# Survival Analysis

### Course content

Survival analysis or failure time data analysis means the statistical analysis of data, where the response of interest is the time from a well-defined time origin to the occurrence of some given event (end-point). In biomedicine the key example is the time from randomization to a given treatment for some patients until death occurs leading to the observation of survival times for these patients. The objective may be to compare different treatment effects on the survival time possibly correcting for information available on each patient such as age and disease progression indicators. This leaves us with a statistical regression analysis problem. Standard methods will, however, often be inappropriate because survival times are frequently incompletely observed with the most common example being right censoring. The event time T is said to be right censored if it is only known that T is larger than an observed right censoring time. This may be because the patient is still alive at the point in time where the study is closed and the data are to be analyzed, or because the subject is lost for follow-up due to other reasons.

The course gives an introduction to concepts and methods in survival and event history analysis and tools for establishing their theoretical properties using counting process and martingale techniques. The exercises are mostly theoretical, but some involve programming in R where properties of the methods are investigated. Topics include counting processes, intensities, filtrations and martingales; the nonparametric Nelson-Aalen and Kaplan-Meier estimators; the log-rank test; hazard regression models including parametric models and the semiparametric Cox proportional hazards model; competing risk; statistical computing in R.

Education

MSc Programme in Mathematics-Economics

MSc Programme in Statistics

Learning outcome

Knowledge:

• Distinguish methods for analysis of time-to-event data from other types of measurements.
• Understand the concepts of censoring and truncation and explain the independent censoring assumption.
• Explain survival analysis concepts such as hazard, survival and cumulative incidence and their relationships.
• Understand how to represent time-to-event data in terms of counting processes.

Skills: Ability to

• demonstrate how common estimators, score equations and test statistics based on time-to-event data can be expressed as martingale integrals
• apply martingale techniques to establish asymptotic properties for commonly used censored data estimators, e.g., by Robelledo’s martingale central limit theorem, Lenglart’s inequality and Duhamel’s equation.
• model covariate effects on event time distributions and explain the underlying assumptions
• implement estimators for censored data in R and assess the accuracy of the estimators based on simulations.

Comptences: Ability to

• explain and understand when methods for censored data and competing risks are needed.
• formulate estimands to answer new questions from censored data and suggest and implement estimators of these. Establish the asymptotic distribution of the estimators.

4 hours of lectures and 3 hours of exercises per week for 7 weeks.

The exercises will consider both theoretical problems as well as implementation in R. Here the students will have to participate actively, that is take active part in working on the problems in class, and take turns demonstrating the solutions to the different problems to the rest of the class.

Regression corresponding to NMAK11022U Regression (Reg) or NMAB22011U Regression for Actuaries (RegAct) or similar. Continuous time stochastic processes corresponding to NMAK24007U Brownian motion (BM) or NMAK24001U Mathematical Finance (MathFin) or similar.

Academic qualifications equivalent to a BSc degree is recommended.

ECTS
7,5 ECTS
Type of assessment
Written assignment, 3 days
Type of assessment details
A take home exam combining theoretical and practical work.
Aid
All aids allowed
Marking scale
7-point grading scale
Censorship form
No external censorship
One internal examiner
Re-exam

30 min oral exam with 30 min preparation time, several internal examiners and all aids allowed.

##### Criteria for exam assessment

The student should convincingly and accurately demonstrate the knowledge, skills and competences described under Intended learning outcome.

Single subject courses (day)

• Category
• Hours
• Lectures
• 28
• Preparation
• 125
• Exercises
• 21
• Exam
• 32
• English
• 206

### Kursusinformation

Language
English
Course number
NMAK16019U
ECTS
7,5 ECTS
Programme level
Full Degree Master
Duration

1 block

Placement
Block 2
Schedulegroup
A
Capacity
No limitation – unless you register in the late-registration period (BSc and MSc) or as a credit or single subject student.
Studyboard
Study Board of Mathematics and Computer Science
##### Contracting department
• Department of Mathematical Sciences
• Department of Public Health
##### Contracting faculty
• Faculty of Science
##### Course Coordinator
• Frank Eriksson   (8-67746b6d757571704275777066306d7730666d)
##### Teacher

Frank Eriksson
Brice Ozenne

Saved on the 26-04-2024

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