Biostatistics Collaboration of Australia
1
Principles of Statistical Inference (PSI)
Semester 1, 2020
Assignment 1
Due 11:59pm, Sunday April 26th, 2020
Copyright © NHMRC Clinical Trials Centre, University of Sydney
Biostatistics Collaboration of Australia
2
Question 1
14 marks
In a group of ?? patients who have undergone a new de
idement technique for small
diabetic ulcers, the time in days ???? for patient ?? to achieve wound healing can be
described by a Weibull distribution with probability density function:
??(????) = ??????????−1??−??????
??
, where ???? ≥ 0,?? > 0 and ?? > 0
In the special case that ?? = 1, the Weibull distribution reduces to the exponential
distribution.
Here we will assume ?? is a known constant (but not necessarily equal to 1) and we
are interested in estimating the rate parameter ??.
a) 4 marks Write down the likelihood function and the log-likelihood function for
the parameter ?? in the sample of n patients. Omit terms that do not depend on
??.
Consider three possible estimators of ??:
�̂�?1 =
??
∑ ??????????=1
�̂�?2 = �
??
∑ ????????=1
�
??
�̂�?3 =
log (2)
????
where m is the sample median
The times for ?? = 150 patients are in the Excel file “Weibull data Q1.xlsx”. For this
example, we will assume ?? = 2.
) 3 marks Calculate the value of each of these estimators for the cu
ent study.
c) 3 marks Calculate the value of the log-likelihood for each of these estimators
for the cu
ent study.
d) 2 marks Which of the estimates is most supported by the data?
e) 2 marks Use the factorisation criterion to find a sufficient statistic for the
parameter ??.
Biostatistics Collaboration of Australia
3
Question 2
15 marks
Continuing from the study described in question 1, and working in general terms for
any value of ??, ???? and ?? …
a) 1 mark Write down the likelihood equation for the parameter ??
) 2 marks Solve the likelihood equation to determine the maximum likelihood
estimator �̂�?. Discuss this in relation to your results for Question 1.
c) 3 marks Determine
i. The observed information
ii. The expected information
iii. The inverse of the expected information
d) 2 marks The investigators are interested in the median time to healing. The
median for the Weibull distribution is �log (2)
??
�
1 ??�
i. Propose an estimate of the median distance.
ii. What property are you using to allow you to do this?
Using the observed times in the spreadsheet …
e) 1 mark Calculate the estimate �̂�?
f) 2 marks Using you results above, obtain a 95% confidence interval for ??
g) 2 marks Derive a 95% confidence interval for the median time to healing by
transforming the confidence interval for ??
h) 2 marks It is known that the median time using the previous standard
technique was 8.9 days. What can you conclude about the new technique
compared to standard?
Biostatistics Collaboration of Australia
4
Question 3
11 marks
A team of researchers wish to test a new vaccine that is hoped to provide protection
against an additional strain of influenza beyond the strains included in the cu
ent
vaccine.
They propose a two-arm study where participants are randomised to receive either
the cu
ent vaccine or the new vaccine, with equal numbers in each treatment arm.
The primary outcome of the study is the proportion of participants who exhibit at
least a four-fold increase in serum antibody level from pre-vaccination to 28 days
post-vaccination.
a) 2 marks Write down the null and alternative hypotheses to test whether the
proportion differs between treatments. Discuss
iefly whether you think a
one- or two-sided hypothesis would be more appropriate for this study.
) 2 marks Prior experience with the standard vaccine shows that the proportion
who exhibit the required increase in antibody level is expected to be 0.67.
Approximately 450 participants are available for the study. Assuming ?? =
0.05,
i. What is the power of this study to detect a difference of 0.1 in
proportions between the two vaccines?
ii. What sample size is required to achieve 80% power?
c) 1 marks The investigators decide that a difference in proportion of 0.07
etween vaccines is more plausible. What sample size is required to detect
that difference, assuming ?? = 0.05.
d) 2 mark What options would you suggest to the investigators to design their
study?
e) 2 marks Not all participants will return for their 28-day post-vaccination
measure. By what factor would you need to increase the sample size to
account for an anticipated 5% loss to follow-up and still achieve the required
statistical power?
f) 2 marks Due to problems during the study with the manufacture of the new
vaccine, some participants allocated to the new vaccine may receive the
standard vaccine instead. By what factor would you need to increase the
sample size to account for an anticipated 5% of participants randomised to
new vaccine who actually receive the standard vaccine and still achieve the
equired statistical power?
Principles of Statistical Inference (PSI)
Semester 1, 2020
Assignment 1
Due 11:59pm, Sunday April 26th, 2020
Copyright © NHMRC Clinical Trials Centre, University of Sydney
Question 1
Question 2
Question 3