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This paper studies the optimality of matched-pair designs in randomized controlled trials (RCTs). Matched-pair designs are examples of stratified randomization, in which the researcher partitions a set of units into strata based on their observed covariates and assign a fraction of units in each stratum to treatment. A matched-pair design is such a procedure with two units per stratum. Despite the prevalence of stratified randomization in RCTs, implementations differ vastly. We provide an econometric framework in which, among all stratified randomization procedures, the optimal one in terms of the mean-squared error of the difference-in-means estimator is a matched-pair design that orders units according to a scalar function of their covariates and matches adjacent units. Our framework captures a leading motivation for stratifying in the sense that it shows that the proposed matched-pair design additionally minimizes the magnitude of the ex-post bias, i.e., the bias of the estimator conditional on realized treatment status. We then consider empirical counterparts to the optimal stratification using data from pilot experiments and provide two different procedures depending on whether the sample size of the pilot is large or small. For each procedure, we develop methods for testing the null hypothesis that the average treatment effect equals a prespecified value. Each test we provide is asymptotically exact in the sense that the limiting rejection probability under the null equals the nominal level. We run an experiment on the Amazon Mechanical Turk using one of the proposed procedures, replicating one of the treatment arms in Dellavigna and Pope (2018), and find the standard error decreases by 29%, so that only half of the sample size is required to attain the same standard error.

The partial potential impact fraction describes the proportion of disease cases that can be prevented if the distribution of modifiable continuous exposures is shifted in a population, while other risk factors are not modified. It is a useful quantity for evaluating the burden of disease in epidemiologic and public health studies. When exposures are measured with error, the partial potential impact fraction estimates may be biased, which necessitates methods to correct for the exposure measurement error. Motivated by the health professionals follow-up study, we develop a Bayesian approach to adjust for exposure measurement error when estimating the partial potential impact fraction under the main study/internal validation study design. We adopt the reclassification approach that leverages the strength of the main study/internal validation study design and clarifies transportability assumptions for valid inference. We assess the finite-sample performance of both the point and credible interval estimators via extensive simulations and apply the proposed approach in the health professionals follow-up study to estimate the partial potential impact fraction for colorectal cancer incidence under interventions exploring shifting the distributions of red meat, alcohol, and/or folate intake.

This article discusses regression analysis of clustered interval-censored failure time data in the presence of a cured fraction or subgroup. Such data often occur in many areas, including epidemiological studies, medical studies, and social sciences. For the problem, a class of semiparametric transformation nonmixture cure models is presented and for estimation, the maximum likelihood estimation procedure is derived. For the implementation of the proposed method, we develop a novel EM algorithm based on a Poisson variable-based augmentation. An extensive simulation study is conducted and suggests that the proposed approach works well in practical situations. Finally the method is applied to an example that motivated this study.

In the last five years, interest in close-kin mark-recapture (CKMR), a variant of mark-recapture that uses genetically inferred kin as 'recaptures', has grown dramatically. However, understanding the basis of CKMR, and properly implementing it, remains challenging. This paper describes an R package, CKMRpop, for simulating age-structured populations with user-specified demography, overdispersed variance in reproductive success (allowing for different ratios of effective to census size) and random sampling of individuals. Using compiled code for the simulation makes it feasible to simulate populations of millions of individuals. From the simulation output, pairs of sampled individuals related within a user-specified number of generations are found. Such pairs form the foundation for CKMR inference, and simulating them provides insight for understanding the statistical basis for CKMR and for assessing the feasibility of CKMR in different scenarios. We predict that CKMRpop will serve as an important tool for researchers contemplating CKMR estimation of population size. Furthermore, the methods presented here for identifying and categorizing relationships beyond half-siblings allow a more complete picture of the wide variety of kin pairs encountered in populations. This identifies the fraction of kin pairs that may not be the target of a CKMR experiment, but may be inadvertently mistaken for a more closely related 'target' kin pair. Additionally, as more distant kin categories will likely be accurately inferred from increasingly available and inexpensive whole genome resequencing, understanding the distributions of more distant relationships in populations is a first step towards broadening the scope of CKMR to include them.

Causal classification concerns the estimation of the net effect of a treatment on an outcome of interest at the instance level, i.e., of the individual treatment effect (ITE). For binary treatment and outcome variables, causal classification models produce ITE estimates that essentially allow one to rank instances from a large positive effect to a large negative effect. Often, as in uplift modeling (UM), one is merely interested in this ranking, rather than in the ITE estimates themselves. In this regard, we investigate the potential of learning to rank (L2R) techniques to learn a ranking of the instances directly. We propose a unified formalization of different binary causal classification performance measures from the UM literature and explore how these can be integrated into the L2R framework. Additionally, we introduce a new metric for UM with L2R called the promoted cumulative gain (PCG). We employ the L2R technique LambdaMART to optimize the ranking according to PCG and show improved results over the use of standard L2R metrics and equal to improved results when compared with state-of-the-art UM. Finally, we show how L2R techniques can be used to specifically optimize for the top-k fraction of the ranking in a UM context, however, these results do not generalize to the test set. ispartof: IEEE Transactions on Knowledge and Data Engineering vol:34 issue:10 pages:4888-4904 status: published

Missing data are ubiquitous in psychological research. They may come about as an unwanted result of coding or computer error, participants' non-response or absence, or missing values may be intentional, as in planned missing designs. We discuss the effects of missing data on χ²-based goodness-of-fit indices in Structural Equation Modeling (SEM), specifically on the Root Mean Squared Error of Approximation (RMSEA). We use simulations to show that naive implementations of the RMSEA have a downward bias in the presence of missing data and, thus, overestimate model goodness-of-fit. Unfortunately, many state-of-the-art software packages report the biased form of RMSEA. As a consequence, the scientific community may have been accepting a much larger fraction of models with non-acceptable model fit. We propose a bias-correction for the RMSEA based on information-theoretic considerations that take into account the expected misfit of a person with fully observed data. The corrected RMSEA is asymptotically independent of the proportion of missing data for misspecified models. Importantly, results of the corrected RMSEA computation are identical to naive RMSEA if there are no missing data.

We formulate and theoretically analyze a mathematical model of COVID-19 transmission mechanism incorporating vital dynamics of the disease and two key therapeutic measures—vaccination of susceptible individuals and recovery/treatment of infected individuals. Both the disease-free and endemic equilibrium are globally asymptotically stable when the effective reproduction numberR0vis, respectively, less or greater than unity. The derived critical vaccination threshold is dependent on the vaccine efficacy for disease eradication wheneverR0v>1, even if vaccine coverage is high. Pontryagin’s maximum principle is applied to establish the existence of the optimal control problem and to derive the necessary conditions to optimally mitigate the spread of the disease. The model is fitted with cumulative daily Senegal data, with a basic reproduction numberR0=1.31at the onset of the epidemic. Simulation results suggest that despite the effectiveness of COVID-19 vaccination and treatment to mitigate the spread of COVID-19, whenR0v>1, additional efforts such as nonpharmaceutical public health interventions should continue to be implemented. Using partial rank correlation coefficients and Latin hypercube sampling, sensitivity analysis is carried out to determine the relative importance of model parameters to disease transmission. Results shown graphically could help to inform the process of prioritizing public health intervention measures to be implemented and which model parameter to focus on in order to mitigate the spread of the disease. The effective contact rateb, the vaccine efficacyε, the vaccination ratev, the fraction of exposed individuals who develop symptoms, and, respectively, the exit rates from the exposed and the asymptomatic classesσandϕare the most impactful parameters.

Standard survival methods are inappropriate for mismeasured outcomes. Previous research has shown that outcome misclassification can bias estimation of the survival function. We develop methods to accurately estimate the survival function when the diagnostic tool used to measure the outcome of disease is not perfectly sensitive and specific. Since the diagnostic tool used to measure disease outcome is not the gold standard, the true or error-free outcomes are latent, they cannot be observed. Our method uses the negative predictive value (NPV) and the positive predictive values (PPV) of the diagnostic tool to construct a bridge between the error-prone outcomes and the true outcomes. We formulate an exact relationship between the true (latent) survival function and the observed (error-prone) survival function as a formulation of time-varying NPV and PPV. We specify models for the NPV and PPV that depend only on parameters that can be easily estimated from a fraction of the observed data. Furthermore, we conduct an in-depth study to accurately estimate the latent survival function based on the assumption that the biology that underlies the disease process follows a gamma process. We examine the performance of our method by applying it to the Viral Resistance to Antiviral Therapy of Chronic Hepatitis C (VIRAHEP-C) data. To show the broader relevance of our research, we apply our proposed methodology to a dataset from the Alzheimer’s Disease Neuroimaging Initiative (ADNI).

Mixture cure models have been developed as an effective tool to analyze failure time data with a cure fraction. Used in conjunction with the logistic regression model, this model allows covariate-adjusted inference of an exposure effect on the cured probability and on the hazard of failure for the uncured subjects. However, the covariate-adjusted inference for the overall exposure effect is not directly provided. In this paper, we describe a Cox proportional hazards cure model to analyze interval-censored survival data in the presence of a cured fraction and then apply a post-estimation approach by using model-predicted estimates difference to assess the overall exposure effect on the restricted mean survival time scale. For baseline hazard/survival function estimation, simple parametric models as fractional polynomials or restricted cubic splines are utilized to approximate the baseline logarithm cumulative hazard function, or alternatively, the full likelihood is specified through a piecewise linear approximation for the cumulative baseline hazard function. Simulation studies were conducted to demonstrate the unbiasedness of both estimation methods for the overall exposure effect estimates over various baseline hazard distribution shapes. The methods are applied to analyze the interval-censored relapse time data from a smoking cessation study.

To improve the accuracy of bifacial gain estimation, recent radiative models of solar energy systems have abandoned the traditional assumption of isotropic ground-reflected radiance. However, surface reflectance itself is still commonly considered as a constant — partly because of the recommendations of previous evaluations of reflectance models. This paper presents the findings of a new model evaluation based on a large database of measurements from 26 sites, which are representative of major land covers and climates. Both novel and previously reviewed formulations are validated with the data. On a global average, data-based estimation reduces mean absolute error by 22%, 29%, and 39% with constant, univariate, and bivariate models, respectively, compared to literature-based estimates. Only at the urban and snow-and-ice sites does time-variant estimation not notably improve accuracy. Arid sites tend to favour univariate models based on solar elevation, and diffuse fraction adds little value as the second predictor. By contrast, bivariate estimation clearly improves accuracy at vegetated and water sites. When considering the best-performing model for each site, the global average mean absolute error is 11%. Two novel formulations, univariate and bivariate, provide superior performance at many sites. The proposed 3-parameter bivariate model is one of the top performers at 19 out of the 26 considered sites.

The genetic effect-size distribution describes the number of variants that affect disease risk and the range of their effect sizes. Accurate estimates of this distribution would provide insights into genetic architecture and set sample-size targets for future genome-wide association studies. We developed Fourier Mixture Regression (FMR) to estimate common-variant effect-size distributions from GWAS summary statistics. We validated FMR in simulations and in analyses of UK Biobank data, using interim-release summary statistics (max N=145k) to predict the results of the full release (N=460k). Analyzing summary statistics for 10 diseases (avg Neff=169k) and 22 other traits, we estimated the sample size required for genome-wide significant SNPs to explain 50% of SNP-heritability. For most diseases the requisite number of cases is 100k-1M, an attainable number; ten times more would be required to explain 90% of heritability. In well-powered GWAS, genome-wide significance is a conservative threshold, and loci at less stringent thresholds have true positive rates that remain close to 1 if confounding is controlled. Analyzing the shape of the effect-size distribution, we estimate that heritability accumulates across many thousands of SNPs with a wide range of effect sizes: the largest effects (at the 90th percentile of heritability) are 100 times larger than the smallest (10th percentile), and while the midpoint of this range varies across traits, its size is similar. These results suggest attainable sample size targets for future GWAS, and they underscore the complexity of genetic architecture.

The first case of COVID-19 in India detected on January 30, 2020, after its emergence in Wuhan, China, in December 2019. The lockdown was imposed as anemergency measure by the Indian government to prevent the spread of COVID-19 but gradually eased out due to its vast economic consequences. Just 15?days after the relaxation of lockdown restrictions, Delhi became India's worst city in terms of COVID-19 cases. In this paper, we propose a variable-order fractional SIR (susceptible, infected, removed) model at state-level scale. We introduce a algorithm that uses the differential evolution algorithm in combination with Adam?Bashforth?Moulton method to learn the parameters in a system of variable-order fractional SIR model. The model can predict the confirm COVID-19 cases in India considering the effects of nationwide lockdown and the possible estimate of the number of infliction inactive cases after the removal of lockdown on June 1, 2020. A new parameter p is introduced in the classical SIR model representing the fraction of infected people that get tested and are thereby quarantined. The COVID-19 trajectory in Delhi, as per our model, predicts the slowing down of the spread between January and February 2021, touching a peak of around 5 lakh confirmed cases.

Aims Cancer incidence varies across England, which affects the local-level demand for treatments. The magnetic resonance-linac (MR-linac) is a new radiotherapy technology that combines imaging and treatment. Here we model the demand and demand variations for the MR-linac across England. Materials and methods Initial clinical indications were provided by the MR-linac consortium and introduced into the Malthus radiotherapy clinical decision trees. The Malthus model contains Clinical Commissioning Group (CCG) population, cancer incidence and stage presentation data (for lung and prostate) and simulated the demand for the MR-linac for all CCGs and Radiotherapy Operational Delivery Networks (RODN) across England. Results Based on the initial target clinical indications, the MR-linac could service 16% of England's fraction burden. The simulated fractions/million population demand/annum varies between 3000 and 10 600 fractions/million at the CCG level. Focussing only on the cancer population, the simulated fractions/1000 cancer cases demand/annum ranges from 1028 to 1195 fractions/1000 cases. If a national average for fractions/million demand was then used, at the RODN level, the variation from actual annual demand ranges from an overestimation of 8400 fractions to an underestimation of 5800 fractions. When using the national average fractions/1000 cases, the RODN demand varies from an overestimation of 3200 fractions to an underestimation of 3000 fractions. Conclusions Planning cancer services is complex due to regional variations in cancer burden. The variations in simulated demand of the MR-linac highlight the requirement to use local-level data when planning to introduce a new technology., Highlights • Variations in local-level cancer incidence drive variations in treatment demand. • Using national averages can cause large under-/overestimation of treatment demand. • MR-linac demand is shown at the new Radiotherapy Operation Delivery Network level. • Local-level data should be used when planning to introduce a new technology.

This paper investigates the impact of top-distribution undercoverage on the Gini coefficient. First, we show that failing to correct for underreporting and nonresponse at the top does not necessarily result in an underestimated Gini coefficient. Then, we establish analytical conditions under which the Gini coefficient of the uncorrected distribution is higher than the Gini coefficient of the true distribution, i.e. the distribution that incorporates underreporting and/or nonresponse. In addition, we propose a Gini approximation based on the Atkinson approximation $$G=G_{1-p}\cdot (1-S_{p})+S_{p}$$ to correct for underreporting at the top. Under plausible assumptions, the approximation proposed is very close to the real Gini coefficient. We also show that before correcting for underreporting we need to first address the issue of nonresponse as otherwise the approximation may be strongly upper biased. Finally, this work proposes a procedure to estimate the fraction of nonrespondents at the very top under the assumption that underreported incomes belong to individuals with incomes higher than those reported in a household survey. To evaluate the proposed methodologies, this paper uses Chile and Canada as examples, where we include undistributed business profits—an underreported source of income—to measure income inequality.

We propose group sequential methods for cluster randomized trials (CRTs) with time-to-event endpoint. The alpha spending function approach is used for sequential data monitoring. The key to this approach is determining the joint distribution of test statistics and the information fraction at the time of interim analysis. We prove that the sequentially computed log-rank statistics in CRTs do not have independent increment property. We also propose an information fraction for group sequential trials with clustered survival data and a corresponding sample size determination approach. Extensive simulation studies are conducted to evaluate the performance of our proposed testing procedure using some existing alpha spending functions in terms of expected sample size and maximal sample size. Real study examples are taken to demonstrate our method.

In this work, we develop a ternary diagram approach for visualizing the progression of a disease outbreak. Based on cumulative epidemic statistics, the fraction of patients classified as active cases (A), recovered (R), and deceased (D) at any given time is plotted on a ternary diagram. The location of the point gives the relative proportions of cases in these three categories. The trajectory of this point over time provides an alternative scale-free indication of the state of the outbreak. This technique offers an alternative to the mainstream epidemic curves that plot the number of cases against time and can reveal trends not easily detected in conventional data displays. Because it relies on relative proportions rather than absolute numbers of patients in different categories, this method is more robust to errors caused by low diagnosis and detection rates. Such insights allow better prioritization of areas where more intensive measures are needed.

Planned missing data designs allow researchers to have highly-powered studies by testing only a fraction of the traditional sample size. In two-method measurement planned missingness designs, resea...

This paper is focused on estimating the parameter of Exponential distribution under right-censored data with cure fraction. The maximum likelihood estimation and Bayesian approach were used. The Bayesian method is implemented using gamma, Jeffreys, and extension of Jeffreys priors with two loss functions, which are; squared error loss function and Linear Exponential Loss Function (LINEX). The methods of the Bayesian approach are compared to maximum likelihood counterparts and the comparisons are made with respect to the Mean Square Error (MSE) to determine the best for estimating the parameter of Exponential distribution under right-censored data with cure fraction. The results show that the Bayesian with gamma prior under LINEX loss function is a better estimation of the parameter of Exponential distribution with cure fraction based on right-censored data.

Peaks in the spectral density estimates of seasonally adjusted data are indicative of an inadequate adjustment. Spectral peaks are currently assessed in the X-13ARIMA-SEATS program via the visual significance (VS) approach; this paper provides a rigorous statistical foundation for VS by defining measures of uncertainty for spectral peak measures, allowing for formal hypothesis testing, using the framework of fixed-bandwidth fraction asymptotics for taper-based spectral density estimates. The simulation results show that the test has good size and power properties for a variety of peak features.

When comparing survival times of treatment and control groups under a more realistic nonproportional hazards scenario, the standard log-rank (SLR) test may be replaced by a more efficient weighted log-rank (WLR) test, such as the Fleming-Harrington (FH) test. Designing a group-sequential clinical trial with one or more interim looks during which a FH test will be performed, necessitates correctly quantifying the information fraction (IF). For SLR test, IF is defined simply as the ratio of interim to final numbers of events; but for FH test, it can deviate substantially from this ratio. In this article, we separate the effect of weight function (of FH test) alone on IF from the effect of censoring. We have shown that, without considering the effect of censoring, IF can be derived analytically for FH test using information available at the design stage and the additional effect due to censoring is relatively smaller. This article intends to serve two major purposes: first, to emphasize and rationalize the deviation of IF in weighted log-rank test from that of SLR test which is often overlooked (Jiménez, Stalbovskaya, and Jones); second, although it is impossible to predict IF for a weighted log-rank test at the design stage, our decomposition of effects on IF provides a reasonable and practically feasible range of IF to work with. We illustrate our approach with an example and provide simulation results to evaluate operating characteristics.

Clustered interval-censored failure time data occur in many areas and many methods have been proposed for their analysis. In practice, in addition to the clustering and interval censoring, such dat...

Novel cancer therapies are associated with survival patterns that differ from established therapies, which may include survival curves that plateau after a certain follow-up time point. A fraction of the patient population is then considered statistically cured and subject to the same mortality experience as the cancer-free general population. Mixture cure models have been developed to account for this characteristic. As compared to standard survival analysis, mixture cure models can often lead to profoundly different estimates of long-term survival, required for health economic evaluations. This tutorial is designed as a practical introduction to mixture cure models. Step-by-step instructions are provided for the entire implementation workflow, i.e., from gathering and combining data from different sources to fitting models using maximum likelihood estimation and model results interpretation. Two mixture cure models were developed to illustrate (1) an "uninformed" approach where the cure fraction is estimated from trial data and (2) an “informed” approach where the cure fraction is obtained from an external source (e.g., real-world data) used as an input to the model. These models were implemented in the statistical software R, with the freely available code on GitHub. The cure fraction can be estimated as an output from (“uninformed” approach) or used as an input to (“informed” approach) a mixture cure model. Mixture cure models suggest presumed estimates of long-term survival proportions, especially in instances where some fraction of patients is expected to be statistically cured. While this type of model may initially seem complex, it is straightforward to use and interpret. Mixture cure models have the potential to improve the accuracy of survival estimates for treatments associated with statistical cure, and the present tutorial outlines the interpretation and implementation of mixture cure models in R. This type of model will likely become more widely used in health economic analyses as novel cancer therapies enter the market. Supplementary Information The online version contains supplementary material available at 10.1007/s41669-021-00260-z.

We consider comparative effectiveness research (CER) from observational data with two or more treatments. In observational studies, the estimation of causal effects is prone to bias due to confounders related to both treatment and outcome. Methods based on propensity scores are routinely used to correct for such confounding biases. A large fraction of propensity score methods in the current literature consider the case of either two treatments or continuous outcome. There has been extensive literature with multiple treatment and binary outcome, but interest often lies in the intersection, for which the literature is still evolving. The contribution of this article is to focus on this intersection and compare across methods, some of which are fairly recent. We describe propensity-based methods when more than two treatments are being compared, and the outcome is binary. We assess the relative performance of these methods through a set of simulation studies. The methods are applied to assess the effect of four common therapies for castration-resistant advanced-stage prostate cancer. The data consist of medical and pharmacy claims from a large national private health insurance network, with the adverse outcome being admission to the emergency room within a short time window of treatment initiation.

Estimating bird and bat mortality at wind facilities typically involves searching for carcasses on the ground near turbines. Some fraction of carcasses inevitably lie outside the search plots, and accurate mortality estimation requires accounting for those carcasses using models to extrapolate from searched to unsearched areas. Such models should account for variation in carcass density with distance, and ideally also for variation with direction (anisotropy). We compare five methods of accounting for carcasses that land outside the searched area (ratio, weighted distribution, non-parametric, and two generalized linear models (glm)) by simulating spatial arrival patterns and the detection process to mimic observations which result from surveying only, or primarily, roads and pads (R&P) and applying the five methods. Simulations vary R&P configurations, spatial carcass distributions (isotropic and anisotropic), and per turbine fatality rates. Our results suggest that the ratio method is less accurate with higher variation relative to the other four methods which all perform similarly under isotropy. All methods were biased under anisotropy; however, including direction covariates in the glm method substantially reduced bias. In addition to comparing methods of accounting for unsearched areas, we suggest a semiparametric bootstrap to produce confidence-based bounds for the proportion of carcasses that land in the searched area.

In a joint analysis of longitudinal quality of life (QoL) scores and relapse-free survival (RFS) times from a clinical trial on early breast cancer conducted by the Canadian Cancer Trials Group, we observed a complicated trajectory of QoL scores and existence of long-term survivors. Motivated by this observation, we proposed in this paper a flexible joint model for the longitudinal measurements and survival times. A partly linear mixed effect model is used to capture the complicated but smooth trajectory of longitudinal measurements and approximated by B-splines and a semiparametric mixture cure model with the B-spline baseline hazard to model survival times with a cure fraction. These two models are linked by shared random effects to explore the dependence between longitudinal measurements and survival times. A semiparametric inference procedure with an EM algorithm is proposed to estimate the parameters in the joint model. The performance of proposed procedures are evaluated by simulation studies and through the application to the analysis of data from the clinical trial which motivated this research.

This study aims to determine the criteria for errors made by students at SDN 7 Sindue in solving story problems on fraction operations based on Watson's theory. The research subjects consisted of 3 students, each of whom had high, medium, and low abilities according to the ranking obtained. Based on the results of research conducted, the criteria for errors that are mostly carried out by students based on Watson's theory are error type Id (incorrect data), Ip (incorrect procedure), Od (missing data), Oc (missing conclusion), Rlc (level conflict response), Um (Indirect manipulation), Shp (Skills hierarchy). And the most common mistakes made by subjects 1, 2, and 3 are IP (Inappropriate Procedure), and RLC (Response Level Conflict) errors.

Scientifically and accurately forecasting of future shale gas output tends is very important in making energy policies, especially for China whose historical data of shale gas output is very limited. The existing grey shale gas output prediction model does not perform well in prediction due to its defects. To overcome these shortcoming, this paper, based on the principle of “new information priority”, combined with the contradiction between model prediction results and qualitative data analysis conclusions, designs a grey prediction model combining new initial conditions and original data reprocessing. Then the new model’s accumulative order is optimized by fraction accumulation generation operation and its properties is discussed. Finally, the new model is used to simulate and forecast shale gas output in China from 2012 to 2018 and compared it with the existing shale gas prediction model. The comparison results show that the new model reduces by 84.34%, 68.96% and 75.60% of the mean relative simulation percentage error (MRSPE), mean relative prediction percentage error (MRPPE) and comprehensive mean relative percentage error (CMPPE) respectively. This paper not only has theoretical innovations but also provides a good mathematical method for predicting shale gas output.

High-quality processes with very low defect rates are usually modeled using a Poisson distribution. Considering any correlation between the number of defects on a unit and the fraction of nonconfor...

Diabetes is a chronic metabolic disease that causes blood glucose (BG) concentration to make dangerous excursions outside its physiological range. Measuring the fraction of time spent by BG outside this range, and, specifically, the time-below-range (TBR), is a clinically common way to quantify the effectiveness of therapies. TBR is estimated from data recorded by continuous glucose monitoring (CGM) sensors, but the duration of CGM recording guaranteeing a reliable indicator is under debate in the literature. Here we framed the problem as random variable estimation problem and studied the convergence of the estimator, deriving a formula that links the TBR estimation error variance with the CGM recording length. Validation is performed on CGM data of 148 subjects with type-1-diabetes. First, we show the ability of the formula to predict the uncertainty of the TBR estimate in a single patient, using patient-specific parameters; then, we prove its applicability on population data, without the need of parameters individualization. The approach can be straightforwardly extended to other similar metrics, such as time-in-range and time-above-range, widely adopted by clinicians. This strengthens its potential utility in diabetes research, e.g., in the design of those clinical trials where minimal CGM monitoring duration is crucial in cost-effectiveness terms.

Although many persons in the United States have acquired immunity to COVID-19, either through vaccination or infection with SARS-CoV-2, COVID-19 will pose an ongoing threat to non-immune persons so long as disease transmission continues. We can estimate when sustained disease transmission will end in a population by calculating the population-specific basic reproduction number ℛ0, the expected number of secondary cases generated by an infected person in the absence of any interventions. The value of ℛ0 relates to a herd immunity threshold (HIT), which is given by 1 − 1/ℛ0. When the immune fraction of a population exceeds this threshold, sustained disease transmission becomes exponentially unlikely (barring mutations allowing SARS-CoV-2 to escape immunity). Here, we report state-level ℛ0 estimates obtained using Bayesian inference. Maximum a posteriori estimates range from 7.1 for New Jersey to 2.3 for Wyoming, indicating that disease transmission varies considerably across states and that reaching herd immunity will be more difficult in some states than others. ℛ0 estimates were obtained from compartmental models via the next-generation matrix approach after each model was parameterized using regional daily confirmed case reports of COVID-19 from 21-January-2020 to 21-June-2020. Our ℛ0 estimates characterize infectiousness of ancestral strains, but they can be used to determine HITs for a distinct, currently dominant circulating strain, such as SARS-CoV-2 variant Delta (lineage B.1.617.2), if the relative infectiousness of the strain can be ascertained. On the basis of Delta-adjusted HITs, vaccination data, and seroprevalence survey data, we find that no state has achieved herd immunity as of 20-September-2021.Significance StatementCOVID-19 will continue to threaten non-immune persons in the presence of ongoing disease transmission. We can estimate when sustained disease transmission will end by calculating the population-specific basic reproduction number ℛ0, which relates to a herd immunity threshold (HIT), given by 1 − 1/ℛ0. When the immune fraction of a population exceeds this threshold, sustained disease transmission becomes exponentially unlikely. Here, we report state-level ℛ0 estimates indicating that disease transmission varies considerably across states. Our ℛ0 estimates can also be used to determine HITs for the Delta variant of COVID-19. On the basis of Delta-adjusted HITs, vaccination data, and serological survey results, we find that no state has yet achieved herd immunity.

Multiple imputation (MI) inference handles missing data by imputing the missing values $m$ times, and then combining the results from the $m$ complete-data analyses. However, the existing method for combining likelihood ratio tests (LRTs) has multiple defects: (i) the combined test statistic can be negative, but its null distribution is approximated by an $F$-distribution; (ii) it is not invariant to re-parametrization; (iii) it fails to ensure monotonic power owing to its use of an inconsistent estimator of the fraction of missing information (FMI) under the alternative hypothesis; and (iv) it requires nontrivial access to the LRT statistic as a function of parameters instead of data sets. We show, using both theoretical derivations and empirical investigations, that essentially all of these problems can be straightforwardly addressed if we are willing to perform an additional LRT by stacking the $m$ completed data sets as one big completed data set. This enables users to implement the MI LRT without modifying the complete-data procedure. A particularly intriguing finding is that the FMI can be estimated consistently by an LRT statistic for testing whether the $m$ completed data sets can be regarded effectively as samples coming from a common model. Practical guidelines are provided based on an extensive comparison of existing MI tests. Issues related to nuisance parameters are also discussed., Comment: To appear in Statistica Sinica

Background. Several mathematical representations of contagious disease COVID-19 were evolved in order to capture the pragmatic aspect of unfurling of the disease. It is learned that individuals who became receptive were infected with a rate proportional to the fraction of the individuals affected by the infection, in the comprehensive population as well as the infected individuals recuperate at a sustained rate. It is also observed that in the SIR model, all contacts impart the disease with an identical probability. Objective. We will estimate the dynamic epidemic behaviour of inflected population for India with the use of fractional-order SIR simulations and compare our results with the results obtained for extrapolated actual cases of the infected people. Methods. We have obtained the approximate solutions of the fractional-order Susceptible-Infectious-Recovered model within the framework of the modified Riemann–Liouville fractional differential operator using a new iterative fractional complex transform technique. Results. The optimal values of the fractional-order SIR model parameters were identified with the use of the New Iterative Method. The dynamic incident rate with high and low reproduction number is predicted as well as the illustrated graphical with actual data is provided. To sum, the fractional calculus model for a complex system proposed here is just an indication to show what might happen if we do not control the reproduction number in the community. Conclusions. The control measures that have already been found like swift surveillance, quarantine and social distancing means, such as face masks and closures, assisted in curtailing coronavirus transmission – estimated by the average number of people each infected individual infects, or reproduction number, to close to the level of 1 in each month.

Background The population attributable fraction (PAF) is the fraction of disease cases in a sample that can be attributed to an exposure. Estimating the PAF often involves the estimation of the probability of having the disease given the exposure while adjusting for confounders. In many settings, the exposure can interact with confounders. Additionally, the exposure may have a monotone effect on the probability of having the disease, and this effect is not necessarily linear. Methods We develop a semiparametric approach for estimating the probability of having the disease and, consequently, for estimating the PAF, controlling for the interaction between the exposure and a confounder. We use a tensor product of univariate B-splines to model the interaction under the monotonicity constraint. The model fitting procedure is formulated as a quadratic programming problem, and, thus, can be easily solved using standard optimization packages. We conduct simulations to compare the performance of the developed approach with the conventional B-splines approach without the monotonicity constraint, and with the logistic regression approach. To illustrate our method, we estimate the PAF of hopelessness and depression for suicidal ideation among elderly depressed patients. Results The proposed estimator exhibited better performance than the other two approaches in the simulation settings we tried. The estimated PAF attributable to hopelessness is 67.99% with 95% confidence interval: 42.10% to 97.42%, and is 22.36% with 95% confidence interval: 12.77% to 56.49% due to depression. Conclusions The developed approach is easy to implement and supports flexible modeling of possible non-linear relationships between a disease and an exposure of interest.

In many human surveys, information is in most cases not obtained from all the units in the survey even after some call backs. To cope with this problem, survey statisticians usually consider and adopt the non-respondents sub sampling schemes introduced by Hansen and Hurwitz (1946, Jour. Amer. Statist. Assoc., 41, 517–529). The topic which is of practical importance is the estimation of population mean at the current occasion in two- occasion successive sampling in presence of non-response. In this paper, an alternative to Satici and Kadilar (2011, Hacettepe Jour. Math. Statist. 40(1), 115–124) estimator for the population mean at the current occasion in successive sampling having two occasions is suggested under the missing data case. The minimum mean squared error and optimum fraction of a fresh sample of suggested estimator are obtained. The suggested estimator is compared with the estimator due to Singh and Priyanka (2007, Statistics in Transition. 8(2), 273–292) and Satici and Kadilar (2011, Hacettepe Jour. Math. Statist. 40(1), 115–124) and the efficient conditions for the suggested estimator are obtained. It has been shown theoretically that the proposed class of estimators is more efficient than the Singh and Priyanka (2007) and Satici and Kadilar (2011) estimators. The performance of the suggested class of estimators is examined over Singh and Priyanka (2007) and Satici and Kadilar (2011) estimators with the aid of an empirical study.

Inherently, the survival function of a distribution converges to zero as time approaches infinity. The medical interpretation of this definition is that all the patient populations are susceptible ...

In medical studies, it is often observed that a portion of subjects will never experience the event of interest and thus can be treated as cured or long-term survivors. Many populations of early-stage cancer patients contain both uncured and cured individuals that should be modeled using cure models. In prognostic studies, the cure status (uncure or cure) is an issue of interest for medical practitioners, and the disease status (death or alive) of an individual is not a fixed characteristic and it varies along the time. These statuses are usually predicted by a prognostic risk score. The time-dependent receiver operating characteristic (ROC) curve is a powerful tool to evaluate these predicting performances dynamically. In the context with a cure fraction, quantifying and estimating the predictive performances of the risk score is a challenge since the disease status and cure status are both unknown among individuals who are censored. In this paper, to assess the predictive accuracy for the survival outcome with a cure fraction, we propose a time-dependent ROC curve semi-parametric estimator based on the sieve maximum likelihood (ML) estimation under the mixture cure model. We also apply a Bernstein-based smoothing method in the estimation procedure, and this estimator can lead to substantial gain in efficiency. In addition, we derive the time-dependent area under the ROC curve (AUC) to summarize the discriminatory capacity of the risk score globally. Finally, we evaluate the finite sample performance of the proposed methods by extensive simulations and illustrate the estimation using two real data sets, one from a melanoma study and the other from stomach cancer.

Objectives When populations contain mixtures of cured and uncured patients, the use of traditional parametric approaches to estimate overall survival (OS) can be biased. Mixture cure models may reduce bias compared with traditional parametric models, but their accuracy is subject to certain conditions. Importantly, mixture cure models assume that that there is enough follow-up to identify individuals censored at the end of the follow-up period as cured. The purpose of this article is to describe biases that can occur when mixture cure models are used to estimate mean survival from data with limited follow-up. Methods We analyzed 6 trials conducted by the SWOG Cancer Research Network Leukemia Committee. For each trial, we analyzed 2 data sets: the data released to the committee when the results of the trial were unblinded and a second data set with additional follow-up. We estimated mean OS using parametric survival models with and without a cure fraction. Results When using mixture cure models, in 4 trials, estimates of mean OS were higher with the first analysis (with limited follow-up) compared with estimates from data with longer follow-up. In 1 trial, the reverse pattern was observed. In 1 trial, the cure estimate changed little with additional follow-up. Conclusions Caution should be taken when using mixture cure models in scenarios with limited follow-up. The biases resulting from fitting these models may be exacerbated when the models are being used to extrapolate OS and estimate mean OS.

Traditionally, Inventory Holding Cost (IHC) is assumed to be a combination of several costs and determined by the summation of these cost components. Several authors have suggested that the value of IHC ranges between 12-50% of the procurement cost of an item. However, due to the absence of a generally acceptable methodology, many practitioners still determine this percentage based on estimates, benchmarks and intuition. Giving considerations to this reality, a mathematical model to determine the value of IHC using systems dynamics approach was developed. IHC was viewed holistically to identify relevant quantities, their interactions (static or dynamics), behaviour and consequences. A Causal Loop Diagram (CLD) was developed to establish the relationship among these quantities. Thereafter, CLD was transformed into a Flow Diagram (FD). FD was used to formulate a set of systems dynamics equations to obtain IHC. The interaction among fraction of goods ordered per month (FOM), fraction sold per month (FSM) and fraction damaged per month (FDM) was simulated to obtain percentage values of IHC. The value of IHC obtained from the model and simulation analysis ranges between 22.58-25.39% of the item held in stock. Based on these results, it is concluded that the developed model can be used for simulation and system analysis of the holding cost component of an inventory system under different contextual settings.

For a Phase III randomized trial that compares survival outcomes between an experimental treatment versus a standard therapy, interim monitoring analysis is used to potentially terminate the study early based on efficacy. To preserve the nominal Type I error rate, alpha spending methods and information fractions are used to compute appropriate rejection boundaries in studies with planned interim analyses. For a one-sided trial design applied to a scenario in which the experimental therapy is superior to the standard therapy, interim monitoring should provide the opportunity to stop the trial prior to full follow-up and conclude that the experimental therapy is superior. This paper proposes a method called total control only (TCO) for estimating the information fraction based on the number of events within the standard treatment regimen. Based on theoretical derivations and simulation studies, for a maximum duration superiority design, the TCO method is not influenced by departure from the designed hazard ratio, is sensitive to detecting treatment differences, and preserves the Type I error rate compared to information fraction estimation methods that are based on total observed events. The TCO method is simple to apply, provides unbiased estimates of the information fraction, and does not rely on statistical assumptions that are impossible to verify at the design stage. For these reasons, the TCO method is a good approach when designing a maximum duration superiority trial with planned interim monitoring analyses.

Geometric charts have an important role in monitoring fraction nonconforming in high-yield processes where the rate of nonconforming is very low, say, parts per million. Currently, the average numb...

Many product characteristics are qualitative in nature, e.g. colour, brightness, surface finish etc. The manufacturing process of such products is usually described in terms of fraction nonconforming or conforming which is assumed to follow binomial distribution. Measuring capability of a binomial process implies assessing to what extent the fraction nonconforming or conforming in the continuous stream of lots conform to the specification limits. The Cp or Cpl of a binomial process can be estimated using several approaches. However, these approaches generally give widely varying assessment about the capability of a given binomial process. Consequently, a user of the index may inadvertently be led to erroneous decision making based on an inaccurate estimate of the index. In this paper, a procedure is proposed for assessing accuracies of estimates of Cpu or Cpl obtained by different methods. Subsequently, the best method for evaluating capability of a binomial process is identified based on analysis of multiple case studies, and also the methods giving inaccurate estimates are highlighted. Keywords: Process capability index, binomial process, fraction nonconforming, nonconforming lot (NL), predicted NL%, prediction error

A three-arm trial including an experimental treatment, an active reference treatment and a placebo is often used to assess the non-inferiority (NI) with assay sensitivity of an experimental treatment. Various hypothesis-test-based approaches via a fraction or pre-specified margin have been proposed to assess the NI with assay sensitivity in a three-arm trial. There is little work done on confidence interval in a three-arm trial. This paper develops a hybrid approach to construct simultaneous confidence interval for assessing NI and assay sensitivity in a three-arm trial. For comparison, we present normal-approximation-based and bootstrap-resampling-based simultaneous confidence intervals. Simulation studies evidence that the hybrid approach with the Wilson score statistic performs better than other approaches in terms of empirical coverage probability and mesial-non-coverage probability. An example is used to illustrate the proposed approaches.

Many clinical research studies evaluate a time-to-event outcome, illustrate survival functions, and conventionally report estimated hazard ratios to express the magnitude of the treatment effect when comparing between groups. However, it may not be straightforward to interpret the hazard ratio clinically and statistically when the proportional hazards assumption is invalid. In some recent papers published in clinical journals, the use of restricted mean survival time (RMST) or τ-year mean survival time is discussed as one of the alternative summary measures for the time-to-event outcome. The RMST is defined as the expected value of time to event limited to a specific time point corresponding to the area under the survival curve up to the specific time point. This article summarizes the necessary information to conduct statistical analysis using the RMST, including the definition and statistical properties of the RMST, adjusted analysis methods, sample size calculation, information fraction for the RMST difference, and clinical and statistical meaning and interpretation. Additionally, we discuss how to set the specific time point to define the RMST from two main points of view. We also provide developed SAS codes to determine the sample size required to detect an expected RMST difference with appropriate power and reconstruct individual survival data to estimate an RMST reference value from a reported survival curve.

Cure rate models have been developed for the analysis of time to event data with cured fraction. In this paper we propose a proportional hazards competing risks regression model for the analysis of survival data with cured fraction. Estimation of the proposed model parameters is done by maximum likelihood method via EM algorithm. Simulation work is carried out to examine influence of sample size on bias and variability of estimators. We applied the model to real life time data.

In radiation oncology, Machine Learning classification publications are typically related to two outcome classes, e.g. the presence or absence of distant metastasis. However, multi-class classification problems also have great clinical relevance, e.g., predicting the grade of a treatment complication following lung irradiation. This work comprised two studies aimed at making work in this domain less prone to statistical blindsides. In multi-class classification, AUC is not defined, whereas correlation coefficients are. It may seem like solely quoting the correlation coefficient value (in lieu of the AUC value) is a suitable choice. In the first study, we illustrated using Monte Carlo (MC) models why this choice is misleading. We also considered the special case where the multiple classes are not ordinal, but nominal, and explained why Pearson or Spearman correlation coefficients are not only providing incomplete information but are actually meaningless. The second study concerned surrogate biomarkers for a clinical endpoint, which have purported benefits including potential for early assessment, being inexpensive, and being non-invasive. Using a MC experiment, we showed how conclusions derived from surrogate markers can be misleading. The simulated endpoint was radiation toxicity (scale of 0-5). The surrogate marker was the true toxicity grade plus a noise term. Five patient cohorts were simulated, including one control. Two of the cohorts were designed to have a statistically significant difference in toxicity. Under 1000 repeated experiments using the biomarker, these two cohorts were often found to be statistically indistinguishable, with the fraction of such occurrences rising with the level of noise.

To ensure reproducibility in research quantifying episodic migraine attacks, and identifying attack onset, a sound theoretical model of a migraine attack, paired with a uniform standard for counting them, is necessary. Many studies report on migraine frequencies—e.g. the fraction of migraine-days of the observed days—without paying attention to the number of discrete attacks. Furthermore, patients’ diaries frequently contain single, migraine-free days between migraine-days, and we argue here that such ‘migraine-locked days’ should routinely be interpreted as part of a single attack. We tested a simple Markov model of migraine attacks on headache diary data and estimated transition probabilities by mapping each day of each diary to a unique Markov state. We explored the validity of imputing migraine days on migraine-locked entries, and estimated the effect of imputation on observed migraine frequencies. Diaries from our patients demonstrated significant clustering of migraine days. The proposed Markov chain model was shown to approximate the progression of observed migraine attacks satisfactorily, and imputing on migraine-locked days was consistent with the conceptual model for the progression of migraine attacks. Hence, we provide an easy method for quantifying the number and duration of migraine attacks, enabling researchers to procure data of high inter-study validity.

Geometric charts are effective in monitoring the fraction nonconforming in high-quality processes. The in-control fraction nonconforming is unknown in most actual processes; therefore, it should be estimated using the Phase I sample. However, if the Phase I sample size is small the practitioner may not achieve the desired in-control performance because estimation errors can occur when the parameters are estimated. Therefore, in this paper, we adjust the control limits of geometric charts with the bootstrap algorithm to improve the in-control performance of charts with smaller sample sizes. The simulation results show that the adjustment with the bootstrap algorithm improves the in-control performance of geometric charts by controlling the probability that the in-control average run length has a value greater than the desired one. The out-of-control performance of geometric charts with adjusted limits is also discussed.

The accurate measure of fetal fraction is important to ensure the results of noninvasive prenatal testing. However, measuring fetal fraction could require a substantial amount of data and additional costs. Therefore, this study proposes an alternative method of measuring fetal fraction with a limited sample size and low sequencing reads. Adaptive machine-learning algorithms customized to each laboratory's environment, were used to measure fetal fraction. Pregnant women with female fetuses were tested to exclude the bias caused by training data of women carrying male fetuses. The accuracy of fetal DNA fraction prediction was enhanced by increasing the training sample size. When trained with 1,000 samples (males) and tested with 45 samples (females), the optimal bin sizes using the read count and size features were 300 kb and 800 kb, respectively. Comparing the new 300-kb bin to the 50-kb bin used by SeqFF with 4,000-5,000 training samples, the correlation was approximately 3-5% higher with the 300-kb bin. Therefore, we propose an effective and tailored method to measure fetal fraction in individual laboratories with limited sample collecting conditions and relatively low-coverage sequencing data.

The focus of this study was to investigate the students' ability in ordering four types of proper fractions. Type I: same numerators; Type II: same denominators; Type III: different numerators and different compatible denominators; Type IV: different numerators and different incompatible denominators. The researchers used library search and survey methodology through document search and questionnaire methods. The respondents of this study were 100 fourteen-year old students in two secondary schools from two districts. The results revealed that (i) students could easily order Type I and Type II proper fractions; (ii) students had less difficulty ordering Type II proper fractions compared with Type I proper fractions; (iii) students had difficulty ordering Type III and Type IV proper fractions; and (iv) when presented with tasks on ordering of Type III and Type IV proper fractions, students had most difficulty when ordering Type IV fractions. This showed that students still struggle with determining magnitudes of Type III and Type IV proper fractions. The findings of this study revealed that an appropriate method of instruction should be developed to teach students to order fractions of Type III and IV because the skill of ordering proper fractions is necessary for the development of fraction sense and overall mathematical development.

An important objective of experimental biology is the quantification of the relationship between predictor and response variables, a statistical analysis often termed variance partitioning (VP). In this paper, a series of simulations is presented, aiming to generate quantitative estimates of the expected statistical uncertainty of VP analyses. We demonstrate scenarios with considerable uncertainty in VP estimates, such that it significantly reduces the statistical reliability of the obtained results. Especially when a predictor variable of a dataset shows a low between-group variance, VP estimates may show a high margin of error. This becomes particularly important when the respective predictor variable only explains a small fraction of the overall variance, or the number of replicates is particularly small. Moreover, it is demonstrated that the expected error of VP estimates of a dataset can be approximated by bootstrap resampling, giving researchers a tool for the quantification of the uncertainty associated with an arbitrary VP analysis. The applicability of this method is demonstrated by a re-analysis of the Oribatid mite dataset introduced by Borcard and Legendre in 1994 and the Barro Colorado Island tree count dataset by Condit and colleagues. We believe that this study may encourage biologists to approach routine statistical analyses such as VP more critically, and report the error associated with them more frequently., Comment: 19 pages, 7 figures