Making the transition from calculus to advanced calculus/real analysis can be challenging for undergraduate students. Part of this challenge lies in the shift in the focus of student activity, from a ...focus on algorithms and computational techniques to activities focused around definitions, theorems, and proofs. The goal of Realistic Mathematics Education (RME) is to support students in making this transition by building on and formalizing their informal knowledge. There are a growing number of projects in this vein at the undergraduate level, in the areas of abstract algebra (TAAFU: Larsen, 2013; Larsen & Lockwood, 2013), differential equations (IO-DE: Rasmussen & Kwon, 2007), geometry (Zandieh & Rasmussen, 2010), and linear algebra (IOLA: Wawro, et al., 2012). This project represents the first steps in a similar RME-based, inquiry-oriented instructional design project aimed at advanced calculus. The results of this project are presented as three journal articles. In the first article I describe the development of a local instructional theory (LIT) for supporting the reinvention of formal conceptions of sequence convergence, the completeness property of the real numbers, and continuity of real functions. This LIT was inspired by Cauchy’s proof of the Intermediate Value Theorem, and has been developed and refined using the instructional design heuristics of RME through the course of two teaching experiments. I found that a proof of the Intermediate Value Theorem was a powerful context for supporting the reinvention of a number of the core concepts of advanced calculus. The second article reports on two students’ reinventions of formal conceptions of sequence convergence and the completeness property of the real numbers in the context of developing a proof of the Intermediate Value Theorem (IVT). Over the course of ten, hour-long sessions I worked with two students in a clinical setting, as these students collaborated on a sequence of tasks designed to support them in producing a proof of the IVT. Along the way, these students conjectured and developed a proof of the Monotone Convergence Theorem. Through this development I found that student conceptions of completeness were based on the geometric representation of the real numbers as a number line, and that the development of formal conceptions of sequence convergence and completeness were inextricably intertwined and supported one another in powerful ways. The third and final article takes the findings from the two afroementioned papers and translates them for use in an advanced calculus classroom. Specifically, Cauchy's proof of the Intermediate Value Theorem is used as an inspiration and touchstone for developing some of the core concepts of advanced calculus/real analysis: namely, sequence convergence, the completeness property of the real numbers, and continuous functions. These are presented as a succession of student investigations, within the context of students developing their own formal proof of the Intermediate Value Theorem.
Patients should be prescribed medication based on their medical condition, without prejudice because of their race, gender, or primary language. However, previous research has shown that men are ...prescribed more medication than women, patients who are White are prescribed more medications than patients who are non-White, and English-speaking people are prescribed more medications than non-English-speaking patients. However, it is unclear whether these differences also occur in pediatric orthopaedic populations.
We asked: (1) Was the amount of opiates prescribed at discharge associated with patient age, gender, race, or primary language? (2) Did the amount of opiates prescribed to patients at discharge change from 2010 to 2020?
In a single center, between January 2010 and December 2019, we treated 331 patients younger than 18 years surgically for upper and lower long-bone extremity fractures. Patients were considered eligible if they had a nonpathologic fracture. Femur fractures were not included. Based on these criteria, all patients were eligible. The mean age was 12 ± 4 years. The mean weight was 57 ± 33 kg. Among these patients, 76% (253 of 331) were boys and 24% (78 of 331) were girls. From the hospital discharge records, we recorded the amount of opiates prescribed at the time of discharge as measured by morphine milligram equivalents (MMEs). We examined the association of age, gender, race, primary language, weight, and year of treatment using this measurement. We determined a patient's race retrospectively by information given by their parents at time of admission. We did not attempt to contact patients to obtain more nuanced information about their racial background. These data were obtained from the electronic health record. The Wilcoxon rank sum test, t-test, or chi-square test was used to assess associations depending on the distribution of variables, as appropriate. Because opioids as measured in MMEs is zero-inflated, a two-part model analysis was used to adjust for confounding variables. One component of the model was for the probability of having any opiate prescription and another was for the mean number of opioids received. Findings were considered statistically significant if p values were < 0.05.
In total, 57% (189 of 331) of children were prescribed opiates at discharge after surgery for long-bone fractures. Opiate MMEs increased with patient age (r = 0.38; p < 0.01). Boys and girls showed no difference in the amounts of pain medication (adjusted odds ratio OR 1.38 95% confidence interval (CI) 0.80 to 2.39; p = 0.71; adjusted opioid difference: 0.35 MME 95% CI -51.7 to 52.4; p = 0.99), nor were there differences between patients who were White and those who were non-White (adjusted OR 0.78 95% CI 0.49 to 1.23; p = 0.28; adjusted opioid difference: 21.5 MME 95% CI -19.3 to 62.4; p = 0.30), or between patients for whom English was there primary language and those for whom English was not their primary language (adjusted OR 1.16 95% CI 0.52 to 2.57; p = 0.71; adjusted opioid difference: 22.7 MME 95% CI -55.7 to 101.3; p = 0.57) when adjusted for age and weight. There was no change in opioid prescription amounts from 2010 to 2020 after adjusting for changes in patient age across years (Spearman r = -0.08; p = 0.16).
Fairness in opioid prescribing based on race, gender, or primary language is possible. Additional research is needed to determine what factors in our institution led to this result. We suggest that prescribers should apply consistent protocols based on factors such as weight or injury type rather than making individual decisions for each patient. This will lead to fairer opioid prescribing to patients from different race and gender groups.
Level III, therapeutic study.
These three essays focus on various manifestations of risk: collective institutional investor risk, individual retail investor risk, and risk generated by government debt. In the first essay, we look ...at how institutional investor concentration can impact downside risk. The U.S. equities market price process is largely driven by the information set and actions of large institutional investors, not individual retail investors. Using quarterly 13-F holdings, we construct the Herfindahl-Hirschman Index (HHI) of institutional investor concentration as a measure of granularity. Our contributions are both empirical and theoretical. We provide a comprehensive study of how granularity affects: (1) the cross-section of returns, (2) conditional variances across stocks and (3) downside risk. We find that constructing a low-HHI minus high-HHI portfolio produces an annualized return of 5.6%. Using an approach advocated by Koijen and Yogo, we document that the cross-section of HHI portfolios can be explained by a conditional asset pricing model involving heterogeneous investor demands driven by time-varying beliefs over asset characteristics. We document the adverse impact that investor ownership concentration has on both conditional volatility, and critically, a robust set of downside risk measures at both the portfolio and the firm level. In the second essay, we study how individual retail investor risk could be mitigated via the robo-advising industry. Artificial intelligence, or AI, enhancements are increasingly shaping our daily lives. Financial decision-making is no exception to this. We introduce the notion of AI Alter Egos, which are shadow robo-investors, and use a unique data set covering brokerage accounts for a large cross-section of investors over a sample from January 2003 to March 2012, which includes the 2008 financial crisis, to assess the potential benefits of robo-investing. We have detailed investor characteristics and records of all trades. Our data set consists of investors typically targeted for robo-advising. We explore robo-investing strategies commonly used in the industry, including some involving advanced machine learning methods. The man versus machine comparison allows us to shed light on possible benefits the emerging robo-advising industry may provide to certain segments of the population, such as low income and/or low education investors. In the third essay, we study how government debt can impact future growth via a specific risk channel. Elevated levels of government debt raise concerns about their effects on long-term growth prospects. Using the cross section of US stock returns, we show that (i) high-R&D firms are more exposed to government debt and pay higher expected returns than low-R&D firms, and (ii) higher levels of the debt-to-GDP ratio predict higher risk premiums for high-R&D firms. Furthermore, rises in the cost of capital for innovation-intensive firms predict declines in subsequent productivity and economic growth. We propose a production-based asset pricing model with endogenous innovation and fiscal policy shocks that can rationalize key aspects of the empirical evidence. Our study highlights a novel and distinct risk channel shaping the link between government debt and future growth.
The answer to this question can, at its simplest, be achieved by the analysis of data at two timepoints; baseline (prior to initiation of therapy) and one additional timepoint, some days, weeks or ...months later, when the pharmacokinetics and pharmacodynamics of the drug indicate it should have reached peak efficacy. In reality, longer-term trials are usually carried out to ensure maintenance of an effect over time, with data collection visits occurring at regular intervals (normally one to six months apart). In Figure 1 (see facing page), a graph shows the effect of a hypothetical experimental medication (medication A) compared to two active comparator medications (medications B and C) on physical functioning in rheumatoid arthritis. Assessment of physical functioning is made using a well-established patient-reported outcome (PRO) questionnaire, the SF-36. Items on this questionnaire are measured and scored using interval data, and lower scores are indicative of less impairment. In this trial, all three medications are efficacious, in that they have improved physical functioning over the three months of the study. They have done so to a "similar" (non-inferior) degree. In response to the two questions above, one must conclude based on the data presented in Figure 1, that (1) yes, there was a significant improvement in physical functioning from baseline to trial endpoint (3 months) with the experimental medication, but (2) no, the benefit was not greater than with standard of care (using traditional inferential statistical analysis). The apparent similarities in the impact of these medications on physical functioning may be a function of the simplicity of the study design; more specifically, the infrequent measurement of a concept which may change frequently may be masking important differences between the medications. Imagine that, in fact, the medications behaved as per Figure 2, captured in the same trial using twice-weekly diary data rather than just once at the end of the trial. This shows clear differences between the medications. Although our answer to the first question ("did the experimental therapy achieve clinical benefit in patients who enrolled in the study?") remains yes, the answer to the second question ("is the benefit greater than with standard of care?") is now not so clear. Utilizing area-under-the-curve analyses (in addition to change-from-baseline analyses) would suggest medication A differs significantly to medication B, but not to medication C. However, the data now provides a basis for hypothesizing that these three drugs are fundamentally different. The differences are in the patient experience. It is too simplistic to suggest that more data points are more informative. Rather, the frequency of data capture must relate to the dependent variable (outcome) of interest, and be captured in a manner that reduces measurement error. For example, daily collection of "quality-of-life" data may decrease the variability in the measurement owing to participant burden and generalizations by participants when completing the scales. This would reduce the sensitivity of the assessment to identify a meaningful change from baseline within and between cohorts. On the other hand, researchers have shown reduced error variance by collecting episodic data in trials of overactive bladder, largely as a function of reduced recall bias and heterogeneity within participants over the course of a trial. The timing of assessment is also key; while random momentary assessment may be appropriate and beneficial for capturing "in-the-moment" fatigue in multiple sclerosis (which is known to follow a diurnal pattern), it is appropriate to standardize collection of pain and stiffness data once during a 24-hour period among rheumatoid arthritis patients.
The rise of materials informatics, coupled with an abundance of calculated materials property data and ever-growing supercomputing power, has invited the reexamination of existing empirical models ...through the lens of modern machine learning techniques. Large datasets and user-friendly programming libraries accelerate the creation of surrogate models with, in many cases, little human parametrization effort. However, materials design and discovery often require physical interpretability that can be difficult to extract from complex machine-learned models. Moreover, the integration of physical insight during the fitting process, a prerequisite of traditional modeling, is sometimes overlooked with machine learning. This work discusses a new approach to fitting interatomic potentials using a cubic B-spline basis set and regularized linear regression. The introduced ultra-fast (UF) potentials, based on two-body distances and three-body angles, are as interpretable and fast to evaluate as traditional empirical potentials. In addition, UF potentials are nearly as accurate as state-of-the-art machine-learning potentials that are two to four orders of magnitude slower. Benchmarks based on first-principles calculations of bulk configurations, point defects, surfaces, and dislocations in bcc tungsten demonstrate the speed and accuracy of UF potentials. Furthermore, melting point predictions involving thousands of atoms demonstrate their extrapolative capacity and applicability for larger simulations. This work also explores the use of machine learning to generate interpretable analytic equations to model feature-property relationships. Using symbolic regression, we identify new equations to address the problem of predicting the critical temperature of new electron-phonon paired superconductors. Using Eliashberg theory and data from real materials, augmented by artificially generated models, we learn new equations that correct for the systematic underprediction of the critical temperature associated with widely-used, existing models. We also demonstrate the use of the framework to learn a classifier for octahedral tilting in oxide perovskites. In both applications, we find that the new equations are comparable in prediction error to machine-learning models of higher complexity, like artificial neural networks, and are more interpretable.
Historically, there was a tendency amongst electronic clinical outcome assessment (eCOA) service providers and their sponsor clients to prevent changes to eCOA data, particularly for electronic ...patient-reported outcome (ePRO) measures, or to allow changes to eCOA data but only after receiving sponsor approval to do so. Further, sponsors were also approving or denying data change requests (DCRs).From 2018 to 2022, 45 representatives of Critical Path (C-Path) Institute’s Patient-Reported Outcome (PRO) Consortium, eCOA Consortium (previously ePRO Consortium), and the eClinical Forum collaborated to develop guidelines on changes to eCOA data. These were compiled into the present manuscript, which outlines best practices aligned to latest health authority guidance, and based on expertise from all three organizations.This work coincided with a shift in clinical trial industry practices and emergence of regulations and quality functions that require that investigative sites, not sponsors, be responsible for the maintenance of accurate source records. In other words, the question was not whether, but how, PRO data should be changed. The resulting set of core principles can become the foundation upon which sponsors, investigators and eCOA providers can work together on DCRs.To achieve that, eCOA studies must:• be setup in a manner that minimizes potential user errors;• have a documented oversight plan and documented site staff training;• follow clearly defined and standard workflows–but also mitigate undefined change types;• allow trial documentation per ALCOA+ principles; and• induce sponsors into timely reviews and reconciliations of discordant data (i.e., via sensitivity analysis) and not by arbitrarily overruling the investigators.
