Multiple regression interaction terms.
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Welcome to the wonderful world of marginality, the idea that answers the
question posed.
For an account of marginality for the more general class of generalized linear
models see McCullagh & Nelder (1989), Generalized linear models (2nd ed.)
Chapman & Hall, London. The relevant sections are in Ch. 3.
There are two aspects: the first concerns categorical variables (factors) and
their interactions, and the second deals with functional marginality between
continuous terms and their interactions.
Marginality and factorial models: the main point is that it does not make
inferential sense to hypothesise that a marginal effect A, say, is null when
when an interaction, say A.B, is not. Models containing A.B must include the
marginal terms A & B.
Functional marginality and continuous terms. In general it does not make sense
to include a term x1.x2 without including both x1 & x2. Models that do not
obey this rule have the nasty feature that the goodness of fit of a linear model
changes with the scale of x, e.g. if you change from degrees C to degrees F the
goodness of fit changes. There are special cases where it makes sense to
include a cross term without one of its marginal terms; in such cases there is
a special point on the scale where the value of the response is known in some
sense.
For more information see:
(1) Nelder, J.A. (1994) The statistics of linear models: back to basics.
Statistics and Computing, 4, 221-234.
(2) Nelder, J.A. (1997) Functional marginality is important (letter).
J.R.S.S. (C), 46, 281-2.
(3) Nelder,J.A. (1998) How strong is the weak heredity principle?
Amer. Statistician, 52, 315-318.
(4) Nelder, J.A. (19??) Functional marginality and response-surface fitting.
Accepted by J. Appl. Stat.
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