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Allan, there is no set answer to these questions besides "it depends". But
here are some thoughts anyway. They are not exhaustive by any means.

> I'm working with an adjacency matrix that is both polytomous (a 0 to 3
> scale) and asymmetrical. I would like to make the matrix symmetrical and
> dichotomous (using the transformations in UCINET). Are there any
> guidelines for the proper way to apply these tranformations?
>
> For example, it seems intuitively that, in symmetrizing, taking the
> average of two relations would retain more variance than taking the
> minimum or maximum of a relationship. Is the average therefore
> preferable in symmetrizing, or are there other factors to consider in
> the method chosen?

I assume that the relation measured is either (a) logically symmetric (but
empirically contains "errors"), or (b) in constructing the symmetric
version, you are deliberately creating a new relation that has its own,
separate, interpretation. Otherwise, perhaps you should not be symmetrizing.

In the first case, you could view Xij and Xji as two estimates of the same
quantity (an attribute of the relationship between them). Averaging the two
estimates is then a reasonable thing to do (assuming the values reflect a
continuous scale). You might view symmetrizing by minimum as a conservative
choice, making the assumption that, if Xij=3 and Xji=2, it is unclear
whether the relationship is a 3, but it is at least a 2.

The second case arises when, say, you have measured who lends money to whom.
This is not a logically symmetric relation, so if you symmetrize it you
should think of the result as measuring something else since you are truly
transforming your data. For 1/0 data (which is easier to think about), if
you symmetrize by maximum, then you are interested in seeing which pairs are
involved in the kind of relationship that permits (or emerges from) one
lending to the other. If you symmetrize by minimum, you are interested in
which pairs are involved in a reciprocal relationship, a mutual dependence.

>
> Also, in dichotomizing the data, is there a standard method for
> determining the cutoff used? The scale used in our measures have a point
> that, conceptually, makes sense to draw the cutoff, but I'm not sure if
> there's a quantitative method of finding a cutoff that's preferable. I'm
> working with several different but related networks, and I would like to
> keep the cutoff the same in all of them.

If the values have meaning, like 0=never heard of person, 1=know of them,
2=aquaintance, 3=friend, then you will use the cutoff that makes sense for
your research question. Possibly you'll dichotomize at different levels for
different analyses.

The only theoretically-based empirical method of determining a cutoff that I
can think of just now is the freeman-granovetter transitivity method
implemented in very old and very new versions of ucinet (but missing for a
long period in between). It's under Subgroups.

There is also the time-honored results-based method. You dichotomize at each
level and stare at the picture, using your ethnographic intuition to decide
which seems most right. Some would also say choose the cutoff that leads to
the strongest correlations among the variables you eventually create. For
the most part, that is a bad method.

>
> Finally, in both symmetrizing and dichotomizing, which should be applied
> first?

This one is amusing to experiment with, so I won't spoil it by commenting.

steve.

Steve Borgatti, PhD Director
Organization Studies Department
Boston College
Chestnut Hill, MA 02467
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