Choosing A Test For A Covariance Hypothesis

Start With the Null Hypothesis and the Sampling Design

Choosing a covariance test is not a matter of selecting the most familiar name. First specify which covariance parameters the null hypothesis restricts, then identify how the observations were collected. A test for one population's covariance structure does not automatically apply to several independent groups, and a test of equal scalar variances does not test equality of whole covariance matrices.

For a \(p\)-dimensional random vector \(X\), write its covariance matrix as \(\Sigma\). In a one-population problem, the hypotheses may constrain \(\Sigma\) to one fixed matrix, to a structural family such as diagonal matrices, or to a scalar multiple of the identity. In a group comparison, each group has a covariance matrix \(\Sigma_g\), and the null may require some or all of these matrices to agree.

Match the Test to the Restriction

These nulls are not interchangeable. Sphericity is more restrictive than diagonality: it requires both zero off-diagonal entries and equal variances. A diagonal covariance matrix allows the variances to differ. Block diagonality permits covariance within each specified block but rules out covariance between blocks; it is not a general test for independence of every pair of variables.

The group structure matters just as much. Bartlett’s equal-variance test concerns one scalar variance per group. Box’s \(M\) concerns all variances and covariances in each group. If a multivariate test is required but only equality of selected marginal variances is scientifically relevant, the null must be formulated accordingly rather than silently substituted with equality of full covariance matrices.

Likelihood-Ratio Calibration for One Population

For the one-population choices, a common starting point is a sample \(X_1,\ldots,X_n\) of independent observations from \(N_p(\mu,\Sigma)\), with unknown mean \(\mu\). Define the residual sum-of-products matrix \(A=\sum_{i=1}^n(X_i-\bar X)(X_i-\bar X)^\mathsf{T}\), the usual sample covariance \(S=A/(n-1)\), and the full-likelihood covariance estimate \(S_{\mathrm{ML}}=A/n\). Thus \(S_{\mathrm{ML}}=((n-1)/n)S\). The distinction between these two covariance estimates matters for the likelihood coefficient.

Derivation. For a fixed \(\Sigma\), maximizing the normal likelihood over \(\mu\) gives \(\hat\mu=\bar X\). Up to terms independent of \(\Sigma\), the resulting log-likelihood is

Over all positive-definite \(\Sigma\), this is maximized at \(\hat\Sigma=S_{\mathrm{ML}}\). Under the null, the only allowed value is \(\Sigma_0\). Subtracting the unrestricted maximum log-likelihood from the null maximum and multiplying by \(-2\) gives

Since \(\log|\Sigma_0|-\log|S_{\mathrm{ML}}|=-\log|\Sigma_0^{-1}S_{\mathrm{ML}}|\), this is the stated expression. Under the null, the usual Wilks approximation is chi-square with \(p(p+1)/2\) degrees of freedom when the model is regular and the sample is sufficiently large. This is an asymptotic calibration, not an exact finite-sample claim.

A different likelihood can arise if one models \(A\) directly as a Wishart random matrix with \(n-1\) degrees of freedom. Its likelihood-ratio calculation uses the residual covariance estimate \(S=A/(n-1)\) and a multiplier \(n-1\). That is not the profiled full normal-sample likelihood just derived. Before using a formula, check which likelihood and which covariance estimate it describes.

Sphericity Is a Particular Structural Null

For one multivariate population, sphericity means \(\Sigma=\sigma^2I_p\) for some unknown \(\sigma^2>0\). The variance level is not specified in advance; the hypothesis restricts the shape of the covariance matrix. Under the same independent multivariate-normal model with unknown mean, the full-sample likelihood-ratio statistic is

Derivation. Use the profiled likelihood above and write \(T=S_{\mathrm{ML}}\). On the sphericity null, the log-likelihood, apart from its constant, is \(-\frac n2\{p\log(\sigma^2)+\operatorname{tr}(T)/\sigma^2\}\). Differentiating the expression in braces with respect to \(\sigma^2\) gives \(p/\sigma^2-\operatorname{tr}(T)/(\sigma^2)^2\). Its zero occurs at \(\hat\sigma^2=\operatorname{tr}(T)/p\), which is positive when \(T\) is positive definite. Substitution gives the restricted log-likelihood term \(p\log(\operatorname{tr}(T)/p)+p\). Subtracting the unrestricted maximum, whose corresponding term is \(\log|T|+p\), yields

Because \(T=((n-1)/n)S\), the common scale cancels: \(p\log(\operatorname{tr}(T)/p)-\log|T|=p\log(\operatorname{tr}(S)/p)-\log|S|\). This proves the displayed formula. The sphericity null has one free covariance parameter, so its regular Wilks degrees of freedom are \(p(p+1)/2-1\).

Group Tests Require Independent Groups

For equal-variance testing, the data consist of \(k\) independent groups of scalar observations, with a normal distribution within each group and an unknown group mean. The null is \(\sigma_1^2=\cdots=\sigma_k^2\); the alternative permits separate group variances. Bartlett’s test is designed for this setting, and its classical calibration relies on normality. It is not a test of equality of multivariate covariance matrices.

For equal-covariance testing, the observations are independent \(p\)-vectors in each of \(k\) independent groups. The means may differ; the null concerns covariance alone: \(\Sigma_1=\cdots=\Sigma_k\). Box’s \(M\) test compares the separate group covariance estimates with a pooled estimate, with a finite-sample correction commonly used for its approximate chi-square calibration. Its normal-theory approximation can be sensitive to nonnormality. In either group setting, the groups must be independent for these standard procedures; repeated or paired measurements call for a model that represents their dependence.

A Practical Selection Check

A useful final check is the number of restrictions. For a \(p\)-dimensional covariance matrix, there are \(p(p+1)/2\) distinct entries. A fixed-matrix null imposes that many restrictions; a diagonal null imposes \(p(p-1)/2\); and a sphericity null leaves one parameter, imposing \(p(p+1)/2-1\). For \(k\) groups, equality of all covariance matrices imposes \((k-1)p(p+1)/2\) restrictions. These counts provide a check on the usual regular chi-square degrees of freedom; they do not by themselves validate the assumptions or calibration.

Common Mistakes

• Treating diagonality as sphericity. Diagonality permits unequal variances; sphericity requires the variances to be equal as well as the off-diagonal entries to vanish.
• Using an equal-variance test for vector outcomes. Equal marginal variances do not establish equality of full covariance matrices, which also includes cross-variable covariances.
• Ignoring the sampling design. Standard equal-covariance tests assume independent groups; treating paired or repeated observations as independent changes the model.
• Mixing likelihood conventions. With an unknown mean in the full normal sample, the profiled likelihood uses coefficient \(n\) and covariance estimate \(A/n\). A coefficient \(n-1\) belongs to the Wishart likelihood for the residual sum-of-products matrix, not that profiled likelihood.
• Assuming a chi-square reference is automatically reliable. Wilks and related calibrations are approximate under their regularity conditions; departures from normality can affect covariance tests, so the model and calibration need scrutiny.