Mathematical Sciences
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Mathematical research at MSU is focused primarily on related topics in pure and applied mathematics. Research programs complement each other and are often applied to problems in science and engineering. Research in statistics encompasses a broad range of theoretical and applied topics. Because the statisticians are actively engaged in interdisciplinary work, much of the statistical research is directed toward practical problems. Mathematics education faculty are active in both qualitative and quantitative experimental research areas. These include teacher preparation, coaching and mentoring for in-service teachers, online learning and curriculum development.
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Item Joint Spatial Modeling Bridges the Gap Between Disparate Disease Surveillance and Population Monitoring Efforts Informing Conservation of At-risk Bat Species(Springer Science and Business Media LLC, 2024-02) Stratton, Christian; Irvine, Kathryn M.; Banner, Katharine M.; Almberg, Emily S.; Bachen, Dan; Smucker, KristinaWhite-Nose Syndrome (WNS) is a wildlife disease that has decimated hibernating bats since its introduction in North America in 2006. As the disease spreads westward, assessing the potentially differential impact of the disease on western bat species is an urgent conservation need. The statistical challenge is that the disease surveillance and species response monitoring data are not co-located, available at different spatial resolutions, non-Gaussian, and subject to observation error requiring a novel extension to spatially misaligned regression models for analysis. Previous work motivated by epidemiology applications has proposed two-step approaches that overcome the spatial misalignment while intentionally preventing the human health outcome from informing estimation of exposure. In our application, the impacted animals contribute to spreading the fungus that causes WNS, motivating development of a joint framework that exploits the known biological relationship. We introduce a Bayesian, joint spatial modeling framework that provides inferences about the impact of WNS on measures of relative bat activity and accounts for the uncertainty in estimation of WNS presence at non-surveyed locations. Our simulations demonstrate that the joint model produced more precise estimates of disease occurrence and unbiased estimates of the association between disease presence and the count response relative to competing two-step approaches. Our statistical framework provides a solution that leverages disparate monitoring activities and informs species conservation across large landscapes. Stan code and documentation are provided to facilitate access and adaptation for other wildlife disease applications.Item Statistical assessment on determining local presence of rare bat species(Wiley, 2022-06) Irvine, Kathryn M.; Banner, Katharine M.; Stratton, Christian; Ford, William M.; Reichert, Brian E.Surveying cryptic, sparsely distributed taxa using autonomous recording units,although cost-effective, provides imperfect knowledge about species presence.Summertime bat acoustic surveys in North America exemplify the challenges with characterizing sources of uncertainty: observation error, inability to census populations, and natural stochastic variation. Statistical uncertainty, if not considered thoroughly, hampers determining rare species presence accurately and/or estimating range wide status and trends with suitable precision. Bat acoustic data are processed using an automated workflow in which proprietary or open-source algorithms assign a species label to each recorded high-frequency echolocation sequence. A false-negative occurs, if a species is actually present but not recorded and/or all recordings from the species are of such poor quality that a correct species identity cannot be assigned to any observation. False positives for a focal species are a direct result of the presence and incorrect identification of a recording from another species. We compare four analytical approaches in terms of parameter estimation and their resulting(in)correct decisions regarding species presence or absence using realistic data-generating scenarios for bat acoustic data within a simulation study. The cur-rent standard for deciding species presence or absence uses a multinomial likelihood-ratio test p value (maximum likelihood estimate [MLE]-metric) that accounts for known species misidentifications, but not imperfect detection and only returns a binary outcome (evidence of presence or not). We found that the MLE-metric had estimated median correct decisions less than 60% for presence and greater than 85% for absence. Alternatively, a multispecies count detection model was equivalent to or better than the MLE-metric for correct claims of rare species presence or absence using the posterior probability a species was present at a site and, importantly, provided unbiased estimates of relative activity and probability of occurrence, creating opportunities for reducing posterior uncertainty through the inclusion of meaningful covariates. Single-species occupancy models with and without false-positive detections removed were insufficient for determining local presence because of substantially biased occurrence and detection probabilities. We propose solutions to potential barriers for integrating local, short-term and range wide, long-term acoustic surveys within a cohesive statistical framework that facilitates determining local species presence with uncertainty concurrent with estimating species–environment relationships.Item Coupling validation effort with in situ bioacoustic data improves estimating relative activity and occupancy for multiple species with cross‐species misclassifications(Wiley, 2022-03) Stratton, Christian; Irvine, Kathryn M.; Banner, Katharine M.; Wright, Wilson J.; Lausen, Cori; Rae, JasonThe increasing complexity and pace of ecological change requires natural resource managers to consider entire species assemblages. Acoustic recording units (ARUs) require minimal cost and effort to deploy and inform relative activity, or encounter rates, for multiple species simultaneously. ARU-based surveys require post-processing of the recordings via software algorithms that assign a species label to each recording. The automated classification process can result in cross-species misidentifications that should be accounted for when employing statistical modelling for conservation decision-making. Using simulation and ARU-based detection counts from 17 bat species in British Columbia, Canada, we investigate three strategies for adjusting statistical inference for species misclassification: (a) ‘coupling’ ambiguous and unambiguous detections by validating a subset of survey events post-hoc, (b) using a calibration dataset on the software algorithm's (in)accuracy for species identification or (c) specifying informative Bayesian priors on classification probabilities. We explore the impact of different Bayesian prior specifications for the classification probabilities on posterior estimation. We then consider how the quantity of data validated post-hoc impacts model convergence and resulting inferences for bat species relative activity as related to nightly conditions and yearly site occupancy after accounting for site-level environmental variables. Coupled methods resulted in less bias and uncertainty when estimating relative activity and species classification probabilities relative to calibration approaches. We found that species that were difficult-to-detect and those that were often inaccurately identified by the software required more validation effort than more easily detected and/or identified species. Our results suggest that, when possible, acoustic surveys should rely on coupled validated detection information to account for false-positive detections, rather than uncoupled calibration datasets. However, if the assemblage of interest contains a large number of rarely detected or less prevalent species, an intractable amount of effort may be required, suggesting there are benefits to curating a calibration dataset that is representative of the observation process. Our findings provide insights into the practical challenges associated with statistical analyses of ARU data and possible analytical solutions to support reliable and cost-effective decision-making for wildlife conservation/management in the face of known sources of observation errors.Item Considerations for assessing model averaging of regression coefficients(2016-08) Banner, Katharine M.; Higgs, Megan D."Model choice is usually an inevitable source of uncertainty in model-based statistical analyses. While the focus of model choice was traditionally on methods for choosing a single model, methods to formally account for multiple models within a single analysis are now accessible to many researchers. The specific technique of model averaging was developed to improve predictive ability by combining predictions from a set of models. However, it is now often used to average regression coefficients across multiple models with the ultimate goal of capturing a variable\'s overall effect. This use of model averaging implicitly assumes the same parameter exists across models so that averaging is sensible. While this assumption may initially seem tenable, regression coefficients associated with particular explanatory variables may not hold equivalent interpretations across all of the models in which they appear, making explanatory inference about covariates challenging. Accessibility to easily implementable software, concerns about being criticized for ignoring model uncertainty, and the chance to avoid having to justify choice of a final model have all led to the increasing popularity of model averaging in practice. We see a gap between the theoretical development of model averaging and its current use in practice, potentially leaving well-intentioned researchers with unclear inferences or difficulties justifying reasons for using (or not using) model averaging. We attempt to narrow this gap by revisiting some relevant foundations of regression modeling, suggesting more explicit notation and graphical tools, and discussing how individual model results arecombined to obtain a model averaged result. Our goal is to help researchers make informed decisions about model averaging and to encourage question-focused modeling over method-focused modeling. "Item Pathogen prevalence and abundance in honey bee colonies involved in almond pollination(2016-03) Cavigli, Ian; Daughenbaugh, Katie F.; Martin, Madison; Lerch, Michael D.; Banner, Katharine M.; Garcia, Emma; Brutscher, Laura M.; Flenniken, Michelle L.Honey bees are important pollinators of agricultural crops. Since 2006, US beekeepers have experienced high annual honey bee colony losses, which may be attributed to multiple abiotic and biotic factors, including pathogens. However, the relative importance of these factors has not been fully elucidated. To identify the most prevalent pathogens and investigate the relationship between colony strength and health, we assessed pathogen occurrence, prevalence, and abundance in Western US honey bee colonies involved in almond pollination. The most prevalent pathogens were Black queen cell virus (BQCV), Lake Sinai virus 2 (LSV2), Sacbrood virus (SBV), Nosema ceranae, and trypanosomatids. Our results indicated that pathogen prevalence and abundance were associated with both sampling date and beekeeping operation, that prevalence was highest in honey bee samples obtained immediately after almond pollination, and that weak colonies had a greater mean pathogen prevalence than strong colonies.