Compare group mean vectors jointly with Wilks lambda, keeping SSCP scaling, eigenvalue geometry, response units and classical reference assumptions clear.
MANOVA and Wilks lambda
Multivariate analysis of variance (MANOVA) tests whether groups have the same population mean vector across several responses. Wilks lambda is a determinant-ratio statistic used for that joint test. A quant comparing execution venues might ask whether mean slippage and mean fill delay differ jointly, allowing for their co-movement rather than treating the two outcomes as unrelated evidence.
The question is about means
For g groups and p response variables, the one-way null is
H0:μ1=μ2=⋯=μg.
The alternative is that at least one group mean vector differs. Correlation between responses within an observation is allowed. Separate unadjusted ANOVAs ask coordinate-by-coordinate questions, omit the joint covariance geometry, and do not automatically control the chance of any false rejection across the collection. MANOVA can detect differences in combinations of responses that are less apparent in individual coordinates; it is not guaranteed to outperform every alternative analysis. A joint rejection does not identify which venues or which responses differ. UCLA's one-way MANOVA example illustrates the distinction between joint and separate tests.
Despite its name, this is not a test that group covariance matrices are equal. Equality of within-group covariance matrices is an assumption of the classical mean test. Nor does an observed venue difference establish a causal venue effect: order size, urgency and instrument selection can also differ across venues.
From sums of squares to matrices
Write observation i in group k as the column vector yki, its group sample mean as yˉk, and the overall sample mean as yˉ. In the one-way model, define
E is the error sum-of-squares-and-cross-products matrix (SSCP), measuring within-group scatter. H is the hypothesis SSCP, measuring between-group scatter here. Their sum is the total centred SSCP. Diagonal entries are sums of squared deviations; off-diagonal entries retain cross-products between responses. In more general multivariate linear models, H must correspond to the particular effect or contrast being tested. Stanford's multivariate regression notes develop the matrix decomposition and hypothesis construction.
These are unnormalised sums, not interchangeable covariance estimates. With N=∑knk, the pooled within-group covariance matrix estimate is E/(N−g) in this one-way model. Substituting separately normalised within- and between-group matrices into the SSCP formula below generally changes the statistic.
For positive-definite E and positive-semidefinite H,
Λ=det(E+H)det(E)=j=1∏p1+λj1,
where λj are the eigenvalues of E−1H, including zero roots. Although that product matrix need not be symmetric, it is similar to the symmetric positive-semidefinite matrix E−1/2HE−1/2, so its roots are real and nonnegative. JMP documents both forms of Wilks lambda.
Thus 0<Λ≤1. Larger hypothesis scatter relative to error scatter pushes lambda down; a larger lambda means less separation in this sense. Lambda is neither a probability that the means are equal nor a p-value. It is also not generally a simple percentage of unexplained variation across the original responses.
A two-response calculation, without a significance claim
whose factors 1/(1+λj) multiply to the same lambda. No sample size, group design or hypothesis/error degrees of freedom accompanies these matrices. They determine the statistic, not a significance conclusion. We cannot attach an F reference distribution or calculate a p-value from the supplied information. JMP's reference transformations explicitly require design ranks and error degrees of freedom.
With error SSCP E fixed, scaling the fictional hypothesis SSCP by t reduces Wilks lambda from 1 at t=0 to 0.7236 at t=1 and 0.5538 at t=2. No significance cutoff is shown.
Holding E fixed, increasing the hypothesis SSCP lowers lambda. This toy sensitivity curve illustrates the statistic's direction, not statistical significance or an observed market relationship.
Units cancel; assumptions do not
Changing slippage from basis points to percentage points must not change the joint conclusion. More generally, apply the same nonsingular response transformation y∗=Ay to every observation. Then E∗=AEA⊤ and H∗=AHA⊤, giving
This derivation establishes invariance to nonsingular changes of response units or linear combinations. Dividing E and H by different degrees of freedom is a different operation and does not share that cancellation.
Classical reference calibration assumes independent observation vectors, a correctly specified mean model, multivariate normal errors and a common positive-definite within-group covariance. Enough residual degrees of freedom and nonredundant responses are needed for E to be invertible. In the one-way normal model, N−g≥p permits full rank with probability one; exact redundancy still makes actual summaries singular. These normal linear-model conditions underpin the classical calculation, as Stanford explains.
Serially dependent trading observations, heavy tails, changing covariance and extreme outliers can make the classical calibration unreliable. Adding a duplicated response can make the determinant ratio undefined. A pseudoinverse or resampling scheme requires its own justified procedure; computing a number does not restore the classical reference law. Wilks lambda is also distinct from Wilks' theorem, the asymptotic likelihood-ratio result.
For the scalar starting point, One-Way Analysis of Variance develops between/within sums of squares and follow-up comparisons (public preview; full lesson requires membership).
