4th year Flashcards
(12 cards)
Define the hazard h(t) as an instantaneous failure rate at time t
The hazard at time t is defined as h(t) = lim (δ→0^+) P(T ≤ t + δ|T > t) δ [2] with the interpretation as instantaneous failure rate
Give an example of a lifetime distribution where the hazard function is constant.
The (negative) exponential distribution Exp(λ) with pdf
f(t) = λe^(−λt), t > 0
has survival function S(t) = e
^(−λt) and constant hazard h(t) = λ.
Write down the Cox proportional hazards (PH) model for a general hazard function h(t) given
some covariate x, clearly stating any assumptions associated with it.
The Cox proportional hazards model is
h(t) = h0(t)e^(βx), where h_0(t) is an unspecified hazard function and β is a constant.
Define the survival function S(t)
S(t) = P(T > t), t > 0.
Explain proportional hazards and comment on the hazard ratio in a Cox model.
The hazard h(t; x) is proportional to h0(t) and the hazard ratio
h(t; x)/h(t; x∗) = e^[β(x−x∗)]
does not depend on t.
Explain partial likelihood used in the fitting of a Cox model. How are tied cases dealt with?
Partial likelihood is based on the order in which failures occur and relative risk. It is constructed as a
product of risk ψ = e
βx divided by total risk just before each failure.
When d observations are tied, their contribution becomes the product of the d risks divided by the
sum of all possible products of d from the subset at risk.
Why is the Cox model semi-parametric?
The Cox model is semi-parametric because it has a nonparametric part h0(t) and a parametric part e^(βx)
The Kaplan-Meier estimator is given by
Sˆ(t) =∏(ti ≤t) (1−di/ri),
Relationship between survival and hazard
h(t) = −S′(t)/S(t) S(t) = exp(−H(t))
If T ∼ W (α,β),
, then V = (T /β)^α ∼ Exp(1)
what formula?
Partial log-likelihood for Cox’s model
Lp = = P(I1 = i1)P(I2 = i2|I1 = i1)P(I3 = i3|I2 = i2,I1 = i1) ···P(In = in|In−1 = in−1,...,I1 = i1)
The conditional probability for Cox’s model at tj is
ψij/[SUM(k≥j) ψik]
