16.3 Sample size and design for continuous outcomes

• For 80% probability of 95% confidence, same principles apply: 2.8 standard errors

• For very small samples, use t-distribution.

• The standard error for comparison of means is $\sqrt{\sigma_1^2/n_1 + \sigma_2^2/n_2}$.

If the groups are equal and the standard errors are expected to be equal, then $\text{s.e.} = 2\sigma/\sqrt{n}$. If the goal is 80% power, then the effect must be 2.8 times this standard error. For example, if the effect is expected to 0.5 standard deviations, we require $0.5\sigma/\text{se} = 2.8$ :

$$0.5 \sqrt{n}/2 = 2.8\\ n=(2.8*4)^2 = 125.4$$