IDL Analyst Reference Guide: Regression

IMSL_HYPOTH_TEST

The IMSL_HYPOTH_TEST function performs tests for a multivariate general linear hypothesis HbU = G given the hypothesis sums of squares and crossproducts matrix S_H.

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 S_H, the sums of squares and crossproducts attributable to the hypothesis.

Keywords

DOUBLE

If present and nonzero, double precision is used.

HOTELLING_TRACE

Named variable into which the one-dimensional array containing the Hotelling's trace and p-value is stored.

PILLAI_TRACE

Named variable into which the one-dimensional array containing the Pillai's trace and p-value is stored.

ROY_MAX_ROOT

Named variable into which the one-dimensional array containing the Roy's maximum root criterion and p-value is stored.

U

Two-dimensional array of size n_dependent by nu containing the U matrix for the test H_pbU = G_p 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

Named variable into which the one-dimensional array containing the Wilk's lamda and p-value is stored.

Discussion

IMSL_HYPOTH_TEST computes test statistics and p-values for the general linear hypothesis HbU = 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 R^TC = H and where D is a diagonal matrix with diagonal elements:

See the section Linear Dependence and the R Matrix.

Error sum of squares and crossproducts matrix for model Y = Xb + e is:

which is input in IMSL_MULTIREGRESS. The error sum of squares and crossproducts matrix for the hypothesis HbU = G computed by IMSL_HYPOTH_TEST is:

Let p equal the order of the matrices S_E and S_H, 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 l_i (i = 1, 2, ... p) of the generalized eigenvalue problem S_Hx = lS_Ex. 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 l _i 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 u + 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 u £ p + 1, the approximation is not valid, and p_value = Hotelling_Trace(1) is set to NaN.

These three test statistics are valid when S_E is positive definite. A necessary condition for S_E to be positive definite is u ³ p. If S_E is not positive definite, a warning error message is issued, and both value and p_value are set to NaN.

Because the requirement u ³ p can be a serious drawback, IMSL_HYPOTH_TEST computes a fourth test statistic based on eigenvalues q_i (i = 1, 2, ..., p) of the generalized eigenvalue problem S_Hw = q(S_H + S_E) w. This test statistic requires a less restrictive assumption—S_H + S_E is positive definite. A necessary condition for S_H + S_E to be positive definite is u + q ³ p. If S_E is positive definite, IMSL_HYPOTH_TEST avoids the computation of the generalized eigenvalue problem from scratch. In this case, the eigenvalues q_i are obtained from l_i 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(u - 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'  
P-value  
     0.000010  

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)  
Wilk value = 0.00314861  p-value =  9.89437e-06    
PRINT, 'Roy value = ', roy_max_root(0), '  p-value =', $  
   roy_max_root(1)  
Roy value = 316.601  p-value =  9.89437e-06  
PRINT, 'Hotelling value = ', hotelling_trace(0), '  p-value =', $  
   hotelling_trace(1)  
Hotelling value = 316.601  p-value =  9.89437e-06  
PRINT, 'Pillai value = ', pillai_trace(0), '  p-value =', $  
   pillai_trace(1)  
Pillai value = 0.996851  p-value =  9.89437e-06