STAT 432/532 – Survival Analysis Midterm
STAT 432/532 – Survival Analysis
Lingzhu Li
Instructions: Answers must be accomplished by adequate justification. Good luck!
1. (a) (b) (c)
2. (a)
(b)
Explain what is meant by censored observations in survival data.
State the difference between the right and left censoring.
Describe the interval censoring and illustrate it by a practical example.
Determine the survival function and density function for survival random variable with hazard function
b(x)= 2x . 1+x2
Suppose X is the time to an event and S(x) is its survival function. Show that E(X) = ∞ S(t)dt.
0
3. A clinical trial to evaluate the efficacy of maintenance chemotherapy for acute myelogenous leukemia (AML) was conducted. The following table shows times of remission (i.e. freedom from symptoms in a precisely defined sense) of AML patients received chemotherapy.
9, 13, 13+, 18, 23, 28+, 31, 34, 45+, 48, 161+ Observations with + are right censored.
(a) Calculate the Kaplan-Meier estimate for the survival probability S(48). (b) Find a 95% log-transformed confidence interval for S(48).
(c) Sketch a plot for estimated S(t).
Winter 2020 1/2
STAT 432/532 – Survival Analysis Midterm
4. Let X have a uniform distribution on 0 to 100 days with probability density function
5.
f(x) =
Find the mean residual lifetime at 25, 50, and 75 days.
1/100, for 0 < x < 100, 0, otherwise .
(a) Write down Nelson–Aalen estimator H ̃ (t) of the cumulative hazard rate func- tion H(t).
(b) DeriveanapproximateexpressionforthevarianceofS ̃(t)estimatedbyH ̃(t). [Hint: Delta method.]
Winter 2020 2/2