Reference no: EM132310781

PRINCIPLES OF STATISTICAL INFERENCE (PSI)

Exercise : This exercise is based on the Extended Example. Before attempting the questions below, make sure you have worked through the details of the Extended Example, including the checking of calculations where indicated. In your solution show any formula that you have used.

Now suppose that the study described were continued until the sample size had doubled, with the resulting data shown in the following

Table:

Table. Observed data for (continued) randomised trial of HIV therapies

	Mono therapy	Combination therapy	Total
Response	x0 = 90	x1 = 114	204
No response	108	88	196
Total	n0 = 198	n1 = 202	n = 400

(a) Work out the 3 (approximate) posterior distributions for the difference in response rates, as in the Extended Example notes, based on the new data, and obtain the posterior probability of the difference being greater than 0, 0.05 and 0.1, as in Table 2 of the notes. How do the interpretations change, from the perspective of the "sceptic" and the "enthusiast"?

(b) Suppose you wish to adopt what you regard as a "realistic" prior distribution, which is normal in shape, but allows a priori for a 20% chance that the difference in rates is negative (i.e. favours the control monotherapy) and a 20% chance that it is greater than 0.1.

Figure out the appropriate parameters for this normal prior distribution, and work these through to obtain the corresponding posterior distribution and posterior probabilities as before. How different are the conclusions from this fourth prior distribution to those obtained under each of the previous three prior distributions?

HINT: For this exercise, it will be convenient to set up a spreadsheet in MS Excel or similar or code in a software package to implement the formulas required to perform the various calculations.

A final comment: you should have observed that with the larger sample size of these new data, the differences between the results under the range of prior distributions is less pronounced than it was for the smaller sample size. This illustrates a very important general fact: the more data that accumulate, the less important the prior distribution. All approaches to statistics agree that large sample sizes are the best strategy for accumulating reliable evidence!