keeping interaction terms

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Hello,<?xml:namespace prefix = o ns = "urn:schemas-microsoft-com:office:office" /><o:p></o:p>

while doing my thesis in habitat modelling I??ve come across a problem with interaction terms. My question concerns the usage of interaction terms for linear regression modelling with R. If an interaction-term (predictor) is chosen for a multiple model, then, according to <?xml:namespace prefix = st1 ns = "urn:schemas-microsoft-com:office:smarttags" /><st1:place w:st="on">Crawley</st1:place> its single term has to be added to the multiple model: lrm(N~a*b+a+b).<o:p></o:p>

This nearly always leads to high correlation rates between the interaction term a*b and its single term a or b. With regards to the law of colinearity modelling should not include correlated variables with an Spearman index >0,7. Does this mean that the interaction term has to be discarded or can the variables stay within the model when correlated? I do not necessarily want to do a PCA on this issue.<o:p></o:p>

Thanks for helping<o:p></o:p>

Christian<o:p></o:p>

Your query opens up many issues.  First, the statement that a main 
effect has to be added if an interaction term is chosen assumes that an 
interaction has meaning without adjustment for main effects.  This is 
not the case.  The hierarchy principle needs to be executed in a forward 
manner.  Second, you are implying that you are not fitting a single 
pre-specified model but are doing variable selection based on p-values. 
  This creates a host of problems.  Third, you imply that correlations 
between main effects and interactions are not to be tolerated.  Again 
this is not the case.  It is a fact of life that we must accomodate. 
[Some people like to center main effects to reduce this correlation but 
that is an artificial and not helpful approach.]

Frank

Frank E Harrell Jr   Professor and Chair           School of Medicine
                      Department of Biostatistics   Vanderbilt University

Adding a bit to Frank Harrell's good comments.

1. Regarding HTML infection: I rolled up my sleeves, washed
   my hands carefully, took a fine sharp knife, cut it all
   out, and then sowed up the incisions.

2. For the rest, see below.

Hello,

while doing my thesis in habitat modelling I??ve come across a
problem with interaction terms. My question concerns the usage
of interaction terms for linear regression modelling with R.
If an interaction-term (predictor) is chosen for a multiple model,
then, according to Crawley its single term has to be added to the
multiple model: lrm(N~a*b+a+b).

This nearly always leads to high correlation rates between the
interaction term a*b and its single term a or b. With regards to
the law of colinearity modelling should not include correlated
variables with an Spearman index >0,7. Does this mean that the
interaction term has to be discarded or can the variables stay
within the model when correlated?
I do not necessarily want to do a PCA on this issue.

There's more than a suggestion in your statements that you tend
to be drawn along by people's prescriptions. Instead, try to
think simply about it.

If, after fitting "a+b", you make a "significant difference" by
further including "a:b", then the interaction between a and b
matters, even if you observe high correlations. The latter should
not lead you to ignore the former.

How much it matters is of course another question. You could
examine this, in R, by comparing the predicted values from the
"a+b" model with the predicted values from the "a*b" model.
Though they will be different, you will have to judge whether
the amount of difference is large enough to be of real importance
in your application. (It is possible to get highly "significant"
results, i.e. small P-values, from small effects).

Even if it does matter, in real terms, you are left with the
fundamental difficulty, indicated by Frank, that interpreting
interaction between variables a and b is simple only when the
variables a and b are orthogonal in the data (either by accident
or by design). If they are non-orthogonal, then you have to
think carefully about how to interpret it, and this does depend
on what it all means.

Maybe we could help more with this if we knew more about your
investigation (perhaps off-list, if you prefer).

Best wishes,
Ted.


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E-Mail: (Ted Harding) <Ted.Harding at nessie.mcc.ac.uk>
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Date: 08-Oct-05                                       Time: 14:14:48
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