Class Activity, September 26

Confidence intervals with the Titanic data

In this class activity, we will construct confidence intervals for quantities of interest in the titanic data. Let \(Y_i\) denote whether passenger \(i\) survived. We fit the model

\[Y_i \sim Bernoulli(p_i)\]

\[\log \left( \dfrac{p_i}{1 - p_i} \right) = \beta_0 + \beta_1 Sex_i + \beta_2 Age_i + \beta_3 SecondClass_i + \\ \hspace{3cm} \beta_4 FirstClass_i + \beta_5 Sex_i \cdot Age_i\]

Questions

  1. Run the code below to construct a 95% confidence interval for \(\beta_4 - \beta_3\), the difference in log odds between first and second class passengers with the same sex and age.
# read in the data, and convert passenger class with a factor
# specify that the order is third, second, first
titanic <- read.csv("https://sta214-f22.github.io/labs/Titanic.csv")
titanic <- titanic %>%
  drop_na() %>%
  mutate(Pclass = factor(Pclass, levels = c(3, 2, 1))) 

# fit a logistic regression model
m1 <- glm(Survived ~ Sex*Age + Pclass, data = titanic, family = binomial)

# now we want to construct a confidence interval for beta_4 - beta_3
# first, create a vector to specify the linear combination of coefficients
# (note that by default, R puts interaction coefficients at the end)
a <- c(0, 0, 0, -1, 1, 0)

# calculate lower and upper bounds
t(a) %*% coef(m1) - qnorm(0.975) * sqrt(t(a) %*% vcov(m1) %*% a)
t(a) %*% coef(m1) + qnorm(0.975) * sqrt(t(a) %*% vcov(m1) %*% a)

  1. Modify the code from Question 1 (you just need to change the a vector) to create a 95% confidence interval for the log odds of survival for a 20 year old, female passenger in second class.

  2. Using your confidence interval from Question 2, create a 95% confidence interval for the probability of survival for a 20 year old, female passenger in second class.