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survival analysis - Introduction to Survival Analysis and...

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Introduction to Survival Analysis and Censored Data Original applications in biometry were to survival times in cancer clinical trails Many other applications in biometry: eg. Disease onset ages Other applications in industrial life- testing Interest centered not only on average or median survival time but also on probability of surviving beyond 2 years, 5 years, 10 years etc. Best describe with a survival function S(t) o For T= a subject’s survival time, S(t)= P[T>t] o Characterizes the distribution of survival times T o Gives useful information for each t Horizontal axis: time (in years) Vertical axis: Survival function At t=0; s(t) = 1 At t =1; s(t) = .95 At t = 4 ; s(t) = .22 At t =6; s(t)= .14 Probability distribution for survival time They are continuous like normal distribution But, they only give p for greater than 0; unlike normal distribution It will not allow and probability for negative number Survival time between two points is just area under curve For continuous random variable, density function and survival function corresponds Knowing the exact shape of survival function tells everything about distribution of survival time What is that point in time, when half of patient will live and half of them will die From survival function, median is the point in time when survival function is 0.5 P [T> median] = 0.5 P [T≤ median] = 0.5
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With density function, P [T> median] = 0.5 Illustrative Data Survival function estimate This approximation based on six observation, looks like step function If we have more and more observation, this will closer to continuous So, non-parametric is always steps, but with larger sample size, it become closer to continuous It based on proportion of person die and figure out what is the probability of living beyond that time P [T>3.5 years] = 3/6 = 1/2 Median Estimate If we have odds number, it is easy
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