COVID-19 cases and deaths in the United States follow Taylor's law for heavy-tailed distributions with infinite variance.

The spatial and temporal patterns of severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) cases and COVID-19 deaths in the United States are poorly understood. We show that variations in the cumulative reported cases and deaths by county, state, and date exemplify Taylor's law of fluctuation scaling. Specifically, on day 1 of each month from April 2020 through June 2021, each state's variance (across its counties) of cases is nearly proportional to its squared mean of cases. COVID-19 deaths behave similarly. The lower 99% of counts of cases and deaths across all counties are approximately lognormally distributed. Unexpectedly, the largest 1% of counts are approximately Pareto distributed, with a tail index that implies a finite mean and an infinite variance. We explain why the counts across the entire distribution conform to Taylor's law with exponent two using models and mathematics. The finding of infinite variance has practical consequences. Local jurisdictions (counties, states, and countries) that are planning for prevention and care of largely unvaccinated populations should anticipate the rare but extremely high counts of cases and deaths that occur in distributions with infinite variance. Jurisdictions should prepare collaborative responses across boundaries, because extremely high local counts of cases and deaths may vary beyond the resources of any local jurisdiction.

Blinded sample size re-estimation for comparing over-dispersed count data incorporating follow-up lengths.

Blinded sample size re-estimation (BSSR) is an adaptive design to prevent the power reduction caused by misspecifications of the nuisance parameters in the sample size calculation of comparative clinical trials. However, conventional BSSR methods used for overdispersed count data may not recover the power as expected under the misspecification of the working variance function introduced by the specified analysis model. In this article, we propose a BSSR method that is robust to the misspecification of the working variance function. A weighted estimator of the dispersion parameter for the BSSR is derived, where the weights are introduced to incorporate the difference in the distribution of follow-up length between the interim analysis with BSSR and the final analysis. Simulation studies demonstrated the power of the proposed BSSR method was relatively stable under misspecifications of the working variance function. An application to a hypothetical randomized clinical trial of a treatment to reduce exacerbation rate in patients with chronic obstructive pulmonary disease is provided.

Random regression test-day models to describe milk production and fatty acid traits in first lactation Walloon Holstein cows.

We investigated the use of different Legendre polynomial orders to estimate genetic parameters for milk production and fatty acid (FA) traits in the first lactation Walloon Holstein cows. The data set comprised 302,684 test-day records of milk yield, fat and protein contents, and FAs generated by mid-infrared (MIR) spectroscopy, C16:0 (palmitic acid), C18:1 cis-9 (oleic acid), LCFAs (long-chain FAs), SFAs (saturated FAs) and UFAs (unsaturated FAs) were studied. The models included random regression coefficients for herd-year of calving (h), additive genetic (a) and permanent environment (p) effects. The selection of the best random regression model (RRM) was based on the deviance information criterion (DIC), and genetic parameters were estimated via a Bayesian approach. For all analysed random effects, DIC values decreased as the order of the Legendre polynomials increased. Best-fit models had fifth-order (degree 4) for the p effect and ranged from second- to fifth-order (degree 1-4) for the a and h effects (LEGhap: LEG555 for milk yield and protein content; LEG335 for fat content and SFA; LEG545 for C16:0 and UFA; and LEG535 for C18:1 cis-9 and LCFA). Based on the best-fit models, an effect of overcorrection was observed in early lactation (5-35 days in milk [DIM]). On the contrary, third-order (LEG333; degree 2) models showed flat residual trajectories throughout lactation. In general, the estimates of genetic variance tended to increase over DIM, for all traits. Heritabilities for milk production traits ranged from 0.11 to 0.58. Milk FA heritabilities ranged from low-to-high magnitude (0.03-0.56). High Spearman correlations (>0.90 for all bulls and >0.97 for top 100) were found among breeding values for 155 and 305 DIM between the best RRM and LEG333 model. Therefore, third-order Legendre polynomials seem to be most parsimonious and sufficient to describe milk production and FA traits in Walloon Holstein cows.

Modeling the rate of HIV testing from repeated binary data amidst potential never-testers.

Many longitudinal studies with a binary outcome measure involve a fraction of subjects with a homogeneous response profile. In our motivating data set, a study on the rate of human immunodeficiency virus (HIV) self-testing in a population of men who have sex with men (MSM), a substantial proportion of the subjects did not self-test during the follow-up study. The observed data in this context consist of a binary sequence for each subject indicating whether or not that subject experienced any events between consecutive observation time points, so subjects who never self-tested were observed to have a response vector consisting entirely of zeros. Conventional longitudinal analysis is not equipped to handle questions regarding the rate of events (as opposed to the odds, as in the classical logistic regression model). With the exception of discrete mixture models, such methods are also not equipped to handle settings in which there may exist a group of subjects for whom no events will ever occur, i.e. a so-called "never-responder" group. In this article, we model the observed data assuming that events occur according to some unobserved continuous-time stochastic process. In particular, we consider the underlying subject-specific processes to be Poisson conditional on some unobserved frailty, leading to a natural focus on modeling event rates. Specifically, we propose to use the power variance function (PVF) family of frailty distributions, which contains both the gamma and inverse Gaussian distributions as special cases and allows for the existence of a class of subjects having zero frailty. We generalize a computational algorithm developed for a log-gamma random intercept model (Conaway, 1990. A random effects model for binary data. Biometrics46, 317-328) to compute the exact marginal likelihood, which is then maximized to obtain estimates of model parameters. We conduct simulation studies, exploring the performance of the proposed method in comparison with competitors. Applying the PVF as well as a Gaussian random intercept model and a corresponding discrete mixture model to our motivating data set, we conclude that the group assigned to receive follow-up messages via SMS was self-testing at a significantly lower rate than the control group, but that there is no evidence to support the existence of a group of never-testers.

Linking parasite populations in hosts to parasite populations in space through Taylor's law and the negative binomial distribution.

The spatial distribution of individuals of any species is a basic concern of ecology. The spatial distribution of parasites matters to control and conservation of parasites that affect human and nonhuman populations. This paper develops a quantitative theory to predict the spatial distribution of parasites based on the distribution of parasites in hosts and the spatial distribution of hosts. Four models are tested against observations of metazoan hosts and their parasites in littoral zones of four lakes in Otago, New Zealand. These models differ in two dichotomous assumptions, constituting a 2 × 2 theoretical design. One assumption specifies whether the variance function of the number of parasites per host individual is described by Taylor's law (TL) or the negative binomial distribution (NBD). The other assumption specifies whether the numbers of parasite individuals within each host in a square meter of habitat are independent or perfectly correlated among host individuals. We find empirically that the variance-mean relationship of the numbers of parasites per square meter is very well described by TL but is not well described by NBD. Two models that posit perfect correlation of the parasite loads of hosts in a square meter of habitat approximate observations much better than two models that posit independence of parasite loads of hosts in a square meter, regardless of whether the variance-mean relationship of parasites per host individual obeys TL or NBD. We infer that high local interhost correlations in parasite load strongly influence the spatial distribution of parasites. Local hotspots could influence control and conservation of parasites.

Benefits and risks of diversification for individual fishers.

Individuals relying on natural resource extraction for their livelihood face high income variability driven by a mix of environmental, biological, management, and economic factors. Key to managing these industries is identifying how regulatory actions and individual behavior affect income variability, financial risk, and, by extension, the economic stability and the sustainable use of natural resources. In commercial fisheries, communities and vessels fishing a greater diversity of species have less revenue variability than those fishing fewer species. However, it is unclear whether these benefits extend to the actions of individual fishers and how year-to-year changes in diversification affect revenue and revenue variability. Here, we evaluate two axes by which fishers in Alaska can diversify fishing activities. We show that, despite increasing specialization over the last 30 years, fishing a set of permits with higher species diversity reduces individual revenue variability, and fishing an additional permit is associated with higher revenue and lower variability. However, increasing species diversity within the constraints of existing permits has a fishery-dependent effect on revenue and is usually (87% probability) associated with increased revenue uncertainty the following year. Our results demonstrate that the most effective option for individuals to decrease revenue variability is to participate in additional or more diverse fisheries. However, this option is expensive, often limited by regulations such as catch share programs, and consequently unavailable to many individuals. With increasing climatic variability, it will be particularly important that individuals relying on natural resources for their livelihood have effective strategies to reduce financial risk.

Analysis of longitudinal adherence data: Hitting the target with consistency.

Typical descriptions of partial adherence summarize to what extent a patient follows a prescribed treatment plan. Another aspect of adherence relates to how consistently a patient behaves. We distinguish these two notions as "adherence target" and "adherence consistency." Most research on adherence focuses on the former concept, but the latter can be a useful idea in assessing patients' compliance. Quantifying these two notions goes beyond typical summaries of center and spread, because they may be related to each other. That is, the mean is often related to the standard deviation via the variance function. Our approach defines a parameter that measures consistency separate from the adherence target. For exponential dispersion or quasi-likelihood families, the parameter corresponds to the dispersion parameter. Assessing adherence consistency to medical treatment requires longitudinal data, which can be collected with new technology. Two examples from pediatric medicine demonstrate that detangling the notion of consistency from the adherence target is useful.

Critique of methods for estimating heats in isothermal titration calorimetry.

Isothermal titration calorimetry data recorded on a MicroCal/Malvern VP-ITC instrument for water-water blanks and for dilution of aqueous solutions of BaCl2 and Ba(NO3)2 are analyzed using Origin software, the freeware NITPIC program, and in-house algorithms, to compare precisions for estimating the heat per injection q. The data cover temperatures 6-47 °C, injection volumes 4-40 µL, and average heats 0-200 µcal. For water-water blanks, where baseline noise limits precision, NITPIC and the in-house algorithm achieve precisions of 0.05 µcal, which is better than Origin by a factor of 4. The precision differences decrease with increasing |q|, becoming insignificant for |q| > 200 µcal. In its default mode, NITPIC underestimates |q| for peaks with incomplete return to baseline, but the shortfall can be largely corrected by overriding the default injection time parameter. The variance estimates from 26 dilution experiments are used to assess the data variance function. The results determine the conditions under which weighted least squares should be used to estimate thermodynamic parameters from ITC data.

Power and sample size calculation incorporating misspecifications of the variance function in comparative clinical trials with over-dispersed count data.

Over-dispersed count data are frequently observed in clinical trials where the primary endpoint is occurrence of clinical events. Sample sizes of comparative clinical trials with these data are typically calculated under negative binomial models or quasi-Poisson models with specified variance functions, or under the assumption that the specified "working" variance functions are correctly specified. In this article, we propose a sample size formula anticipating misspecifications of the working variance function. We derived a method based on the asymptotic distribution of a Wald test statistic with a sandwich-type robust variance estimator under quasi-Poisson models. Under misspecifications of the working variance function, the asymptotic variance of the estimator of the treatment effect is expressed as a form involving both true and working variance functions. Our sample size formula includes several existing formulas as special cases when the working variance function is correctly specified as the true variance function. We also consider a sensitivity analysis for possible misspecifications of the "true" variance function when estimating sample sizes using our formula. A simulation study demonstrated the adequacy of our formulas in finite sample size settings. An application to a clinical trial to evaluate the treatment effect on prevention of COPD exacerbation is provided.

Estimation of an inter-rater intra-class correlation coefficient that overcomes common assumption violations in the assessment of health measurement scales.

BACKGROUND: Intraclass correlation coefficients (ICC) are recommended for the assessment of the reliability of measurement scales. However, the ICC is subject to a variety of statistical assumptions such as normality and stable variance, which are rarely considered in health applications. METHODS: A Bayesian approach using hierarchical regression and variance-function modeling is proposed to estimate the ICC with emphasis on accounting for heterogeneous variances across a measurement scale. As an application, we review the implementation of using an ICC to evaluate the reliability of Observer OPTION5, an instrument which used trained raters to evaluate the level of Shared Decision Making between clinicians and patients. The study used two raters to evaluate recordings of 311 clinical encounters across three studies to evaluate the impact of using a Personal Decision Aid over usual care. We particularly focus on deriving an estimate for the ICC when multiple studies are being considered as part of the data. RESULTS: The results demonstrate that ICC varies substantially across studies and patient-physician encounters within studies. Using the new framework we developed, the study-specific ICCs were estimated to be 0.821, 0.295, and 0.644. If the within- and between-encounter variances were assumed to be the same across studies, the estimated within-study ICC was 0.609. If heteroscedasticity is not properly adjusted for, the within-study ICC estimate was inflated to be as high as 0.640. Finally, if the data were pooled across studies without accounting for the variability between studies then ICC estimates were further inflated by approximately 0.02 while formerly allowing for between study variation in the ICC inflated its estimated value by approximately 0.066 to 0.072 depending on the model. CONCLUSION: We demonstrated that misuse of the ICC statistics under common assumption violations leads to misleading and likely inflated estimates of interrater reliability. A statistical analysis that overcomes these violations by expanding the standard statistical model to account for them leads to estimates that are a better reflection of a measurement scale's reliability while maintaining ease of interpretation. Bayesian methods are particularly well suited to estimating the expanded statistical model.

Random sampling of skewed distributions implies Taylor's power law of fluctuation scaling.

Taylor's law (TL), a widely verified quantitative pattern in ecology and other sciences, describes the variance in a species' population density (or other nonnegative quantity) as a power-law function of the mean density (or other nonnegative quantity): Approximately, variance = a(mean)(b), a > 0. Multiple mechanisms have been proposed to explain and interpret TL. Here, we show analytically that observations randomly sampled in blocks from any skewed frequency distribution with four finite moments give rise to TL. We do not claim this is the only way TL arises. We give approximate formulae for the TL parameters and their uncertainty. In computer simulations and an empirical example using basal area densities of red oak trees from Black Rock Forest, our formulae agree with the estimates obtained by least-squares regression. Our results show that the correlated sampling variation of the mean and variance of skewed distributions is statistically sufficient to explain TL under random sampling, without the intervention of any biological or behavioral mechanisms. This finding connects TL with the underlying distribution of population density (or other nonnegative quantity) and provides a baseline against which more complex mechanisms of TL can be compared.

OPTIMAL SHRINKAGE ESTIMATION OF MEAN PARAMETERS IN FAMILY OF DISTRIBUTIONS WITH QUADRATIC VARIANCE.

This paper discusses the simultaneous inference of mean parameters in a family of distributions with quadratic variance function. We first introduce a class of semi-parametric/parametric shrinkage estimators and establish their asymptotic optimality properties. Two specific cases, the location-scale family and the natural exponential family with quadratic variance function, are then studied in detail. We conduct a comprehensive simulation study to compare the performance of the proposed methods with existing shrinkage estimators. We also apply the method to real data and obtain encouraging results.

Testing for parallelism in the heteroscedastic four-parameter logistic model.

For bioassay data in drug discovery and development, it is often important to test for parallelism of the mean response curves for two preparations, such as a test sample and a reference sample in determining the potency of the test preparation relative to the reference standard. For assessing parallelism under a four-parameter logistic model, tests of the parallelism hypothesis may be conducted based on the equivalence t-test or the traditional F-test. However, bioassay data often have heterogeneous variance across dose levels. Specifically, the variance of the response may be a function of the mean, frequently modeled as a power of the mean. Therefore, in this article we discuss estimation and tests for parallelism under the power variance function. Two examples are considered to illustrate the estimation and testing approaches described. A simulation study is also presented to compare the empirical properties of the tests under the power variance function in comparison to the results from ordinary least squares fits, which ignore the non-constant variance pattern.

Stochastic population dynamics in a Markovian environment implies Taylor's power law of fluctuation scaling.

Taylor's power law of fluctuation scaling (TL) states that for population density, population abundance, biomass density, biomass abundance, cell mass, protein copy number, or any other nonnegative-valued random variable in which the mean and the variance are positive, variance=a(mean)(b),a>0, or equivalently log variance=loga+b×log mean. Many empirical examples and practical applications of TL are known, but understanding of TL's origins and interpretations remains incomplete. We show here that, as time becomes large, TL arises from multiplicative population growth in which successive random factors are chosen by a Markov chain. We give exact formulas for a and b in terms of the Markov transition matrix and the values of the successive multiplicative factors. In this model, the mean and variance asymptotically increase exponentially if and only if b>2 and asymptotically decrease exponentially if and only if b<2.

Methods and results for small area estimation using smoking data from the 2008 National Health Interview Survey.

The National Health Interview Survey, conducted by the National Center for Health Statistics, is designed to provide reliable design-based estimates for a wide range of health-related variables for national and four major geographical regions of the USA. However, state-level or substate-level estimates are likely to be unreliable because they are based on small sample sizes. In this paper, we compare the efficiency of different area-level models in estimating smoking prevalence for the 50 US states and the District of Columbia. Our study is based on survey data from the 2008 National Health Interview Survey in conjunction with a number of potentially related auxiliary variables obtained from the American Community Survey, an ongoing large complex survey conducted by the US Census. A major portion of this study is devoted to the investigation of several methods for estimating survey sampling variances needed to implement an area-level hierarchical model. Based on our findings, a hierarchical Bayesian method that uses a survey-adjusted random sampling variance model to capture the complex survey sampling variability appears to be somewhat superior to the other considered area-level models in accounting for small sample behavior of estimated survey sampling variances. Several diagnostic procedures are presented to compare the proposed methods.

Single-step genomic BLUP with many metafounders.

Single-step genomic BLUP (ssGBLUP) model for routine genomic prediction of breeding values is developed intensively for many dairy cattle populations. Compatibility between the genomic (G) and the pedigree (A) relationship matrices remains an important challenge required in ssGBLUP. The compatibility relates to the amount of missing pedigree information. There are two prevailing approaches to account for the incomplete pedigree information: unknown parent groups (UPG) and metafounders (MF). unknown parent groups have been used routinely in pedigree-based evaluations to account for the differences in genetic level between groups of animals with missing parents. The MF approach is an extension of the UPG approach. The MF approach defines MF which are related pseudo-individuals. The MF approach needs a Γ matrix of the size number of MF to describe relationships between MF. The UPG and MF can be the same. However, the challenge in the MF approach is the estimation of Γ having many MF, typically needed in dairy cattle. In our study, we present an approach to fit the same amount of MF as UPG in ssGBLUP with Woodbury matrix identity (ssGTBLUP). We used 305-day milk, protein, and fat yield data from the DFS (Denmark, Finland, Sweden) Red Dairy cattle population. The pedigree had more than 6 million animals of which 207,475 were genotyped. We constructed the preliminary gamma matrix (Γ pre ) with 29 MF which was expanded to 148 MF by a covariance function (Γ 148). The quality of the extrapolation of the Γ pre matrix was studied by comparing average off-diagonal elements between breed groups. On average relationships among MF in Γ 148 were 1.8% higher than in Γ pre . The use of Γ 148 increased the correlation between the G and A matrices by 0.13 and 0.11 for the diagonal and off-diagonal elements, respectively. [G]EBV were predicted using the ssGTBLUP and Pedigree-BLUP models with the MF and UPG. The prediction reliabilities were slightly higher for the ssGTBLUP model using MF than UPG. The ssGBLUP MF model showed less overprediction compared to other models.

Mean and Covariance Estimation for Functional Snippets.

We consider estimation of mean and covariance functions of functional snippets, which are short segments of functions possibly observed irregularly on an individual specific subinterval that is much shorter than the entire study interval. Estimation of the covariance function for functional snippets is challenging since information for the far off-diagonal regions of the covariance structure is completely missing. We address this difficulty by decomposing the covariance function into a variance function component and a correlation function component. The variance function can be effectively estimated nonparametrically, while the correlation part is modeled parametrically, possibly with an increasing number of parameters, to handle the missing information in the far off-diagonal regions. Both theoretical analysis and numerical simulations suggest that this hybrid strategy is effective. In addition, we propose a new estimator for the variance of measurement errors and analyze its asymptotic properties. This estimator is required for the estimation of the variance function from noisy measurements.

Spatial heterogeneity of soil heavy metal content in Baiyinhua mining area, Inner Mongolia, China.

To evaluate the spatial distribution characteristics of soil heavy metal content in Xiwuzhumuqin Banner, Inner Mongolia, we analyzed the spatial heterogeneity of soil Cu, Cr, Pb, and Mn contents within 8 km distance of the mining area. Results showed that the contents of Cu, Cr, Pb and Mn in soil were 12.7, 32.6, 29.9 and 201.3 mg·kg-1, and their coefficients of variation were 26.8%, 33.9%, 27.1% and 45.7%, respectively. According to the model fitting by semi-variance function, the spatial distribution of Cu, Cr, Pb and Mn were consistent with the Gaussian model, Gaussian model, Gaussian model and linear model, respectively. Results of the spatial distribution pattern analysis showed that the spatial correlation levels of Mn, Cr and Cu were high, which were mainly affected by structural factors, but little affected by random factors. The spatial correlation level of Pb element was moderate, which was affected by both structural factors and random factors. Results of the fractal dimension analysis showed that the spatial distribution of four heavy metal contents was simple. Combined with 2D and 3D views, the four types of heavy metals all presented gradient distribution, which decreased with the increases of distance. The contents of Cu and Mn were mainly concentrated within 1.5 km from the mining area, while Cr and Pb were mainly concentrated within 2.0 km and 3.0 km from the mining area, respectively.

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We conducted an experiment to test the characteristics and differences of the spatial distribution of constructive species Stipa breviflora at different scales under different stocking rates in the S. breviflora desert steppe in Siziwang Banner, Inner Mongolia. The spatial distribution of S. breviflora under four treatments (control, light grazing, moderate grazing, and heavy grazing) at different scales (small scale as 1 m×1 m and mesoscale as 5 m×10 m) were analyzed. The results showed that the population density of S. breviflora at mesoscale in the control and light grazing was significantly lower than that at the small scale. Grazing significantly increased the population density of S. breviflora in the meso- and small scales. At the small scale, the population distribution of S. breviflora in the control, light grazing, moderate grazing, and heavy grazing treatments conformed to linear, exponential, exponential and exponential models, respectively, and Gaussian, exponential, Gaussian and exponential models at mesoscale fitted by semi-variance function. The spatial distribution pattern at small scales in the control was simple and better but was more complex and poorer under the heavy grazing. At the mesoscale, it was simple and better under the heavy grazing but complex and poor under the moderate grazing. The spatial heterogeneity of S. breviflora population reduced and were more uniform under the moderate and heavy grazing at meso- and small scales. In addition, the trend of population distribution in the enclosure, moderate and heavy grazing were generally the same, while light grazing showed inconsistent trend at different scales.

Long-term frailty modeling using a non-proportional hazards model: Application with a melanoma dataset.

The semiparametric Cox regression model is often fitted in the modeling of survival data. One of its main advantages is the ease of interpretation, as long as the hazards rates for two individuals do not vary over time. In practice the proportionality assumption of the hazards may not be true in some situations. In addition, in several survival data is common a proportion of units not susceptible to the event of interest, even if, accompanied by a sufficiently large time, which is so-called immune, "cured," or not susceptible to the event of interest. In this context, several cure rate models are available to deal with in the long term. Here, we consider the generalized time-dependent logistic (GTDL) model with a power variance function (PVF) frailty term introduced in the hazard function to control for unobservable heterogeneity in patient populations. It allows for non-proportional hazards, as well as survival data with long-term survivors. Parameter estimation was performed using the maximum likelihood method, and Monte Carlo simulation was conducted to evaluate the performance of the models. Its practice relevance is illustrated in a real medical dataset from a population-based study of incident cases of melanoma diagnosed in the state of São Paulo, Brazil.