BISM 6051

Temple University

BISM

### Question Description

I need help with a Statistics question. All explanations and answers will be used to help me learn.

Please find the following assignment attached below! You must have SAS and know how to use the software to proceed with the assignment. Thank you!

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## Final Answer

Attached.

Mathematical Statistics II

(BISM 6051)

Homework #10

1. Data in the table below represent the lifetimes in days of two groups of rats

(distinguished by a pretreatment regimen) following exposure to the carcinogen

DMBA (from Kalbfleisch and Prentice (1980)). Time in parenthesis indicates the

right censored data.

Group 1 (n=21)

142, 156, 163, 198, (204), 205, 232,

232, 233, 233, 233, 233, 239, 240,

261, 280, 280, 296, 296, 323, (344)

Group 2

(n=19)

143, 164, 188, 188, 190, 192, 206,

209, 213, 216, (216), 220, 227, 230,

234, (244), 246, 265, 304

Using Kaplan-Meier method,

a) Calculate the estimates of the survival function, 𝑆̂(𝑡𝑖) at 𝑡𝑖.

S(ti) =

𝑁𝑢𝑚𝑏𝑒𝑟 𝑙𝑖𝑣𝑖𝑛𝑔 𝑎𝑡 𝑠𝑡𝑎𝑟𝑡−𝑐𝑢𝑚𝑢𝑙𝑎𝑡𝑖𝑣𝑒 𝑁𝑢𝑚𝑏𝑒𝑟 𝑑𝑒𝑎𝑑

𝑁𝑢𝑚𝑏𝑒𝑟 𝑙𝑖𝑣𝑖𝑛𝑔 𝑎𝑡 𝑠𝑡𝑎𝑟𝑡

When there is censoring at ti eg t1: at S(t2)

Number of start changes to the current number at risk whenever there is a censor,

and also cumulative dead starts again. For example, number of start becomes

number at risk at t2

S(t2)= (number at start- cumulative number dead)/ number at risk * S(t1)

Group 1

Number at start = 21

number

time (ti) n.risk

dead

142

21

156

20

163

19

198

18

204

18

205

16

232

15

233

13

239

9

240

8

261

7

280

6

296

4

323

2

344

1

1

1

1

1

0

1

2

4

1

1

1

2

2

1

0

cumulative

dead

censored

1

0

2

0

3

0

4

0

4

1

1

0

3

0

7

0

8

0

9

0

10

0

12

0

14

0

15

0

15

1

Formula: survival

survival

(S(ti))

(S(ti))

=(21-1)/21

0.9524

=(21-2)/21

0.9048

=(21-3)/21

0.8571

=(21-4)/21

0.8095

=(21-4)/21

0.8095

=(16-1)/16 * 0.8095

0.7589

=(16-3)/16 * 0.8095

0.6577

=(16-7)/16 * 0.8095

0.4554

=(16-8)/16 * 0.8095

0.4048

=(16-9)/16 * 0.8095

0.3542

=(16-10)/16 * 0.8095

0.3036

=(16-12)/16 * 0.8095

0.2024

=(16-14)/16 * 0.8095

0.1012

=(16-15)/16 * 0.8095

0.0506

=(16-15)/16 * 0.80...