Performs tests for a multivariate general linear hypothesis HβU = G given the hypothesis sums of squares and crossproducts matrix SH.

The type double function is imsls_d_hypothesis_test.

The p-value corresponding to Wilks’ lambda test.

Function imsls_f_hypothesis_test computes test statistics and p-values for the general linear hypothesis HβU = G for the multivariate general linear model.

The hypothesis sum of squares and crossproducts matrix input in scph is

where C is a solution to RTC = H,(CTDC)- denotes the generalized inverse of CTDC, and D is a diagonal matrix with diagonal elements

For a detailed discussion, see Linear Dependence and the R Matrix in the Usage Notes.

The error sum of squares and crossproducts matrix for the model Y = Xβ + ɛ is

which is input in regression_info. The error sum of squares and crossproducts matrix for the hypothesis HβU = G computed by imsls_f_hypothesis_test is

Let p equal the order of the matrices SE and SH, i.e.,

Let q (stored in dfh) be the degrees of freedom for the hypothesis. Let v (input in regression_info) be the degrees of freedom for error. Function imsls_f_hypothesis_test computed three test statistics based on eigenvalues λi (i = 1, 2, …, p) of the generalized eigenvalue problem SHx = λSEx. These test statistics are as follows:

The associated p-value is based on an approximation discussed by Rao (1973, p. 556). The statistic

has an approximate F distribution with pq and ms − pq ∕ 2 + 1 numerator and denominator degrees of freedom, respectively, where

and

The F test is exact if min (p, q) ≤ 2 (Kshirsagar, 1972, Theorem 4, p. 299−300).

where c is output as value. The p-value is based on the approximation

where s = max (p, q) has an approximate F distribution with s and ν + q − s numerator and denominator degrees of freedom, respectively. The F test is exact if s = 1; the p-value is also exact. In general, the value output in p_value is lower bound on the actual p-value.

U is output as value. The p-value is based on the approximation of McKeon (1974) that supersedes the approximation of Hughes and Saw (1972). McKeon’s approximation is also discussed by Seber (1984, p. 39). For

the p-value is based on the result that

has an approximate F distribution with pq and b degrees of freedom. The test is exact if min (p, q) = 1. For ν ≤ p + 1, the approximation is not valid, and p_value is set to NaN.

These three test statistics are valid when SE is positive definite. A necessary condition for SE to be positive definite is ν ≥ p. If SE is not positive definite, a warning error message is issued, and both value and p_value are set to NaN.

Because the requirement ν ≥ p can be a serious drawback, imsls_f_hypothesis_test computes a fourth test statistic based on eigenvalues θi (i = 1, 2, …, p) of the generalized eigenvalue problem SHw = θ(SH + SE) w. This test statistic requires a less restrictive assumption—SH + SE is positive definite. A necessary condition for SH + SE to be positive definite is ν + q ≥ p. If SE is positive definite, imsls_f_hypothesis_test avoids the computation of the generalized eigenvalue problem from scratch. In this case, the eigenvalues θi are obtained from λi by

The fourth test statistic is as follows:

V is output as value. The p-value is based on an approximation discussed by Pillai (1985). The statistic

has an approximate F distribution with s(2m + s + 1) and s(2n + s + 1) numerator and denominator degrees of freedom, respectively, where

The F test is exact if min (p, q) = 1.

The data for this example are from Maindonald (1984, p. 203−204). A multivariate regression model containing two dependent variables and three independent variables is fit using function imsls_f_regression and the results stored in the structure regression_info. The sum of squares and crossproducts matrix, scph, is then computed with a call to imsls_f_hypothesis_scph for the test that the third independent variable is in the model (determined by specification of h). Finally, function imsls_f_hypothesis_test is called to compute the p-value for the test statistic (Wilk’s lambda).

This example is the same as the first example, but more statistics are computed. Also, the U matrix, u, is explicitly specified as the identity matrix (which is the same default configuration of U).