survival analysis
TRANSCRIPT
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SURVIVAL ANALYSIS
Presented by
Sampa Baidya
III Ph.D
DFK 1201
Dept of Aquaculture
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SURVIVAL ANALYSIS
Branch of statistics that focuses on time-to-event data and
their analysis.
deals with analysis of time duration to until one or more
events happen
e.g. 1. death in biological organisms
2. failure in mechanical systems.
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Contd…
In engineering- reliability
analysis
In economics – duration
analysis
In sociology- event history
analysis
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Objectives of survival analysis?
• Estimate probability that an individual surpasses
some time-to-event for a group of individuals.
– Ex) probability of surviving longer than two months until second heart
attach for a group of MI patients.
• Compare time-to-event between two or more groups.
– Ex) Treatment vs placebo patients for a randomized controlled trial.
• Assess the relationship of covariates to time-to-event.– Ex) Does weight, BP, sugar, height influence the survival time for a
group of patients?
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Situations when we can use survival
analysis
“Time-to-Event” include:
– Time to death
– Time until response to a treatment
– Time until relapse of a disease
– Time until cancellation of service
– Time until resumption of smoking by someone who had quit
– Time until certain percentage of weight loss
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What is Survival Time?
• Survival time refers to a variable which measures
the time from a particular starting time (e.g., time
initiated the treatment) to a particular endpoint of
interest.
• It is important to note that for some subjects in the
study a complete survival time may not be
available due to censor.
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SURVIVAL DATA
• It can be one of two types:
– Complete Data
– Censored Data
• Complete data – the value of each sample unit is observed or
known.
• Censored data – the time to the event of interest may not be
observed or the exact time is not known.
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Censored data can occur when
– The event of interest is death, but the patient is stillalive at the time of analysis.
– The individual was lost to follow-up without havingthe event of interest.
– The event of interest is death by cancer but the patientdied of an unrelated cause, such as a car accident.
– The patient is dropped from the study without havingexperienced the event of interest due to a protocolviolation.
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ILLUSTRATION OF SURVIVAL DATA
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Survival Function or Curve
Let T denote the survival time
S(t) = P(surviving longer than time t )
= P(T > t)
The function S(t) is also known as the cumulative survival
function. 0 S( t ) 1
Ŝ(t)= number of patients surviving longer than t
total number of patients in the study
The function that describes the probability distribution that an
animal survives to at least time t.
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Empirical survivor function
For the case in which there are no censored individuals
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But usually there is censoring. Therefore
we can estimates S(t) using the Kaplan
Meier estimator
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If there is censoring, the Kaplan meier estimate of survival
is defined as
• ti is the set of observed death times
• ni is the number of individuals at risk at time ti
ni = number known alive at time ti-1 minus those individuals known
dead or censored at time ti-1)
• di is the number of individuals known dead at time ti.
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LOG-RANK TEST
Comparing the survival curves of two
treatment groups
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Use probiotic Control
Survival rate Survival rate
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COX REGRESSION MODEL
Incorporating Covariates
Covariate: independent variable.
This model produces a survival function that predicts the
probability that an event has occurred at a given time t, for
given predictor variables (covariates).
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Cox regression model
𝜆 𝑡, 𝑥𝑖 = 𝜆0 𝑡 𝑒𝛽′𝑥𝑖
• 𝑡 is the time
• 𝑥𝑖 are the covariates for the 𝑖th individual
• 𝜆0 𝑡 is the baseline hazard function. This is the function when all the covariates equal to zero.
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Hazard function
• The hazard function:
𝜆 𝑡 = limΔ𝑡 →0
𝑃 𝑡 < 𝑇 < 𝑡 + Δ𝑡 𝑇 ≥ 𝑡)
∆ 𝑡
This is the risk of failure immediately after time 𝑡, given they have survived past time t.
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