## Example

We will calculate sample sizes for a new hypertension study to compare three treatments. Assuming that DBP is again the primary endpoint, the variance components obtained in Section 2.5 for centre-treatment effects (ac2t = 4.10) and the residual (a2 = 68.4) can be used to estimate sample sizes. A difference of 5 mmHg is to be detected at the 5% significance level with 90% power.

1. Number of centres specified It has been decided that the new study will use four centres. Therefore, the DF for the t distribution are (4 — 1) x (3 — 1) = 6 and we have

(2 x (2.45 + 1.44)2 x 68.4) ~ (4 x 52 - 2 x (2.45 + 1.44)2 x 4.10) = —86.0.

Since r is negative it is not possible to obtain the required power when only four centres are used. If the number of centres is increased to six, then DF = (6 — 1) x (3 — 1) = 10 and

(2 x (2.23 + 1.37)2 x 68.4) ~ (6 x 52 - 2 x (2.23 + 1.37)2 x 4.10) = 40.5.

Thus, 41 x 3 = 123 patients would be required in each of the six centres and the total number of patients is 123 x 6 = 738.

2. Number of patients per centre specified It has been decided that an average of only 15 patients per centre will be used so that the study can be completed quickly. Using r = 5 we obtain an initial estimate of c using z statistics as c = 2 x (1.96 + 1.28)2 x (68.4 + 5 x 4.10)/(5 x 52) = 14.9.

With 15 centres the t distribution DF for use in the original formula would be (15 — 1) x (3 — 1) = 28. On recalculating using t28 statistics c becomes 16.1, which should be rounded up to 17. Therefore, 17 centres should be used with five patients per treatment group (17 x 5 x 3 = 255 patients in total).

3. Neither number of centres n or average patients per centre specified The study will use centres from different countries and therefore the cost of sampling centres compared with patients is high. If we set the relative cost of sampling centres compared with sampling patients at g = 100, then r = V[100 x 68.4/(3 x 4.10)] = 23.6.

Rounding r to 24 we obtain an initial estimate of c:

c = 2 x (1.96 + 1.28)2 x (68.4 + 24 x 4.10)/(24 x 52) = 5.8.

This value of c is rounded to six. As c is small, it should be recalculated more accurately using DF = (6 — 1) x (3 — 1) = 10 as c = 2 x (t10,0.975 + t10,0.90)2 x (68.4 + 24 x 4.10)/(24 x 52) = 2 x (2.23 + 1.37)2 x (68.4 + 24 x 4.10)/(24 x 52) = 7.21.

However, we can be more precise, because the necessity to have an integer number of centres means that in moving from 7.21 centres to eight centres, our value for r can be recalculated. In this instance it is reduced appreciably from 23.6 to 16.6. We can therefore obtain our desired power from 17 patients per treatment per centre, using eight centres. We could also consider the alternative of using seven centres and recalculating r. This gives r = 23.6. We can therefore compare the cost of seven centres with 24 patients per treatment per centre, with the design with eight centres.