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

SURVIVAL ANALYSIS

Presented by

Sampa Baidya

III Ph.D

DFK 1201

Dept of Aquaculture

Page 2: Survival  analysis

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.

Page 3: Survival  analysis

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.

Page 18: Survival  analysis

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.

Page 19: Survival  analysis