three-way interaction

[Jason Yi]

I have a question regarding three-way interactions in regression analysis. There are four variables (A, B, C, D, all of these are continuous variables) and I am interested in two 2-way interactions (A*B and A*C) out of the six possible interaction combinations. Then, I further have some theoretical rationale for the possibility that the interaction pattern of the two 2-way interactions of my interest will differ depending on the value of D, which can be tested by including two 3-way interactions (A*B*D and A*C*D). To test these 3-way interactions, do I need to include all possible 2-way interactions b/w original main effect variables consisting of the 3-way interactions? Or, is it okay to fit the following regression model that has only two 2-way interactions and two 3-way interactions of my interest: Y=a + b1*A + b2*B + b3*C + b4*D + b5*A*B + b6*A*C + b7*A*B*D + b8*A*C*D + e?

Thank you in advance!!

Jason  

[Ornelas, Fermin]

It should be OK to model your interactions as specified. However, it is
likely that your interaction effects will be collinear; thus, conducting
modeling diagnostics should be in order. Be aware that having collinear
variables in the model will render statistical inferences questionable.

Fermin Ornelas, Ph.D.
Management Analyst III, AZ DES
Tel: [phone]
E-mail: [hidden email]


-----Original Message-----
From: SPSSX(r) Discussion [mailto:[hidden email]] On Behalf Of
Jason Yi
Sent: Thursday, April 12, 2007 10:11 AM
To: [hidden email]
Subject: three-way interaction

I have a question regarding three-way interactions in regression
analysis.
There are four variables (A, B, C, D, all of these are continuous
variables)
and I am interested in two 2-way interactions (A*B and A*C) out of the
six
possible interaction combinations. Then, I further have some theoretical
rationale for the possibility that the interaction pattern of the two
2-way
interactions of my interest will differ depending on the value of D,
which
can be tested by including two 3-way interactions (A*B*D and A*C*D). To
test
these 3-way interactions, do I need to include all possible 2-way
interactions b/w original main effect variables consisting of the 3-way
interactions? Or, is it okay to fit the following regression model that
has
only two 2-way interactions and two 3-way interactions of my interest:
Y=a +
b1*A + b2*B + b3*C + b4*D + b5*A*B + b6*A*C + b7*A*B*D + b8*A*C*D + e?

Thank you in advance!!

Jason

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[Burleson,Joseph A.]

It would be best to include any lower order interactions that are
antecedent to the higher-order; hence,

1. if you are interested in b7*A*B*D, you should also include bx*A*D and
bx*B*D, as well as the b5*A*B that is already in.

2. if you are interested in b7*A*C*D, you should also include bx*A*D and
bx*C*D, as well as the b5*A*C that is already in.

Insofar as the other comment about collinearity, follow Kenny's
admonition to "center" all continuous measures about the mean, BEFORE
creating the interactions. One cannot "remove" collinearity between A,
B, C, and D. But you CAN keep the interactions as orthogonal to the main
effects as possible by centering.

Also, if there is collinearity among main and/or interactions, consider
using the Type III (hierarchical) modeling procedure and check the
unique variance associated with the entry of each main AND interaction
as it enters the model. If there is no logic to the entry, one might
play around with entering all main effects first, then all 2-way
interactions in a second block, then the particular 3-way interactions
in a third block.

See Tabachnick & Fidell, or, better yet, Aiken, L. S., & West, S. G.
(1991). Multiple Regression: Testing and Interpreting Interactions. Sage

Joe Burleson


-----Original Message-----
From: SPSSX(r) Discussion [mailto:[hidden email]] On Behalf Of
Jason Yi
Sent: Thursday, April 12, 2007 1:11 PM
To: [hidden email]
Subject: three-way interaction

I have a question regarding three-way interactions in regression
analysis.
There are four variables (A, B, C, D, all of these are continuous
variables)
and I am interested in two 2-way interactions (A*B and A*C) out of the
six
possible interaction combinations. Then, I further have some theoretical
rationale for the possibility that the interaction pattern of the two
2-way
interactions of my interest will differ depending on the value of D,
which
can be tested by including two 3-way interactions (A*B*D and A*C*D). To
test
these 3-way interactions, do I need to include all possible 2-way
interactions b/w original main effect variables consisting of the 3-way
interactions? Or, is it okay to fit the following regression model that
has
only two 2-way interactions and two 3-way interactions of my interest:
Y=a +
b1*A + b2*B + b3*C + b4*D + b5*A*B + b6*A*C + b7*A*B*D + b8*A*C*D + e?

Thank you in advance!!

Jason

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