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Bayesian techniques are being quickly adopted in sports settings. The volume and variety of data produced in sports activities over the past years and the availability of software packages for Bayesian computation have contributed positively to this growth. This article provides a brief review of the latest advances in Bayesian statistics in sports, covering methods and applications. In the literature surveyed, we found a similar number of publications between 2013 and 2017 and those published in the three previous decades, which is an indication of the rapid adoption rate of Bayesian methods in sports.

This paper discusses statistical and computational aspects of inverse or parameter estimation problems based on Ordinary Least Squares (OLS) and Generalized Least Squares (GLS) with appropriate corresponding data noise assumptions of constant variance and nonconstant variance (relative error), respectively. Among the topics addressed are mathematical model, statistical model and data assumptions, and some techniques (residual plots, sensitivity analysis, model comparison tests) for verifying these. The ideas are illustrated throughout with the popular logistic growth model of Verhulst and Pearl as well as with a recently developed population-level model of pneumococcal disease spread., Prepared in cooperation with the Center for Quantitative Sciences in Biomedicine, North Carolina State University, Raleigh, NC and the Department of Mathematics and Statistics, Arizona State University, Tempe, AZ.

A system of ordinary differential equations is formulated to describe the pathogenesis of HIV infection, wherein certain important features that have been shown important by recent experimental research are incorporated in the model. These include the role of CD4+ memory cells that serve as a major reservoir of latently infected cells, a critical role for T-helper cells in the generation of CD8 memory cells capable of efficient recall response, and stimulation by antigens other than HIV. A stability analysis illustrates the capability of this model in admitting multiple locally asymptotically stable (locally a.s.) off-treatment equilibria. The phenomenon of "viral blips" experienced by some patients on therapy with viral load levels suppressed below the detection limit is also investigated. Censored clinical data is used to demonstrate that this model provides reasonable fits to all the patient data available for this study and, moreover, that it exhibits impressive predictive capability., Prepared in cooperation with Massachusetts General Hospital and Harvard Medical School, Boston, MA. Sponsored in part by National Institute of Allergy and Infectious Disease (NIAID), National Institutes of Health (NIH), grant no. 9R01AI071915-05. The original document contains color images.

A robust two-stage approach is used to reanalyze the repeated measurements from an experiment of airway responsiveness in rats randomized to long-term exposure at four ozone doses. The concentration-response data generated for each rat may be represented as a hierarchical nonlinear model encompasing the sources of variation within and between individuals, leading to the possibility of outliers at each sampling level. By estimating an individual profile for each rat, the conditional modeling approach can assess the adequacy of an assumed mean model, a fundamental advantage not intrinsic to marginal techniques. The two-stage population inference is based on the estimated individual parameters, thus maintaining an intuitive appeal to the toxicologists who traditionally have fitted a separate curve for each animal and then applied ANOVA to the summary statistics. However, we formally adjust the standard errors for the extra variability due to the initial estimation of the individual parameters and also allow for their within-rat correlation. The robust two-stage method appropriately downweights the aberrant responses arising sporadically within individuals and, more importantly, the rats which may be outlying relative to the usual population variation. The true effect of chronic ozone exposure, including a significant gender interaction, may be masked by a few rats which exert undue influence on the population estimates in a nonrobust analysis. © 2003 American Statistical Association and the International Biometric Society.

Assay data are often fit by a nonlinear regression model incorporating heterogeneity of variance, as ion radioimmunoassay, for example. Typically, the standard deviation of the response is taken to be proportional to a power theta of the mean. There is considerable empirical evidence suggesting that for assays of a reseasonable size, how one estimates the parameter theta does not greatly affect how well one estimates the the mean regression function. An additional component of assay analysis is the estimation of auxillary constructs such as the minimum detectable concentration, for which many definitions exist; we focus on one such definition. The minimum detectable concentration depends both on theta and the mean regression function. We compare three standard method of estimating the parameter theta due to Rodbard (1978), Raab (1981a) and Carroll and Ruppert (1982b). When duplicate counts are taken at each concentration, the first method is only 20% efficient asymptotically in comparison to the third, and the resulting estimate of the minimum detectable concentration is asymptotically 3.3 times more variable for first than the third. Less dramatic results obtain for the second estimator compared to the third; this estimator is still not efficient, however. Simulation results and an example are supportive of the asymptotic theory. Keywords: Least squares method., Prepared in cooperation with Eli Lilly & Company.

We study the method of extended quasi-likelihood estimation and inference of a variance function recently proposed by Nelder & Pregibon (1987). The estimates are inconsistent in general, and the test levels can be biased, but in many cases such as the exponential family the inconsistency and bias will not be a major concern. Extended quasi-likelihood is compared with Carroll & Ruppert's (1982) pseudo-likelihood method, which gives consistent estimates and, when slightly modified, asymptotically unbiased tests. We quantify the showing in this instance that the two estimates are closely related and may be asymptotically equivalent in many important cases. However, in some cases outside the exponential family, an asymptotic bias can persist. Heteroscedastic regression model.

Assay data are often fitted by a nonlinear regression model incorporating heterogeneity of variance. Typically, the standard deviation of the response is taken to be proportional to a power Theta of the mean. There is considerable empirical evidence suggesting that for assays of a resonable size, how one estimates the parameter Theta does not greatly affect how well one estimates the mean regression function. An additional component of assay analysis is the estimation of auxillary constructs such as the minimum detectable concentration, for which many definitions exist; we focus on one such definition. The minimum detectable concentration depends both on Theta and the mean regression function. We compare standard methods of estimating the parameter Theta. When duplicate counts are taken at each concentration, the first method is only 20% efficient asymptotically in comparison to the fourth for normal data, and in an example the resulting estimate of the minimum detectable concentration is asymptotically 3.7 times more variable. Less dramatic results obtain for the second and third estimators compared to the fourth. Simulation results and an example support the asymptotic theory. The results have implications in applications other than the assay problem in which heterogeneity of variance and issues of calibration arise.