specifying interactions between linear term and cubic term

Discussion in 'Scientific Statistics Math' started by Dr Ad de Jong, Feb 21, 2005.

  1. Dear….,

    We have specified a cubic model to explain the relationship between X
    (customer satisfaction) and Y (customer loyalty). The estimation of
    the quadratic and cubic relationships might be subject to
    multicollinearity because the quadratic and cubic terms are
    mathematical manipulations of the variables under study. To address
    this concern, we first mean centered the first order variables and
    then developed the quadratic and cubic terms (Aiken and West 1991).
    Using the following regression equation:

    ,

    we find an S-shaped, cubic relationship between customer satisfaction
    and customer loyalty. Specifically, we find a significant positive
    linear term, a significant negative quadratic term, and a significant
    positive cubic term.


    However, we assume that the shape of this relationship may be
    dependent on the level of customer involvement. Therefore, we want to
    include an interaction term of customer satisfaction (X) and
    involvement (Z) into the regression equation:


    Y= â0 + â1X + â2X2 + â3X3 + â4Z + â5X3Z+ e.

    MY QUESTIONS:
    1) Is it possible to specify an interaction between a linear variable
    (Z) and the cubic term (X + X2 + X3)? And if so, how should you
    specify and interpret such an interaction?

    2) Is it possible to specify an interaction between the linear
    variable (Z) and the linear term (X)? And how should you interpret
    such an interaction?

    Y= â0 + â1X + â2X2 + â3X3 + â4Z + â5XZ+ e.


    Kind regards,


    Dr. Ad de Jong
    Department of Organisation Science & Marketing
    Eindhoven University of Technology
    Faculty of Technology Management
    Den Dolech 2, Tema 0.04
    PO Box 513, 5600 MB Eindhoven, the Netherlands
    phone : + 31 (0) 40 247 2423/2170, fax : + 31 (0) 40 246 5949
    e-mail : [email protected]
    http://fp.tm.tue.nl/medewerk/a.d.jong/
     
    Dr Ad de Jong, Feb 21, 2005
    #1
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  2. On the face of it, that sounds like a pretty silly thing
    to do. From the results, below, it sounds like Y is (indeed)
    a dichotomous variable, and that the underlying
    relationship follows a logistic curve.

    Read up on logistic regression.

    That's reasonable.
    Difficulty in interpretation, I would guess, is why
    people don't bother to do cubic fits very often;
    and, especially, would not try to extend them that way.
    Interactions are usually modeled as the product of
    the centered variables. Interpret CAREFULLY.
     
    Richard Ulrich, Feb 22, 2005
    #2
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