used to establish the reference distribution. Each data set contained responses generated via Gibbs sampler under the same dimension with the observed one. Let ( x b $_{1}$, . . . , x b $_{N}$) denote the data set obtained from the b -th bootstrapping. The log of the joint likelihood for the bootstrap data is computed as The last term, log z ( ˆ B , ˆ Φ ), in both the l o and l b expressions is fixed at a constant and hence can be ignored when comparing the log-likelihoods across the resamples. As such, the present study reports the unnormalized log-likelihood values by discounting the last term. For comparison purpose, the standard DCM without the graphical component was also considered and fitted to the observed data. Applying the same procedure of parametric bootstrapping, the empirical distribution of ( l $^{1}$, . . . , l$^{500}$) was obtained under the DCM and used as a reference for evaluating the plausibility of the model. l b = log N ∏ i =1 f ( x b i | ˆ B , ˆ Φ , ˆ π ) = N ∑ i =1 log$^{(}$ ∑ α ˆ π$_{α}$ exp$^{(}$ x b i $^{⊤}$ˆ B α + 1 2 x b i $^{⊤}$ˆ Φx b i ) − log z ( ˆ B , ˆ Φ ) . (a) GDCM (b) DCM Figure 2: Parametric Bootstrap for Checking Model Fit 22