The IMSL_HYPOTH_TEST function performs tests for a multivariate general linear hypothesis HβU = G given the hypothesis sums of squares and crossproducts matrix SH.
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IMSL_HYPOTH_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 and where D is a diagonal matrix with diagonal elements:
Error sum of squares and crossproducts matrix for model Y = Xβ + ε is:
which is input in IMSL_MULTIREGRESS. The error sum of squares and crossproducts matrix for the hypothesis HβU = G computed by IMSL_HYPOTH_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 info_v) be the degrees of freedom for error. The IMSL_HYPOTH_TEST function 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:
Wilk’s lambda:
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. 2994300).
Roy’s maximum root:
c = max λi over all i
where c is output as value = ROY_MAX_ROOT(0). The p-value is based on the approximation:
where s = max(p, q) has an approximate F distribution with s and υ + 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 = ROY_MAX_ROOT(1) is lower bound on the actual p-value.
Hotelling’s trace:
U is output as value = HOTELLING_TRACE(0). 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 υ ≤ p + 1, the approximation is not valid, and p-value = HOTELLING_TRACE(1) 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 υ ≥ p. If SE is not positive definite, a warning error message is issued, and both value and the p-value are set to NaN.
Because the requirement υ ≥ p can be a serious drawback, IMSL_HYPOTH_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 υ + q ≥ p. If SE is positive definite, IMSL_HYPOTH_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:
Pillai’s trace:
V is output as value = PILLAI_TRACE(0). 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:
s = min(p, q)
m = 1/2(|p - q| – 1)
n = 1/2(υ - p – 1)
The F test is exact if min(p, q) = 1.
Examples
Example 1
The data for this example are from Maindonald (1984, p. 20310204). A multivariate regression model containing two dependent variables and three independent variables is fit using IMSL_MULTIREGRESS and the results stored in Info_v. The sum of squares and crossproducts matrix, scph, is then computed using HYPOYH_SCPH for the test that the third independent variable is in the model (determined by specification of h). Finally, IMSL_HYPOTH_TEST is used to compute the p-value for the test statistic (Wilk’s lambda).
x = TRANSPOSE([[7.0, 5.0, 6.0], [2.0, -1.0, 6.0], $
[7.0, 3.0, 5.0], [-3.0, 1.0, 4.0], [2.0, -1.0, 0.0], $
[2.0, 1.0, 7.0], [-3.0, -1.0, 3.0], [2.0, 1.0, 1.0], $
[2.0, 1.0, 4.0]])
y = TRANSPOSE([[7.0, 1.0], [-5.0, 4.0], [6.0, 10.0], $
[5.0, 5.0], [5.0, -2.0], [-2.0, 4.0], [0.0, -6.0], $
[8.0, 2.0], [3.0, 0.0]])
h = FLTARR(1, 4)
h(*) = 0
h(0, 3) = 1.0
coefs = IMSL_MULTIREGRESS(x, y, Predict_Info = p)
scph = IMSL_HYPOTH_SCPH(p, h, Dfh = dfh)
pvalue = IMSL_HYPOTH_TEST(p, dfh, scph)
PM, pvalue, format = '(F10.6)', Title = 'P-value'
IDL prints:
P-value
0.000010
Example 2
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).
x = TRANSPOSE([[7.0, 5.0, 6.0], [2.0, -1.0, 6.0], $
[7.0, 3.0, 5.0], [-3.0, 1.0, 4.0], [2.0, -1.0, 0.0], $
[2.0, 1.0, 7.0], [-3.0, -1.0, 3.0], [2.0, 1.0, 1.0], $
[2.0, 1.0, 4.0]])
y = TRANSPOSE([[7.0, 1.0], [-5.0, 4.0], [6.0, 10.0], $
[5.0, 5.0], [5.0, -2.0], [-2.0, 4.0], [0.0, -6.0], $
[8.0, 2.0], [3.0, 0.0]])
h = FLTARR(1, 4)
h(*) = 0
h(0, 3) = 1.0
u = [[1, 0], [0, 1]]
coefs = IMSL_MULTIREGRESS(x, y, Predict_Info = p)
scph = IMSL_HYPOTH_SCPH(p, h, Dfh = dfh)
pvalue = IMSL_HYPOTH_TEST(p, dfh, scph, U = u, $
Wilk_Lambda = wilk_lambda, Roy_Max_Root = roy_max_root, $
Hotelling_Trace = hotelling_trace, $
Pillai_Trace = pillai_trace)
PRINT, 'Wilk value = ', wilk_lambda(0), ' p-value =', $
wilk_lambda(1)
IDL prints:
Wilk value = 0.00314861
p-value = 9.89437e-06
Continue:
PRINT, 'Roy value = ', roy_max_root(0), ' p-value =', $
roy_max_root(1)
IDL prints:
Roy value = 316.601
p-value = 9.89437e-06
Continue:
PRINT, 'Hotelling value = ', hotelling_trace(0), ' p-value =', $
hotelling_trace(1)
IDL prints:
Hotelling value = 316.601
p-value = 9.89437e-06
Continue:
PRINT, 'Pillai value = ', pillai_trace(0), ' p-value =', $
pillai_trace(1)
IDL prints:
Pillai value = 0.996851
p-value = 9.89437e-06
Errors
Warning Errors
STAT_SINGULAR_1: "u "* "scpe "* "u " is singular. Only Pillai’s trace can be computed. Other statistics are set to NaN.
Fatal Errors
STAT_NO_STAT_1: "scpe " + "scph " is singular. No tests can be computed.
STAT_NO_STAT_2: No statistics can be computed. Iterations for eigenvalues for the generalized eigenvalue problem "scph "*x = (lambda)*( "scph "+ "scpe ")*x failed to converge.
STAT_NO_STAT_3: No statistics can be computed. Iterations for eigenvalues for the generalized eigenvalue problem "scph "*x = (lambda)*( "scph "+ "u "* "scpe "* "u ")*x failed to converge.
STAT_SINGULAR_2: "u "* "scpe "* "u " + "scph " is singular. No tests can be computed.
STAT_SINGULAR_TRI_MATRIX: The input triangular matrix is singular. The index of the first zero diagonal element is equal to #.
Syntax
Result = IMSL_HYPOTH_TEST(Info_v, dfh, scph [, /DOUBLE] [, HOTELLING_TRACE=variable] [, PILLAI_TRACE=variable] [, ROY_MAX_ROOT=variable] [, U=array] [, WILK_LAMBDA=variable])
Return Value
The p-value corresponding to Wilks’ lambda test.
Arguments
dfh
Degrees of freedom for the sums of squares and crossproducts matrix.
Info_v
One-dimensional array of type BYTE containing information about the regression fit. See IMSL_MULTIREGRESS.
scph
Two-dimensional array of size nu by nu containing SH, the sums of squares and crossproducts attributable to the hypothesis.
Keywords
DFH (optional)
Named variable containing the degrees of freedom for the sums of squares and crossproducts matrix. This is equal to the rank of input matrix h.
DOUBLE (optional)
If present and nonzero, double precision is used.
HOTELLING_TRACE (optional)
Named variable in the form of a one-dimensional array containing the Hotelling’s trace and p-value .
PILLAI_TRACE (optional)
Named variable in the form of a one-dimensional array containing the Pillai’s trace and p-value.
ROY_MAX_ROOT (optional)
Named variable in the form of a one-dimensional array containing the Roy’s maximum root criterion and p-value.
U (optional)
Two-dimensional array of size n_dependent by nu containing the U matrix for the test HpβU = Gp where nu is the number of linear combinations of the dependent variables to be considered. The value nu must be greater than 0 and less than or equal to n_dependent. Default: nu = n_dependent and U is the identity matrix.
WILK_LAMBDA (optional)
Named variable in the form of a one-dimensional array containing the Wilk’s lamda and p-value.
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