1.19.10 Lec 3

# 1.19.10 Lec 3 - Lecture 3 Time to Event Data Probability...

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1 Lecture 3 Time to Event Data Probability concepts and Probability concepts and distributions 1 Lecture 3 Outline 1. Exploratory Data Analysis: Time to Event Data 2. Basic Probability Concepts 3. Rules of Probability 4. Other Rules of Probability 5. Examples using Probability 6. Bayes' Rule 7. Probability Distributions 2

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2 1. Time to Event Data Also known as “survival” data •T i : time to an event for person i •E x a m ples ¾ Time from enrollment in study until myocardial infarction ¾ Age at first marriage • What is special about time-to-event data? ¾ T i >0 ¾ We do not observe the actual times for people who do not have the event; we only know that they did not have the event up to some time when follow-up ceased; i.e. T the event up to some time when follow up ceased; i.e. T is greater than some value ¾ Write T+ for greater than T when the value is censored 3 1.1 Typical Clinical Study • Enrollment over calendar time X Event Dropped out Censored Start Enrollment End Enrollment End Follow-up Calendar time 4
3 1.1 Typical Clinical Study (cont’d) • Change to time from enrollment to event X 0 Time in Study 5 1.2 Thinking About Possibly Censored Times-to- Events Censoring by drop - out or study termination Censoring by drop out or study termination presents problems • Simple example: ¾ Times-to-events: 1, 2+, 2, 3, 4+, 5+, 6, 8 where 2+ means > 2 What is the mean? • Histogram: where to put 2+ , 4 + and 5+ ? 6

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4 1.2.1 Thinking About Possibly Censored Times-to-Events (cont’d) • Data: 1, 2+, 2, 3, 4+, 5+, 6, 8 Translate times T into a sequence o 0s and 1s Translate times T i into a sequence of 0s and 1s by letting 0 = no event; 1 = event Time to event Time (t) 12345678 11 2+ 0 0 20 1 3 001 4 + 0000 5 + 00000 6 000001 8 00000001 7 1.2.2 Thinking About Possibly Censored Times -to-Events (cont’d) Time to event Time (t) 1 1 2+ 0 0 1 30 0 1 4 + 5 + 6 1 8 0 0 0 0 0 0 0 1 # at r i s k 87543211 # e v e n t s 11100101 Fraction of events 1/8 1/7 1/5 0/4 0/3 1/2 0/1 1/1 Fraction without events 7/8 6/7 4/5 4/4 3/3 1/2 1/1 0/1 Fraction Surviving beyond t .875 .75 .6 .6 .6 .3 .3 0 8
5 ¾ Survival function at t : Basic Descriptors of Time_to_Event Data Cumulative distribution function of T ( ) ( ) SP r o b 1 ( ) tT t F t => = ¾ Hazard function at t : among those who survive the event by time t , the proportion who develop the event in the next unit of time (i.e., instant, click) Probability density function of T () rate of change of S at ( ) value of S at f t ht tS t t == ¾ Percentile function: time at which p % develop the event, i.e., t ( p ) is the time such that ( the median survival time is t (0.5) ( ) ( ) Prob Time_to_event p tp =≤ 9 1.0 1.3 Survival Function • S(t) = Probability of survival beyond t = Pr (T > t) • Pr(survive beyond 6 years) =1 .25=.75 0.4 0.6 0.8

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## This note was uploaded on 02/24/2010 for the course BIOS 600 taught by Professor Staff during the Spring '08 term at UNC.

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1.19.10 Lec 3 - Lecture 3 Time to Event Data Probability...

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