Approaches for sample size determination

1. n=3 for each combination of treatments
2. as large as budget allows
3. SE/CI width for quantity of interest
4. 80% power (for 5% Type I error) for quantity of interest

CI width depends primarily on the SE while power also depends on a hypothesized value (or pilot study) for the quantity of interest.

Sample size calculations based on power approach

We need the client to tell us

• Type I error rate (a)
• Power (b)
• Within treatment variability (s)
• Hypothesized difference for quantity of interest (d)

Shifted t approximation

d = (t_{1-a/2,df} + t_{b,df})*se

where df and se are functions of n. For example, if n is the number of observations per group, then

• one sample (paired) t-test: se = s/sqrt(n), df=n-1
• two sample t-test: se = s*sqrt(2/n), df=2n-2
• interaction for 2x2 factorial: se = s*sqrt(4/n), df=4n-4
• main effect in a 2x3 factorial: se = s*sqrt(2/3n), df = 6n-6
• iteraction in a 2x3 factorial: se = s*sqrt(3/4n), df = 6n-6

With multiple comparisons can adjust Type I error using Bonferroni. For multiple outcomes, use most important or outcome with the smallest d/s.