data points (xi
, yi) for
i = 1, 2, . . . n. Simply put, find the value of β to minimize the function
f(β) = Xn
i=1
(yi − βxi)
2
.
Then, write a function get_beta_no_int that takes input:
• x – A predictor variable
• y – A response variable
The function should then output the βˆ you derived for a given set of data.
(b) Write your derivation in your .Rmd file using TeX. Or write your derivation by hand, scan or photograph
your work, and insert it into the .Rmd as an image. See the RMarkdown documentation for working with
images.
(c) Test your function on the cats data using body weight as x and heart weight as y. What is the estimate
for β for this data?
(d) Check your work in R. The following syntax can be used to fit a model without an intercept:
lm(response ~ 0 + predictor, data = dataset)
Use this to fit a model to the cat data without an intercept. Output the coefficient of the fitted model. It
should match your answer to (c).
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