Constructs an equivalent completely testable multivariate general linear hypothesis H BU = G from a partially testable hypothesis HpBU = Gp.

Once a general linear model y = X β + ɛ is fitted, particular hypothesis tests are frequently of interest. If the matrix of regressors X is not full rank (as evidenced by the fact that some diagonal elements of the R matrix output from the fit are equal to zero), methods that use the results of the fitted model to compute the hypothesis sum of squares (see routine RHPSS) require one to specify in the hypothesis only linear combinations of the regression parameters that are estimable. A linear combination of regression parameters cT β is estimable means that there exists some vector a such that cT = aTX, i.e., cT is in the space spanned by the rows of X. For a further discussion of estimable functions, see Maindonald (1984, pages 166‑168) and Searle (1971, pages 180 ‑ 188). Routine CESTI is only useful in the case of nonfull rank regression models, i.e., when the problem of estimability arises.

Peixoto (1986) noted that the customary definition of testable hypothesis in the context of a general linear hypothesis test H β = g is overly restrictive. He extended the notion of a testable hypothesis (a hypothesis composed of estimable functions of the regression parameters) to include partially testable and completely testable hypotheses. A hypothesis H β = g is partially testable means that the intersection of the row space of H (denoted by R(H)) and the row space of X(R(X)) is not essentially empty and is a proper subset of R(H), i.e., {0} ⊂ R(H) ∩ R(X) ⊂ R(H). A hypothesis H β = g is completely testable means that {0} ⊂ R(H) ⊆ R(X). Peixoto also demonstrated a method for converting a partially testable hypothesis to one that is completely testable so that the usual method for obtaining the sum of squares for the hypothesis from the results of the fitted model can be used. The method replaces Hp in the partially testable hypothesis Hp β = gp by a matrix H whose rows are a basis for the intersection of the row space of Hp and the row space of X. A corresponding conversion of the null hypothesis values from gp to g is also made. A sum of squares for the completely testable hypothesis can then be computed (see routine RHPSS). The sum of squares that is computed for the hypothesis H β = g equals the difference in the error sums of squares from two fitted models the restricted model with the partially testable hypothesis Hp β = gp adjoined to the model as linear equality restrictions (see routine RLEQU) and the unrestricted model.

Routines RGLM, RGIVN, RLEQU, and RCOV can be used to compute the fit of the general linear model prior to invoking CESTI. The R matrix is required for input to CESTI. After converting a partially testable hypothesis to a completely testable hypothesis, RHPSS can be invoked to compute the sum of squares for the hypothesis.

For the general case of the Multivariate General Linear Model Y = XB + E with possible linear equality restrictions on the regression parameters, CESTI converts the partially testable hypothesis Hp BU = Gp to a completely testable hypothesis H BU = G. For the case of the linear model with linear equality restrictions, the definitions of estimable functions, nontestable hypotheses, partially testable hypotheses, and completely testable hypothesis are similar to those previously given for the unrestricted model with the exception that R(X) is replaced by R(R) where R is the upper triangular matrix output from RLEQU. The nonzero rows of R form a basis for the rowspace of the matrix (XT, AT)T. The rows of H form an orthonormal basis for the intersection of two subspaces: the subspace spanned by the rows of Hp and the subspace spanned by the rows of R. The algorithm used by CESTI for computing the intersection of these two subspaces is based on an algorithm for computing angles between linear subspaces due to to Bjorck and Golub (1973). (See also Golub and Van Loan 1983, pages 429‑430). The method is closely related to a canonical correlation analysis discussed by Kennedy and Gentle (1980, 56‑565). The algorithm is as follows:

A one-way analysis-of-variance model discussed by Peixoto (1986) is fitted to some data. The model is

The model is fitted using routine RGLM. Next, the partially testable hypothesis

is converted to a completely testable hypothesis using CESTI. Sum of squares associated with the hypothesis are computed using routine RHPSS. Finally, the F statistic is computed along with the associated p‑value using routine FDF (see Chapter 17, "Probability Distribution Functions and Inverses").