PM599 Counting Process and Martingale Theory Methods with Applications in Epidemiology and Clinical Studies

Bryan Langholz, Instructor

January 9

Readings:

Overview of likelihood theory (Langholz notes)

Chapter on likelihood and estimating equation asymptotics

January 19

Lecture notes:

Data examples and course outline.

Process notation for cohort data

Intensity definition

Homework:

1. Read over examples in Chapter I of ABGK and think about how the data would be described using process notation.

2. Read Sections 10.1 and 10.2 in:

Intro to Counting Processes and Martingale Theory

January 23 and 26:

Langholz out of town. Meet and go over likelihood theory note questions. Discuss readings.

January 30:

Lecture notes:

Processes and martingales

1. Read Sections 10.3-10.4 in Intro to Counting Processes and Martingale Theory. Work on problems 10.9.1-10.9.11.

February 2:

Lecture notes:

1. Read Sections 10.5-10.6 in Intro to Counting Processes and Martingale Theory. Work on problems 10.9.1-10.9.13.

2. Below is the program that clinical trials-like data using the time-interval approximation.  The output is a data set with the processes at each time interval. Write a program that converts the process data into "standard" clinical trials data with:

Subject id, treatment, exit time, failure indicator.

processes for clinical trials data program

February 6:

Lecture notes: Brownian motion simulation program

1. Read Sections 10.7-10.10 in Intro to Counting Processes and Martingale Theory.

2. Read (Browse Sections II.1-II.4 of ABGK)

3. Experiment with "Martingale Central Limit Theorem":

    a. Alter processes for clinical trials data program to generate homogeneous Poisson processes with rate lambda=1 (i.e. Set Y(t)=1 for all t, no covariates)

    b.Compute square root n normalized "Z scores" within each increment.

    c. Sum the increments

    d. Does this process look like Brownian motion?

February 9:

Homework:

Apply Lenglart's inequality and Rebolledo CTL to show asymptotic mean function and distribution for the following situations.

Check your results by computer simulation...

Homogeneous Poisson processes (in class)

Exponential failure time data (in class)

Homogeneous Poisson process with constant rate (exponential) censoring

Clinical trials data with constant rate censoring as in the program (on website, 2/2 notes)

Clinical trials data as described on p. 53 of ABKG.

February 13: Langholz out of town. Meet (3-4 CHP-224) to go over Feb 6 and 9 assignments.

February 22: Lebesque integration, Stochastic integration

Homework

February 27: Brownian motion for cumulative hazard

Asymptotic Brownian motion example

March 2: Asymptotics for cumulative hazard

March 9: Intensity models.

March 20: Partial likelihood for Cox model.

Andersen and Gill 1982