Power Calculations in the Single-Root Case

From an Alternative to Power

A MANOVA test’s significance level describes its rejection probability under the null hypothesis. Power asks a different question: if a specified alternative is true, how likely is the test to reject? In a single-root setting, the signal can be summarized by one noncentrality parameter, but the route from that signal to power depends on whether the test has an exact noncentral-F distribution or uses an F approximation.

The distinction matters in practice. A sample generalized root is a random statistic, while the noncentrality parameter describes the alternative that generates its distribution. Neither is itself a power. Power is the probability that the statistic crosses a rejection threshold, calculated under that alternative.

Definition: For a test with rejection region \(R\), its power at an alternative parameter value \(\theta\) is \(P_\theta(\text{test statistic}\in R)\). If the test rejects for \(F>c_\alpha\), where \(c_\alpha\) is a level-\(\alpha\) upper-tail critical value, its power is \(P_\theta(F>c_\alpha)\). The probability of not rejecting under that alternative is the type II error probability, \(\beta=1-\text{power}\).

For a single tested multivariate contrast, let \(p\) be the number of responses and \(\nu\) the error degrees of freedom. Under the multivariate normal linear model, the contrast’s error-standardized estimate can be represented as a vector \(h\sim N_p(\eta,I_p)\), independent of an error sum-of-squares-and-cross-products matrix \(E\sim W_p(I_p,\nu)\). Here \(I_p\) is the \(p\)-dimensional identity matrix, and the alternative’s noncentrality parameter is \(\delta=\eta'\eta\). In an unstandardized model, the same parameter is the squared length of the contrast mean after standardization by its covariance.

Definition: In this rank-one model, the hypothesis matrix is \(H=hh'\). It has at most one nonzero sample generalized root, denoted \(r\), obtained from the matrix pair \((H,E)\). Under the stated standardization, \(r=h'E^{-1}h\). The parameter \(\delta=\eta'\eta\) is the noncentrality of the alternative; it is not the observed root \(r\).

The rank-one structure makes the exact calculation possible. Wilks’ Lambda, Pillai’s trace, the Hotelling–Lawley trace, and Roy’s largest root all reduce to functions of the same root: respectively \(1/(1+r)\), \(r/(1+r)\), \(r\), and \(r\). Their rejection regions can therefore be expressed as a threshold on \(r\). For the exact rank-one test, the usual F transformation of that root gives a central F distribution under the null and a noncentral F distribution under the alternative.

Exact Noncentral-F Result for a Rank-One Hypothesis

Theorem: Suppose \(h\sim N_p(\eta,I_p)\) and \(E\sim W_p(I_p,\nu)\) are independent, with \(\nu\ge p\), and let \(r=h'E^{-1}h\). Define \[ F=\frac{\nu-p+1}{p}\,r. \] Then \(F\) has a noncentral F distribution with numerator degrees of freedom \(p\), denominator degrees of freedom \(\nu-p+1\), and noncentrality parameter \(\delta=\eta'\eta\). When \(\eta=0\), this is the central \(F_{p,\nu-p+1}\) distribution.

Proof. The squared length \(X=h'h\) has a noncentral chi-square distribution with \(p\) degrees of freedom and noncentrality \(\delta\). Write \(Y=(e_1'E^{-1}e_1)^{-1}\), where \(e_1\) is any fixed unit vector. For a Wishart matrix with identity scale, \(Y\sim\chi^2_{\nu-p+1}\). The distribution of \(E\) is invariant under orthogonal transformations. Consequently, conditional on any nonzero value of \(h\), rotating \(h/\|h\|\) to \(e_1\) shows that \(\|h\|^2 h_0'E^{-1}h_0\), with \(h_0=h/\|h\|\), has the same distribution as \(X/Y\). The conditional distribution of \(Y\) does not depend on \(h\), so \(X\) and \(Y\) are independent. Thus \(r=X/Y\), and

\frac{\nu-p+1}{p}r = \frac{X/p}{Y/(\nu-p+1)} \sim F_{p,\nu-p+1}(\delta).

This is the definition of the noncentral F distribution as a ratio of independent scaled noncentral and central chi-square variables. Under the null, \(\eta=0\), so \(X\sim\chi^2_p\) and \(\delta=0\). \(\square\)

The exact power calculation follows directly. Let \(c_\alpha\) be the upper-tail \(\alpha\) quantile of the central \(F_{p,\nu-p+1}\) distribution. The test rejects when \(F>c_\alpha\). Under the alternative with noncentrality \(\delta\), its power is the noncentral-F upper tail at that same critical value.

Exact power formula: For the rank-one test under the theorem’s assumptions, \[ \operatorname{Power}(\delta) = P\{F_{p,\nu-p+1}(\delta)>c_\alpha\} = 1-G_{p,\nu-p+1,\delta}(c_\alpha), \] where \(G_{p,\nu-p+1,\delta}\) is the noncentral-F cumulative distribution function. The critical value is central because the test is calibrated under the null; the power calculation uses the alternative distribution.

Worked Example: Calculating the Alternative’s Noncentrality

Suppose a single multivariate contrast has two responses. After standardization by the contrast’s covariance, its estimated vector has distribution \(h\sim N_2(\eta,I_2)\), with \(\eta=(1.2,0.5)'\). The error degrees of freedom are \(\nu=24\). Find the noncentrality parameter and write the exact power at level \(0.05\).

The alternative noncentrality is the squared standardized length of the mean vector:

\delta=\eta'\eta=(1.2)^2+(0.5)^2=1.44+0.25=1.69.

Here \(p=2\), so the exact F degrees of freedom are \(2\) and \(24-2+1=23\). Let \(c_{0.05}\) be the upper \(0.05\) quantile of the central \(F_{2,23}\) distribution. Then the exact power is

P\{F_{2,23}(1.69)>c_{0.05}\} = 1-G_{2,23,1.69}(c_{0.05}).

This expression specifies the full calculation: obtain the central critical value, then evaluate the alternative noncentral-F tail there. The noncentrality is \(1.69\), not the realized root from a data set. A numerical value for power requires evaluating the stated distribution function.

Evaluating the Noncentral-F Tail

The noncentral-F distribution can be evaluated using a Poisson mixture of central F distributions. Let \(J\) have a Poisson distribution with mean \(\delta/2\). A noncentral chi-square variable with \(d_1\) degrees of freedom and noncentrality \(\delta\) has the same distribution as a central chi-square variable with \(d_1+2J\) degrees of freedom, conditional on \(J\). Since the denominator chi-square in the F ratio is independent of the numerator, the same mixture gives the noncentral-F tail:

P\{F_{d_1,d_2}(\delta)>c\} = \sum_{j=0}^{\infty} e^{-\delta/2}\frac{(\delta/2)^j}{j!} P\{F_{d_1+2j,d_2}>c\}.

To see the mixture weights, the moment-generating function of a noncentral chi-square variable is

(1-2t)^{-d_1/2} \exp\left(\frac{\delta t}{1-2t}\right) = e^{-\delta/2} \sum_{j=0}^{\infty} \frac{(\delta/2)^j}{j!}(1-2t)^{-(d_1+2j)/2}.

The right side is the moment-generating function of a Poisson mixture of central chi-square distributions, which gives the stated tail formula after division by the independent denominator. The weights sum to one. In a numerical calculation, terms can be added until the remaining Poisson probability is negligible; this also provides a check on truncation error.

Worked Example: Expressing Exact Power as a Mixture

Consider the rank-one model with \(p=1\), \(\nu=20\), level \(\alpha=0.05\), and alternative noncentrality \(\delta=4\). The exact F degrees of freedom are \(1\) and \(20\). The central upper-tail critical value is approximately \(c_{0.05}=4.351\). The Poisson mixture has mean \(\delta/2=2\), so the power is

\sum_{j=0}^{\infty} e^{-2}\frac{2^j}{j!} P\{F_{1+2j,20}>4.351\}.

The first three weights are \(e^{-2}\approx0.1353\), \(2e^{-2}\approx0.2707\), and \(2e^{-2}\approx0.2707\), for \(j=0,1,2\), respectively. Thus the first three contributions are those weights multiplied by the central F upper tails with degrees of freedom \((1,20)\), \((3,20)\), and \((5,20)\), each evaluated at \(4.351\). The remaining terms use the same rule. In particular, the weights are not themselves powers; each must be multiplied by its corresponding central-F rejection probability.

At \(\delta=0\), only the \(j=0\) weight remains, and the mixture reduces to \(P\{F_{1,20}>4.351\}=0.05\), as required by the level of the test. This is a useful check that the mixture and the critical-value convention are being applied consistently.

When the F Calculation Is an Approximation

The exact theorem requires a rank-one hypothesis of the stated multivariate normal form. A single nonzero root describing an alternative does not, by itself, guarantee that the observed statistic has an exact noncentral-F distribution. For example, when the hypothesis has more than one degree of freedom, its sample hypothesis matrix need not be \(hh'\). F calibrations for other MANOVA criteria can also be approximations, even when their null calibration is useful.

In those settings, an approximate power calculation uses the same basic steps but must label the distributional approximation. First use the chosen criterion’s F calibration to specify numerator and denominator degrees of freedom and its rejection threshold. Then use the noncentrality parameter supplied by the alternative and the applicable approximation to evaluate a noncentral-F upper tail. The resulting value is an approximate power, not an exact probability unless the sampling result establishes that exact law.

Worked Example: Using an F Calibration for Approximate Power

Suppose a calibrated MANOVA criterion is compared with an \(F_{4,60}\) reference distribution at level \(0.05\), and the alternative is represented by one nonzero standardized root with noncentrality \(\delta=3\). Denote the central upper-tail critical value by \(c\), where \(P\{F_{4,60}>c\}=0.05\). Under the noncentral-F approximation, the estimated power is

P\{F_{4,60}(3)>c\} = 1-G_{4,60,3}(c).

The calculation uses the central \(F_{4,60}\) distribution to set the rejection threshold and the noncentral \(F_{4,60}(3)\) distribution to estimate rejection under the alternative. It is approximate because the criterion’s F calibration is approximate in this setting. The single nonzero alternative root supplies the signal summary here, but it does not turn the statistic’s full finite-sample distribution into an exact noncentral F law.

Interpreting Power and Avoiding Common Errors

Power is specific to a design and an alternative. It changes with the noncentrality, the response dimension, the error degrees of freedom, the significance level, and—when using an approximation—the selected calibration. A larger \(\delta\) shifts the alternative distribution toward larger F values, increasing the probability of exceeding a fixed upper-tail threshold. But the effect of changing sample size is not captured by merely changing the critical value: the error degrees of freedom and, usually, the standardized signal also change.

A common mistake is to treat the noncentrality parameter as though it were a realized sample root, or to insert it directly as the F statistic. The correction is to keep the roles separate: \(\delta\) parameterizes the distribution under the alternative; \(c_\alpha\) defines the rejection region; and power is the probability mass of that alternative distribution above the threshold. Another mistake is to call every noncentral-F power calculation exact. Check the sampling model and rank condition first. If the F law comes from a criterion calibration rather than an exact rank-one reduction, report the result as approximate.

Key takeaway: For a rank-one hypothesis under the stated normal-Wishart model, transform the sole generalized root to \(F\), find the central-F critical value, and evaluate its noncentral-F upper tail using \(\delta\). Outside that exact setting, the same tail calculation is useful only when its F approximation and noncentrality parameter are justified and clearly identified.

Check Your Understanding

Answer these questions using the definitions and formulas above.

For a rank-one hypothesis with \(p=3\) responses and \(\nu=18\) error degrees of freedom, what are the exact F numerator and denominator degrees of freedom?
If the standardized alternative mean is \(\eta=(0.6,-0.8)'\), calculate its noncentrality parameter.
Why is the critical value central F even though the power is computed under a noncentral F distribution?
In the Poisson-mixture formula, what distribution determines the mixture weights, and what do the weights multiply?
What condition distinguishes an exact noncentral-F power calculation from an F-calibrated approximation?