Study Design & Sample Planning
What Is Statistical Power?
Statistical Power ($1 - \beta$) is the probability that a statistical test will correctly reject a false null hypothesis—that is, the probability of detecting a genuine true effect when one actually exists.
A study with low statistical power risks committing a Type II error ($\beta$) (a "false negative"), potentially discarding a truly effective clinical treatment or meaningful empirical finding. In scientific research, power of **80% ($\ge 0.80$)** or **90%** is the universally recognized benchmark.
The Mathematical Formulas
$$\text{Two-Sample t-Test: } \delta = d \sqrt{\frac{n_1 n_2}{n_1 + n_2}}, \quad \text{Power} = 1 - \Phi(z_{\text{crit}} - \delta)$$
$$\text{ANOVA: } \lambda = N f^2, \quad \text{Power} = 1 - F_{\text{central}}\left(\frac{F_{\text{crit}}}{1 + \lambda/df_1}; df_1, df_2\right)$$
$$\text{Correlation: } z_\beta = \text{arctanh}(|r|)\sqrt{n - 3} - z_{\text{crit}}, \quad \text{Power} = \Phi(z_\beta)$$
Where $\delta$ and $\lambda$ represent non-centrality parameters, $d$ is Cohen's $d$, $f$ is Cohen's $f$, and $z_{\text{crit}}$ is the critical standard normal quantile corresponding to chosen significance level $\alpha$.
Worked Example
Scenario: Independent two-sample t-test with medium effect size $d = 0.50$, $n_1 = 64, n_2 = 64$ ($N = 128$), and $\alpha = 0.05$ (two-tailed).
Step 1: Compute Non-Centrality Parameter ($\delta$):
$\delta = 0.50 \times \sqrt{\frac{64 \times 64}{64 + 64}} = 0.50 \times \sqrt{32} = 0.50 \times 5.6569 = 2.8284$.
Step 2: Determine Critical Value ($df = 126$):
$t_{\text{crit}} = t_{0.975, 126} \approx 1.9790 \quad (z_{\text{crit}} \approx 1.9600)$.
Step 3: Calculate Power ($1 - \beta$):
$\text{Power} \approx 1 - \Phi(1.9790 - 2.8284) = 1 - \Phi(-0.8494) = \Phi(0.8494) \approx 0.8015 \text{ (80.2\%)}$.
Conclusion: The design achieves **80.2% statistical power**, successfully meeting the standard $\ge 80\%$ benchmark with a 19.8% Type II error risk.
