STA 475 ****Assignment ****#1 ****( Fall ****2024)
Instructions
Due ****date: ****Friday September 27th at 11:59pm
• No-questions-asked grace period until Monday September 30th at 5pm
• No late submissions will be accepted after this time
Where ****to ****submit:
• Crowdmark: Submit your answers to each question in correct space on Crowdmark
• MarkUs (link: markus.teach.cs.toronto.edu/markus/cour…): Submit the .qmd file with your answers to Q1 to MarkUs - please just save a copy of the ques-tions and enter your answers in the space provided. You don’t have to use the .qmd file to answer the other questions if you don’t want to.
• You can submit as many times as you like before the deadline
• Email submissions will NOT be accepted
• Let me know if you’re not able to access the Crowdmark and/or MarkUs submission sites by September 23rd.
Other ****notes:
• For some questions, you will need to use R; please include all code and relevant output in the pdf submission (make sure the code is visible in the pdf and doesn’t run out of the margins). If the question asks you to answer a question based on R output, make sure your answer is easy for your TA to find (i.e. start with your answer, then have the 代写STA475 R to Simulate Data code and output afterwards for reference). Your TA should be able to understand your answer without looking at your and output (as appropriate), but may refer to these to better understand what you did.
• While you may discuss questions with your classmates, you MUST submit independent work that you did yourself. Students submitting identical solutions (e.g. identical sen- tences, derivations steps, or chunks of code) will be investigated for violations of academic integrity.
• If you believe you’ve found a typo or error in the assignment, please email sta475@utoronto.ca so I can look into it and get back to the class as quickly as possible.
Question ****1 ****[10 ****points ]
Note : **Parts (a) to **(d) will **be **graded **on **MarkUs **and **part (e ) **will **be **graded **manually **on Crowdmark. Please **su bmit **all **answers **to Crowdmark **(including **parts **a **- **d)
Understanding the shelf life of fresh fruits and vegetables is key for optimizing storage, pack- aging, and transportation, helping reduce food waste and ensure fresh products for consumers. In this question, we’ll assume the time to spoilage of a fresh produce item follows an exponen- tial distribution with mean 5 days. Suppose we consider a sample of 2000 fresh produce items under this distribution, and observe them until 60% (1200 items) have spoiled and become unfit for sale, at which point the observation process will end.
(a) ****Use ****R ****to ****simulate ****data ****for ****the ****process ****described ****above, ****generating ****a ****tibble ****named ****sim ****with ****the ****following ****columns: T ****and ****delta . Make ****sure ****your ****code is commented ****so ****that ****someone ****reading ****your ****code ****can ****understand ****what ****you ****were ****trying ****to ****do. Sort ****the ****sim ****tibble ****in ****incr easing ****order ****of ****T ****(hint: you ****may ****find ****the ****arrange () ****function ****useful ****for ****this ).
library(tidyverse)
SEED <- 1 # DO NOT CHANGE THIS LINE
set.seed(SEED) # DO NOT CHANGE THIS LINE
n <- 2000
## Q1a
sim <- NULL # REPLACE NULL WITH YOUR ANSWER
Q1a_dataframe_with_T_and_delta <- sim # DO NOT CHANGE THIS LINE head(Q1a_dataframe_with T and_delta) # DO NOT CHANGE THIS LINE
NULL
(b) ****Modify ****the ****tibble ****you ****generated ****in ****the ****previous ****part ****to ****add ****a ****column ****named ****X ****which ****records ****the ****observed ****response ****for ****each ****observati on. Your ****updated ****sim tibble ****should ****now ****have ****3 ****columns: T , ****delta ****and ****X . ****Don’t ****change ****the ****order ****of ****the ****rows, ****compared ****to ****part ****(a).
Q1b: Modify the sim tibble
Q1b_dataframe_with_X <- sim # DO NOT CHANGE THIS LINE
head(Q1b_dataframe_with_X) # DO NOT CHANGE THIS LINE
NULL
(c) ****Based ****on ****your ****simulated ****values, ****how ****long ****do ****you ****need ****to ****mo nitor ****the
process ****to ****observe ****1200 ****failures? Save ****your ****exact ****answer ****(no ****round ing) ****to ****the ****variable ****Q1c_study_duration_1200 _failures .
# Q1c
Q1c_study_duration_1200_failures <- NULL # Replace NULL with your answer
Q1c_study_duration_1200_failures # DO NOT CHANGE THIS LINE
NULL
(d) ****Based ****on ****your ****simulated ****values, ****compar e ****the ****means ****of ****x and ****T in ****one ****sentence ****(you ****should ****report ****the ****numerical ****values ****for ****t hese ****means). Save ****the ****exact ****values ****of ****each ****mean ****(no ****rounding) ****in ****the ****variables ****Q1d_ mean_X ****and
Q1d_mean_T .
# Q1d
Q1d_mean_X <- NULL # Replace NULL with your answer
Q1d_mean_T <- NULL # Replace NULL with your answer
Q1d_mean_X # DO NOT CHANGE THIS LINE
NULL
Q1d_mean_T # DO NOT CHANGE THIS LINE
NULL
(e) ****Using ****properties ****of ****the ****exponential ****distribution, ****der ive ****the ****expected ****monitoring ****time ****required ****to ****observe ****1500 ****failures.
Question ****2 ****[10 ****points ]
To answer the questions below, refer to the article titled “Similar rate of return to sports activity between posterior-stabilised and cruciate-retaining primary total knee arthroplasty in young and active patient”. This paper describes a study comparing two types of knee replacement surgury: posterior-stabilized and cruciate-retaining. Note: You don’t need to read the whole article, but should focus on relevant aspects of the Statistical analysis and results sections.
(a) ****This ****paper ****describes ****several ****response ****variables ****and ****analyses. One ****of ****the
analyses ****is ****a ****time-to-event ****anal ysis ****(see ****Figure ****4). ****What ****is ****the ****response ****in ****this ****time-to-event ****analysis? Be ****as ****specific ****as ****po ssible ****(including ****units).
(b) ****Based ****on ****Figure ****4, ****when ****did ****the ****las t ****observed ****failure ****event ****occur ****in ****the ****PS ****group ****(it’s ****OK ****to just ****say ****between ****1 0 ****and ****20 ****months, ****for ****example, ****based ****on ****the ****tickmarks ****shown ****on ****the ****x-axis). Explain ****how ****can ****answer ****this ****questio n ****based ****on ****the ****K-M ****plot ****only.
(c) ****What ****is ****potentially ****misleading ****with ****the ****visualization ****in ****Figure ****4? Do ****you ****agree ****with ****the ****authors’ ****choice?
(d) ****Consider ****the ****Kaplan-Meier ****curves ****in ****Figure ****4. Interpret ****and ****compare ****the
estimates ****of ****s(t) for ****the ****posterior-stabilized ****and ****cruciate-retaining ****groups ****at ****90 ****months. Use ****R ****to ****obtain K aplan-Meier ****estimates ****for ****each ****group ****separately ****for ****the ****data ****in ****the ****knees ****tibble, ****and ****provide ****exact ****values ****of ****the ****estimates ****for ****e ach ****group ****to ****use ****in ****your ****interpretat ion.
knees <- read_csv("knees.csv", col_types = "cdddcccdddcddddddcdddd")
(e) ****Based ****on ****the ****data ****in ****the ****knees ****tibble ****you ****used ****in ****the ****previous ****part, ****calculate ****the ****Nelson-Aalen ****estimate ****for ****the ****5-year ****cumulative ****h azard ****for ****the posterior-stabilized ****group. Compare ****this ****to ****the ****estimate ****of ****H(t) obtained ****by ****transforming ****the ****K-M ****estimate ****of ****the ****5-year ****survival ****probability. Which approach ****would ****you ****prefer ****t o ****use ****in ****this ****case ****and ****why?
Question ****3 ****[11 ****points ]
The data below summarises the number of weeks 5 individuals spend collecting unemployment insurance before starting a new job. Some of the individuals are lost to followup before starting a new job, and their follow-up times are denoted with +.
| ID Weeks collecting EI | |
|---|---|
| 1 | 2 |
| 2 | 3+ |
| 3 | 10 |
| 4 | 11 |
| 5 | 14+ |
(a) ****In ****this ****part, ****you ****will ****calculate ****K-M ****e stimates ****and ****their ****variances ****by ****hand ****(not ****using R). ****Construct ****a ****table ****reporting ****tj , ****rj , ****dj , ****S(t) , ****and ****ur(S(t)) . Show ****your ****work, ****not just ****your ****final ****answers.
(b) ****Use ****R ****to ****produce ****a ****table ****showing ****the ****K-M ****estima tes ****of ****the ****survival ****probabilities ****and ****estimated ****standard ****errors ****of ****these ****estimates ****to ****check ****your ****work ****from ****part ****(a).
(c) ****Estimate ****the ****33d ****percentile ****( t0.33 ) ****of ****the ****time ****spent ****on ****unemployment ****based ****on ****our
sample. Explain ****how ****you ****obtaine d ****your ****estimate.
(d) ****Use ****the ****log-log ****transformation ****to ****calculate ****an ****approximate ****95% ****CI ****for ****S(10) . Show ****your ****work, ****specifical ly ****(i) ****how ****you ****substitue ****values ****into ****the ****appropriate ****variance ****formula, ****(ii) ****calculating ****a ****CI ****of ****the ****transformed ****es timand, ****and ****(iii) ****the ****final ****confidence ****interval. ****Note: it ****is ****true ****that ****the ****sample ****size ****h ere ****is ****very ****small, ****but ****construct ****the ****confidence interval ****nonetheless.
Question ****4 ****[5 ****points ]
Consider a discrete failure time response T , with distinct failure times a1 < a2 < … < an (no ties), where aj < t ≤ aj+1 and a0 = 0.
(a) ****Prove ****that ****P (T > aj |T > aj — 1 ) = 1 — h(aj ) , ****where ****h(t) is ****the ****hazard ****function .
(b) ****Prove ****that ****S(t) = Πaj≤t [1 — h(aj )]
(c) ****Prove ****that ****P (t = am ) = h(am ) Πaj≤am — 1 (1 — h(aj ))
Question ****5 ****[5 ****points ****+ ****1 ****bonus ****(max ****100% ****on ****this ****assignmen t)]
Find a peer-reviewed journal article where the response of interest is a time-to-event response. This should not be an article mentioned in any of our course materials or examples. For one bonus mark on this question, find an article where the data underpinning the article is also available to download. The University of Toronto Libraries have a number of databases you can use to search for research articles, for example WebOfScience
(a) ****Provide ****a ****reference ****for ****the ****article ****you ****selected ****(including ****the ****title , ****authors’ ****names, journal, ****and ****date ****of ****publi cation). You ****must ****also ****submit ****either ****a ****link ****to ****the ****article ****(make ****sure ****it ****will ****work ****for ****the ****grader ****by ****testing ****it ****in ****an ****incognito
browser) ****or ****pdf ****of ****the ****article. In ****each ****of ****the ****parts ****below, ****indicate ****where ****you ****found ****your ****answer ****in ****the ****article ****(page/section/paragraph ****num ber).
(b) ****What ****is ****the ****time-to-event ****re sponse? Be ****as ****specific ****as ****you ****can ****(include ****units).
(c) ****What ****kind ****of ****censoring ****are ****the ****responses ****subject ****to. Give ****a ****specific ****example ****of ****a ****reason ****an ****observation ****would ****be ****considered ****censored ****in ****th is ****study.
(d) ****What ****method(s) ****are ****used ****to ****analyze ****the ****time-to-event ****response ****in ****your ****article?
(e) ****What ****is ****one ****thing ****mentioned ****in ****the ****methods ****section ****of ****your ****article ****that ****you ****found ****interesting ****and ****would ****like ****to ****learn ****more ****about.
Bonus: Is ****data ****available ****for ****the ****article ****you ****found ****in ****Question ****5 ? If ****yes, ****indicate ****where ****it ****can ****be ****found ****(URL ****or ****instructions) ****and ****submit ****the ****data ****to ****Quercus ****at
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