Survival Analysis Notes

Ht p tj y tj j y j j since nj nj 05cj is

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Unformatted text preview: 1 1 1 1 1 1 12/13 11/12 10/11 9/10 8/9 7/8 6/7 5/6 4/5 3/4 2/3 1/2 0/1 1 12/13 11/13 10/13 9/13 8/13 7/13 6/13 5/13 4/13 3/13 2/13 1/13 0/13 UNM Estimating the hazard function Recall that the hazard function is h (t ) = f (t ) P (t ≤ Y ≤ t + ∆|Y ≥ t ) ≈ S (t ) ∆ For tj ≤ t < tj +1 , if τj = tj +1 − tj is the j − th interval length. h(t ) ≈ P (tj ≤ Y ≤ tj + τj |Y ≥ τj ) τj Since nj = nj − 0.5cj is the average no. at risk in the j − th interval, nj − 0.5dj is the average no. alive in the j − th interval. UNM Then P (tj ≤ Y ≤ tj + τj |Y ≥ τj ) ≈ dj /(nj − 0.5dj ) so the estimator for the hazard is h ∗ (t ) = dj nj − 0.5dj 1 τj with actuarial correction. The Kaplan-Meier estimator (no corr...
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