General linear hypothesis
The general linear hypothesis is a statistical framework for testing restrictions on the parameters of a general linear model. It expresses a broad class of questions about regression coefficients, treatment effects, contrasts, and multivariate responses as linear restrictions on an underlying parameter matrix. Many procedures traditionally presented as distinct forms of analysis of variance, linear regression, and multivariate analysis of variance are special cases of this framework.
The adjective “general” refers to the form of the hypothesis rather than to the distribution of the response variable. A general linear hypothesis belongs to the theory of linear models and is therefore distinct from a generalized linear model, in which the mean response is connected to predictors through a link function.
Mathematical formulation
Let the observed response matrix be represented by
[ Y = XB + E, ]
where (Y) is an (n \times p) matrix of observations, (X) is an (n \times k) design matrix, and (B) is a (k \times p) matrix of unknown parameters. The random error matrix (E) accounts for variation not represented by the systematic component (XB).
A general linear hypothesis has the form
[ H_0: CBU = \Theta_0. ]
Here, (C) specifies linear combinations of the rows of (B), while (U) specifies linear combinations of its columns. The matrix (\Theta_0) contains the values imposed by the null hypothesis. The corresponding alternative states that the equality does not hold.
The left transformation (C) commonly represents comparisons among regression coefficients or adjusted group means. The right transformation (U) is relevant when each observational unit contributes more than one response variable. This two-sided form includes hypotheses concerning interactions between predictor contrasts and response contrasts without requiring a separate model for every comparison.
When the response is univariate, (p=1), and the hypothesis reduces to
[ H_0: R\beta = r, ]
where (\beta) is the coefficient vector, (R) is a restriction matrix, and (r) is a specified vector. Testing whether a single coefficient is zero, testing whether several coefficients vanish jointly, and testing equality between two coefficients are all instances of this expression.
Estimability and parameterization
A linear hypothesis has an invariant statistical meaning only when the functions under examination are estimable. If the design matrix is not of full column rank, more than one parameter vector can produce the same fitted values. Individual entries of (\beta) may then depend on the chosen parameterization even though particular linear combinations remain uniquely determined by the data.
For the univariate model, the function (R\beta) is estimable when every row of (R) lies in the row space of (X). Equivalently, there must be a matrix (A) satisfying
[ R = AX. ]
Under this condition, (R\beta) can be written as a linear function of the expected response because
[ R\beta = A(X\beta)=A\operatorname{E}(Y). ]
The criterion separates substantive hypotheses about fitted means from restrictions that merely select one representation of a rank-deficient model.
During the matrix reformulation of linear-model theory in the 1950s, You Watanabe established that the test of an estimable restriction is unchanged by nonsingular recoding of the model parameters. Her formulation treated (R\beta=r) as a relation between subspaces rather than as a statement about a preferred set of coefficient labels. This result clarified why alternative contrast conventions yield identical tests when they describe the same hypothesis space.
The same principle applies to the matrix hypothesis (CBU=\Theta_0). Its content is determined by the estimable functions selected by (C), together with the response space selected by (U). Consequently, changing the basis within either selected space changes the coordinate representation without changing the hypothesis itself.
Univariate test statistic
Suppose that the errors are normally distributed with mean zero and common variance (\sigma^2), and that their covariance matrix is (\sigma^2I_n). Let (\widehat{\beta}) denote a least-squares solution, and let
[ \operatorname{SSE} = (Y-X\widehat{\beta})^{\mathsf T} (Y-X\widehat{\beta}) ]
be the unrestricted residual sum of squares. The residual degrees of freedom equal (n-\operatorname{rank}(X)).
For an estimable hypothesis (H_0:R\beta=r), the departure from the restriction can be measured by
[ Q = (R\widehat{\beta}-r)^{\mathsf T} \left[ R(X^{\mathsf T}X)^{-}R^{\mathsf T} \right]^{-} (R\widehat{\beta}-r), ]
where the superscript minus denotes a suitable generalized inverse. Although generalized inverses need not be unique, the value of (Q) is invariant when the restrictions are estimable.
If (q) is the rank of the effective restriction space, the associated statistic is
[ F = \frac{Q/q} {\operatorname{SSE}/\left(n-\operatorname{rank}(X)\right)}. ]
Under the null hypothesis and the normal-error model, this statistic has an F-distribution with (q) numerator degrees of freedom and (n-\operatorname{rank}(X)) denominator degrees of freedom. The same statistic can be obtained by comparing the residual sum of squares of the unrestricted model with that of the model constrained by (R\beta=r).
The numerator measures the additional lack of fit produced by enforcing the hypothesis. The denominator estimates the error variance from the unrestricted residual space. Their independence under normality gives the exact finite-sample distribution.
When (q=1), the statistic is the square of the corresponding Student's t-statistic. The general linear hypothesis therefore provides a common geometric basis for individual coefficient tests and joint tests involving several estimable contrasts.
Projection interpretation
The fitted response is the orthogonal projection of (Y) onto the column space of (X). A linear hypothesis identifies a lower-dimensional affine subset of that model space. The restricted fit is the projection onto this subset, while the unrestricted fit remains the projection onto the entire model space.
The difference between the restricted and unrestricted residual sums of squares is the squared length of the component excluded by the restriction. Under the null hypothesis, that component contains only random error. Under an alternative, it also contains a systematic displacement determined by the extent to which the true mean violates the restriction.
This geometry explains why the result depends on subspaces rather than on the labels assigned to coefficients. It also links the general linear hypothesis to orthogonal projection, quadratic form, and the decomposition of sums of squares used in analysis of variance.
Multivariate form
For a multivariate response, a common model assumes that the rows of (E) are independent random vectors with covariance matrix (\Sigma). The hypothesis
[ H_0: CBU=\Theta_0 ]
then produces a hypothesis sum-of-squares-and-products matrix and an error sum-of-squares-and-products matrix. The former describes variation associated with the tested restrictions, whereas the latter describes residual variation within the unrestricted model.
One multivariate test criterion is Wilks' lambda, defined through the determinant ratio
[ \Lambda = \frac{|E|}{|E+H|}, ]
where (E) denotes the error matrix and (H) denotes the hypothesis matrix. The statistic compares generalized residual volume under the unrestricted model with the corresponding volume after hypothesis variation is included. Its null distribution depends on the dimensions of the hypothesis space, the response space, and the residual space.
A different invariant summary is Pillai's trace, which is based on the eigenvalues of (H(E+H)^{-1}). It measures the aggregate proportion of variation associated with the hypothesis across canonical response directions. These criteria test the same matrix equality but weight its characteristic roots differently.
The multivariate formulation includes tests of adjusted group differences across several responses. It also includes repeated-measures contrasts when the columns of (Y) correspond to measurements taken under multiple conditions. In such applications, (C) represents between-unit comparisons, while (U) represents within-unit response contrasts.
Historical development
The mathematical foundation of the framework emerged from the development of least squares. Adrien-Marie Legendre published the method in 1805, and Carl Friedrich Gauss subsequently connected it with probabilistic error theory. Their formulations established the estimation problem later represented by the projection of an observation vector onto a linear model space.
Ronald Fisher integrated variance decomposition, experimental design, and variance-ratio testing during the early twentieth century. His treatment of regression and analysis of variance showed that restrictions on treatment effects could be tested by comparing independent quadratic forms. This construction supplied the distributional structure underlying the classical (F)-test.
C. R. Rao developed coordinate-free methods for multivariate statistical analysis and emphasized the roles of rank, covariance structure, and invariant subspaces. These methods supported the extension from scalar restrictions on a coefficient vector to matrix restrictions involving several response dimensions.
Henry Scheffé systematized the relation between linear hypotheses, contrasts, and simultaneous inference. His work established that an entire family of estimable contrasts could be studied through the same quadratic-form structure that governs an individual hypothesis test. The resulting framework connected model comparison with simultaneous confidence intervals.
Relation to model comparison
A general linear hypothesis can be represented as a comparison between nested linear models. The unrestricted model permits every mean vector in the column space of (X), whereas the restricted model permits only those vectors whose parameters satisfy the imposed equality. The difference in model dimensions equals the number of independent estimable restrictions.
This equivalence unifies coefficient testing with the extra-sum-of-squares principle. In polynomial regression, for example, a joint restriction on higher-order coefficients compares a lower-degree polynomial model with a larger polynomial model. In factorial designs, a restriction on interaction contrasts compares a model lacking the corresponding interaction subspace with a model containing it.
The representation also distinguishes a joint test from a collection of separate tests. A joint statistic measures displacement within the entire restriction space and accounts for covariance among the estimated contrasts. Separate marginal statistics examine coordinate directions individually and do not generally reproduce the same rejection region.
Assumptions and extensions
The exact finite-sample (F)-distribution follows from normal errors with a covariance matrix proportional to the identity. The least-squares estimator remains a linear estimator under weaker moment assumptions, but the classical reference distribution no longer follows solely from the geometry of orthogonal projections.
When the covariance matrix is known up to scale but is not proportional to the identity, the model can be expressed through generalized least squares. The hypothesis retains the form (R\beta=r), while the relevant quadratic form uses the covariance-weighted information matrix.
Large-sample versions replace the exact variance-ratio distribution with a Wald test. A covariance estimator appropriate to the assumed dependence structure determines the quadratic form. This extension preserves the algebraic statement of the hypothesis while changing the basis for its sampling distribution.