Table 13 Difference between empirical variance and the model-based variance calculation for OTE \({\theta }_{1}=0\) and HTE \({\theta }_{3}=0\). The number of clusters \(I=8\), the total time steps is \(J=5\), the cluster size is \(m=40\) for each cluster, the prevalence of \({X}_{ijk}\) is 50% when estimating HTE, \({\gamma }_{j}=0.1\left(j-1\right)\) for \(j=\mathrm{1,2},\mathrm{3,4},5\), and \({\theta }_{0}={\text{log}}\left(0.15/0.85\right)\)

\(\widehat{{\mathbb{V}}ar}\left({\widehat{\theta }}_{1,m}\right)-\widetilde{{\mathbb{V}}ar}\left({\widehat{\theta }}_{1,m}\right)\)

\(\widehat{{\mathbb{V}}ar}\left({\widehat{\theta }}_{3,m}\right)-\widetilde{{\mathbb{V}}ar}\left({\widehat{\theta }}_{3,m}\right)\)

\(\widehat{{\mathbb{V}}ar}\left({\widehat{\theta }}_{3,m}\right)-\widetilde{{\mathbb{V}}ar}\left({\widehat{\theta }}_{3,m}\right)\)

Data generating Procedure

\({\text{logit}}\left({\mu }_{ijk}\right)={\theta }_{0}+{\gamma }_{j}\)

\({\text{logit}}\left({\mu }_{ijk}\right)={\theta }_{0}+{\gamma }_{j}\)

\({\text{logit}}\left({\mu }_{ijk}\right)={\theta }_{0}+{\gamma }_{j}+{\text{log}}\left(1.68\right){W}_{ij}+{\text{log}}\left(1.5\right){X}_{ijk}\)