## TMA4275 Lifetime analysis, Spring 2017

### Messages

June 6: 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.

June 6: I have reserved room R60 in Realfagsbygget between 9:00 and 16:00 on Thursday June 8 and will be available for questions most of the day (except 1415-1500). If I'm not present, send me and email or a text message (99705519).

June 3: 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"

May 8: Solution proposal Obligatory II, I wrote a solution proposal so that you can understand why and where you lost points. You can come and pick up your assignments on Fridays after 2:00 pm.

April 19: Two days before the exam on thursday June 8 between 10 and 15 I'll be available for questions. Permitted aids on the exam will be approved calculator, one yellow a4 sheet with your own handwritten notes and "Tabeller og formler i statistikk, Tapir forlag".

April 18: I have graded the second obligatory assignment, and uploaded the grades, please check that everything is in order. I suspect that some of you may have given their student number the first time and their candidate number the second. screen_shot_2017-04-18_at_09.29.22.png.

April 3: The next lecture is after the easter holiday on April 20 (no lecture on thursday April 6).

March 28: You do not need "pass" the obligatory exercises (that is, obtain some minimum score) to be allowed to take the final exam, but the score obtained on the two obligatory exercises counts 20% on the final grade so it is of course highly recommended that you do your best.

March 24: If you want to prove that the density in point 8 in obligatory exercise 2 is not a Weibull distribution, here are some hints: Conditional on $\ln x_1$ and $x_2$, we are asumming that $\ln T$ is Gumbel with a known moment generating function (mgf). You also know the mgf of $\ln x_1$. You can then derive the mgf of $\ln T$ unconditional on $\ln x_1$ using the law of total expectation and the definition of mgfs to prove the marginal distributions of $\ln T$ is not a Gumbel distribution.

March 21: There was a small update to obligatory exercise 2 yesterday (use 0.2, 1 and 5 as values for x1).

March 20: Guidance for obligatory exercise 2 takes place in Computer Lab A-293 at 14:15-16:00 Tuesday March 21 and Tuesday March 28.

March 19: No lecture tommorow Monday March 20. Work on obligatory exercise 2.

March 8: I have graded your hand ins, you can come pick it up in my office ( after 12 on wednesdays, thursdays, and fridays). Mean: 8.8, Median: 9,5. ( office 1001)

Feb. 17: An additional question (point 7) has been added to obligatory exercise 1. Also, to understand the behaviour of the TTT-plot and the Barlow-Proschan test under different deviations from exponentiality you need to consider how these depend on the observed data after the last failure time.

Feb. 15 : PROJECT : Should you have any questions regarding the project, feel free to come by my office ( 1001 Sentralbygg II). I am available after 12 on Wednesdays, Thursdays and Fridays.

Feb. 10: Obligatory exercise 1 is now available. Guidance for the exercise takes place at Computer Lab A-293 in the Elektro A building on tuesday 14 and 21 between 15:15 and 17:00 (both Maxime and I will be present).

Feb. 6: In the derivation of the exact confidence interval for type II censoring: The pivotal quantity becomes $2r\hat\theta/\theta$ and not $2\hat\theta/(r\theta)$. This is $\sim\text{Gamma}(r,1/2) = \text{Chisquare}(2r)$. Correcting for this, this confidence interval becomes equal to the Bayesian $(1-\alpha)$ credible interval following from a non-informative scale prior, see Berger 1986, p. 85.

Jan. 31: If you plan to only retake the exam, you may use results on the obligatory exercices from last year. In that case, you just need to me an email with your name, candidate number and/or student id number and the year in which you took the exercise.

Jan. 16: Note that the tutorials are on Tuesdays 15:15-16:00 and not Mondays.

### 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 need not submit new reports this year. You will then need to verify your score from the earlier year.

* Obligatory I. Out: Friday 10 February. Deadline: Sunday 26 February.

* Obligatory II. Out: Thursday, March 16Friday 1 April. Deadline: Friday, March 31Friday 15 April.

These are tentative dates. Submit your report to the teaching assistent.

### 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
12.01 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
19.01 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
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.
26.01 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.
02.02 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.
09.02 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 NO LECTURE - work on Obligatory 1
16.02 NO LECTURE - work on Obligatory 1
20.02 11.4.5 Parametric inference for the Weibull model Slides 9, p. 1-15
23.02 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
27.03 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
02.03 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
09.03 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
20.03 No lecture Work with obligatory exercise 2
23.03 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
27.03 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
30.03 7.4.3 Nonparametric estimation in repairable systems. Slides 14, p. 24-35
03.03 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
06.04 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
20.04
24.04
Go through and discuss topics from exam exercises (in this order, as far as we get):
2009 Problems 2,3 (20.04)
2012 Problems 1 (20.04), 2 (24.04)
2014 Problem 2 (24.04)
2010 Problem 2,3 (24.04)
2014 Problem 3
2014 Problem 1
Chi-square critical values

### Earlier exams

Exam English Norwegian Solution
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

### 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 17.01, S4 pdf Solution1 2 24.01 pdf Solution2 3 31.01 pdf, Solution3,SolutionwithRcode 4 07.02 pdf + help with downloading MINITAB Solution4,Rcodesolution 14.02 Guidance Oblig 1 21.02 Guidance Oblig 1 5 28.02 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

### General information

• Lectures: Mondays 08:15-10:00 in F2 and Thursdays 10:15-12:00 in F2. Exercises: TuesdayMonday 15:15-16:00 in S4 or 14:15-16:00 in the computer lab.
• Exam: June 10, 2016. Written, 4 hours (9.00-13.00). Permitted Aids: Approved Calculator, One yellow sheet with your own formulae and notes. (You can get this page from the department office in 7th floor of Sentralbygg 2. You may write on both sides!)

### 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.

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

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