Question

Assuming that the number (X) of patients experiencing a MLT in a group of six patients follows a binomial distribution with parameter p

X~Bin(6,p)

a) Write down the probability that at most one patient experiences a MLT

b) Tabulate this probability for values of p = 0.05, 0.10, …, 0.95.

c) What conclusions can we make about the underlying rate of MLT if we observe at most 1 patient with MLT in a cohort of 6 patients?

We want to be confident that the population rate of MLT (pt) is at most 10%. This event is unlikely in a cohort of 6 patients if pt is as low as 10%. The investigators therefore propose a Bayesian monitoring rule for the next study. This is designed to strop the study if the posterior probability pt>0.1 exceeds 75%.

d) Why was a Beta prior distribution selected?

e) What is the prior probability at pt>0.1?

f) What is the posterior distribution for pt if d deaths are observed in the first n patients?

g) If no deaths are observed in the first 10 patients, what is the posterior probability that pt>0.1? Suppost the next 2 patents (11 and 12) both experience death. Should you consider stopping this stuyd?

h) Tabulate the posterior probability that pt>0.1 for 2/20, 4/40, 6/60, 8/80 and 10/100 observed deaths.

i) If the study continued to observe 10/100 observed deaths, what could you conclude about the prevalence of death in this study?

Answer 1

X~Bin(6,p)

1. P(at most one patient experiences MLT)

=P(X<=1)=P(X=0)+P(X=1)=(1-p)⁶+6p(1-p)⁵.

2. p P(X<=1)

0.05 0.967226

0.10 0.885735

0.15   0.776484

0.20 0.655360

0.25   0.533936

0.30   0.420175

0.35   0.319080

0.40   0.233280

0.45   0.163567

0.50   0.109375

0.55   0.069198

1. 0.60 0.040960
2. 0.65 0.022322

0.70   0.010935

0.75   0.004639

0.80   0.001600

1. 0.85 0.000399

                   0.90   0.000055

0.95   0.000002

c) If atmost 1 patient is observed with MLT out of 6, the sample proportion of patients with MLT is 1/6= 0.1666667. Therefore, the underlying rate of MLT is expected to be around 0.1666667, on an average.

d) Since the rate of MLT lies between 0 and 1 and a Beta distribution has support (0,1), a Beta distribution is used as a prior.