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STAT 432/532 - Survival Analysis

创建日期：2024-04-09 18:06:25

所属分类：统计学statistics

标签： STAT 432

Survival Analysis

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STAT 432/532 - Survival Analysis
Intructions: The questions are equally weighted. If you have multiple PDFs or
photos, try to combine them into one document. When you answer the questions, please
present enough evidence. If you use R, please also include the R code in .r or .rmd,
and highlight/indicate your answers clearly in your handwritten copy or the
PDF generated by R markdown. Good Luck!
1. (a) What is the difference between Type I censoring and the Type II censoring.
(b) Suppose X is a continuous positive random variable. Its survival function,
hazard function are denoted by S(x) and b(x).
i. Show that E(X) =
∫∞
0
S(x)dx.
ii. Obtain b(x) in term of the mean residual life.
2. Consider a small study with 8 subjects. The event times were recorded as follows:
1, 4, 2+, 4+, 2, 6+, 7+, 4
Here “+” denotes the censoring times.
(a) Calculate the Kaplan-Meier estimate for the survival function S(t), and draw
a plot of the estimated Sˆ(t).
(b) Derive a 100(1 − α)% confidence interval for Sˆ(2.5) using the delta method
with a logit transformation of survival function. A logit function is defined as
logit(x) = log( x
1−x).
(c) Suppose you want to test whether these 8 subjects are from a population with
hazard function
h0(t) =
{
1, 0 < t ≤ 3;
2, t > 3.
Winter 2021 1/3
STAT 432/532 - Survival Analysis Final
How to do this test? What is your conclusion? [ Please state clearly the null
hypothesis, the alternatives, and the test statistics you use. ]
3. To explore the efficacy of triple-drug combinations of antiretroviral therapy for
treatment of HIV-infected patients, investigators performed a randomized study
comparing AZT + zalcitabine (ddC) versus AZT+ zalcitabine (ddC)+ saquinavir.
The data, time from administration of treatment (in days) until the CD4 count
reached a prespecified level, is given below for the two groups.
AZT + zalcitabine (ddC): 85, 32, 38+, 45, 4+, 84, 49, 180+, 87, 75, 102, 39, 12, 11, 80, 35, 6
AZT + zalcitabine (ddC) + saquinavir: 22, 2, 48, 85, 160, 238, 56+, 94+, 51+, 12, 171
80, 180, 4, 90, 180+, 3
(a) Use the log rank statistic to test if there is a difference in the distribution of
the times at which patient’s CD4 reaches the prespecified level for the two
treatments.
(b) Build a Cox proportional model, then test again. Compare the result with
(3a).
4. Suppose we are interested in relationship between the disease-free survival func-
tions of 137 patients given a bone marrow transplant. Three groups were con-
sidered: Group 1 consisting of 38 ALL patients; Group 2 consisting of 54 AML
low-risk patients and Group 3 consisting of 45 AML high-risk patients. We want
to test the hypothesis that the disease-free survival functions of these three pop-
ulations are the same over the range of observation, τ < 2204 days, versus the
alternative that at least one of the populations has a different survival rate.
(a) Denote Zj(τ) =
∑D
i=1W (ti)
{
dij − Yij diYi
}
, j = 1, 2, . . . , K. If one wants to
test if there is any differences in the survival rates for the long term, which
weight function will you suggest?
(b) What test statistic you can use for this test? Write down the formula and
calculate its value.
(c) If you choose 0.05 as the accepted type I error for this test, will you reject
the null hypothesis? Interpret the result.
5. Suppose Z1, . . . , Zp are some factors in a data set and you are now interested in
building a Cox model based on the data.
Winter 2021 2/3
STAT 432/532 - Survival Analysis Final
(a) If you do not have any preference for some factors at the beginning, state
your steps and criteria in detail. [Do not copy from the slides. Use your own
words.]
(b) Is it possible to build a model using other methods? Would the models
selected by different methods be the same?
6. A study was performed to determine the effiacy of boron neutron capture therapy
(BNCT) in treating the therapeutically refractory F98 glioma, using boronopheny-
lalanine (BPA) as the capture agent. F98 glioma cells were implanted into the
brains of rats. Three groups of rats were studied. One group went untreated,
another was treated only with radiation, and the third group received radiation
plus an appropriate concentration of BPA.
You can find the data by “library(KMsurv);data(bnct)” in R.
(a) Using the group factor, fit a Cox proportional model. What method is used
to estimate the parameters in the model? What is the reference group in your
answer?
(b) Find an estimate and a 95% confidence interval for the relative risk of death
for a radiation plus BPA animal as compared to a radiated only animal.
(c) Is delta-method needed in (6b)? Why?
(d) Using an appropriate set of time-dependent covariates, test that the hazard
rates of the three groups are proportional.

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FE 825 IERG 5130 STAT 8178 EE161 ADS2 G54FUZ CS326 113B ECN 226 QBUS6860 ELEC3202 COMP9334 MGT11B MET CS 777 STA238 00099F Python SMM306 CMT206 EC116 SP MAST90084 CS480/680 BCPM0091 CMPT 300 ECON2209 FY20 STAT 432 CVEN9625 MATH1712 CMP9771M CMP3750M CIS 325 AI506 MAT344H1S MGT7180 PSTAT 160B CSYS5010 1D STAT 453/558 ME152B STA304H1F CSCI-128 ME428 CO3002 FINE 7640 COMP3234 ECSE 223 EE497 – 3D MATH5905 ECS648U ENG5009 MATH10242 EC665 A CPI221 ISOM5220 MACE40402 MGT B456F COMP9315 AFM 113 STAT 778 CS 564 MA231 CS 230 CS 124 COMP5511 CSC317 ITEC1010 ELEC207 INFT 3019 7CCMFM06 EE 161 5G ECLT5810 MATH 7349 CS526 O2 COMP90054 ELE00063M ACCT5034 ARC1015 CSCI 40 CSCI E40 FIN 524B CS 462 389COM CMT309 MFIN703 COMP4250 STA304 STA304H1S STA 1003 FIN 448 BUFN 732 CS520 CVEN3701 ECMT6006 PHAS0100 CSIT314 MDIA 5028 CSCI 2134 MAT 451 CMPT 459 STAT 457 INFO20003 COMP4141 IELE8066 F540 ECM9 CSE 6321 FIN 430 MATH3911 BFC5916 COMP1011 EEE 6209 STATS 240 CW3 COMP206 COMP1003 4331W ISE 530 CMPT 295 MATH 239 QF5203 INFR08010 IT114105 FIT2093 ECON3124 ECON1195 MATH 337 STAT 3006 COMP202 COMP3130 MATH3609 COMP 250 CS6140 IEEE ELE00041I FIN40090 CIV 5899 MATH3066 IGNMENT MATH 2009 STAT3405 HIST 101 MATH 3281 CEGE0042 ICT532 LAB 7 LAB 5 MCO455 CSIT214 Q4 IFB104 A1 ECO220Y5Y DA5020 CMPSC473 FIN9007 DSCI553 MSCI534 COMP3308 CSE-112 CMPT 310 EE104 F 2C P20 ALY2010 BUSM060 STA258 ECO00030M R BIOSTAT 274 COS6012-B FIT3155 CMPUT 340 EC421 INFS4205 ECOM135 NATS 1740 COMP809 MBE 6005 STA465 CSSE2310 CIVL2700 DATA4700 CMPSC 473 ISE 529 CS5044 MAST30020 COMP2207 ENGG1001 COMP0046 FIT3165 STAT GR5241 MECH0007 vATHK1001 ANALYTIC ATHK1001 MATH352001 HW3 S1 BMAN 24662 COMP 636 Stat-461 STAT 302 STAT 2132 STAR534 ENG4137 CMP3752M CS3103 BISM3222 STA 135 ME155A IIMT3601 ECON3371 ECON 3371 HW4 FINA 5840 DT211C CTEC3902 MAT005 MATH2148 S261F Econ 332 MATH3734 MIS41270 7CCMFM13 SYST 664 CIS434 CENG10014 MT4111 MT4527 ECMM424 CMT219 Math 136 MATH 3120 COMP10002 MAST20009 MATH0058 BEEM117 ISYE 6420 MGMTMFE 405 25579 1ECE 273 M122 COMP4434 1ELEC ELEC5620M STAT0010 BFF2401 MXB361 MATH3811 CS330 CMT307 FIT1033 CptS 583 UP 431 MATH6005 AF5343 CS2435 ECOM30002 ES50061 CS 2506 ECMT6002 CODE 00099F MT4539 ESSAY ICM317 MATH1231 MFIT5010 BMA547 MATH334 CIVL2201 MATH39032 T-04 CIVE97119 EMESTER CAB302 CAB301 59PM STAT802 ISYS3412 RM3 COSC 2123 AcF 311 MATH2022 1FINAL 4CCS1PHC ICM 142 ENVS3023 Math 4023 I1 ACSE8 FIN B385F CS-275 STA 4109-01 A31021 EE140 MT1 EE124 ECO-6004B Inf2-IADS CS126 IS2184 UL18 MATH5092 ECOM009 COMP 6211 ELEC202 MKT3019 IAB230 FIT5046 584-S21 COMP3210 CO250 A29401 MATH 136 MGT6174 ECMT6007 DATA2001 MAT 443 MATH10222 PHDE0071 ELE00123M COSC1295 MAT00045H MATH32032 1004GRC MAS223 COMP10001 DATA3404 CIVL2110 STAT2008 FNCE10002 EEE6431 COM6515 COMP9120 NBS8185 ECTE942 ECON 405 MATH314301 STAT 2011 COMPSCI 350 COMP20007 CMPSC-132 ECON5002 STA 35 MTH6108 FIT5147 MTH6150 100C FINA 5260 DESC9115 ECON4998 NBS8257 GMM ECON 6070 BSA2 ECE599 STATS 102B STA120 INFS601 BFC3241 MATH20602 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1300A ENGG 1300 OLET1139 STAT 3503 STAT 206 FEM11149 ELEN 4720 COMP20003 SOLA1070 CS574 C11CS LINB29H3 MAST90083 ECON7200 STAT 610 LINB29 DSM-5 COMP3161 EEEN40010 MATH 4431 K353 STA 3386 IERG4300 COMS W4167 ENG5298 AMME 2302 CS3910 COMPSCI 320 STATS 369 GR5242 COSC 328 CSCI435 MGTS1301 INFS2200 COMP3900 CIS 375 Module AE03 COMP3121 CS260 HAM503 MGMT5050 INT3075 COSC 285 GEOL0029 GEGE2X01 STAT1201 MATH1061 MMF1941H STAT 5060 CS 520 COMP 4320 MSIN0104 000T FINS3648 ECMT2160 ECON4003 Phys 129L CSE 142 CS 3130 MGEB01 EECS 340 Finance (20443/20444) ECMM149 COMP5310 EEEN60180 CSC 111 AT2 CSCI 4430 STAB57 MA/CSC 427 MATH322 D100 CSCI544 CITS3004 CI6227 CSE 565 APCO 1P01 852L1 ELEC S411F COMP3207 STATS330 ECON7530 FIT5149 CSI4104 CMPT 371 COMP201 SEHS4696 FI4003 MATH6182 CG1 WS MATH401 MGT 205 QTM 110 COMP 0137 MACE12201 FIT5210 ECE 544 BISM7206 HUDM 6055 ECON3203 STATS 210 MSCI 570 MATH6173 COMPSCI4039 Accounting COM3503 AOE2024 ECON213 PSYC10004 Control Data 100/200A PHIL2617 PHYS3116 MIE250 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