The　application　of　network　meta-analysis　is　becoming　increasingly　widespread,　and　for　a　successful　implementation,　it　requires　that　the　direct　comparison　result　and　the　indirect　comparison　result　should　be　consistent.　Because　of　this,　a　proper　detection　of　inconsistency　is　often　a　key　issue　in　network　meta-analysis　as　whether　the　results　can　be　reliably　used　as　a　clinical　guidance.　Among　the　existing　methods　for　detecting　inconsistency,　two　commonly　used　models　are　the　design-by-treatment　interaction　model　and　the　side-splitting　models.　While　the　original　side-splitting　model　was　initially　estimated　using　a　Bayesian　approach,　in　this　context,　we　employ　the　frequentist　approach.　In　this　paper,　we　review　these　two　types　of　models　comprehensively　as　well　as　explore　their　relationship　by　treating　the　data　structure　of　network　meta-analysis　as　missing　data　and　parameterizing　the　potential　complete　data　for　each　model.　Through　both　analytical　and　numerical　studies,　we　verify　that　the　side-splitting　models　are　specific　instances　of　the　design-by-treatment　interaction　model,　incorporating　additional　assumptions　or　under　certain　data　structure.　Moreover,　the　design-by-treatment　interaction　model　exhibits　robust　performance　across　different　data　structures　on　inconsistency　detection　compared　to　the　side-splitting　models.　Finally,　as　a　practical　guidance　for　inconsistency　detection,　we　recommend　utilizing　the　design-by-treatment　interaction　model　when　there　is　a　lack　of　information　about　the　potential　location　of　inconsistency.　By　contrast,　the　side-splitting　models　can　serve　as　a　supplementary　method　especially　when　the　number　of　studies　in　each　design　is　small,　enabling　a　comprehensive　assessment　of　inconsistency　from　both　global　and　local　perspectives.