Solution to question 11.4
I will use the 'magic number' method of calculating the sample size.
Here we are dealing with binary data, we are concerned with the proportion of 'successes', i.e. what proportion of patients showed an improvement of at least 2 units.
Successes with Fairlycheapo = 0·65
Successes required with Fluoriwonder = 0·80
α = 0·05 (5%)
Power = 80%
From table, magic number = 7·8
We need:
(0·65 x 0·35 + 0·80 x 0·20) ÷ (0·65 - 0·80)2 x 7·8 = 135 patients per group
(Remember to round up to the next whole number.) So, we need 270 patients for the whole study.
If the conditions we altered, as so:
Successes with Fairlycheapo = 0·45
Successes required with Fluoriwonder = 0·65
α = 0·05 (5%)
Power = 80%
From table, magic number = 7·8
We need:
(0·45 x 0·55 + 0·60 x 0·40) ÷ (0·45 - 0·60)2 x 7·8 = 169 patients per group
(Remember to round up to the next whole number.) So, we need 338 (or 340, if we round up) patients for the whole study. Note that even though we are still looking for a 15% difference we need more patients. This is typical; if we are looking at proportions near 0·5 we need more patients than if we are looking at proportions near 0 or 1.
