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This thesis project focuses on algorithms that generate good sampling points for function approximation. In one dimension, polynomial interpolation using equispaced points is unstable, with high Oscillations near the endpoints of the interpolated interval. On the other hand, Chebyshev nodes provide both stable and highly accurate points for polynomial interpolation. In higher dimensions, optimal sampling points are unknown. This project addresses this problem by finding algorithms that are robust in various domains for polynomial interpolation and least-squares. To measure the quality of the nodes produced by said algorithms, the Lebesgue constant will be used. In the algorithms, a number of numerical techniques will be used, such as the Gram-Schmidt process and the pivoted-QR process. In addition, concepts such as node density and greedy algorithms will be explored.
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Pelvic Circumferential Compression Devices (PCCDs), an important medical device when caring for patients with pelvic fractures, play a crucial role in the stabilization and reduction of the fracture. During pelvic fracture cases, control of internal bleeding through access to the femoral artery is of utmost importance. Current designs of PCCDs do not allow vital access to this artery and in attempts to gain access, medical professionals and emergency care providers choose to cut into the PCCDs or place them in suboptimal positions with unknown downstream effects. We researched the effects on surface pressure and the overall pressure distribution created by the PCCDs when they are modified or placed incorrectly on the patient. In addition, we investigated the effects of those misuses on pelvic fracture reduction, a key parameter in stabilizing the patient during critical care. We hypothesized that incorrectly placing or modifying the PCCD will result in increased surface pressure and decreased fracture reduction. Our mannequin studies show that for SAM Sling and T-POD, surface pressure increases if a PCCD is incorrectly placed or modified, in support of our hypothesis. However, opposite results occurred for the Pelvic Binder, where the correctly placed PCCD had higher surface pressure when compared to the incorrectly placed or modified PCCD. Additionally, pressure distribution was significantly affected by the modification of the PCCDs. The cadaver lab measurements show that modifying or incorrectly placing the PCCDs significantly limits their ability to reduce the pelvic fracture. These results suggest that while modifying or incorrectly placing PCCDs allows access to the femoral artery, there are potentially dangerous effects to the patient including increased surface pressures and limited fracture reduction.
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Data Sources: I use the Healthcare Cost and Utilization Project’s Nationwide Inpatient Sample (NIS) from 2000 to 2011. The NIS is a 20% sample of all inpatient claims. The Manhattan Institute supplied data on the availability of health savings accounts in each state. State PTR implementation dates were gathered by Hans Christensen, Eric Floyd, and Mark Maffett of University of Chicago’s Booth School of Business by contacting the health department, hospital association, or website controller in each state.
Study Design: The NIS data was collapsed by procedure, hospital, and year providing averages for the dependent variable, Cost, and a host of covariates. Cost is a product of Total Charges within the NIS and the hospital’s Cost to Charge ratio. A new binary variable, PTR, was defined as ‘0’ if the year was strictly less than the disclosure website’s implementation date, ‘1’ for afterwards, and missing for the year of implementation. Then, using multivariate OLS regression with fixed effect modeling, the change in cost from before to after the year of implementation is estimated.
Principal Findings: The analysis estimates the effect of PTR to decrease the average cost per procedure by 7%. Specifications identify within state, within hospital, and within procedure variation, and reports that 78% of the cost decrease is due to within-hospital, within-procedure price discounts. An additional model includes the interaction of PTR with the prevalence of health savings accounts (hereafter, HSAs) and procedure electivity. The results show that PTR lowers costs by an additional 3 percent with each additional 10 percentage point increase in the availability of HSAs. In contrast, the cost reductions from PTR were much smaller for procedures more frequently coded as elective.
Conclusions: The study concludes price transparency regulations can lead to a decrease in a procedure’s costs on average, primarily through price discounts and slightly through lower cost procedures, but not due to patients moving to cheaper hospitals. This implies that hospitals are taking initiative and lowering prices as the competition’s prices become publically available suggesting that hospitals – not patients – are the biggest users of price transparency websites. Hospitals are also finding some ways to provide cheaper alternatives to more expensive procedures. State regulators should evaluate if a better metric other than charge prices, such as expected out-of-pocket payments, would evoke greater patient participation. Furthermore, states with higher prevalence of HSAs experience greater effects of PTR as expected since patients with HSAs have greater incentives to lower their costs. Patients should expect a shift towards plans that offer these types of savings accounts since they’ve shown to have a reduction of health costs on average per procedure in states with higher prevalence of HSAs.