2. Consider the following censored lifetime data: 12 13 14+ 15+ 23 23+ 25 25 16+ 26 18 18 19 21 26 26+ 27+ 29 22+ 30 Recall that the + sign corresponds to censoring: for e.g., 14+ means exit by censoring at age 14. (a) Find estimates of mortality rate λ(t), cumulative mortality rate A(t), and the survival probability (Nelson-Aalen and Kaplan-Meier) S (t). Provide your answers in 4 decimal digits. ktk dk Ук λκ Ак SKM SNA k 123 + 4 where tk denotes event times, dk the number of exit by the event at tk, yk the num- ber of individuals at risk at tk, λk := λ(tk), Ak = (tk), while SM and SNA respectively Kaplan-Meier and Nelson-Aalen estimates of S (tk). Complete the table in your assignment submission. KM are (b) Use R to plot estimated cumulative hazard rate ^(t) and survival probability s(t). Also write down each expression of ^(t) and s(t) as a function of time t. ΝΑ (c) It is known that the estimated variance Var(SNA (t)) of the estimate SNA (t) of survival probability S (t) for the Nelson-Aalen estimator is given by Var(S^(t)) = [S^(t)]² Σ dk(yk - dk) k:tk≤t Yk Use the formula to estimate the variance of S (t) at age t = 18. (d) Derive a 95% confidence interval for S (t) aged t = 18 using SNA (t).
2. Consider the following censored lifetime data: 12 13 14+ 15+ 23 23+ 25 25 16+ 26 18 18 19 21 26 26+ 27+ 29 22+ 30 Recall that the + sign corresponds to censoring: for e.g., 14+ means exit by censoring at age 14. (a) Find estimates of mortality rate λ(t), cumulative mortality rate A(t), and the survival probability (Nelson-Aalen and Kaplan-Meier) S (t). Provide your answers in 4 decimal digits. ktk dk Ук λκ Ак SKM SNA k 123 + 4 where tk denotes event times, dk the number of exit by the event at tk, yk the num- ber of individuals at risk at tk, λk := λ(tk), Ak = (tk), while SM and SNA respectively Kaplan-Meier and Nelson-Aalen estimates of S (tk). Complete the table in your assignment submission. KM are (b) Use R to plot estimated cumulative hazard rate ^(t) and survival probability s(t). Also write down each expression of ^(t) and s(t) as a function of time t. ΝΑ (c) It is known that the estimated variance Var(SNA (t)) of the estimate SNA (t) of survival probability S (t) for the Nelson-Aalen estimator is given by Var(S^(t)) = [S^(t)]² Σ dk(yk - dk) k:tk≤t Yk Use the formula to estimate the variance of S (t) at age t = 18. (d) Derive a 95% confidence interval for S (t) aged t = 18 using SNA (t).
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
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Please assist with the following questions with the answers. Use R code as requested
![2. Consider the following censored lifetime data:
12 13
14+ 15+
23 23+ 25 25
16+
26
18 18 19
21
26 26+ 27+ 29
22+
30
Recall that the + sign corresponds to censoring: for e.g., 14+ means exit by censoring at
age 14.
(a) Find estimates of mortality rate λ(t), cumulative mortality rate A(t), and the survival
probability (Nelson-Aalen and Kaplan-Meier) S (t).
Provide your answers in 4 decimal digits.
ktk
dk
Ук
λκ
Ак
SKM
SNA
k
123 +
4
where tk denotes event times, dk the number of exit by the event at tk, yk the num-
ber of individuals at risk at tk, λk := λ(tk), Ak = (tk), while SM and SNA
respectively Kaplan-Meier and Nelson-Aalen estimates of S (tk).
Complete the table in your assignment submission.
KM
are
(b) Use R to plot estimated cumulative hazard rate ^(t) and survival probability s(t).
Also write down each expression of ^(t) and s(t) as a function of time t.
ΝΑ
(c) It is known that the estimated variance Var(SNA (t)) of the estimate SNA (t) of survival
probability S (t) for the Nelson-Aalen estimator is given by
Var(S^(t)) = [S^(t)]² Σ
dk(yk - dk)
k:tk≤t
Yk
Use the formula to estimate the variance of S (t) at age t = 18.
(d) Derive a 95% confidence interval for S (t) aged t = 18 using SNA (t).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fba18de34-fc06-47a6-b1ea-c54726b84874%2F220edf23-90b8-48e2-bc97-2f618d2c9c1f%2Faxj50fj_processed.png&w=3840&q=75)
Transcribed Image Text:2. Consider the following censored lifetime data:
12 13
14+ 15+
23 23+ 25 25
16+
26
18 18 19
21
26 26+ 27+ 29
22+
30
Recall that the + sign corresponds to censoring: for e.g., 14+ means exit by censoring at
age 14.
(a) Find estimates of mortality rate λ(t), cumulative mortality rate A(t), and the survival
probability (Nelson-Aalen and Kaplan-Meier) S (t).
Provide your answers in 4 decimal digits.
ktk
dk
Ук
λκ
Ак
SKM
SNA
k
123 +
4
where tk denotes event times, dk the number of exit by the event at tk, yk the num-
ber of individuals at risk at tk, λk := λ(tk), Ak = (tk), while SM and SNA
respectively Kaplan-Meier and Nelson-Aalen estimates of S (tk).
Complete the table in your assignment submission.
KM
are
(b) Use R to plot estimated cumulative hazard rate ^(t) and survival probability s(t).
Also write down each expression of ^(t) and s(t) as a function of time t.
ΝΑ
(c) It is known that the estimated variance Var(SNA (t)) of the estimate SNA (t) of survival
probability S (t) for the Nelson-Aalen estimator is given by
Var(S^(t)) = [S^(t)]² Σ
dk(yk - dk)
k:tk≤t
Yk
Use the formula to estimate the variance of S (t) at age t = 18.
(d) Derive a 95% confidence interval for S (t) aged t = 18 using SNA (t).
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