In this presentation we're going to introduce survival analysis, it's use and functions and so on and so far ...

1. 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
Branch of statistics that focuses on time-to-event data and
their analysis.
Survival
Analysis

2. Cont’d…
In engineering- reliability
analysis
In economics – duration
analysis
In sociology- event history
analysis

3. • Estimate time-to-event probability for a group of
individuals.
– Ex) probability of surviving longer than two months until second heart
attack 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?
Of survival analysis

4. “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
when we can use
survival analysis

5. What is Survival Time?
It is important to note that for some
subjects in the study a complete survival
time may not be available due to censor.
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.

6. 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.

7. When censored data can occur
– 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.

8. ILLUSTRATION OF
SURVIVAL DATA

9. 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.
Survival

10. Empirical survivor
For the case in which there are no censored
individuals
But usually there is censoring. Therefore we
can estimates S(t) using the Kaplan Meier
estimator

11. 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.
Kaplan Meier

12. Log-Rank Test
Comparing the survival curves of two treatment groups

13. 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).

14. 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.

15. 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.