quiz.qmd

---
title: "10-Minute In-Class Quiz 1: Survival Analysis (Chapters 1–4)"
format: pdf
---

## **Instructions**
- You have **10 minutes** to complete this quiz.
- Answer all questions *concisely*.
- Show all relevant calculations where applicable.

## **Question 1 (4 points)**
The table below shows survival data for **5 individuals** in a study. The event/censoring times and status
(1: event; 0: censoring) information are given:

| Time     | Status | 
|----------|------------------------------|
| 2        | 1                              | 
| 4        | 0                              | 
| 6        | 1                              | 
| 7.5        | 0                              | 
| 8        | 1                              | 

(a) Compute the Kaplan-Meier estimator $\hat{S}(t)$ at times $t = 2, 6, 7.5$, and $8$.  
(b) Interpret $\hat{S}(6)$ in words.

## **Question 2 (2 points)**
The hazard function $h(t)$ is defined as:
$$
\lambda(t) = \lim_{\Delta t \to 0} \frac{P(t \leq T < t+\Delta t \mid T \geq t)}{\Delta t}
$$
(a) Explain what the hazard function represents in survival analysis.  
(b) How is it related to the survival function $S(t)$?  

## **Question 3 (2 points)**
You fit a **Kaplan-Meier estimator** to a dataset and obtain the following estimated survival probabilities at two time points:
$$
\hat{S}(3) = 0.80, \quad \hat{S}(5) = 0.65.
$$
For a subject who has already survived to 3 (years), what is their probability of surviving to 5 (years)?