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I'm using the R package statnet to fit some ERGMs to the Faux Dixon High simulated network data provided with the package. The first model I fit is almost identical to the model used to generate the simulated data, and is shown below:

model.fdh <- ergm(fdh ~ edges 
                  + mutual 
                  + absdiff("grade") 
                  + nodefactor("race") 
                  + nodefactor("grade")
                  + nodefactor("sex") 
                  + nodematch("race", diff = TRUE, levels=c("B","O","W")) 
                  + nodematch("grade", diff = TRUE) 
                  + nodematch("sex", diff = FALSE) 
                  + idegree(0:1) 
                  + odegree(0:1)
                  + gwesp(0.1,fixed=T), 
                  constraints = ~bd(maxout=10), 
                  control =
                    control.ergm(MCMLE.steplength = 1, MCMC.burnin = 100000, MCMC.interval =
                                   10000, MCMC.samplesize = 2500, MCMLE.maxit = 100), verbose=T)

Unsurprisingly, since this model is almost exactly like the one used to generate the simulated data in the first place, the model fits the data extremely well. Below are some diagnostic plots.

enter image description here

In addition, the log-likelihood of model is 0.2368874, and AIC = -11801 and BIC = -11557.

Next, I tried fitting a more complicated model to the data, shown below:

model.fdh2 <- ergm(fdh ~ edges 
               + mutual 
               + absdiff("grade") 
               + nodefactor("race") 
               + nodefactor("grade")
               + nodefactor("sex") 
               + nodematch("race", diff = TRUE, levels=c("B","O","W")) 
               + nodematch("grade", diff = TRUE) 
               + nodematch("sex", diff = FALSE) 
               + gwidegree(0.1, fixed=F)
               + gwodegree(0.1, fixed=F)
               + istar(2)
               + ostar(2)
               + gwesp(0.1,fixed=F)
               + transitive, 
               constraints = ~bd(maxout=10), 
               control =
                 control.ergm(MCMLE.steplength = 1, MCMC.burnin = 100000, MCMC.interval =
                                10000, MCMC.samplesize = 2500, MCMLE.maxit = 100), verbose=T)

Based on diagnostic plots, this model appears to fit the data less precisely.

enter image description here

Obviously I'm not showing diagnostic plots for all of the statistics that statnet returns, but all of them more or less look like this: the simulated graphs tend to overstate the statistics in this second model. However, this second model has a higher log-likelihood than the previous one at 1.120519, with AIC= -11781 and BIC = -11501. Does this make sense? Is it possible to have an ERGM with a higher log-likelihood but an apparent worse level of fit to the data?

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