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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:

# Load the package
library(ergm)
# Load the data
data(sampson)
# What is the triad census?
> summary(samplike ~ triadcensus)
 triadcensus.012 triadcensus.102 triadcensus.021D triadcensus.021U
             205 190 12 24
triadcensus.021C triadcensus.111D triadcensus.111U triadcensus.030T
              24 68 34 5
triadcensus.030C triadcensus.201 triadcensus.120D triadcensus.120U
               0 35 15 6
triadcensus.120C triadcensus.210 triadcensus.300
               5 18 8
#So no 030C triads. Fit a model without them, conditioning on the number of
edges:

>summary(ergm(samplike ~ triadcensus(c(1:8,10:15)),constraints=~edges))

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

Formula: samplike ~ triadcensus(c(1:8, 10:15))

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.021U 0.32146 0.26072 0.004 0.218592
triadcensus.021C 0.29910 0.21705 0.003 0.169254
triadcensus.111D 0.92183 0.26354 0.003 0.000542 ***
triadcensus.111U -0.48680 0.18529 0.002 0.009061 **
triadcensus.030T -0.86231 0.53972 0.007 0.111191
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

which clearly indicates triadic structure.

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
internet: [log in to unmask]



On 6/26/09 7:39 AM, "Richard Heidler" <[log in to unmask]> wrote:

> ***** 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
> Mail: <mailto:[log in to unmask]> [log in to unmask]
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>
>
> <http://www.foev-speyer.de/Ueberuns/Mitarbeiter/heidler.asp>
> http://www.foev-speyer.de/Ueberuns/Mitarbeiter/heidler.asp
>
>
>
>
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