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1. SURVIVAL ANALYSIS
Presented by
Sampa Baidya
III Ph.D
DFK 1201
Dept of Aquaculture

2. SURVIVALANALYSIS
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.

3. Contd…
In engineering- reliability
analysis
In economics – duration
analysis
In sociology- event history
analysis

4. 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?

5. 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

6. 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.

7. 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.

8. Censored data can occur when
– The event of interest is death, but the patient is still
alive at the time of analysis.
– The individual was lost to follow-up without having
the event of interest.
– The event of interest is death by cancer but the patient
died of an unrelated cause, such as a car accident.
– The patient is dropped from the study without having
experienced the event of interest due to a protocol
violation.

9. ILLUSTRATION OF SURVIVAL DATA

10. 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.

11. Empirical survivor function
For the case in which there are no censored individuals

12. But usually there is censoring. Therefore
we can estimates S(t) using the Kaplan
Meier estimator

13. 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.

14. LOG-RANK TEST
Comparing the survival curves of two
treatment groups

15. Use probiotic Control
Survival rate Survival rate

16. 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).

17. Cox regression model
𝜆 𝑡, 𝑥𝑖 = 𝜆0 𝑡 𝑒 𝛽′ 𝑥 𝑖
• 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.

18. Hazard function
• The hazard function:
𝜆 𝑡 = lim
Δ𝑡 →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.