Introduction and Useful Properties Flashcards
Give relationship between survival function and cdf
S(t) = 1 - F(t)
Give relationship between survival function and pdf
f(u) = - d/du S(u)
If a≥b, what is S(b) - S(a)
∫(b to a) f(u) du ≥ 0
What is Shatn(t)
(number of observations > t) / n
What is a hazard function and how is it found?
Hazard function is probability of event occurring in time (t, t + δt) given that is has not already happened
h(t) = f(t) / S(t)
How can S(t) be obtained from the hazard function?
exp( - ∫(0 to t) h(u) du )
What is the integrated hazard function
H(t) = ∫(0 to t) h(u) du
How are the survival function and integrated hazard function related?
S(t) = exp( - H(t) )
What is the expected survival time?
E(t) = ∫(0 to infinity) S(t) dt
What is the residual life expectancy?
r(t) = ∫(t to infinity) S(u) du / S(t)
How can mean life expectancy be found from residual life expectancy?
r(t) + t
What the empirical distribution function and what is it used for?
Fhat(t) = number of observations ≤ t / n
plotting against t gives indication whether reasonable to assume certain parent distribution
What is the risk set?
number of people known to be alive at time t
How do you find the estimated survival function?
mulitply each prob not dead for each time event, also known as the Kaplan-Meier estimate
How are the empirical distribution function and Kaplan-Meier estimate linked?
EDF complement of K-M
