The polar cap, defined as a region of open magnetic flux, is an ideal region in which to investigate how properties of the solar wind directly affect the magnetosphere. For such studies, the polar cap (PC) index provides a useful characterization of the state of the polar ionosphere. In this thesis, we use statistical models to study how polar cap properties, quantified by the PC index, depend on solar wind parameters and other geomagnetic indices during intervals of exceptionally large (10mV/m) merging electric field by applying regression analysis to the PC index and its potential drivers. By examining 53 one to two-day intervals that include such extreme fields, we find that the PC index correlates strongly with the modified electric field (EKR) proposed by Kivelson and Ridley except in anomalous events during which the auroral oval expanded poleward to the latitude of the PC index station and the index increased because of proximity to auroral zone currents. It is found that nightside magnetospheric processes, as represented by AL index, make a significant contribution to the PC index. A linear regression analysis shows that the portion of the PC index directly driven by the solar wind electric field outweighs the contribution arising from energy release in the magnetotail by roughly a factor of 2. Using a model selection approach, we found neither the solar wind dynamic pressure (Pdyn) nor jumps in Pdyn to directly contribute to the PC index. However, there exists some correlation between the PC index and Pdyn, because of the common dependence of EKR and Pdyn on the solar wind number density.