Introduction to Survival Analysis and
Censored Data
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Original applications in biometry were
to survival times in cancer clinical trails
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Many other applications in biometry:
eg. Disease onset ages
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Other applications in industrial life
testing
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Interest centered not only on average or
median survival time but also on
probability of surviving beyond 2 years,
5 years, 10 years etc.
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Best describe with a survival function
S(t)
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For T= a subject’s survival time, S(t)=
P[T>t]
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Characterizes the distribution of
survival times T
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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
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Probability distribution for survival time
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They are continuous like normal
distribution
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But, they only give p for greater than 0;
unlike normal distribution
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It will not allow and probability for
negative number
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Survival time between two points is just
area under curve
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For continuous random variable, density
function and survival function
corresponds
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Knowing the exact shape of survival
function tells everything about
distribution of survival time
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What is that point in time, when half of
patient will live and half of them will die
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From survival function, median is the
point in time when survival function is 0.5
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P [T> median] = 0.5
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P [T≤ median] = 0.5
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With density function, P [T> median] =
0.5
Illustrative Data
Survival function estimate
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This approximation based on six
observation, looks like step function
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If we have more and more observation,
this will closer to continuous
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So, nonparametric is always steps, but
with larger sample size, it become closer
to continuous
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It based on proportion of person die and
figure out what is the probability of living
beyond that time
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P [T>3.5 years] = 3/6 = 1/2
Median Estimate
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If we have odds number, it is easy
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 Spring '11
 BarbaraMc.Knight
 Normal Distribution, probability density function, Survival analysis, survival function

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