Detection of dependence between two groups of variables in the presence of another group

Let F
₁
, F
₂
and F
₃
be three groups of variables some of which may be nominal. A method is proposed to test whether F
₂
is dependent on F
₃
in the presence of Fj. An area of application of the test is multivariate model building. The representation of nominal variable is by means of binary codes. The construction of the test begins with the computation of an orthogonal matrix based on the vectors of observations on the variables in F
₁
. The vectors of observations on the variables in F
₂
and F
₃
are rotated via pre-multiplication by the orthogonal matrix. The components of the rotated vectors which are orthogonal to the space spanned by the vectors of observations on the variables in F
₁
are investigated for dependence between F
₂
and F
₃
using a method reported in the literature. The test is applied to a polychotomous regression example. The performance of the test is characterized by its good discerning ability when there is a dependence between the two groups under study, and satisfactory Type I error when there is no dependence between the two groups.