MSDS 490: Healthcare Analytics and Decision Making
Homework Assignment 3
Due Date: 12/02/2024 (Monday Midnight)
Submission Instructions: zip all your solution files (data, R-code, Word document, figures, etc.) into one, following the file naming convention Last Name First Name HW#.zip Use online submission tools in Canvas to submit this homework.
Total Score: 100. Each Problem has an Equal Weight.
1. (Basic understanding of Survival Analysis). Table 1 provides data on ten patients which enrolled in a clinical study that was conducted for 20 months.
Table 1: Dataset for Kaplan-Meier Survival Curve Analysis
(i) Provide a table showing the desired calculations (patients at risk) to plot the Kaplan-Meier curve for patients in Group A, and Group B.
(ii) Provide confidence intervals for the KM survivor curves.
(iii) Perform. the log-rank test testing if the survival curve for patients in Group A is statistically different from that of patients in group B.
(iv) Develop the partial likelihood function to train for the coefficients of a Cox-Proportional Hazard model for patients in Group A.
2. (Bias in Survival Analysis). The article [1] discusses two types of potential bias associated with Survival Analysis. In your own words, describe these two types of biases. Use one or more examples from papers cited in this section for illustration.
3. (Discrete Time Markov Chain) . Progression of CD4 count of an HIV positive patient is described by a Markov chain with three states: (0, 200), [200, 500), [500, ∞ ). These states are labeled as 1, 2 and 3 respectively. The probability transition matrix for patients transitioning from one state to another every three months is given in Table 2.
Table 2: Probability Transition Matrix of CD4 counts in Standard Care
Assume that an individual is recently diagnosed as HIV positive, and has his CD4 count in [500 , ∞ ). (i) What is the probability that this person’s CD4 count will be in the range (0 , 200) in the fourth quarters after the initial diagnosis.
(ii) What is the probability that this patient’s CD4 count will be in the range (0 , 200) for the first time in the fourth quarter after the initial diagnosis. (iii) Assume that it costs 2 , 000 per quarter to care for a HIV+ patient in State 3, 5, 000 per quarter to care for the patient in State 2, and 10 , 000 per quarter to care for the patient in State 1. What is the expected cost of caring for this patient during the first year after diagnosis? (iv) What is the steady state probability of finding the patient in States 1, 2, 3; and the corresponding steady state expected yearly cost of patient care.
(v) Now assume that a new medication has become available in the market. The medication adds $2,000 quarterly to the cost of standard care. Clinical trial has shown that those on this medication have an improved Probability Transition Matrix of CD4 count. This transition matrix is given in Table 3. Calculate the steady state yearly cost of
Table 3: Probability Transition Matrix of CD4 counts in Improved Care
care for the patient(s) who receive the new medication.
4. Use the data in the above question and validate your answers for each question using a simulator. Your simulation should perform. at least 100 replications when estimating the desired confidence intervals. Note that your simulation should not perform. matrix-matrix product or matrix inversion calculations, i.e., the only calculations you will perform. will involve matrix-vector products to keep track of patient transitioning. For achieving “steady state”, you can run the patient transition for 25 quarters (3-month time intervals) starting from any state.
5. (Partially Observable Markov Decision Process) . The article [2] discusses the concept of Partially Observable Markov Decision Processes (POMDP) and its role in Medical Decision Making. It discusses this concept with a prostate cancer screening example. You goal is to describe the concept of POMDP, with the necessary equations and subsequently its use by the authors in the cancer screening example.
References
[1] Peter Groves, Basel Kayyali, David Knott, and Steve Van Kuiken. A practical overview and reporting strategies for statistical analysis of survival studies. CHEST, 158:S39–S48, 2020.
[2] Lauren N. Steimle and Brian T. Denton. Markov Decision Processes for Screening and Treatment of Chronic Diseases, pages 189–222. Springer International Publishing, Cham, 2017.