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Unformatted text preview: rvival function is S (t) = prob(T ≥ t), which satisﬁes S (0) = 1, S ′ (t) ≤ 0, and limt→∞ S (t) =
0. The hazard rate is given by λ(t) = −S ′ (t)/S (t) ∈ R+ , and has the following interpretation: For
small δ > 0, λ(t)δ is approximately the probability of the event occurring in [t, t + δ ], given that it
has not occurred up to time t. The survival function can be expressed in terms of the hazard rate:
t S (t) = exp −
57 λ(τ ) dτ .
0 (The hazard rate must have inﬁnite integral over [0, ∞).) The Cox proportional hazards model gives the hazard rate as a function of some features or explanatory variables (assumed constant in time) x ∈ Rn . In particular, λ is given by
λ(t) = λ0 (t) exp(wT x),
where λ0 (which is nonnegative, with inﬁnite integral) is called the baseline hazard rate, and w ∈ Rn
is a vector of model parameters. (The name derives from the fact that λ(t) is proportional to
exp(wi xi ), for each i.)
Now suppose that we have observed a set of independent samples, with event tim...
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- Fall '13
- The Aeneid