The smallest effect a test can catch

Question

You have n = 100 observations, known \sigma, and will run a one-sided test at \alpha = 0.025. You want at least 80\% power. What is the smallest effect size \delta (the minimum detectable effect) this test can reliably catch?

Accepted Answer

The minimum detectable effect (MDE) is the smallest true shift \delta for which power reaches your target. Starting from the requirement that the standardized effect clears both the critical value and the power point, \frac{\delta\sqrt n}{\sigma} \ge z\alpha + z\beta, solve for \delta: \delta \ge \frac{(z\alpha + z\beta)\,\sigma}{\sqrt n}. With z\alpha = z{0.025} = 1.96 and z\beta = z{0.20} = 0.84 (the 80\% power point), \delta \ge \frac{(1.96 + 0.84)\,\sigma}{\sqrt{100}} = \frac{2.80\,\sigma}{10} = 0.28\,\sigma. So this test can reliably catch effects of about 0.28\sigma or larger. Anything …

The smallest effect a test can catch

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