## The theory underlying sample size calculations using the Normal distribution

Suppose we wish to design a trial in which the main outcome measure is the difference in means of a particular variable in a group receiving standard therapy and a group receiving an experimental therapy. We will assume there will be n patients in each group, that the significance level has been set to be a (one-sided for simplicity) and the power 1 — ¡3. We will also assume that the SD of the distributions from which the two means are calculated is approximately the same, S, hence the SD of the means is S/^/n. This quantity is referred to as the Standard Error of the mean (SE).

Under the null hypothesis of no difference (Ho), the difference in means (d = m1 — m2) is assumed to follow a Normal distribution with mean zero and standard deviation

Under the alternative hypothesis (H1), the difference d is assumed to follow a Normal distribution with mean S and standard deviation, x S/^Jn, as under the null hypothesis.

To determine the sample size which satisfies both error constraints, we need to find the critical value, D, such that:

under H0, p(d < D) = 1 — a, under H1, p(d > D) = 1 — 3.

That is,

(D — 0)/( J2 x S/Jn) = Z— therefore D = ZX—a x (J2 x S/Jn), (D — S)/( J2 x S/Jn) = —Z1—3 therefore D also = S — (Z1—3 x (J2 x S/Jn)). Hence: Z1—a x (^/2 x S/ jri) = S — (Z1—p x (*J2 x S/*Jn)).

This can be rearranged to give the number of patients, n, required in each group to satisfy the defined error rates as

For a 2-sided test, a slight approximation is involved, but the equation can simply be written as

The quantity (Z1—a/2 + Z1—p)2 is fundamental to many sample size calculations, and the most commonly used values are given in Table 5.2. In general, in this chapter we

Significance level (a) |
Power (1 — p) | ||||

2-sided |
1-sided |
0.80 |
0.85 |
0.90 |
0.95 |

0.01 |
0.005 |
11.679 |
13.048 |
14.879 |
17.814 |

0.02 |
0.01 |
10.036 |
11.308 |
13.017 |
1 5.770 |

0.05 |
0.025 |
7.849 |
8.978 |
10.507 |
12.995 |

0.1 |
0.05 |
6.183 |
7.189 |
8.564 |
10.822 |

present sample size formulae in the simplest terms, which will often mean that slight approximations are involved. These should therefore be used only as a rough guide to sample size, and more complete tables or software should be used for final calculations.

## Post a comment