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Unformatted text preview: 1 KaplanMeier methods and Parametric Regression methods Kristin Sainani Ph.D. http://www.stanford.edu/~kcobb Stanford University Department of Health Research and Policy 2 More on KaplanMeier estimator of S(t) (“productlimit estimator” or “KM estimator”) When there are no censored data, the KM estimator is simple and intuitive: Estimated S(t)= proportion of observations with failure times > t. For example, if you are following 10 patients, and 3 of them die by the end of the first year, then your best estimate of S(1 year) = 70%. When there are censored data, KM provides estimate of S(t) that takes censoring into account (see last week’s lecture). If the censored observation had actually been a failure: S(1 year)=4/5*3/4*2/3=2/5=40% KM estimator is defined only at times when events occur! (empirically defined) 3 KM (productlimit) estimator, formally ] 1 [ ) ( at time event the have number who the is risk at s individual are there , event time each at s event time distinct k : 1 ∏ ≤ = < < < t t j j j j j j j k j j n d t S t d n t t ... t t 4 KM (productlimit) estimator, formally S(t) represents estimated survival probability at time t: P(T>t) Observed event times Typically d j = 1 person, unless data are grouped in time intervals (e.g., everyone who had the event in the 3 rd month). The risk set n j at time t j consists of the original sample minus all those who have been censored or had the event before t j This formula gives the productlimit estimate of survival at each time an event happens. d j /n j =proportion that failed at the event time t j 1 d j /n j =proportion surviving the event time Multiply the probability of surviving event time t with the probabilities of surviving all the previous event times. ] 1 [ ) ( at time event the have number who the is risk at s individual are there , event time each at s event time distinct k : 1 ∏ ≤ = < < < t t j j j j j j j k j j n d t S t d n t t ... t t 5 Example 1: timetoconception for subfertile women “Failure” here is a good thing. 38 women (in 1982) were treated for infertility with laparoscopy and hydrotubation. All women were followed for up to 2years to describe timetoconception. The event is conception, and women "survived" until they conceived. Example from: BMJ , Dec 1998; 317: 1572  1580. Data from: Luthra P, Bland JM, Stanton SL. Incidence of pregnancy after laparoscopy and hydrotubation. BMJ 1982; 284: 10131014 Raw data: Time (months) to conception or censoring in 38 subfertile women after laparoscopy and hydrotubation (1982 study) 1 2 1 3 1 4 1 7 1 7 1 8 2 8 2 9 2 9 2 9 2 11 3 24 3 24 3 4 4 4 6 6 9 9 9 10 13 16 Conceived (event) Did not conceive (censored) 7 Corresponding KaplanMeier Curve S(t) is estimated at 9 event times....
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 Fall '11
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 Survival analysis, event time

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