Noncentrality Parameters and Power Curves

A t-statistic under the null hypothesis follows the ordinary, symmetric t-distribution centered on zero. But if the true effect isn't zero, the same statistic still follows a t-shaped curve — just shifted away from zero and slightly reshaped. That shifted curve is a noncentral distribution, and the amount of shift is the noncentrality parameter, usually written $\delta$ or $\lambda$.

For a one-sample t-test, the noncentrality parameter is $\delta = \frac{\mu - \mu_0}{\sigma / \sqrt{n}}$ — the true gap from the null, measured in standard errors. Notice this is exactly the t-statistic you'd expect to see on average if the alternative is true. When $\delta = 0$, the noncentral t-distribution collapses back to the ordinary central one, which is why the null case is really just the special case $\delta = 0$.

Worked example. A strategy's true daily edge is 0.05% with a daily return standard deviation of 1%, and a backtest covers 400 days. The standard error is $1\%/\sqrt{400} = 0.05\%$, so $\delta = 0.05\%/0.05\% = 1.0$. A noncentrality of 1.0 is modest — plugging it into the noncentral t-distribution with the test's critical value shows power around 30–40% at the usual 5% significance level, meaning a real, tradeable edge this small would still be missed by the test more often than not.

Power curves plot exactly this: power on the y-axis against noncentrality (or equivalently, sample size, since $n$ sits inside $\delta$) on the x-axis. They flatten out near 100% once the noncentrality parameter is large enough that overlap between the null and alternative distributions is negligible, which is the visual explanation for why power analysis asks "how big a sample before this curve gets steep enough."