# 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,

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

2. ### Richard UlrichGuest

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.

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