Local Alternatives and Asymptotic Power in Generalized Models

Why Local Alternatives Matter

A test’s significance level describes its behavior when the null hypothesis is true; power describes its ability to detect departures from the null. For generalized models, a useful way to compare tests is to ask how they behave when the parameter moves away from the null by an amount proportional to $1/\sqrt{n}$. Such departures shrink with sample size, but remain statistically detectable. They reveal the tests’ local asymptotic power and provide a common scale for comparing Wald, score, and likelihood-ratio procedures.

This tutorial studies a regular likelihood model with a finite-dimensional parameter. The information matrix determines which local departures are detectable and how strongly: departures in poorly informed directions have less power than equally sized departures in well-informed directions.

Setup and the Information-Based Distance

Let $\theta\in\mathbb{R}^d$ be the model parameter, and consider testing the $q$ linear restrictions $H_0:A\theta=a$, where $A$ is a $q\times d$ matrix of rank $q$. Let $\theta_0$ be an interior point satisfying $A\theta_0=a$. The model may include nuisance parameters; they are components of $\theta$ not fixed by these restrictions.

Definition: A sequence of local alternatives, also called contiguous alternatives in this setting, has the form $\theta_n=\theta_0+h/\sqrt{n}$, for a fixed vector $h\in\mathbb{R}^d$. It departs from the null when $Ah\ne 0$. If $Ah=0$, the local shift remains within the null restrictions.

Assume independent observations and a regular likelihood: the true parameter is interior, the model is identifiable and differentiable in quadratic mean, the score satisfies a central limit theorem, and the usual likelihood expansion and information identity hold locally. Assume also that the normalized Fisher information converges to a positive-definite matrix $I$, and that the unrestricted and restricted maximum-likelihood estimators have the usual local asymptotic normality properties. For generalized linear models, these conditions require, among other things, a nonsingular limiting design information matrix; any estimated nuisance or dispersion parameters must be included in the regular parameterization when relevant. The tests below use the likelihood and information for the full parameter vector.

The matrix $I$ is information per observation in the limit: the total information is asymptotically $nI$. The matrix $M=AI^{-1}A'$ measures uncertainty in the tested contrasts after accounting for the other parameters. It is positive definite because $I$ is positive definite and $A$ has full row rank.

Local Likelihood and the Common Limit

Write $\ell_n(\theta)$ for the log-likelihood and let $\Delta_n=n^{-1/2}\ell_n'(\theta_0)$ be the normalized score at the null point. Local asymptotic normality gives, for bounded $t$,

\ell_n(\theta_0+t/\sqrt{n})-\ell_n(\theta_0) =t'\Delta_n-\frac{1}{2}t'It+o_p(1).

Under the local alternative $\theta_n=\theta_0+h/\sqrt{n}$, the normalized score satisfies $\Delta_n\xrightarrow{d}N(Ih,I)$. This follows from the central limit behavior under the null together with the likelihood-ratio change of measure for the local shift (Le Cam’s third lemma). Thus the local likelihood is asymptotically a quadratic function with random linear term having mean $Ih$.

Theorem: Under the stated regularity conditions, if $Ah\ne 0$, each of the Wald, score, and likelihood-ratio statistics for $H_0:A\theta=a$ converges in distribution under $\theta_n=\theta_0+h/\sqrt{n}$ to a noncentral chi-square distribution with $q$ degrees of freedom and noncentrality parameter $$ \lambda=(Ah)'(AI^{-1}A')^{-1}(Ah). $$

Here a noncentral chi-square variable with $q$ degrees of freedom and parameter $\lambda$ can be defined as $\sum_{j=1}^q(Z_j+\mu_j)^2$, where the $Z_j$ are independent standard normal variables and $\sum_j\mu_j^2=\lambda$. Under the null, $h=0$, so $\lambda=0$ and the limit is the usual central chi-square distribution.

Deriving the Three Test Limits

The local log-likelihood, omitting its constant term, is the concave quadratic $Q(t)=t'\Delta_n-\tfrac12t'It$. Its unrestricted maximizer is $\hat t=I^{-1}\Delta_n$. The restricted maximizer under the null satisfies $At=0$. The following calculation derives the common quadratic statistic and shows why all three tests have the same local limit.

Find the constrained maximizer.
Using a Lagrange multiplier for $At=0$, the maximizer is $\bar t=I^{-1}\Delta_n-I^{-1}A'(AI^{-1}A')^{-1}AI^{-1}\Delta_n$. Direct substitution gives $A\bar t=0$.

Form the likelihood-ratio statistic.
The statistic is twice the difference between the unrestricted and restricted maxima of $Q$. Completing the square yields $2\{Q(\hat t)-Q(\bar t)\}=(AI^{-1}\Delta_n)'(AI^{-1}A')^{-1}(AI^{-1}\Delta_n)$.

Identify the score statistic.
The constrained score residual is $\Delta_n-I\bar t=A'(AI^{-1}A')^{-1}AI^{-1}\Delta_n$. Its quadratic form using $I^{-1}$ is exactly the expression in Step 2, the efficient score statistic in local coordinates.

Identify the Wald statistic.
The unrestricted estimate has local coordinate $\hat t=I^{-1}\Delta_n$. Its tested contrast is $A\hat t$, with covariance $AI^{-1}A'$; the Wald quadratic is therefore the same expression.

To obtain the distribution, set $V=AI^{-1}A'$. Under the local alternative, $AI^{-1}\Delta_n$ converges to a normal vector with mean $Ah$ and covariance $V$. Therefore its quadratic form using $V^{-1}$ is noncentral chi-square with $q$ degrees of freedom and noncentrality $(Ah)'V^{-1}(Ah)$. The likelihood, score, and Wald calculations above have this same limiting quadratic form, proving the theorem. The finite-sample statistics need not be equal; their equivalence here is asymptotic. $\square$

Worked Examples

Worked Example: A Local Shift in a Logistic Intercept

Suppose $Y_1,\ldots,Y_n$ are independent Bernoulli observations with $\operatorname{logit}\{P(Y_i=1)\}=\theta$. Test $H_0:\theta=0$ against local alternatives $\theta_n=4/\sqrt{n}$. At the null, $p=1/2$, and the per-observation Fisher information for the logit is $p(1-p)=1/4$.

Here $d=q=1$, $A=1$, $I=1/4$, and $h=4$. Thus $\lambda=h^2/I=16/(1/4)=64$ would be incorrect: the general formula is $h^2/(I^{-1})$, or equivalently $h^2 I$, when there is one tested parameter. Directly, $AI^{-1}A'=4$, so $\lambda=4^2/4=4$.

At a two-sided 5% level, the limiting critical value is the 95th percentile of $\chi^2_1$, approximately $3.841$. Since a $\chi^2_1(4)$ variable has the distribution of $(Z+2)^2$, its limiting rejection probability is $P(|Z+2|>1.96)=\Phi(-3.96)+1-\Phi(-0.04)\approx 0.516$. Thus all three tests have approximately 51.6% local asymptotic power for this shift.

Worked Example: A Treatment Effect With an Unknown Intercept

Consider logistic regression with an intercept $\alpha$ and treatment coefficient $\beta$, using treatment coding $x=0,1$ and equal numbers in the two groups. Test $H_0:\beta=0$. At $(\alpha,\beta)=(0,0)$, each response probability is $1/2$. The limiting information per observation, averaging over the two treatment groups, is

I=\frac14 \begin{pmatrix} 1&1/2\\ 1/2&1/2 \end{pmatrix}, \qquad I^{-1}= \begin{pmatrix} 8&-8\\ -8&16 \end{pmatrix}.

Take $A=(0,1)$ and local shift $h=(0,8)'$, so $\beta_n=8/\sqrt{n}$. The intercept is nuisance, and its uncertainty is reflected in $AI^{-1}A'=16$. The noncentrality parameter is $\lambda=(Ah)^2/16=64/16=4$, giving the same limiting two-sided 5% power, approximately 0.516, as in the previous example.

For comparison, if the intercept were known and fixed, the information for $\beta$ would be the lower-right entry of $I$, $1/8$, giving $\lambda=8^2(1/8)=8$. Estimating the intercept reduces information about the treatment effect and therefore reduces local power.

Worked Example: Joint Test of Two Poisson Log-Rates

Let independent counts satisfy $Y_j\sim\operatorname{Pois}((n/2)e^{\theta_j})$ for $j=1,2$. Test the joint null $H_0:\theta_1=\theta_2=0$. At the null, the information per $n$ is $I=\operatorname{diag}(1/2,1/2)$. Take local shift $h=(1,1)'$. Since $A$ is the $2\times2$ identity, $\lambda=h'Ih=1/2+1/2=1$, and the limiting statistic is $\chi^2_2(1)$.

At level 0.05, the central $\chi^2_2$ critical value is $5.991$. The noncentral chi-square mixture representation gives the power as $\sum_{j=0}^{\infty} e^{-1/2}(1/2)^j/j!\;P(\chi^2_{2+2j}>5.991)$. The first terms are $0.6065(0.05)+0.3033(0.1998)+0.0758(0.4241)+0.0126(0.6482)+\cdots$, which sum to approximately $0.133$. Thus the joint test has about 13.3% local asymptotic power for this shift.

Calculating Power and Avoiding Common Errors

For a level-$\alpha$ test, let $c$ be the $1-\alpha$ quantile of the central $\chi^2_q$ distribution. The local asymptotic power is

P\{\chi^2_q(\lambda)>c\}.

The critical value is central because the test is calibrated under the null. The power uses the noncentral distribution because the alternative shifts the limiting contrast away from zero. For one restriction, this tail probability can be written in terms of a normal tail; for multiple restrictions, the noncentral chi-square distribution or its Poisson mixture provides the calculation.

Using the wrong information distance. The expression $h'Ih$ applies directly when the whole parameter vector is tested. With nuisance parameters, use $(Ah)'(AI^{-1}A')^{-1}(Ah)$; the restriction and the nuisance uncertainty determine the relevant distance.
Using a central chi-square distribution for power. A central chi-square tail gives the null rejection probability, not power under a nonzero local shift.
Treating local power as fixed-alternative power. A shift $h/\sqrt{n}$ shrinks as $n$ grows. A fixed nonzero departure is a different asymptotic regime; the noncentral chi-square limit derived here is for contiguous shifts.
Assuming exact agreement of the tests. Wald, score, and likelihood-ratio statistics can differ in finite samples. Their shared noncentral limit is a regular large-sample result.

Key takeaway: Under regularity, local power is governed by the tested shift $Ah$ measured in the covariance metric $AI^{-1}A'$. This produces a noncentral chi-square limit shared by Wald, score, and likelihood-ratio tests.

Check Your Understanding

Use the restriction matrix and limiting information to answer each question.

For a single tested parameter with information per observation $I=2$ and local shift $h=1$, what is the noncentrality parameter?
Why does the noncentrality parameter for a test with nuisance parameters generally differ from $h'Ih$?
For a level-$\alpha$ test, which distribution supplies the critical value, and which supplies local asymptotic power?
In the two-group logistic example, how did estimating the intercept affect information about the treatment coefficient?
What happens to the noncentrality parameter if a local shift satisfies $Ah=0$?