Sample/practice exam 2016, questions PDF

Preview

Summary

2016 mid-semester exam questions, first semester...

Description

ANU$ID:$_________________________________________$ $

Research School of Finance, Actuarial Studies and Applied Statistics

Mid-Semester Examination, 1st Semester 2016

SURVIVAL MODELS / BIOSTATISTICS (STAT3032/4072/7042/8003)

Writing period: 90 Minutes Study period: 5 Minutes Permitted materials in room: One (1) A4 sheet of double-sided notes, non-electronic dictionary, calculator (any type), non-electronic stationery.

Total Marks: 45 marks Redeemable policy: The entire weighting of this mid-semester exam will be moved towards the final exam if this reweighting is advantageous to the student’s final course grade.

Instructions to candidates: •! Enter your ANU ID on the top left of this cover page and each subsequent page. •! You must attempt all questions WITHIN THIS EXAM BOOKLET. All pages of this booklet must be submitted, including computer output. •! Questions should be attempted in whichever order you see fit. •! For multiple-choice and True-or-False questions: o! Showing work is optional but recommended. o! If your final answer is incorrect, then any work that you show could be given part marks. If your final answer is correct, then no marks will be deducted even if the work shown is incorrect. •! For open-ended questions: full marks will only be given if all the steps involved in the calculation and/or logical argument are correct; marks will be deducted for failure to show appropriate calculations, formulae, and/or logical argument.

ANU ID: u______________________________________

Part I

ANU Stat3032/4072/7042/8003 Mid-Sem Exam March 2016: Page 1 of 9

[19 pts]

Background A follow-up study was conducted on six patients labelled A to F over 12 calendar weeks. Each horizontal line segment in the figure below covers the calendar weeks from the start of the patient’s follow-up to the time of the patient’s death (marked

X) / withdrawal / loss of contact to the study.

For all questions in this section, let

S(t) = P(survival beyond time t) denote the survival function for the

population that is represented by patients A–F.

Questions Let tj denote the jth patient’s time until death. For these data, write out the values of t A , t B , … , t F to the nearest 0.5 week. Mark the value of tj with a + sign if it is censored, e.g., t G =

1. [3 pts]

7.5+ for a hypothetical patient G.

tA =

tB =

tC =

tD =

tE =

tF =

ANU ID: u______________________________________

2. [5 pts]

ANU Stat3032/4072/7042/8003 Mid-Sem Exam March 2016: Page 2 of 9

Consider a lifetable estimate of

S(t) based on arranging your t A , … , t F in fortnightly

temporal bins. (A fortnight contains 14 days.) Let

ni di ci l ∗i pi Si ltab

S i

= number at risk at the ith bin’s start = number of deaths in the ith bin = number of censored observations in the ith bin = number at risk in the ith bin, adjusted for censoring = proportion that survive the ith bin relative tol ∗i = S(t) for t = start of ith bin = lifetable estimate of Si

Fill in the blanks below to complete the corresponding life table entries.

i (fortnights) 0 1 2 3 4 5 6

ni

di

ci

l ∗i =

pi =

ltab Si

1

ANU ID: u______________________________________

3. [3 pts]

Sketch the graph of

ltab

S

ANU Stat3032/4072/7042/8003 Mid-Sem Exam March 2016: Page 3 of 9

ltab (t) as a step function based on your S i above. The graph

must be on a continuous time axis measured in weeks.

4. [3 pts]

state and explain one difference between the answer this question without actually producing

ltab

S

K-M

S

K-M

S

(t)for these data. Briefly K-M (t) above and Dr. Zeuss' S (t). You must

Suppose Dr. Zeuss will compute the Kaplan-Meier estimate

for these data.

ANU ID: u______________________________________

5. [5 pts]

ANU Stat3032/4072/7042/8003 Mid-Sem Exam March 2016: Page 4 of 9

Suppose Prof. Harrington will consider the Nelson-Aalen approach and compute

N-A

S

(t)

instead. Let

t(j) = jth smallest time until an observed death rj = number at risk immediately before t (j) fj = number of deaths at t(j) fj qj = rj g(x) = e−x Apply the Taylor series expansion to

S

N-A

K-M

g(q j ) about the value 0 to show that Dr. Zeuss' S 

(t) cannot

(t) anywhere on the temporal axis. You must answer this question K-M N-A without actually producing either S or S for these data. exceed Prof. Harrington’s

ANU ID: u______________________________________

Part II

ANU Stat3032/4072/7042/8003 Mid-Sem Exam March 2016: Page 5 of 9

[18 pts]

Background Consider the dataset (partially shown below) to address the questions: “How long can AA batteries work if purchased after their expiry date on the package? Does it matter whether the batteries are rechargeable or not (hence, regular )? Does it matter how much time has passed ( expir ) since its expiry date?” surv death

store

expir

type

past

1 Albi 1 Woodies 0 Woodies

3.60 4.43 3.20

regular regular regular

yes yes yes

6

1

Albi

3.26 rechargeable

yes

5

1

Albi

3.49 rechargeable

yes

23

1

Albi

2.57

regular

kindof

20 21 ...

22 13

1 Albi 1 Woodies

2.32 2.88

regular regular

kindof kindof

35 ... 39

25

0

1.78

regular

no

12

1 Woodies

1.50 rechargeable

no

40

11

1 Woodies

2.12 rechargeable

no

1 2 3 ... 10 11 ... 19

16 7 6

Albi

This dataset is from a study in which the stores Albi and Woodies provide expired packages of their store-brand rechargeable and regular AA batteries; the study receives 40 packages altogether. One battery is randomly selected from each package, and all 40 selected batteries are connected to a single discharging machine that operates on all of them simultaneously. A battery’s survival time surv is defined as the number of hours (integer) the battery takes to reach a 0 charge ( death =1). However, a battery’s charge could decrease to a very low but non-zero value, then stop discharging; its survival is considered censored ( death =0). Note that expir is in log(# of days). The practical severity of the product’s expiry is considered in the variable past with three categories: no , kindof , and yes , respectively corresponding to expir taking on a value inside [0, 2.3], (2.3, 3], and (3,

∞).

On the next page are results of some survival analyses on this dataset:

ANU ID: u______________________________________

ANU Stat3032/4072/7042/8003 Mid-Sem Exam March 2016: Page 6 of 9

R output for log-rank test among categories of past (some info purposely obliterated with *** ): Call: survdiff(formula = surv ~ past, rho = 0) N Observed Expected (O-E)^2/E (O-E)^2/V past=kindof 14 past=no 9 past=yes 17

10 2 16

Chisq= 23.8

10.56 10.66 6.77

0.03 7.04 12.57

0.051 13.007 19.583

***

R output for Cox PH regression on type and expir (some info purposely obliterated with *** ): Call: coxph(formula = surv ~ type + expir) coef exp(coef) typeregular -1.756 0.173

*** ***

expir

***

1.376 3.958 *** n= 40, number of events= 28

ANU ID: u______________________________________

ANU Stat3032/4072/7042/8003 Mid-Sem Exam March 2016: Page 7 of 9

Questions: TRUE or FALSE [2 pts each]

Based on the information and output given under Background, circle one option between

T and F for each statement below. Label

Statement Consider “ H0 : survival and battery type are independent”. If we conduct a log-

T

F

1.

rank test of H 0 , then we have enough information to deduce that the p-value is 20 hours. d. cannot be determined due to the product limit nature of the Kaplan-Meier estimate. e. cannot be determined because the number of rechargeable batteries at risk reaches 0 midway through the observed timeline.

ANU ID: u______________________________________

ANU Stat3032/4072/7042/8003 Mid-Sem Exam March 2016: Page 8 of 9

7. The above usage of the Cox proportional hazards model is a. inappropriate because of excessive censoring. b. inappropriate because the sample size is too small for each battery type. c. inappropriate because a key assumption required by the model is violated. d. appropriate because there is no obvious parametric model suitable for these data. e. none of the above.

Part III [4 pts each]

[8 pts] Stand-alone questions: for each question below, circle the best option among the five

choices. Each wrong answer is worth 0, 1, or 2 points as part marks.

S(t) defined on a continuous temporal axis measured in years has the feature that “death rate is 1 per year for all t ≥ 8.3.” Let h(t) and H(t) respectively denote the corresponding hazard and cumulative hazard, and let H (t) denote the Nelson-Aalen estimate of H(t). The given

1. A survival function

information implies that a.

h(t) = 1 for all t ≥ 0 .

b.

H( ) has the feature that H ′ (t) = 1 for all t ≥ 8.3.

c.

H ( ) has the feature that observed # of deaths in t-th year = 1 for all t ≥ 8.3. observed # at risk in t-th year

d. e.

H( ) has the feature that H (t + 6) − H (t) = 6 for all t ≥ 8.3. none of the above.

ANU ID: u______________________________________

ANU Stat3032/4072/7042/8003 Mid-Sem Exam March 2016: Page 9 of 9

2. A professional soccer team CB conducts a long-term survival study on their players, for which followup begins/began upon being employed by CB, and failure is defined as leaving CB to join another soccer team (resulting from either the player’s own decision or management’s decision). Retirement from the soccer profession while still under employment by CB is regarded as censoring. Suppose we know that in general, the longer the soccer player has been with the same team, the less likely she will leave to join another soccer team. Thus, CB’s Kaplan-Meier estimate of the survival function tends to a.

underestimate survival only towards the start of the follow-up study.

b.

underestimate survival only towards the end of the follow-up study.

c.

overestimate survival only towards the end of the follow-up study.

d.

underestimate survival across the entire follow-up study.

e.

overestimate survival across the entire follow-up study.

Congratulations! This is the end of the exam.

——————————————————————————————————– ——————————–...