### POWER AND SAMPLE SIZE CALCULATIONS:

#### STATISTICAL POWER CALCULATION:

If we know $$\sigma$$ and $$n$$ is large, and with $$\beta$$ being the type II error rate, the power is $$1-\beta$$

\begin{align} 1-\beta &= \Pr\left(\frac{\bar X -\mu_0}{\sigma/\sqrt{n}} > z_{1-\alpha} \mid \mu = \mu_a \right)\\[3ex] &= \Pr\left(\frac{\bar X-\mu_a +\mu_a -\mu_0}{\sigma/\sqrt{n}} > z_{1-\alpha} \mid \mu = \mu_a \right)\\[3ex] &=\Pr\left(\frac{\bar X -\mu_a}{\sigma/\sqrt{n}} > z_{1-\alpha} - \frac{\mu_a-\mu_0}{\sigma/\sqrt{n}} \mid \mu = \mu_a \right)\\[3ex] &= \Pr\left(Z > z_{1-\alpha} - \frac{\mu_a-\mu_0}{\sigma/\sqrt{n}} \mid \mu = \mu_a \right) \end{align}

Suppose that we wanted to detect an increase in mean of the RDI (respiratory disturbance index) in the context of sleep apnea of at least $$2\small \text{ events/hour}$$ above $$30$$. Assume normality and that the sample in question has a standard deviation of $$4$$. What would be the power if we took a sample of $$16?$$

$Z_{1-\alpha}=1.645$ or…

qnorm(0.95)
## [1] 1.644854

and with $$\mu_a$$ being the true mean under the alternative hypothesis (i.e. sleep-apnea carries along a higher number of RDI with a mean of 32):

$\frac{\mu_a - 30}{\sigma/\sqrt{n}}=\frac{2}{4/\sqrt{16}}=2$

Therefore,

$\Pr(Z>1.645-2)=\Pr(Z>-0.355)=64\%$

or…

1 - pnorm(qnorm(0.95) - 2/(4 / sqrt(16)))
## [1] 0.63876

#### STATISTICAL SAMPLE SIZE CALCULATION:

What $$n$$ sample size would be required to get a power of $$80\,\%$$ (a common benchmark in the sciences)?

For a one-sided test ($$H_a: \mu_a > \mu_0$$):

$0.8=\Pr\left(Z> \, z_{1-\alpha} -\frac{\mu_a -\mu_o}{s/\sqrt{n}} \mid \mu=\mu_a\right)$

which implies that

$z_{1-\alpha} - \frac{\mu_a -\mu_o}{s/\sqrt{n}} = z_{0.20}$

We set $$z_{1-\alpha} - \frac{\mu_a -\mu_o}{s/\sqrt{n}} = z_{0.2}$$ and solve for $$n$$ for any value of $$\mu_a$$:

$n=\left( \sigma \frac{z_{1-\alpha} - z_{0.20}}{\mu_a -\mu_0} \right)^2$

We pick $$\mu_a$$ as the smallest effect that we would reasonably like to detect.

In the cases of $$H_a:\mu_a \neq \mu_0$$ we can just take one of the sides but with $$\alpha/2$$.

For the example above:

(n <- (4*(qnorm(0.95)-qnorm(0.2))/2)^2)
## [1] 24.73023

which would indeed carry an $$80\%$$ power:

1 - pnorm(qnorm(0.95) - 2/(4 / sqrt(n)))
## [1] 0.8