What Power Depends On
A dependence test can be well calibrated and still have little power against a particular alternative. The reason is that alternatives differ in how they depart from independence. A test built around linear covariance may be insensitive to a nonlinear relationship; a tail-focused feature may be sensitive to joint extremes but not to dependence concentrated near the center. Power therefore depends not just on sample size, but also on the alternative’s direction in the features used by the statistic and on how the test weights those features.
A useful way to study this is to examine alternatives that approach the null as the sample size grows. These are called local alternatives. Their effects are small enough to compete with sampling noise, so their limiting rejection probabilities describe the test’s sensitivity near the null. The calculations below use a finite-dimensional feature vector. They capture the same shift principle that appears in many dependence statistics, without claiming that any one feature vector detects every possible form of dependence.
The feature condition says that the alternative’s mean signal is of order \(n^{-1/2}\). It does not say that the alternative is literally a location shift in the original variables: \(Z\) can encode nonlinear functions, threshold events, or other dependence summaries. The matrix \(W\) determines which directions of feature space contribute to the statistic and how strongly.
A Local-Power Result
Here is a precise version of the shift calculation. Assume observations are independent and identically distributed within each row \(n\); the distribution may change from row to row. Suppose \(\sqrt n E_{P_n}[Z]\to h\), and suppose the centered feature averages satisfy the triangular-array central limit theorem \(\sqrt n(\bar Z_n-E_{P_n}[Z])\xrightarrow{d}G\), where \(G\sim N_d(0,\Sigma)\). A sufficient condition for this central limit theorem is convergence of the row covariances to \(\Sigma\), together with the Lindeberg condition. Define \(T_n\) using a fixed \(W\) as above.
Proof. Decompose the scaled feature average into its centered fluctuation and its mean: \(\sqrt n\bar Z_n=\sqrt n(\bar Z_n-E_{P_n}[Z])+\sqrt nE_{P_n}[Z]\). The first term converges in distribution to \(G\), and the second converges to \(h\). Slutsky’s theorem gives \(\sqrt n\bar Z_n\xrightarrow{d}G+h\). The map \(x\mapsto x^\mathsf{T}Wx\) is continuous, so the continuous mapping theorem gives the asserted convergence of \(T_n\). Convergence in distribution implies convergence of probabilities of the event \((q,\infty)\) when the limiting distribution has no atom at \(q\). Applying that fact under the null and under the local alternatives gives the stated calibration and power conclusions. \(\square\)
The two continuity conditions have separate roles. Continuity at \(q\) under the null supports the usual limiting critical-value calibration. Continuity at \(q\) under the alternative is also needed to infer convergence of rejection probabilities from convergence in distribution. If the alternative limit has an atom at \(q\), that inference need not hold without additional information about how \(T_n\) approaches the boundary.
In a special case, the power formula has a familiar form. If \(\Sigma=I_d\), \(W=I_d\), and \(G\sim N_d(0,I_d)\), then the null limit is a central chi-square distribution with \(d\) degrees of freedom, while the alternative limit is noncentral chi-square with \(d\) degrees of freedom and noncentrality parameter \(h^\mathsf{T}h\). For general \(W\) and \(\Sigma\), the limit is a Gaussian quadratic form; it need not be an ordinary noncentral chi-square variable.
Worked Examples: Three Different Dependence Signals
Worked Example: Nonlinear Dependence With Zero Linear Covariance
Let \(X\sim N(0,1)\). Under a dependent distribution \(P_1\), let \(Y=X^2+\varepsilon\), where \(\varepsilon\sim N(0,1)\) is independent of \(X\). Under a null distribution \(P_0\), let \(X\) and \(Y\) be independent, with the same respective marginal distributions as under \(P_1\). Consider the mixture \(P_\theta=(1-\theta)P_0+\theta P_1\). The marginals remain fixed throughout this mixture.
Under \(P_1\), \(E[X]=0\), \(E[Y]=2\), and \(E[XY]=E[X^3]+E[X\varepsilon]=0\). Thus the linear covariance is zero, even though \(Y\) depends on \(X\). Use the scalar feature \(Z=(X^2-1)(Y-2)\). Under \(P_0\), independence and the centered factors give \(E_{P_0}[Z]=0\). Under \(P_1\), \(E_{P_1}[Z]=E[(X^2-1)(X^2+\varepsilon-2)]=2\); this is \(\operatorname{Var}(X^2)=3-1=2\), since the independent noise term has mean zero.
Under \(P_0\), \(E[(X^2-1)^2]=2\) and \(\operatorname{Var}(Y)=\operatorname{Var}(X^2)+\operatorname{Var}(\varepsilon)=2+1=3\). Therefore \(\operatorname{Var}_{P_0}(Z)=2\cdot3=6\). Set \(\theta_n=h/\sqrt n\), with \(h\geq0\). The feature mean is \(2\theta_n\), so its standardized local shift is \(\sqrt n E_{P_{\theta_n}}[Z]/\sqrt6=2h/\sqrt6\). The statistic \(T_n=n\bar Z_n^2/6\) consequently has null limit \(G^2\), and local alternative limit \((G+2h/\sqrt6)^2\), where \(G\sim N(0,1)\). This feature detects a nonlinear association that ordinary covariance misses.
Worked Example: Dependence in Conditional Variance
Let \(P_0\) make \(X\) and \(Y\) independent standard normal variables. Under \(P_1\), let \(X\sim N(0,1)\) and set \(Y=S X\), where \(S\) is an independent random sign taking values \(1\) and \(-1\) with equal probabilities. Both marginals are standard normal under both distributions. Under \(P_1\), \(E[Y\mid X]=0\) and \(\operatorname{Var}(Y\mid X)=X^2\), so the conditional variance depends on \(X\), while \(\operatorname{Cov}(X,Y)=E[SX^2]=0\).
Take \(Z=(X^2-1)(Y^2-1)\). Under \(P_0\), its mean is zero and its variance is \(2\cdot2=4\), because \(X^2-1\) and \(Y^2-1\) are independent and each has variance \(2\). Under \(P_1\), \(Y^2=X^2\), so \(E_{P_1}[Z]=E[(X^2-1)^2]=2\).
For \(P_{\theta_n}=(1-\theta_n)P_0+\theta_nP_1\), with \(\theta_n=h/\sqrt n\), the feature mean is \(2h/\sqrt n\). Dividing by its null standard deviation \(2\) gives local shift \(h\). Thus \(T_n=n\bar Z_n^2/4\) has null limit \(G^2\) and local alternative limit \((G+h)^2\). The signal is in squared magnitudes, not in the signed linear covariance.
Worked Example: A Threshold Feature for Upper-Tail Dependence
Let \(U,V\) each have a uniform distribution on \((0,1)\). Under \(P_0\), they are independent; under \(P_1\), set \(V=U\). Again, the mixture \(P_\theta=(1-\theta)P_0+\theta P_1\) preserves both uniform marginals. Fix \(c\in(0,1)\), let \(A=\{U>c,V>c\}\), and write \(Z=\mathbf{1}_A-p_0\), where \(p_0=(1-c)^2=P_0(A)\).
Under \(P_0\), \(E[Z]=0\) and \(\operatorname{Var}(Z)=p_0(1-p_0)=(1-c)^2c(2-c)\). Under \(P_1\), \(P_1(A)=1-c\), so the increase over the null event probability is \((1-c)-(1-c)^2=c(1-c)\).
For \(\theta_n=h/\sqrt n\), the standardized local shift is \[ \frac{h\,c(1-c)}{\sqrt{(1-c)^2c(2-c)}}=h\sqrt{\frac{c}{2-c}}. \] The statistic \(T_n=n\bar Z_n^2/[p_0(1-p_0)]\) therefore has limit \(G^2\) under the null and \((G+h\sqrt{c/(2-c)})^2\) under the local alternative. This calculation shows how a threshold feature can target joint upper-tail behavior. It does not establish that this threshold is best for every tail alternative; the result depends on the chosen threshold and alternative.
Fixed Effects, Local Effects, and Weighting
A local alternative and a fixed alternative behave differently as \(n\) grows. For a local alternative, the limiting shift \(h\) is fixed, so the limiting power is fixed as well. Increasing \(n\) does not make the effect \(h/\sqrt n\) stop shrinking. By contrast, suppose a fixed distribution has feature mean \(\mu\), with \(\mu^\mathsf{T}W\mu>0\), and the sample feature average converges in probability to \(\mu\). Then \[ \frac{T_n}{n}=\bar Z_n^\mathsf{T}W\bar Z_n\xrightarrow{p}\mu^\mathsf{T}W\mu>0, \] so \(T_n\xrightarrow{p}\infty\). For any fixed finite critical value, the rejection probability tends to one. This argument applies to the feature statistic under the stated convergence and nonzero-signal conditions; it is not a guarantee for a feature whose mean remains zero under the alternative.
Weighting determines what “nonzero signal” means. For the local limit, the deterministic displacement contributes \(h^\mathsf{T}Wh\), while the Gaussian fluctuation and its interaction with \(h\) also affect the full rejection probability. If \(W\) assigns zero weight to a direction, a shift confined to that direction cannot be detected by this statistic. For example, with \(h=(0,a)^\mathsf{T}\) and \(W=\operatorname{diag}(1,0)\), the quadratic statistic ignores the shift entirely. With \(W=I_2\), it includes both coordinates. More generally, weights may emphasize a targeted feature or downweight one, but changing weights also changes the null distribution and hence the critical value.
Dimension brings a related trade-off. Adding features can represent more kinds of dependence, but it can also add noise to a quadratic statistic. Even when the coordinates have no local shift, their sampling variation contributes to the null distribution. For instance, with \(W=I_d\) and \(\Sigma=I_d\), \(E[G^\mathsf{T}G]=d\), whereas the local shift adds \(h^\mathsf{T}h\) to the expectation of the alternative limit. This expectation comparison is informative, but it is not itself a power calculation: power is the probability of exceeding a critical value, and depends on the entire limit distribution.
Common Pitfalls and Why the Distinction Matters
- Confusing fixed and local effects. A fixed nonzero feature mean can become easier to detect as \(n\) grows. An effect exactly \(h/\sqrt n\) has a fixed limiting shift and fixed limiting local power; it does not keep gaining power merely because the sample size increases.
- Assuming a dependence test detects every alternative. A feature can have zero mean under a genuine dependent distribution. The examples show why signed covariance alone can miss nonlinear or variance-based dependence, and why targeted features are not universal.
- Using only null continuity to claim limiting power. The rejection-probability conclusion requires that the critical value is a continuity point of the alternative limit too. If the alternative limit has an atom at the threshold, convergence in distribution alone does not settle the limiting rejection probability.
- Interpreting a larger signal contribution as automatically larger power. The term \(h^\mathsf{T}Wh\) does not by itself determine rejection probability. The noise distribution, weighting, and critical value matter as well.
These distinctions are useful when interpreting a dependence test’s sensitivity. A calibrated test answers whether its rejection rule controls false positives under the null; power analysis asks which departures from that null are visible to its features and weights. In practice, an alternative should be described structurally—such as nonlinear location, changing conditional variance, or joint tail events—before claims about a test’s sensitivity are made.
Check Your Understanding
Use the local-shift framework to answer the following questions.
- For a statistic with \(\Sigma=I_d\) and \(W=I_d\), what is the null limit, and what parameter describes the noncentral chi-square alternative limit?
- Why can the nonlinear-dependence feature in the first worked example have a nonzero mean when \(\operatorname{Cov}(X,Y)=0\)?
- What happens to limiting local power as \(n\) grows when the feature mean is exactly \(h/\sqrt n\), with fixed \(h\)? Contrast this with a fixed nonzero feature mean.
- Why does a continuity condition under the alternative matter when deriving limiting rejection probabilities?
- If \(W=\operatorname{diag}(1,0)\), which two-dimensional local shifts can the quadratic statistic fail to detect?