study of time to symptoms following exposure to COVID-19, the following results were reported: The median incubation period was estimated to be

INSTRUCTIONS TO CANDIDATES

ANSWER ALL QUESTIONS

Applied Survival Analysis

1. (30 points) In a study of time to symptoms following exposure to COVID-19, the following results were reported: The median incubation period was estimated to be
   • days, and 97.5% of those who develop symptoms will do so within 11.5 days of infection.
     • (10 points) Sketch the time-to-symptom distribution, to the extent possible, based on these

 

• (10 points) Could this be consistent with an exponential distribution? Why or why not?
• (10 points) Is censoring possible in such a study? Please

 

 

2. (30 points) Listed below are values of survival time in years for a set of Right- censored times are denoted with “+” as a superscript.

0.4,1.2,1.2, 3.4, 4.3,4.9,5.0,5.0+, 5.1, 5.1+,6.1, 7.1

• (10 points) Let Sˆ(t) and S˜(t) denote the Kaplan-Meier estimator and the Fleming- Harrington estimator of the survivorship function of the event time, Calculate Sˆ(7) and S˜(7) by hand.
• (10 points) Figure 1 below plots the Kaplan-Meier estimates for two subgroups of these subjects separately. Based on the figure and the times listed in part (a), list the event and censoring times for each group

 

 

Survival Time (in Years)

Figure 1: Kaplan-Meier estimates of survival times, by gender

• (10 points) Calculate any version of the logrank test and the Wilcoxon test by hand to formally compare these
• (30 points) Consider survival data from a study conducted on n = 13 subjects. Suppose there are 6 unique failure times and k censored observations, occuring at k unique censoring times. Let τ1 <  τ2  <  · · <  τ6  denote the ordered failure times,  and let   C1 < C2 <     < Ck denote the k ordered  censoring times.  There may  be ties  among  the failure times, but suppose that there are no ties among censored observations (i.e.,   no two or more subjects share the same censoring time).

Let  Sˆ that

denote the Kaplan-Meier estimator of the failure time distribution. Suppose

Sˆ(τ1)   =   (10/11)

Sˆ(τ2)   =   (10/11) × (9/10)

Sˆ(τ3)   =   (10/11) × (9/10) × (7/8)

Sˆ(τ4)   =   (10/11) × (9/10) × (7/8) × (4/6)

Sˆ(τ5)   =   (10/11) × (9/10) × (7/8) × (4/6) × (2/4)

Sˆ(τ6)   =   0

• (15 points) One possible value of k is 4. For k = 4, give an overall ordering of the combined set of failure and censoring
• (15 points) For k = 4, calculate the Kaplan-Meier estimator of the distribution of the time to censoring by hand (i.e., consider “censoring” as the event of interest, and “failure” as the censoring event).
• (extra credit: 10 points) Find another value of k that also satisfies the description of the data and give an overall ordering of the combined set of failure and censoring times for this value of k (hint: depending on k, you may have to reprsent 4/6 as 2/3).
• (extra credit: 10 points) Find a third value of k that also satisfies the description of the data and give an overall ordering of the combined set of failure and censoring times for this value of k (hint: depending on k, you may have to reprsent 4/6 as 2/3).

4. (30 points) Consider the MI study that you have analyzed in past homework Suppose that we are interested in time to drop-out (i.e., time to censoring) from the study as the event of interest.
   • 10 points) Is time to drop-out censored in this study? Please explain why or why not, and if so, by what is it censored?

• (10 points) Please estimate the distribution of time to drop out, i.e., P (D > t), where D denotes time to drop Please include 95% confidence intervals of any type and please identify which type of approximation you are resenting. Please include the code that you used to estimate this.
• (10 points) Suppose you want to compare time to drop-out for those who are obese versus those who are not Please display the Kaplan Meier estimates for each group and using a test of your choosing, formally test whether the distributions differ between these groups.

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