Sphericity Tests for Covariance Structure

What “Sphericity” Tests

In principal component analysis, covariance eigenvalues describe how much variation lies along different directions. A covariance matrix is spherical when no direction has a different variance from any other: its eigenvalues are all equal. Testing this condition is therefore a direct way to assess whether the covariance structure has a preferred direction.

Sphericity is not the same hypothesis as a known identity covariance. Under sphericity, the common variance is an unknown positive parameter; under identity covariance, every variance is fixed at one. A related test asks whether the population correlation matrix is identity, which means the variables are uncorrelated but does not require their variances to be equal.

For the Gaussian tests below, suppose \(X_1,\ldots,X_n\) are independent and identically distributed as \(N_p(\mu,\Sigma)\), with unknown mean \(\mu\), fixed dimension \(p\), and \(n>p\). Define the maximum-likelihood covariance estimate after estimating the mean by

With probability one, \(A\) is positive definite under these conditions, so its determinant and log-determinant are defined. The usual unbiased sample covariance is \(S=\frac{1}{n-1}\sum_i(X_i-\bar X)(X_i-\bar X)^\top\); it is a positive scalar multiple of \(A\), and therefore gives the same scale-free sphericity statistic.

Likelihood Ratio for Spherical Covariance

Consider the hypotheses \(H_0:\Sigma=\sigma^2 I_p\) for some \(\sigma^2>0\), against \(H_1:\Sigma\) is any positive-definite covariance matrix. The Gaussian log-likelihood, apart from a constant, is

For any fixed \(\Sigma\), maximizing over \(\mu\) gives \(\hat\mu=\bar X\). In the unrestricted model the covariance maximum-likelihood estimate is \(A\). Under the spherical null, substituting \(\Sigma=\sigma^2I_p\) into the likelihood shows that the maximizing scale is \(\hat\sigma^2=\operatorname{tr}(A)/p\). These estimates give the likelihood ratio and its log form.

To derive the statistic, at the unrestricted maximum the quadratic term in the log-likelihood is \(np\), since \(\sum_i(X_i-\bar X)^\top A^{-1}(X_i-\bar X)=np\). At the spherical maximum it is also \(np\): the quadratic term is \(\operatorname{tr}(nA)/\hat\sigma^2=np\). Thus those terms cancel in twice the difference of maximized log-likelihoods. The determinant term gives

This statistic is unchanged if \(A\) is multiplied by any positive scalar: the determinant is multiplied by that scalar to the power \(p\), and the trace term changes by the same amount on the log scale. It therefore tests equality of eigenvalues, not the value of the common variance. In terms of the eigenvalues \(\hat\lambda_1,\ldots,\hat\lambda_p\) of \(A\),

The arithmetic-geometric mean inequality implies \(Q_{\mathrm{sph}}\geq 0\), with equality exactly when all sample eigenvalues are equal. The statistic is thus large when the observed eigenvalues are unequal relative to their common average.

For the asymptotic calibration, the unrestricted covariance model has \(p(p+1)/2\) free parameters, one for each entry on or above the diagonal of a symmetric matrix. The spherical null has one free covariance parameter, \(\sigma^2\). Their dimension difference is \(p(p+1)/2-1\). The null is an interior, smooth, identifiable submodel of the positive-definite covariance model, and the Gaussian likelihood is regular. Wilks’ theorem therefore gives the stated chi-square limit. This is a large-sample approximation, not an exact finite-sample distribution.

Identity Covariance Is a Different Test

To test \(H_0:\Sigma=I_p\) against an unrestricted covariance matrix, the null has no free covariance parameters, while the alternative has \(p(p+1)/2\). Maximizing under the null and alternative gives the Gaussian likelihood-ratio statistic

Indeed, under the identity null the log-likelihood at \(\hat\mu=\bar X\) contains the quadratic term \(n\operatorname{tr}(A)\). At the unrestricted maximum, the quadratic term is \(np\), and the covariance determinant contributes \(\log|A|\). Subtracting the maximized log-likelihoods yields \(Q_I\). Wilks’ theorem applies with dimension difference \(p(p+1)/2-0\). Unlike \(Q_{\mathrm{sph}}\), this statistic responds to the overall scale as well as to unequal eigenvalues.

Mauchly’s and Bartlett’s Sphericity Tests

The scale-free determinant ratio

is known as Mauchly’s sphericity criterion. It lies between zero and one, equals one when the covariance eigenvalues are equal, and becomes small as they become more dispersed. The likelihood-ratio statistic is \(-n\log W\), with the chi-square calibration derived above. Mauchly’s classical finite-sample approximation uses a Bartlett correction to the multiplier; it is an approximation, not an exact distribution. For the conventional centered-sample form, the adjusted statistic is \(-[n-1-(2p+5)/6]\log W\), referenced to \(\chi^2_{p(p+1)/2-1}\). The unadjusted Wilks statistic is the direct likelihood-ratio form and has the same first-order large-sample limit.

A different classical procedure, Bartlett’s test of sphericity, tests whether the population correlation matrix is identity. This is the null of zero pairwise correlations, not the covariance-sphericity null. Let \(R\) be the sample correlation matrix. Under Gaussian sampling, the likelihood ratio comparing unrestricted covariance with a diagonal covariance matrix has statistic \(-n\log|R|\); its degrees of freedom are the number of off-diagonal covariance parameters, \(p(p-1)/2\). The conventional Bartlett-corrected approximation is

This correlation test allows different variances: a diagonal covariance matrix can have unequal diagonal entries. Consequently, rejection of the correlation-identity null does not by itself establish unequal covariance eigenvalues, and failure to reject it does not establish covariance sphericity.

Assumptions, Interpretation, and a Common Pitfall

The chi-square calibrations above rely on independent multivariate normal observations and a fixed dimension as sample size grows. The covariance estimates must be nonsingular for the displayed determinants and log-likelihood ratios to be defined. When \(p\) is not small relative to \(n\), or when the data are markedly non-Gaussian, the nominal large-sample calibration may be poor; the formulas should not be treated as distribution-free tests.

A common mistake is to read “sphericity” as “identity covariance” and conclude that any rejection implies the data have a non-unit variance. The correction is to specify the null before interpreting the statistic: \(Q_{\mathrm{sph}}\) tests equality of eigenvalues while leaving their common scale free; \(Q_I\) tests a fixed covariance matrix; Bartlett’s correlation test tests zero correlations while allowing unequal marginal variances. These hypotheses are related, but they are not interchangeable.