---
title: Introduction to Analysing Competing risk data
tags:  #[第3回大阪sas勉強会]  
author: [森岡裕[SASユーザー総会世話人]](https://convert.docswell.com/user/6484025)
site: [Docswell](https://www.docswell.com/)
thumbnail: https://bcdn.docswell.com/page/GJWDX9Z872.jpg?width=480
description: Introduction to Analysing Competing risk data by 森岡裕[SASユーザー総会世話人]
published: April 10, 24
canonical: https://convert.docswell.com/s/6484025/5YW373-2024-04-10-111235
---
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Introduction to Analysing
Competing risks data


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Source
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All materials are from :
Using the PHREG Procedure to Analyze Competing-Risks Data
(Ying So, Guixian Lin, and Gordon Johnston, SAS Institute Inc.)
https://support.sas.com/rnd/app/stat/papers/2014/competingrisk2014.pdf


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What is Competing-risks data?
A competing-risks situation arises when
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Individual can experience more than one type of event
One event might hinder the occurrence of other types of events
Only the time to failure for the earliest of these events is observed


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Example of Competing-risks situations
Breast cancer patients can
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die from breast cancer
censored
die from causes other than breast cancer


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Problems?
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The product-limit estimator (K-M method), might yield biased results when
analysing competing-risks data


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What to do?
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One way is to use the cumulative incidence function (CIF)
CIF = the probability subdistribution function of failure from a specific cause.


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What is the CIF?
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The probability of failure due to cause k prior to time t.
In equation
𝐹𝑘 𝑡 = 𝑃 𝑇 ≤ 𝑡, δ = 𝑘 𝑘 = 1, … 𝐾
where T = Failure time, δ = censor (0) or events (1, … K)


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CIF (derivation)
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First define the cause-specific hazard function h(t)
𝑃 𝑡 &lt; 𝑇 &lt; 𝑇+ ∆𝑡 , 𝛿=𝑘 𝑇 &gt; 𝑡)
∆𝑡
∆𝑡 →0
ℎ𝑘 𝑡 = lim
, 𝑘 = 1, … 𝐾
The instantaneous rate of failure due to cause k conditional on survival until time t or later
Can derive following equation
ℎ𝑘 𝑡 =
𝑓𝑘 (𝑡)
𝑆(𝑡)
→ 𝑓𝑘 𝑡 = 𝑆 𝑡 ℎ𝑘 𝑡
𝑡
Integrating both side, we get 𝐹𝑘 𝑡 = ‫׬‬0 𝑆 𝑢 ℎ𝑘 𝑢 𝑑𝑢 , 𝑘 = 1, … 𝐾


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Define the sub-distribution hazard
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The hazard of the CIF
ℎ෨ 𝑘 𝑡 =
𝑑
𝑑𝑡
log(1 − 𝐹𝑘 (𝑡))
Not the same as the cause-specific hazard (difference in risk sets)


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Apply these concepts to data!!
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DATA - Acute leukemia patients treated with bone marrow transplant. Want
to find out if time until relapse is different for disease groups.
This is a competing risks scenario because death is a competing factor for
relapse.


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Data
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Variables
Time-variable: T Time until events or
censor
Censor variable: Status (0 censored, 1
=relapse, 2=death)
Covariates:
LogWaittime (wait time until transplant)
Group (1= acute lymphoblastic leukemia
(ALL), 2= acute myelocytic leukemia
(AML) low-risk, 3= AML high-risk)


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Modelling the Cumulative incidence
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Modelling by defining the hazard of the cumulative incidence function
(subdistribution hazard)
Impose the proportional hazards assumption on the subdistribution hazards
Can fit this model in PHREG using EVENTCODE= option in Model statement


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Thank you!!


