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Emergency department (ED) overcrowding is a national problem that is associated with ambulance diversion, decreased patient and provider satisfaction and poor patient outcomes. This study presents a novel approach to modeling the relationship between time of day, day of week, and ED arrivals using a hierarchical polynomial regression model. A series of hierarchical regression models were created to determine polynomial effects and capture the covariability (defined as R2) of the relationships from the 2009 to 2017 National Hospital Ambulat