33071-33081

# 33071-33081 - 1 Kaplan-Meier methods and Parametric...

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Unformatted text preview: 1 Kaplan-Meier 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 Kaplan-Meier estimator of S(t) (“product-limit 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 (product-limit) 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 (product-limit) 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 product-limit 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: time-to-conception 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 2-years to describe time-to-conception. 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: 1013-1014 Raw data: Time (months) to conception or censoring in 38 sub-fertile 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 Kaplan-Meier Curve S(t) is estimated at 9 event times....
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33071-33081 - 1 Kaplan-Meier methods and Parametric...

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