## TMA4275 Lifetime analysis, Spring 2018

### Messages

June 1: I have posted a preliminary solution to todays exam below.

May 25: To get the grades on project 2, contact Rasmus. On May 30, I will be available for answering questions in R21 in realfagsbygget between 9:00 and 16:00 except between 10:30-11:30. Send me an email or sms (99705519) if I'm not there.

May 7: The syllabus of the course is mainly the slides, but also material covered as part of the exercises and the projects including material referred to in the project (some sections from Moore 2016). The exam will include output from survival analysis in R so you'll need to be familiar with such output and the models behind such output. Writing R code, however, will not be part of the exam.

May 1. Permitted aids on the exam are "Tabeller og formler i statistikk, Tapir Forlag, K. Rottmann: Matematisk formelsamling, Calculator Casio fx-82ES PLUS, CITIZEN SR-270X, CITIZEN SR-270X College or HP30S, one yellow A4-sheet with your own handwritten notes."

April 17. Tentative date for "spørretime" is Wednesday, May 30 from 10:00-15:00. Let me know if you prefer an other date.

April 12. Rasmus (and sometimes me) will be available at the computerlab on the next few Tuesdays at 9:00 to assist with project 2. The last lecture will be on Tuesday 17 (I will continue going through some old exam, problem 2, june 2017 next time).

April 10. I have moved the deadline for project 2 to May 6.

April 4. Fitting the NHPP process using optim in project 2 may be a bit tricky. You should obtain parameter estimates close to $a=-3$, $b=2$ and $c=30$. Although the model is not strictly identifiable without imposing constraints on the parameter (note how several parameter values lead to the same model), the numerical optimisation appears to work best without imposing any constraints on $a$, $b$ and $c$ and without doing any reparameterization. The optimization then appear to converge for almost any choice of starting values including c(0,0,0).

March 12. You can get your graded project 1 back if you visit Rasmus in his office, room 1126 in SII.

Feb. 9: In the coming weeks, for project 1 and 2, Rasmus (and some times me) will be available between 9 and 10 each tuesday in Nullrommet (instead of 8-9 in B3).

Feb. 6: No teaching in week 8 (on february 20 and 22).

Feb. 6: To find a better time to meet for assistance with project 1 and 2 (this will take place in Nullrommet), please fill in suitable times at this google docs sheet at your earliest convenience.

Feb. 5: Project 1 is out, see below.

Jan. 16: You'll need access to the computerlab later on in the semester. If you don't already have access, fill in the necessary information at https://goo.gl/forms/Vg8umJdGLm0ERn9f1 and I'll provide access.

Jan. 1: The first lecture is Tuesday, January 9 at 10:15 in F4.

### General information

• Lectures: Tuesday 10:15-12:00 in F4 and Thursdays 12:15-14:00 in F3.
• Exam: June 1, 9:00.

### Course material

Slides and lecture notes will be provided online along the course.

Relevant books for the course will be:

Meeker & Escobar: Statistical methods for reliability data. Wiley, 1998.

Collett: Modelling Survival Data in Medical Research. 2nd ed. Chapman & Hall/CRC 20003.

For background in basic statistics: Walpole, Myers, Myers and Ye: Probability and Statistics for Engineers and Scientists, Prentice Hall.

For background in stochastic processes: Sheldon M. Ross: Introduction to probability models, Academic Press.

Moore, 2016. Applied Survival Analysis using R (freely available as ebook on springer link) covers much of the same material as and also has detailed explanations on how to do things in R.

### Weekly exercises

Note that the computer based exercises will be updated to include information on how to solve the problems in R.

 # Date & Place Problems Solution 1 16.01 pdf Solution1 2 23.01 pdf Solution2 3 30.01 pdf, Solution3,SolutionwithRcode 4 06.02 pdf + help with downloading MINITAB Solution4,Rcodesolution 13.02 Guidance Oblig 1 20.02 Guidance Oblig 1 5 27.02 pdf Solution5 6 07.03 pdf Solution6 7 14.03 Exam 2004: Problems 2 and 3 Solution (see below) 21.03 Guidance Oblig 2 28.03 Guidance Oblig 2 8 04.04 pdf Solution8 9 25.04 Exam 2005: Problem 2, Exam 2011: Problem 3

### Obligatory exercises

There are two obligatory exercise sets that will be graded, each counting 10% of the final grade in the course. Two students may work together and submit a common report. If you have completed the obligatory exercises in an earlier year, you can optionally use the score from previous years (in this case, send me an email with your student id number).

* Obligatory I. Out: Friday 5 February. Deadline: Friday, 2 March. Submit your report at https://www.dropbox.com/request/ia2cPZxNqWUNubkqnxaE

* Obligatory II. Out: Wednesday, March 14. Deadline: Sunday, May 6. Submit your report at https://www.dropbox.com/request/V7ZD1OrfYMUwGvrWpLEt

These are tentative dates.

We recommend that you write the report using Markdown or LaTeX + knitr, perhaps using tools such as sharelatex, dropbox or git (via bitbucket.org) if you're collaborating. The report should be written as a self-contained document intelligible to your peers. Make sure you include the email and student identity number of all groups members.

### Software and dataset

We recommend R and Rstudio and the survival-package all of which can be downloaded and installed for free. If you're not familiar with R you may want to read and solve some of the problems in "A (very) short introduction to R". A more detailed and technical introduction focusing on R as a language is "An Introduction to R". You may alternatively use https://innsida.ntnu.no/wiki/-/wiki/English/Minitab but then you're mostly on your own.

## Lecture plan and progress

R&H refers to relevant sections in Rausand & Høyland: System Reliability Theory: Models, Statistical Methods, and Applications, 2nd Edition. Wiley 2004.

Date R&H Topic Slides Notes/Supplementary reading
09.01 2.3-2.5 Introduction and motivation. General concepts for lifetime modeling. Slides 1, Slides 2
11.01 2.6, 2.9-2.14 Parametric families of lifetime distributions. Slides 3
16.01 (2.17) Gumbel distribution. Log-location-scale families Slides 4 Extreme value distribution, More on log-location-scale families
18.01 11.1-11.3.3, 11.3.5 Censoring; empirical survival function; Kaplan-Meier estimator. Nelson-Aalen estimator Slides 5,Slides 6, p. 1-6 Alternative derivation of K.-M.-estimator as non-parametric maximum likelihood estimate of R(t). I'll go through this in the lecture.
23.01 11.3.6 Properties of the exponential distribution Slides 6, p. 7-16 About the Exponential Distribution, Poisson Process, Total Time on Test and Barlow-Proschan's Test.
25.01 11.3.7 Derivation of Kaplan-Meier and Nelson-Aalen estimator, TTT-plot Slides 5 and Slides 6 (p. 17-23)
30.01 The logrank test. Slides 6 (p. 44-50). Complete the table on p. 50 Note on the logrank test See e.g sec. 4.1 in Moore 2016.
01.02 11.3.7 More on TTT-plot. Barlow-Proschans test. Introduction to parametric methods Slides 6 (p. 24-43) Algorithm for TTT and BP
06.02 11.4.3, 11.4.4, 11.4.5 Parametric inference for the exponential model. Confidence intervals for the exponential distribution. Slides 8, p. 1-21. The standard confidence interval for positive parameters. Some likelihood theory.
08.02 Likelihood contributions from left truncated, left censored and interval censored observations. Slides 7 Ch. 3.5 and ch. 12.2 in Moore. aftreg in eha R-package does survival regression with left truncation (survreg in the survival-package does not). icfit in the interval package does non-parametric estimation of $R(t)$ for left and interval censored data (survfit in the survival package does not)
13.02 11.4.5 Parametric inference for the Weibull model Slides 9, p. 1-15
15.02 11.4.5 Parametric inference for the Weibull model (cont). Inference in log-location-scale models. Slides 9, p. 15-17, Slides 10, p. 1-20
27.02 Threshold models (3-parameter Weibull). Parametric survival regression. Slides 10, p. 21-27 (p. 28-31 not in curriculum), Slides 11, p. 1-22 Book chapter on survival regression
01.03 Parametric survival regression (cont.), Proportional hazards and Cox-regression Slides 11, p. 23-43, Modelling of covariates and factors
06.03 Proportional hazards and Cox-regression (cont.) Slides 12, p. 1-16 Case study in Cox regression: Medical data
08.03 Proportional hazards and Cox-regression (cont.) Slides 12, p. 17-31 Case study in Cox regression: Reliability, Some notes on ties etc..
13.03 7.1, 7.2.1 Model checking in Cox-regression. Case study of Cox-regression. Slides 12, p. 32-48, Moore 2016, ch. 7
12, 7.3.1, 7.4.1, 7.4.2, 7.4.3 Accelerated life testing. Recurrent events and repairable systems. Slides 13, Slides 14, p. 1-14 Download INSULATE.MTW
7.3.1, 7.4.1, 7.4.2, 7.4.3 Recurrent events and repairable systems. The nonhomogeneous Poisson process (NHPP). Slides 14, p. 15-23
7.4.3 Nonparametric estimation in repairable systems. Slides 14, p. 24-35
7.4.3, 7.4.4 Final example on nonparametric estimation in repairable systems, Parametric estimation in NHPPs. Slides 14, p. 36-41, Slides 15, p. 1-10 Alternative derivation of likelihood function
15.3 7.4.4, 7.4.5, 7.3.1, 7.3.2, 7.3.3 Parametric estimation in NHPPs. Trend testing in NHPPs. Renewal processes. Slides 15, p. 11-23, Slides 16, p. 1-28 (p. 29-31 are not in curriculum)
Easter
10.4, 12.4, 17.4, 19.4 Go through and discuss topics from exam exercises (in this order, as far as we get):
2009 Problems 2,3 (10.04)
2012 Problems 1 (10.04), 2 (12.04)
2014 Problem 2 (12.04)
2010 Problem 2,3 (17.04)
2014 Problem 3
2014 Problem 1
Chi-square critical values

### Earlier exams

Exam English Norwegian Solution
2018V pdf pdf
2017V pdf pdf
2016V pdf pdf pdf
2015V pdf pdf pdf
2014V pdf pdf pdf
2013V pdf pdf pdf
2012V pdf pdf Probl 1-2-4
2011V pdf
2010V pdf Probl 2-3
2009V pdf pdf
2008V pdf pdf
2007V pdf
2006V pdf pdf
2005V pdf pdf
2004V pdf pdf
2003V pdf pdf
2002V pdf pdf
2001V pdf pdf
2000V pdf pdf

### Typos in R&H, 2004, 2nd ed.

Page. 56: First line should read "as $t \rightarrow -\infty$" (instead of "$+\infty$").