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Dear Richard,

Another approach that I like is to use the "triadcensus" term in the R
package "ergm". For example, for a version of Sampson's "liking" data:

library(ergm)
data(sampson)
# What is the triad census?
205              190               12               24
24               68               34                5
0               35               15                6
5               18                8
#So no 030C triads. Fit a model without them, conditioning on the number of
edges:

==========================
Summary of model fit
==========================

Newton-Raphson iterations:  6
MCMC sample of size 10000

Monte Carlo MLE Results:
Estimate Std. Error MCMC s.e.  p-value
triadcensus.012   0.28455    0.12531     0.004 0.023890 *
triadcensus.102   0.68510    0.17680     0.003 0.000132 ***
triadcensus.021D -0.75819    0.27592     0.003 0.006372 **
triadcensus.111D  0.92183    0.26354     0.003 0.000542 ***
triadcensus.111U -0.48680    0.18529     0.002 0.009061 **
triadcensus.201  -0.27605    0.12509     0.002 0.028097 *
triadcensus.120D  0.95660    0.07497     0.001  < 1e-04 ***
triadcensus.120U -1.01444    0.18354     0.002  < 1e-04 ***
triadcensus.120C -0.55847    0.21630     0.002 0.010312 *
triadcensus.210  -0.87899    0.11918     0.001  < 1e-04 ***
triadcensus.300  -1.74996    0.15365     0.002  < 1e-04 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Null  Deviance: 424.21  on 306  degrees of freedom
Residual Deviance: 203.93  on 292  degrees of freedom
Deviance: 220.27  on  14  degrees of freedom

AIC: 231.93    BIC: 284.06

The p-values for each triad type are given. For example, there are
statistically significantly more 102 triads than would be expected under a
Erdos-Renyi model.

As the usual caveat, the p-values are based on the curvature of the
log-likelihood which does not have the same asymptotic justification as in,
for example, logistic regression. One can do an exact test based on the
"ergm" "simulate" command and compute a (Monte-Carlo) exact p-value if you
want to see if they differ much (or at all).

If you just want to test for triangles then I would use:

> summary(ergm(samplike ~ triangle, constraints=~edges))
==========================
Summary of model fit
==========================

Formula:   samplike ~ triangle

Newton-Raphson iterations:  6
MCMC sample of size 10000

Monte Carlo MLE Results:
Estimate Std. Error MCMC s.e. p-value
triangle  0.14196    0.01975     0.007  <1e-04 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Null  Deviance: 424.206  on 306  degrees of freedom
Residual Deviance: 409.821  on 305  degrees of freedom
Deviance:  14.385  on   1  degrees of freedom

AIC: 411.82    BIC: 415.54

Further information can be found at http://statnet.org

Cheers,

Mark

for the statnet development team

-------------------------------------------------
Mark S. Handcock
Professor of Statistics
Department of Statistics, B313 Padelford Hall
University of Washington, Box 354322     Phone:  (206) 543-6774
Seattle, WA  98195-4322.           FAX:   (206) 457-1953
Web:  www.stat.washington.edu/~handcock

> *****  To join INSNA, visit http://www.insna.org  *****
>
> Dear list members,
>
>
>
> Does someone know how to calculate significances for the triads in a triad
> census? I know this should be easy, but neither pajek, nor R is calculating
> the significances of the different triad types. Which software could be used
> (the network is rather big with 2131 actors) ?
>
>
>
> Kind regards
>
>
>
> Richard Heidler
>
>
>
>
> ----------------------------------
>
>
>
> Richard Heidler
> Deutsches Forschungsinstitut für öffentliche Verwaltung Speyer (FÖV)
> Postfach 1409
> D-67324 Speyer
>
>
>
> Tel.: 06232/654-429
> Fax: 06232/654-418
>
>
>
>  <http://www.foev-speyer.de/Ueberuns/Mitarbeiter/heidler.asp>
> http://www.foev-speyer.de/Ueberuns/Mitarbeiter/heidler.asp
>
>
>
>
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