SURVIVAL ANALYSIS

Presented by

Sampa Baidya

III Ph.D

DFK 1201

Dept of Aquaculture

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.

Contd…

In engineering- reliability

analysis

In economics – duration

analysis

In sociology- event history

analysis

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?

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

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.

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.

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.

ILLUSTRATION OF SURVIVAL DATA

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.

Empirical survivor function

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

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.

LOG-RANK TEST

Comparing the survival curves of two

treatment groups

Use probiotic Control

Survival rate Survival rate

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).

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.

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.