Bootstrap-based Confidence Intervals Research Articles

Bootstrap-based confidence intervals for the standard two-sided power distribution

The two-parameter standard two-sided power family of distributions on (0, 1) is considered in this article. We propose bootstrap standard errors for the maximum likelihood estimators, as well as bootstrap confidence intervals for its parameters, once these important statistical measures of accuracy cannot be computed based on first-order asymptotic theory. We consider Monte Carlo simulation experiments to verify the performance of the bootstrap methods, and the numerical results are quite promising. Applications to real data are also considered to illustrate the proposed methodology in practice.

Likelihood-Free Parameter Estimation with Neural Bayes Estimators

Neural Bayes estimators are neural networks that approximate Bayes estimators. They are fast, likelihood-free, and amenable to rapid bootstrap-based uncertainty quantification. In this article, we aim to increase the awareness of statisticians to this relatively new inferential tool, and to facilitate its adoption by providing user-friendly open-source software. We also give attention to the ubiquitous problem of estimating parameters from replicated data, which we address using permutation-invariant neural networks. Through extensive simulation studies we demonstrate that neural Bayes estimators can be used to quickly estimate parameters in weakly identified and highly parameterized models with relative ease. We illustrate their applicability through an analysis of extreme sea-surface temperature in the Red Sea where, after training, we obtain parameter estimates and bootstrap-based confidence intervals from hundreds of spatial fields in a fraction of a second.

Real-time fully automated dosimetric computation for CT images in the clinical workflow: A feasibility study.

BackgroundCurrently, the volume computed tomography dose index (CTDIvol), the most-used quantity to express the output dose of a computed tomography (CT) patient’s dose, is not related to the real size and attenuation properties of each patient. The size-specific dose estimates (SSDE), based on the water-equivalent diameter (D W) overcome those issues. The proposed methods found in the literature do not allow real-time computation of D W and SSDE.PurposeThis study aims to develop a software to compute D W and SSDE in a real-time clinical workflow.MethodIn total, 430 CT studies and scans of a water-filled funnel phantom were used to compute accuracy and evaluate the times required to compute the D W and SSDE. Two one-sided tests (TOST) equivalence test, Bland–Altman analysis, and bootstrap-based confidence interval estimations were used to evaluate the differences between actual diameter and D W computed automatically and between D W computed automatically and manually.ResultsThe mean difference between the D W computed automatically and the actual water diameter for each slice is −0.027% with a TOST confidence interval equal to [−0.087%, 0.033%]. Bland–Altman bias is −0.009% [−0.016%, −0.001%] with lower limits of agreement (LoA) equal to −0.0010 [−0.094%, −0.068%] and upper LoA equal to 0.064% [0.051%, 0.077%]. The mean difference between D W computed automatically and manually is −0.014% with a TOST confidence interval equal to [−0.056%, 0.028%] on phantom and 0.41% with a TOST confidence interval equal to [0.358%, 0.462%] on real patients. The mean time to process a single image is 13.99 ms [13.69 ms, 14.30 ms], and the mean time to process an entire study is 11.5 s [10.62 s, 12.63 s].ConclusionThe system shows that it is possible to have highly accurate D W and SSDE in almost real-time without affecting the clinical workflow of CT examinations.

The number needed to treat adjusted for explanatory variables in regression and survival analysis: Theory and application.

The number needed to treat (NNT) is an efficacy index commonly used in randomized clinical trials. The NNT is the average number of treated patients for each undesirable patient outcome, for example, death, prevented by the treatment. We introduce a systematic theoretically‐based framework to model and estimate the conditional and the harmonic mean NNT in the presence of explanatory variables, in various models with dichotomous and nondichotomous outcomes. The conditional NNT is illustrated in a series of four primary examples; logistic regression, linear regression, Kaplan‐Meier estimation, and Cox regression models. Also, we establish and prove mathematically the exact relationship between the conditional and the harmonic mean NNT in the presence of explanatory variables. We introduce four different methods to calculate asymptotically‐correct confidence intervals for both indices. Finally, we implemented a simulation study to provide numerical demonstrations of the aforementioned theoretical results and the four examples. Numerical analysis showed that the parametric estimators of the NNT with nonparametric bootstrap‐based confidence intervals outperformed other examined combinations in most settings. An R package and a web application have been developed and made available online to calculate the conditional and the harmonic mean NNTs with their corresponding confidence intervals.

On the confidence intervals of process capability index Cpm based on a progressive type‐II censored sample

AbstractThis paper is concerned with the process capability index in the case of the progressively type‐II censored sample. In particular, the asymptotic distribution of maximum likelihood estimators of the process capability index is derived and it is used to construct the approximate confidence intervals with the delta rule. In addition, five parametric bootstrap confidence intervals for the index are discussed, in contrast to approximate confidence intervals, do not require second derivatives of the likelihood function. A Monte Carlo simulation study is performed to observe the coverage probabilities and the average width of these intervals. Simulation results indicate that the approximate confidence interval is better than the bootstrap based confidence intervals in terms of the coverage probabilities. Two practical examples are presented to illustrate the usability of the method discussed here.

Prediction of Severe Acute Graft-Versus-Host Disease (aGVHD) Based on Whole Transcriptome Sequencing of Donor CD4 T Cells

Background: After decades of intensive research, GVHD remains the unrelenting nemesis of patients and physicians involved in allogeneic hematopoietic cell transplantation (AHCT). While some plasma biomarkers can predict response to GVHD therapy at day 14-post AHCT and at the onset of clinical GVHD, an ideal tool would enable prediction of GVHD prior to transplantation. Recent systems immunology analyses of human cohorts have shown that immune parameters are stable over time within a given person, but display substantial inter-individual variations. We therefore hypothesized that inter-individual variations in T-cell function could explain a fundamental question in GVHD pathogenesis: while all allogeneic donor-recipient pairs are histo-incompatible, only a fraction of recipients present severe aGVHD post-AHCT. In order to address this question, we searched for potential correlation between the whole transcriptome of donor CD4 T cells and the occurrence of aGVHD in their recipient.Methods and Patients: We performed RNA-sequencing on purified CD4 T cells from 330 AHCT donors from three centers (Boston, Calgary, Montreal). Donor and recipient were 8/8 HLA matched siblings. CD4 T lymphocytes were isolated from blood samples collected from donors prior to hematopoietic stem cell mobilisation and collection. Recipients with acute myeloid leukemia (AML), acute lymphoid leukemia (ALL), chronic myeloid leukemia (CML), or myelodysplastic syndromes were included. Both myeloablative and reduced intensity conditioning were included. Peripheral blood and marrow graft were used as source of stem cells. Recipients received standard calcineurin based GVHD prophylaxis. Acute(a) and chronic(c) GVHD were graded according to Glucksberg and IBMTR classification. RNA was poly-A enriched and sequenced on an Ion Proton to an average depth of 40M reads. Expression profiles were computed using TopHat and cuffquant. A nested cross-validation protocol was used to train a logistic regression with either L1 or L2 regularization to predict the onset of acute GVHD of IBMTR grade C and above. An outer 10-fold cross-validation was used for the selection of hyper-parameters (number of genes considered, type and weight of the regularization) and an inner leave-one-out cross-validation was used for the predictor training. Area under the ROC curve (AUC) and relative risks were computed based on out-of-sample predicted risk of GVHD.Results: aGVHD developed in 43%. According to IBMTR classification, severe and potentially lethal aGVHD of grade ≥ C was diagnosed in 13.6% of recipients.Figure 1A presents the cross-validated performance of logistic regression classifier using either all annotated transcripts (solid line, AUC = 0.62) or expression profiles in which miRNA and snoRNA were removed (dashed line, AUC = 0.49). Discrimination between high-risk vs. low-risk donors was absolutely contingent upon the inclusion miRNA and snoRNA expression profile in our algorithm. Depending on the selected threshold applied to our predictive score, we can identify donors whose relative risk to trigger severe aGVHD is increased by 1.8- to 4-fold (Figure 1B depicting bootstrap-based confidence intervals (95%) for relative risks derived for all possible thresholds).Conclusion: Transcriptome sequencing of donor CD4 T cells can segregate donors that carry a high- vs. low-risk of causing severe aGVHD in their recipient. The key variable here was the miRNA and snoRNA expression profile. Considering that i) miRNA and snoRNA detected with our method (poly-A enriched RNA) are mostly located in introns, and ii) that intron retention regulates CD4 T-cell activity (PMID: 27369383), the most parsimonious mechanistic explanation is that the extent of intron retention in donor CD4 T cells determines the risk of severe aGVHD in recipients. Transcriptome sequencing of donor CD4 T cells could become a useful tool to optimize donor selection, personalized GVHD prophylaxis and improve AHCT outcome. [Display omitted] DisclosuresBusque:Novartis Canada Inc.: Honoraria; Pfizer: Honoraria; Bristol Myer Squibb: Honoraria; Paladin: Honoraria.

Spatially Varying Coefficient Models with Sign Preservation of the Coefficient Functions

This paper considers the estimation and inference of spatially varying coefficient models, while preserving the sign of the coefficient functions. In practice, there are various situations where coefficient functions are assumed to be in a certain subspace. For example, they should be either nonnegative or nonpositive on a domain by their nature. However, optimization on a global space of coefficient functions does not ensure that estimates preserve meaningful features in their signs. In this paper, we propose sign-preserved and efficient estimators of the coefficient functions using novel bivariate spline estimators under their smoothness conditions. Our algorithm, based on the alternating direction method of multipliers, yields estimated coefficient functions that are nonnegative or nonpositive, consistent, and efficient. Simulation studies are conducted to address the advantages of the sign preservation method for a specific situation, where coefficient functions have sign constraints. Furthermore, we propose residual bootstrap-based confidence intervals for sign preserving coefficient functions over the domain of interest after adjusting the inherent bias of penalized smoothing spline techniques. Finally, we evaluate our method in a case study using air temperature, land surface temperature, and elevation in the USA. Supplementary materials accompanying this paper appear online.

Cognitive Functioning: An Underlying Mechanism of Age and Gender Differences in Self-Assessed Risk Tolerance among an Aging Population

Attitude toward risk plays a vital role in an individual’s financial decision-making and well-being. Past studies have found significant association of age and gender with risk tolerance. However, studies on the factors affecting the underlying mechanism are scant. The purpose of this research is to test whether cognitive functioning mediates the association between age, gender, and self-assessed risk tolerance. Using the 2014 wave of the Health and Retirement Study, path analysis was conducted to test the hypothesized model. Results revealed a negative direct association between age and risk tolerance. Moreover, the study also found a lower level of risk tolerance in women. A bootstrap-based confidence interval revealed that a significant portion of the relationship between age and risk tolerance was mediated by cognition. However, the gender difference in risk tolerance was not explained by cognition. Financial planning practitioners and policymakers should understand the contribution of cognitive functioning toward the difference in risk tolerance in older populations and implement strategies to reinforce cognitive functioning to mitigate the adversity of a low level of risk tolerance.

A method for estimating the transmissibility of influenza using serial cross-sectional seroepidemiological data

BackgroundSeroepidemiological surveillance data has been demonstrated to be useful for estimating the cumulative incidence of influenza, and measures the difference between pre- and post-epidemic seropositive fractions. Despite this, such studies relied on a chosen cut-off value for seropositivity. The aim of the present study is to develop a method to analyze distributions of serial cross-sectional seroepidemiological surveillance datasets using an epidemiological model so that the transmission potential can be estimated without imposing a cut-off value. MethodsA mathematical model of influenza transmission with a discrete antibody titer level was constructed. The final size equation for pre- and post-epidemic titer levels was derived. Subsequently, using the estimated distribution of the dilution increase caused by infection and the measurement error distribution, the model parameters were optimized using the maximum likelihood method. A bootstrap-based confidence interval calculation and sensitivity analysis were also performed. ResultsWithout imposing a cut-off value, the cumulative incidence was quantified, thereby yielding an estimate of the basic reproduction number. For the purpose of exposition, the proposed method was applied to influenza A/Victoria/3/75(H3N2) data, and serological data between 1975 and 1976 were compared. The estimated reproduction number was greater than that using the cut-off value of the hemagglutination inhibition level with titer level 20 (dilution 1:20) or above to define positives. ConclusionThe proposed method without a cut-off value offers an unbiased approach to estimating the cumulative incidence along with the reproduction number. If a cut-off value is required, the results imply that titer level 20 or above may better represent a reasonable cut-off value for calculating the incidence, but it could underestimate the basic reproduction number.

Quantitative Microscopy: Particle Size/Shape Characterization, Addressing Common Errors Using 'Analytics Continuum' Approach.

Particle size/shape characterization of active pharmaceutical ingredient (API) is integral to successful product development. It is more of a correlative property than a decision-making measure. Though microscopy is the only technique that provides a direct measure of particle properties, it is neglected for reasons like non-repeatability and non-reproducibility which is often attributed to a) fundamental error, b) segregation error, c) human error, d) sample randomness, e) sample representativeness etc. Using the "Sucrose" as model sample, we propose "analytics continuum" approach that integrates optical microscope PSD measurements complimented by NIR spectroscopy-based trending analysis as a prescreening tool to demonstrate sample randomness and representativeness. Furthermore, plethora of statistical tests are utilized to infer population statistics. Subsequently, an attribute-based control chart and bootstrap-based confidence interval was developed to monitor product performance. A flowchart to serve as an elementary guideline is developed, which is then extended to handle more complex situations involving API crystallized from two different solvent systems. The results show that the developed methodology can be utilized as a quantitative procedure to assess the suitability of API/excipients from different batches or from alternate vendors and can significantly help in understanding the differences between material even on a minor scale.

Nonparametric bootstrap inference for the targeted highly adaptive least absolute shrinkage and selection operator (LASSO) estimator.

The Highly-Adaptive least absolute shrinkage and selection operator (LASSO) Targeted Minimum Loss Estimator (HAL-TMLE) is an efficient plug-in estimator of a pathwise differentiable parameter in a statistical model that at minimal (and possibly only) assumes that the sectional variation norm of the true nuisance functions (i.e., relevant part of data distribution) are finite. It relies on an initial estimator (HAL-MLE) of the nuisance functions by minimizing the empirical risk over the parameter space under the constraint that the sectional variation norm of the candidate functions are bounded by a constant, where this constant can be selected with cross-validation. In this article we establish that the nonparametric bootstrap for the HAL-TMLE, fixing the value of the sectional variation norm at a value larger or equal than the cross-validation selector, provides a consistent method for estimating the normal limit distribution of the HAL-TMLE. In order to optimize the finite sample coverage of the nonparametric bootstrap confidence intervals, we propose a selection method for this sectional variation norm that is based on running the nonparametric bootstrap for all values of the sectional variation norm larger than the one selected by cross-validation, and subsequently determining a value at which the width of the resulting confidence intervals reaches a plateau. We demonstrate our method for 1) nonparametric estimation of the average treatment effect when observing a covariate vector, binary treatment, and outcome, and for 2) nonparametric estimation of the integral of the square of the multivariate density of the data distribution. In addition, we also present simulation results for these two examples demonstrating the excellent finite sample coverage of bootstrap-based confidence intervals.

Raster Image Correlation Spectroscopy Performance Evaluation

Raster image correlation spectroscopy (RICS) is a fluorescence image analysis method for extracting the mobility, concentration, and stoichiometry of diffusing fluorescent molecules from confocal image stacks. The method works by calculating a spatial correlation function for each image and analyzing the average of those by model fitting. Rules of thumb exist for RICS image acquisitioning, yet a rigorous theoretical approach to predict the accuracy and precision of the recovered parameters has been lacking. We outline explicit expressions to reveal the dependence of RICS results on experimental parameters. In terms of imaging settings, we observed that a twofold decrease of the pixel size, e.g., from 100 to 50 nm, decreases the error on the translational diffusion constant (D) between three- and fivefold. For D = 1 μm2 s−1, a typical value for intracellular measurements, ∼25-fold lower mean-squared relative error was obtained when the optimal scan speed was used, although more drastic improvements were observed for other values of D. We proposed a slightly modified RICS calculation that allows correcting for the significant bias of the autocorrelation function at small (≪50 × 50 pixels) sizes of the region of interest. In terms of sample properties, at molecular brightness E = 100 kHz and higher, RICS data quality was sufficient using as little as 20 images, whereas the optimal number of frames for lower E scaled pro rata. RICS data quality was constant over the nM–μM concentration range. We developed a bootstrap-based confidence interval of D that outperformed the classical least-squares approach in terms of coverage probability of the true value of D. We validated the theory via in vitro experiments of enhanced green fluorescent protein at different buffer viscosities. Finally, we outline robust practical guidelines and provide free software to simulate the parameter effects on recovery of the diffusion coefficient.

A Note on Conditional Versus Joint Unconditional Weak Convergence in Bootstrap Consistency Results

The consistency of a bootstrap or resampling scheme is classically validated by weak convergence of conditional laws. However, when working with stochastic processes in the space of bounded functions and their weak convergence in the Hoffmann–Jorgensen sense, an obstacle occurs: due to possible non-measurability, neither laws nor conditional laws are well defined. Starting from an equivalent formulation of weak convergence based on the bounded Lipschitz metric, a classical circumvention is to formulate bootstrap consistency in terms of the latter distance between what might be called a conditional law of the (non-measurable) bootstrap process and the law of the limiting process. The main contribution of this note is to provide an equivalent formulation of bootstrap consistency in the space of bounded functions which is more intuitive and easy to work with. Essentially, the equivalent formulation consists of (unconditional) weak convergence of the original process jointly with two bootstrap replicates. As a by-product, we provide two equivalent formulations of bootstrap consistency for statistics taking values in separable metric spaces: the first in terms of (unconditional) weak convergence of the statistic jointly with its bootstrap replicates, the second in terms of convergence in probability of the empirical distribution function of the bootstrap replicates. Finally, the asymptotic validity of bootstrap-based confidence intervals and tests is briefly revisited, with particular emphasis on the (in practice, unavoidable) Monte Carlo approximation of conditional quantiles.

Individual Differences in Developmental Change: Quantifying the Amplitude and Heterogeneity in Cognitive Change across Old Age.

It is well known that cognitive decline in older adults is of smaller amplitude in longitudinal than in cross-sectional studies. Yet, the measure of interest rests generally with aggregated group data. A focus on individual developmental trajectories is rare, mainly because it is difficult to assess intraindividual change reliably. Individual differences in developmental trajectories may differ quantitatively (e.g., larger or smaller decline) or qualitatively (e.g., decline vs improvement), as well as in the degree of heterogeneity of change across different cognitive domains or different tasks. The present paper aims at exploring, within the Geneva Variability Study, individual change across several cognitive domains in 92 older adults (aged 59–89 years at baseline) over a maximum of seven years and a half. Two novel, complementary methods were used to explore change in cognitive performance while remaining entirely at the intra-individual level. A bootstrap based confidence interval was estimated, for each participant and for each experimental condition, making it possible to define three patterns: stability, increase or decrease in performance. Within-person ANOVAs were also conducted for each individual on all the tasks. Those two methods allowed quantifying the direction, the amplitude and the heterogeneity of change for each individual. Results show that trajectories differed widely among individuals and that decline is far from being the rule.

Bootstrap-calibrated empirical likelihood confidence intervals for the difference between two Gini indexes

Measuring the difference between two Gini indexes is important in comparing the inequality between two groups. This paper systematically provides confidence intervals for this difference. First, normal approximation- and bootstrap-based confidence intervals are provided for the difference. Second, empirical likelihood (EL)-based confidence intervals are proposed according to a ratio statistic that is proven to have a weighted Chi-squared distribution with one degree of freedom. Third, two calibration approaches are established to improve the undercoverage issue of the EL method; these two approaches are augmented EL and bootstrap-calibrated EL. Monte Carlo simulations show that bootstrap-calibrated EL generally outperforms other methods in constructing confidence intervals. In the simulations, these methods are compared with the permutation method to test the equality of two Gini indexes. Lastly, these methods are applied to real data.

A better confidence interval for the sensitivity at a fixed level of specificity for diagnostic tests with continuous endpoints.

For a diagnostic test with continuous measurement, it is often important to construct confidence intervals for the sensitivity at a fixed level of specificity. Bootstrap-based confidence intervals were shown to have good performance as compared to others, and the one by Zhou and Qin (2005) was recommended as the best existing confidence interval, named the BTII interval. We propose two new confidence intervals based on the profile variance method and conduct extensive simulation studies to compare the proposed intervals and the BTII intervals under a wide range of conditions. An example from a medical study on severe head trauma is used to illustrate application of the new intervals. The new proposed intervals generally have better performance than the BTII interval.

Evaluating the uncertainty and reliability of standardized indices

Standardized indices are widely used in the spatio-temporal monitoring of several hydrological variables. The estimation of these indices is affected by uncertainty which depends on the methods adopted for their quantification and on the characteristics (i.e., size and variability) of the available sample of observations. In this paper various uncertainty measures, applicable to any kind of standardized index, are proposed. These measures derive from bootstrap-based confidence intervals expressed in years of return period and are effective for assessing both the uncertainty and the reliability of the index estimate. In the illustrative case study the indices considered are the Standardized Precipitation Index and the Standardized Precipitation Evapotranspiration Index. Their time series have been quantified by both nonparametric and parametric approaches, using the weather data of a single station in central Italy. For the parametric approach, two possible types of distributions have been assumed for each index. The results are discussed in order to analyze the behavior of the proposed uncertainty measures in relation to: sample size, type of approach (parametric or nonparametric), time scale, type of standardized index, and type of anomaly (excess or deficit).

A Comparison of Confidence Interval Methods of Fixed Effect in Nested Error Regression Model

Linear mixed-effects models are very popular and powerful tools in many scientific fields such as zoology, biology, and education. Estimators of fixed effects do not only depend on the variances of error terms but they also depend on random terms in mixed-effect models. When the distributions of random effects are unknown or enough sample size cannot be obtained, standard methods may fail. This study aims to determine a promising confidence interval method among existing methods in terms of coverage probability of true value of parameter. Standard and parametric bootstrap-based confidence interval methods for nested error regression model were compared in the simulation study under small samples. It is observed that parametric bootstrap-based method provides better coverage rates for small intra-correlation and profile likelihood method usually provides better results for moderate and strong correlation.

Estimation of Stress-Strength Reliability for the Generalized Pareto Distribution Based on Progressively Censored Samples

This paper deals with the estimation of stress-strength reliability parameter, \(R = P\left( Y < X \right) \), based on progressively type II censored samples when stress, strength are two independent generalized Pareto random variables. The maximum likelihood estimators, their asymptotic distributions, asymptotic confidence intervals, bootstrap based confidence intervals and Bayes estimators are derived for \(R\). Using Monte Carlo simulations, the MSE, Bayes risk estimators, credible sets and coverage probabilities are computed and compared.

Quantile rank maps: A new tool for understanding individual brain development

We propose a novel method for neurodevelopmental brain mapping that displays how an individual's values for a quantity of interest compare with age-specific norms. By estimating smoothly age-varying distributions at a set of brain regions of interest, we derive age-dependent region-wise quantile ranks for a given individual, which can be presented in the form of a brain map. Such quantile rank maps could potentially be used for clinical screening. Bootstrap-based confidence intervals are proposed for the quantile rank estimates. We also propose a recalibrated Kolmogorov–Smirnov test for detecting group differences in the age-varying distribution. This test is shown to be more robust to model misspecification than a linear regression-based test. The proposed methods are applied to brain imaging data from the Nathan Kline Institute Rockland Sample and from the Autism Brain Imaging Data Exchange (ABIDE) sample.