# 6b.7 - Sample Size and Power

6b.7 - Sample Size and Power

For a continuous outcome that is approximately normally distributed in an equivalence trial, the number of patients needed in the active control arm, $$n_A$$, where $$AR = \dfrac{n_E}{n_A}$$, to achieve $$100 \left(1 - \beta \right)\%$$ statistical power with an $$\alpha$$-level significance test is approximated by:

$$n_A = \left( \frac{AR+1}{AR}\right) \left(t_{n_{1}+n_{2}-2, 1-\alpha}+ t_{n_{1}+n_{2}-2, 1-\beta}\right)^2 \sigma^2 / \left(\Psi - |\Delta|\right)^2$$

Notice the difference in the t percentiles between this formula and that for a superiority comparison, described earlier. The difference is due to the two one-sided testing that is performed.

Most investigators assume that the true difference in population means, $$\Delta = \mu_E - \mu_A$$, is null in this sample size formula. This is an optimistic assumption and may not be realistic.

NOTE! the formula above simplifies to the formula on p. 189 in the FFDRG text if $$AR =1, \Delta = 0$$ and substituting Z for t )

For a binary outcome, the zone of equivalence for the difference in population proportions between the experimental therapy and the active control, $$p_E - p_A$$, is defined by the interval $$(-\Psi, +\Psi)$$. The number of patients needed in the active control arm, $$n_A$$, where $$AR = \dfrac{n_E}{n_A}$$, to achieve $$100 \left(1 - \beta \right)\%$$ statistical power with an $$\alpha$$ significance test is approximated by:

$$n_A = \left( \frac{AR+1}{AR}\right) \left(z_{1-\alpha}+ z_{1-\beta}\right)^2 \bar{p}(1-\bar{p}) / \left(\Psi - |p_E-p_A|\right)^2$$

where

$$\bar{p}= \left( AR \cdot p_E+p_A\right) / (AR+1)$$

How does this formula compare to FFDRG p. 189? The choice of the value for p in our text is to use the control group value, assuming, that $$p_e - p_a = 0$$.

For a time-to-event outcome, the zone of equivalence for the hazard ratio between the experimental therapy and the active control, $$\Lambda$$, is defined by the interval $$\left(\dfrac{1}{\Psi}, +\Psi\right)$$, where $$\Psi$$ is chosen > 1. The number of patients who need to experience the event to achieve $$100 \left(1 - \beta \right)\%$$ statistical power with an $$\alpha$$-level significance test is approximated by

$$E = \left( \frac{(AR+1)^2}{AR}\right) \left(z_{1-\alpha}+ z_{1-\beta}\right)^2 / \left( log_{e} \left(\Psi / \Lambda \right)\right)^2$$

If $$p_E$$ and $$p_A$$ represent the anticipated failure rates in the two treatment groups, then the sample sizes can be determined from $$n_A = \dfrac{E}{\left(AR\times p_E + p_A\right)}$$ and $$n_E = AR \times n_A$$

If a hazard function is assumed to be constant during the follow-up period [0, T], then it can be expressed as $$\lambda(t) = \lambda = \dfrac{-\text{log}_e(1 - p)}{T}$$. In such a situation, the hazard ratio for comparing two groups is $$\Lambda = \dfrac{\text{log}_e\left(1 - p_E\right)}{\text{log}_e\left(1 - p_A\right)}$$. The same formula can be applied, with different values of $$p_E$$ and $$p_A$$, to determine $$\Psi$$.

For a continuous outcome that is approximately normally distributed in a non-inferiority trial, the number of subjects needed in the active control arm, $$n_A$$, where $$AR = \dfrac{n_E}{n_A}$$, to achieve $$100 \left(1 - \beta \right)\%$$ statistical power with a $$\alpha$$-level significance test is approximated by:

$$n_A = \left( \frac{AR+1}{AR}\right) \left(t_{n_{1}+n_{2}-2, 1-\alpha}+ t_{n_{1}+n_{2}-2, 1-\beta}\right)^2 \sigma^2 / \left(\Psi - |\Delta|\right)^2$$

Notice that the sample size formulae for non-inferiority trials are exactly the same as the sample size formulae for equivalence trials. This is because of the one-sided testing for both types of designs (even though an equivalence trial involves two one-sided tests). Also notice that the choice of $$Z_\alpha$$ in the formulas above have assumed a one-sided test or two one-sided tests, but the requirements of regulatory agencies and the approach in our FFDRG text is to use the Z value that would have been used for a 2-sided hypothesis test. In homework, be sure to state any assumptions and the approach you are taking.

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