Survival 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

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

SurvivalAnalysis

Cont’d…

In engineering- reliability analysis

In economics – duration analysis

In sociology- event history analysis

• 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

“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

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.

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.

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.

ILLUSTRATION OF SURVIVAL DATA

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 ttotal number of patients in the study

The function that describes the probability distribution that an animal survives to at least time t.

Survival

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

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

Log-Rank Test

Comparing the survival curves of two treatment groups

COX REGRESSION MODELIncorporating 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

• is the time• are the covariates for the individual• is the baseline hazard function. This

is the function when all the covariates equal to zero.

Hazard function

• The hazard function:

This is the risk of failure immediately after time given they have survived past time t.