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Notifications

Exam terms will be organized upon email request. The capacity of each exam term will be limited to a single student. Exams are oral and take place in the examinator's office on the 2nd floor.

Schedule 

Lecture
Monday 9:00 - 10:30    
Exercise Class
Monday 15:40 - 17:10   (instructor: Jan Vávra)

Course Materials

Progress of lectures

  1. Week 1 (2 lectures)

    Extended course notes, Sec. 1.1.-1.3, pp. 4-12.

    Audio recordings:

    • lecture_201005_1.mp4: Censoring. (17 min.)
    • lecture_201005_2.mp4: Survival function. Hazard function: definition. (34 min.)
    • lecture_201005_3.mp4: Hazard function: properties. Mean residual lifetime: definition. (34 min.)

    Homework 1:

    Prove corollary to Lemma 1.1. - see p. 8 of the course notes. Send the solution (a scan or a legible photo of a handwritten sheet is OK) by email. Deadline: Tuesday Oct. 13.

  2. Week 2 (2 lectures)

    Extended course notes, Sec. 1.4., 2.1, 2.2, pp. 12-22.

    Audio recordings:

    • lecture_201012_1.mp4: Independent censoring. (15 min.)
    • lecture_201012_2.mp4: Parametric likelihood for independent random censoring. (31 min.)
    • lecture_201012_3.mp4: Exponential distribution with type II censoring. (35 min.)

    Homework 2:

    Prove that Type II censoring satisfies the independent censoring condition - see p. 13-14 of the course notes. Send the solution (a scan or a legible photo of a handwritten sheet is OK) by email. Deadline: Tuesday Oct. 20.

  3. Week 3 (2 lectures)

    Extended course notes, Sec. 2.3., 3, 3.1, 3.2, pp. 23-32.

    Audio recordings:

    • lecture_201019_1.mp4: Exponential regression with arbitrary random censoring. (11 min.)
    • lecture_201019_2.mp4: Counting Processes and Martingales. Doob-Meyer decomposition (38 min.)
    • lecture_201019_3.mp4: Martingale integrals. (31 min.)

    Homework 3:

    Prove Theorem 3.3 by application of the Doob-Meyer Theorem - see p. 29 of the course notes. Send the solution (a scan or a legible photo of a handwritten sheet is OK) by email. Deadline: Tuesday Oct. 27.

  4. Week 4 (2 lectures)

    Extended course notes, Sec. 3.2, 3.3, 4.1, pp. 32-42.

    Audio recordings:

    • lecture_201026_1.mp4: Multivariate counting process. Summary of martingale integral results. (36 min.)
    • lecture_201026_2.mp4: Central Limit Theorems for sums of martingale integrals. Weak convergence of stochastic processes. (43 min.)
    • lecture_201026_3.mp4: Nelson-Aalen and Kaplan-Meier estimators: introduction. (30 min.)

    Homework 4:

    Prove that the Kaplan-Meier estimator is a generalization of the empirical distribution function to censored data - see p. 42 of the course notes. Send the solution (a scan or a legible photo of a handwritten sheet is OK) by email. Deadline: Tuesday Nov. 3.

  5. Week 5

    Extended course notes, Sec. 4.2, pp. 42-46.

    Audio recordings:

    • lecture_201102_1.mp4: Properties of the Nelson-Aalen estimator (part 1) (48 min.)
    • lecture_201102_2.mp4: Properties of the Nelson-Aalen estimator (part 2) (15 min.)

    Homework 5:

    Prove part (ii) of Theorem 4.2 - see p. 43 of the course notes. Send the solution (a scan or a legible photo of a handwritten sheet is OK) by email. Deadline: Tuesday Nov. 10.

  6. Week 6

    Extended course notes, Sec. 4.3, pp. 47-52.

    Audio recordings:

    • lecture_201109_1.mp4: Properties of the Kaplan-Meier estimator (part 1) (27 min.)
    • lecture_201109_2.mp4: Properties of the Kaplan-Meier estimator (part 2) (30 min.)

    Homework 6:

    Derive the Greenwood formula - see p. 50 of the course notes. Send the solution (a scan or a legible photo of a handwritten sheet is OK) by email. Deadline: Tuesday Nov. 17.

  7. Week 7

    Extended course notes, Sec. 4.4, pp. 53-56.

    Audio recordings:

    • lecture_201116.mp4: Confidence bounds for the survival function (35 min.)

    Homework: None.

  8. Week 8

    Extended course notes, Sec. 5.1-5.3, pp. 57-63.

    Audio recordings:

    • lecture_201123_1.mp4: Logrank test: motivation (30 min.)
    • lecture_201123_2.mp4: Linear rank statistics for censored data (30 min.)
    • lecture_201123_3.mp4: Moments of weighted logrank tests (20 min.)

    Homework 7:

    Show that with uncensored data, the Gehan-Wilcoxon test is equivalent to the Wilcoxon two sample rank-sum test. - see p. 61 of the course notes. Send the solution (a scan or a legible photo of a handwritten sheet is OK) by email. Deadline: Sunday Dec. 6.

  9. Week 9

    Extended course notes, Sec. 5.4-5.5, pp. 64-68.

    Audio recordings:

    • lecture_201130_1.mp4: Asymptotic results for weighted logrank statistics (33 min.)
    • lecture_201130_2.mp4: Weighted logrank tests under the alternative (30 min.)

    Homework: None.

  10. Week 10

    Extended course notes, Sec. 6.1-6.2, pp. 69-74.

    Audio recordings:

    • lecture_201207_1.mp4: Cox model: introduction (31 min.)
    • lecture_201207_2.mp4: Development of the partial likelihood (20 min.)
    • lecture_201207_3.mp4: Partial likelihood score statistic (17 min.)

    Homework 8:

    Show that the Cox partial likelihood score statistic evaluated at the true parameter is a sum of martingale integrals - see p. 74 of the course notes. Send the solution (a scan or a legible photo of a handwritten sheet is OK) by email. Deadline: Tuesday Dec. 15.

  11. Week 11

    Extended course notes, Sec. 6.2-6.3, pp. 75-80.

    Audio recordings:

    • lecture_201214_1.mp4: Moments of the partial likelihood score statistic (30 min.)
    • lecture_201214_2.mp4: Weak convergence of the partial likelihood score statistic (25 min.)

    Homework 9 (voluntary):

    Show that the unweighted logrank test can be derived as the score test from the Cox model with a single binary covariate indicating group membership. It you wish to have your solution checked, send it by email. Deadline: None.

    Note: this homework is voluntary but you may be asked at the exam to demonstrate a solution to it.

  12. Week 12

    Extended course notes, Sec. 6.3-6.4, pp. 80-85.

    Audio recordings:

    • lecture_201221_1.mp4: Consistency and as. normality of the MPLE (26 min.)
    • lecture_201221_2.mp4: Estimation of the baseline hazard and conditional survival (12 min.)

    Homework: none.

  13. Week 13  

    Extended course notes, Sec. 6.5-6.6, pp. 85-89.

    Audio recordings:

    • lecture_210104_1.mp4: Generalizations of the Cox model (50 min.)
    • v_bonus.mp4: O čem jsme se nezmínili: bonusové video v češtině.

    Homework: none.

Supplementary course materials

Most of the contents of the course is covered by two monographs. Both are available in the library or can be purchased (e.g., from Amazon).

Course Plan

The course covers methods for analysis of censored data: non-negative random variables with time-to-event interpretation whose values are not always fully observed (survival analysis, reliability theory, risk analysis)

The lecture focuses on the development, theoretical justification, and interpretation of the methods.

The exercise classes will teach how to apply these methods to real problems but may include some theoretical tasks as well.

The following topics will be included:

Prerequisites

This course assumes the knowledge of linear regression theory and, preferably but not necessarily, generalized linear models. Intermediate-level knowledge of probability theory, continuous martingales, and counting process theory is also required.

Master students of "Probability, statistics and econometrics" must have completed courses NMSA407 Linear Regression and NMTP436 Continuous Martingales and Counting Processes before enrolling in this course.

Master students of "Financial and Insurance Mathematics" must have completed courses NMSA407 Linear Regression and NMFM408 Probability for Finance and Insurance before enrolling in this course.

Requirements for Credit/Exam 

Credit:

The credit for the exercise class will be awarded to the student who hands in a satisfactory solution to each assignment by the prescribed deadline.

The nature of these requirements precludes any possibility of repeated attempts to obtain the exercise class credit.

Exam:

The exercise class credit is necessary to sign up for the exam.