Motivated by the questions of risk assessment in climatology (temperature change in North America) and medicine (impact of statin usage and coronavirus disease 2019 on hospitalized patients), we address the problem of estimating the set in the domain of a function whose image equals a predefined subset of the real line. Existing methods require strict assumptions. We generalize the estimation of such sets to dense and nondense domains with protection against inflated Type I error in exploratory data analysis. This is achieved by proving that confidence sets of multiple upper, lower, or interval sets can be simultaneously constructed with the desired confidence nonasymptotically through inverting simultaneous confidence intervals. Nonparametric bootstrap algorithm and code are provided.

Regression modeling is the workhorse of statistics and there is a vast literature on estimation of the regression function. It has been realized in recent years that in regression analysis the ultimate aim may be the estimation of a level set of the regression function, ie, the set of covariate values for which the regression function exceeds a predefined level, instead of the estimation of the regression function itself. The published work on estimation of the level set has thus far focused mainly on nonparametric regression, especially on point estimation. In this article, the construction of confidence sets for the level set of linear regression is considered. In particular, 1 - α $$ 1-\\alpha $$ level upper, lower and two-sided confidence sets are constructed for the normal-error linear regression. It is shown that these confidence sets can be easily constructed from the corresponding 1 - α $$ 1-\\alpha $$ level simultaneous confidence bands. It is also pointed out that the construction method is readily applicable to other parametric regression models where the mean response depends on a linear predictor through a monotonic link function, which include generalized linear models, linear mixed models and generalized linear mixed models. Therefore, the method proposed in this article is widely applicable. Simulation studies with both linear and generalized linear models are conducted to assess the method and real examples are used to illustrate the method.

Given a functional central limit (fCLT) for an estimator and a parameter transformation, we construct random processes, called functional delta residuals, which asymptotically have the same covariance structure as the limit process of the functional delta method. An explicit construction of these residuals for transformations of moment-based estimators and a multiplier bootstrap fCLT for the resulting functional delta residuals are proven. The latter is used to consistently estimate the quantiles of the maximum of the limit process of the functional delta method in order to construct asymptotically valid simultaneous confidence bands for the transformed functional parameters. Performance of the coverage rate of the developed construction, applied to functional versions of Cohen\'s d, skewness and kurtosis, is illustrated in simulations and their application to test Gaussianity is discussed.

Much of the research on multiple comparison and simultaneous inference in the past 60 years or so has been for the comparisons of several population means. Spurrier seems to have been the first to investigate multiple comparisons of several simple linear regression lines using simultaneous confidence bands. In this paper, we extend the work of Liu et al. for finite comparisons of several univariate linear regression models using simultaneous confidence bands to finite comparisons of several multivariate linear regression models using simultaneous confidence tubes. We show how simultaneous confidence tubes can be constructed to allow more informative inferences for the comparison of several multivariate linear regression models than the current approach of hypotheses testing. The methods are illustrated with examples.

In risk assessment, it is often desired to make inferences on the risk at certain low doses or on the dose(s) at which a specific benchmark risk (BMR) is attained. At times, [Formula: see text] dose levels or BMRs are of interest, and some form of multiplicity adjustment is necessary to ensure a valid [Formula: see text] simultaneous inference. Bonferroni correction is often employed in practice for such purposes. Though relative simple to implement, the Bonferroni strategy can suffer from extreme conservatism (Nitcheva et al., 2005; Al-Saidy et al., 2003). Recently, Kerns (2017) proposed the use of simultaneous hyperbolic and three-segment bands to perform multiple inferences in risk assessment under Abbott-adjusted log-logistic model with the dose level constrained to a given interval. In this paper, we present and compare methods for deriving multiplicity-adjusted upper limits on extra risk and lower bounds on the benchmark dose under Abbott-adjusted log-logistic model. Monte Carlo simulations evaluate the characteristics of the simultaneous limits. An example is given to illustrate the use of the methods.

Despite the utility of multiparameter flow cytometry for a wide variety of biological applications, comparing single parameter histograms of fluorescence intensity remains a mainstay of flow cytometric analysis. Even comparisons requiring multiparameter gating strategies often end with single parameter histograms as the final readout. When histograms overlap, analysis relies on comparison of mean or median fluorescence intensities, or determination of percent positive based on an arbitrary cutoff. Earlier attempts to address this problem utilized either simple channel-by-channel subtraction without statistical evaluation, or the Kolmogorov-Smirnov (KS) or Chi-square test statistics, both of which proved to be overly sensitive to small and biologically insignificant differences. Here we present a method for the comparison of two single-parameter histograms based on difference curves and their simultaneous confidence bands generated by bootstrapping raw channel data. Bootstrapping is a nonparametric statistical approach that can be used to generate confidence intervals without distributional assumptions about the data. We have constructed simultaneous confidence bands and show them to be superior to KS and Cox methods. The method constructs 95% confidence bands about the difference curves, provides a P value for the comparison and calculates the area under the difference curve (AUC) as an estimate of percent positive and the area under the confidence band (AUCSCB95), providing a lower estimate of the percent positive. To demonstrate the utility of this new approach we have examined single-color fluorescence intensity data taken from a cell surface proteomic survey of a lung cancer cell line (A549) and a published fluorescence intensity data from a rhodamine efflux assay of P-glycoprotein activity, comparing rhodamine 123 loading and efflux in CD4 and CD8 T-cell populations. SAS source code is provided as supplementary material.