In this paper, two inferential procedures for selecting the significant predictors in the PLS1 regression model are introduced. The significant PLS components are first obtained and the two predictor selection methods, called PLS-Forward and PLS-Bootstrap, are applied to the PLS model obtained. They are also compared empirically to two other methods that exist in the literature with respect to the quality of fit of the model and to their predictive ability. Although none of the four methods is uniformly best, it is seen that PLS-Forward and PLS-Bootstrap perform well and can be very useful in practical situations in identifying the important explanatory variables.

SUMMARYIn this paper we write the PLS multivariate regression model in terms of a redundancy index and obtain some properties of the successive PLS components. We study their significance in the model and build tests of hypotheses to this effect. A stopping rule is given to obtain the right number of PLS components, and a numerical measure is defined to assess the overall quality of the PLS regression. Finally, an algorithm is given explicitly as it was coded in S-Plus and is applied in some chosen examples.

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In this paper we consider a measure of multivariate association between two vectors which generalizes the multiple correlation coefficient and obtain its exact distribution when the parent population is multivariate normal. A test is proposed for the hypothesis that several such measures are zero. This test is an exact test and an explicit expression for its power is obtained. Moreover numerical results are obtained concerning the power of this test in some particular cases. AcademicPress AMS 1991 subject classifications: 62H15; 62E15; 62H20.

redundancy analysis, significant factors, tests of significance, canonical correlation, multivariate linear regression,