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I strongly doubt that permutation tests are an adequate protection
against the dependence between nodal variables in a network. I have not
seen any proofs or simulations that support this, but I would be
interested to be informed of results supporting permutation tests for
nodal variables in networks (if they exist, which would surprise me).
Permutation tests assume permutational invariance of the residuals, and
this is quite close to independence. I expect that in situation without
outliers, a nodal permutation test will have results very close to those
of a test in an OLS regression model.
Let me mention a simple example illustrating why permutation tests may
go wrong under network dependence. Suppose that there is a network with
N nodes, which because of transitivity/clustering is split up into k
clusters that are internally highly connected, and weakly connected with
each other. Suppose that the dependent variable has a strong network
autocorrelation. Then the data is like a sample with k independent
residuals, rather than N. (It will be so in the extreme case of perfect
network autocorrelation and total separation between the clusters.) Note
that k could be as low as 2 (we might even think of k=1, corresponding
to the nodal variable being an emergent phenomenon). But the permutation
test will treat it as a sample with N independent residuals.
Summarizing: I think that regression modeling of nodal variables using
nodal permutations is a seemingly nice idea which, however, does not
offer any protection against network dependence.
Tom A.B. Snijders
Professor of Statistics in the Social Sciences
University of Oxford
Professor of Statistics and Methodology
Department of Sociology
University of Groningen
On 09/09/2011 17:36, Ian McCulloh wrote:
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> Sender: Social Networks Discussion Forum<[log in to unmask]>
> Poster: Ian McCulloh<[log in to unmask]>
> Subject: Re: Using node level regression analysis in Ucinet
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> I think that is exactly why we do permutations. ERGMs can also help control for structural effects. I was more questioning the validity of using a centrality measure as a dependent/independent variable in a multiple regression or something.
> Ian McCulloh
> On Sep 9, 2011, at 9:34 AM, Philip Leifeld<[log in to unmask]> wrote:
>> But that's what the random permutations are good for, isn't it? Or is your point that the permutation approach is per se an inadequate tool for this purpose?
>> Am 09.09.2011 14:36, schrieb Ian McCulloh:
>>> I think the fundamental point is that OLS assumes independence, among
>>> other things. Residual analysis is really a tool to validate whether
>>> those assumptions hold. We know already that independence is not a
>>> valid assumption. The degree to which this may bias findings is
>>> unclear (to me at least). I'd appreciate some discussion on this
>>> topic by the group. Thanks.
>>> Ian McCulloh
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