Statistical Power

Statistical power is the probability that a hypothesis test correctly rejects the null hypothesis when a specific alternative is true, $\text{Power}=1-\beta =P(\text{reject }{H}_0\mid H_1\text{ true})$, evaluated at a particular effect size, since power varies across the range of possible true values covered by a composite alternative. Power depends jointly on the sample size, the significance level α, the true effect size, and the variability of the underlying measurements, a priori power analysis, computing the sample size needed to achieve a target power (conventionally 80%) for a minimum meaningful effect size, is a standard step in planning a study before data collection begins.

• Power is always evaluated at a specific assumed effect size, not as a single property of a test in the abstract, this is why power calculations require the researcher to specify, in advance, the minimum effect size considered scientifically or practically meaningful, an underspecified or overly optimistic assumed effect size is a common source of underpowered studies.
• Post-hoc (observed) power, calculated after a study using the effect size actually observed in the data, is widely considered statistically unsound and is discouraged by most methodologists, it is mathematically just a monotonic transformation of the p-value and doesn't provide any independent diagnostic information beyond it.
• The 80% power convention, like the 5% significance-level convention, traces to influential mid-20th-century methodological writing (notably Jacob Cohen's) rather than any universal statistical law, fields with especially costly errors (e.g. clinical trials with serious safety implications) sometimes target higher power, such as 90%.