AE 06: Exam 02 review

Published

April 9, 2026

Important

There is no repo for this AE.

library(tidyverse)
library(knitr)
library(tidymodels)
library(pROC)
library(Stat2Data)

Exercise 1

Suppose you fit a simple linear regression model.

  1. Draw a scatterplot that contains an observation with large leverage but low Cook’s distance.

  2. Draw a scatterplot that contains an observation with large leverage and high Cook’s distance.

  3. Draw a scatterplot that contains an observation with a large standardized residual.

Exercise 2

  1. Describe what it means for \(\tilde{\boldsymbol{\beta}}\) to be the maximum likelihood estimator.

  2. What are properties of MLEs?

Exercise 3

This exercise is adapted from (casella2024statistical?).

Suppose there are \(n\) observations \((x_1, y_1), \ldots, (x_n, y_n)\), such that the relationship between \(X\) and \(Y\) can be summarized as \[ y_i = \beta x_i^2 + \epsilon_i \hspace{8mm} \epsilon_i \sim N(0,\sigma^2_{\epsilon}) \]

  • Find the maximum likelihood estimator of \(\beta\) .

  • Is the MLE unbiased?

Note

Use this data analysis scenario for Exercises 4 - 6.

The data for this analysis is about credit card customers. It can be found in the file credit.csv. The following variables are in the data set:

  • income: Income in $1,000’s
  • limit: Credit limit
  • rating: Credit rating
  • cards: Number of credit cards
  • age: Age in years
  • education: Number of years of education
  • own: Whether an individual owns their home ( No or Yes )
  • student: Whether the individual was a student ( No or Yes )
  • married: Whether the individual was married (No and Yes)
  • region: Region the individual is from ( South, East, and West)
  • balance: Average credit card balance in $.
credit <- read_csv("data/credit.csv") |>
  mutate(maxed = factor(if_else(balance == 0, 1, 0)))

The objective of this analysis is to predict whether a person has maxed out their credit card, i.e., had $0 average card balance.

We’ll start with a model predicting the odds of maxed = 1 using income, rating, and region.

credit_fit <- glm(maxed ~ income + rating + region, data = credit, 
                  family = "binomial")

tidy(credit_fit) |>
  kable(digits = 3)
term estimate std.error statistic p.value
(Intercept) 9.898 1.449 6.829 0.000
income 0.113 0.021 5.273 0.000
rating -0.057 0.008 -7.397 0.000
regionSouth -0.595 0.604 -0.985 0.324
regionWest -0.082 0.649 -0.126 0.900

Exercise 4

The logistic regression model takes the following form:

\[ \log\big(\frac{\pi_i}{1 - \pi_i}\big) = \beta_0 + \beta_1 ~ income + \beta_2 ~ rating + \beta_3 ~ regionSouth + \beta_4 ~ regionWest \]

  1. Write the interpretation of income in terms of the odds of maxing out a credit card.

  2. What is the odds ratio of the South versus the East regions?

  3. Suppose there are two individuals. Individual 1 has an income of $64,000, a credit rating of 590, and is from the South region. Individual 2 has an income of $135,000, a credit rating of 695, and is from the East region. Use the equation above to show how the odds of maxing out a credit card differ between Individual 1 and Individual 2. Write your answer in terms of \(\beta_0, \beta_1, \beta_2\), etc.

Exercise 5

We consider adding the interaction between region and income to the current model. We use a drop-in-deviance test to determine whether or not to add the interaction term. The output for the drop-in-deviance test is below.

credit_fit_int <- glm(maxed ~ income + rating + region + region * income, data = credit, 
                  family = "binomial")

anova(credit_fit, credit_fit_int, test = "Chisq") |>
  kable(digits = 2)
Resid. Df Resid. Dev Df Deviance Pr(>Chi)
395 107.66 NA NA NA
393 106.53 2 1.13 0.57
  1. State the null and alternative hypotheses in words and using mathematical notation.
  2. Describe what the test statistic \(G\) means in the context of the data.
  3. Show why the degrees of freedom for the test statistic are equal to 2.
  4. State the conclusion in the context of the data.

Exercise 6

Use the model credit_fit that includes the main effects for income, rating, and region. The ROC curve and AUC are below.

credit_fit_aug <- augment(credit_fit, type.predict = "response")

credit_fit_aug |>
  roc_curve(
    maxed, 
    .fitted,
    event_level = "second"
  ) |>
  autoplot()

credit_fit_aug |>
  roc_auc(
    maxed, 
    .fitted,
    event_level = "second"
  )
# A tibble: 1 × 3
  .metric .estimator .estimate
  <chr>   <chr>          <dbl>
1 roc_auc binary         0.983
  1. Based on the AUC, do you think this model sufficiently identifies those who will max out their credit card vs. those who will not? Explain.
  2. Suppose a credit card company uses your model to inform the credit limit to give to new customers. Do you think they would prioritize having high sensitivity or high specificity? Explain.
  3. Based on your response to part(b), would you set a high (closer to 1) or low (closer to 0) classification threshold?