Harrell-Davis Estimator Research Articles

Accuracy and Consistency of Confidence Limits for Monosyllable Identification Scores Derived Using Simulation, the Harrell-Davis Estimator, and Nonlinear Quantile Regression.

Audiological diagnosis and rehabilitation often involve the assessment of whether the maximum speech identification score (PBmax) is poorer than expected from the pure-tone average (PTA) threshold. This requires the estimation of the lower boundary of the PBmax values expected for a given PTA (one-tailed 95% confidence limit, CL). This study compares the accuracy and consistency of three methods for estimating the 95% CL. The 95% CL values were estimated using a simulation method, the Harrell-Davis (HD) estimator, and non-linear quantile regression (nQR); the latter two are both distribution-free methods. The first two methods require the formation of sub-groups with different PTAs. Accuracy and consistency in the estimation of the 95% CL were assessed by applying each method to many random samples of 50% of the available data and using the fitted parameters to predict the data for the remaining 50%. A total of 642 participants aged 17 to 84 years with sensorineural hearing loss were recruited from audiology clinics. Pure-tone audiograms were obtained and PBmax scores were measured using monosyllables at 40 dB above the speech recognition threshold or at the most comfortable level. For the simulation method, 6.7 to 8.2% of the PBmax values fell below the 95% CL for both ears, exceeding the target value of 5%. For the HD and nQR methods, the PBmax values fell below the estimated 95% CL for approximately 5% of the ears, indicating good accuracy. Consistency, estimated from the standard deviation of the deviations from the target value of 5%, was similar for all the methods. The nQR method is recommended because it has good accuracy and consistency, and it does not require the formation of arbitrary PTA sub-groups.

MODELLING NONLINEAR RELATION BY USING RUNNING INTERVAL SMOOTHER, CONSTRAINED B-SPLINE SMOOTHING AND DIFFERENT QUANTILE ESTIMATORS

This paper compares the small-sample properties of two non-parametric regression methods, running interval smoother and constrained b-spline smoothing. The running interval smoother method deals with estimation of a conditional quantile (or a measure of location) using different estimators and here our focus is on Harrell-Davis and newly proposed NO quantile estimators. The constrained b-spline smoothing method uses the quantile regression estimator while obtaining conditional quantile estimates. Constrained b-spline smoothing and running interval smoother methods are compared with a simulation study by using theoretical distributions. Furthermore, the methods are examined graphically to understand how they can model the relationship between variables. Constrained b-spline smoothing and running interval smoother with NO estimator outperformed running interval smoother with Harrell-Davis estimator in terms of mean squared error.

Evaluating instructional designs with mental workload assessments in university classrooms

ABSTRACT Cognitive cognitive load theory (CLT) has been conceived for improving instructional design practices. Although researched for many years, one open problem is a clear definition of its cognitive load types and their aggregation towards an index of overall cognitive load. In Ergonomics, the situation is different with plenty of research devoted to the development of robust constructs of mental workload (MWL). By drawing a parallel between CLT and MWL, as well as by integrating relevant theories and measurement techniques from these two fields, this paper is aimed at investigating the reliability, validity and sensitivity of three existing self-reporting mental workload measures when applied to long learning sessions, namely, the NASA Task Load index, the Workload Profile and the Rating Scale Mental Effort, in a typical university classroom. These measures were aimed at serving for the evaluation of two instructional conditions. Evidence suggests these selected measures are reliable and their moderate validity is in line with results obtained within Ergonomics. Additionally, an analysis of their sensitivity by employing the descriptive Harrell-Davis estimator suggests that the Workload Profile is more sensitive than the Nasa Task Load Index and the Rating Scale Mental Effort for long learning sessions.

Sex-specific 99th percentiles derived from the AACC Universal Sample Bank for the Roche Gen 5 cTnT assay: Comorbidities and statistical methods influence derivation of reference limits

ObjectivesOur purpose was to determine a) overall and sex-specific 99th percentile upper reference limits (URL) and b) influences of statistical methods and comorbidities on the URLs. MethodsHeparin plasma from 838 normal subjects (423 men, 415 women) were obtained from the AACC (Universal Sample Bank). The cobas e602 measured cTnT (Roche Gen 5 assay); limit of detection (LoD), 3ng/L. Hemoglobin A1c (URL 6.5%), NT-proBNP (URL 125ng/L) and eGFR (60mL/min/1.73m2) were measured, along with identification of statin use, to better define normality. 99th percentile URLs were determined by the non-parametric (NP), Harrell-Davis Estimator (HDE) and Robust (R) methods. Results355 men and 339 women remained after exclusions. Overall<50% of subjects had measureable concentrations ≥ LoD: 45.6% no exclusion, 43.5% after exclusion; compared to men: 68.1% no exclusion, 65.1% post exclusion; women: 22.7% no exclusion, 20.9% post exclusion. The statistical method used influenced URLs as follows: pre/post exclusion overall, NP 16/16ng/L, HDE 17/17ng/L, R not available; men NP 18/16ng/L, HDE 21/19ng/L, R 16/11ng/L; women NP 13/10ng/L, HDE 14/14ng/L, R not available. ConclusionsWe demonstrated that a) the Gen 5 cTnT assay does not meet the IFCC guideline for high-sensitivity assays, b) surrogate biomarkers significantly lowers the URLs and c) statistical methods used impact URLs. Our data suggest lower sex-specific cTnT 99th percentiles than reported in the FDA approved package insert. We emphasize the importance of detailing the criteria used to include and exclude subjects for defining a healthy population and the statistical method used to calculate 99th percentiles and identify outliers.

Quantile estimation and comparing two independent groups with an approach based on percentile bootstrap

ABSTRACTAlthough the most common approach for comparing two independent groups is on the basis of some measure of location, determination of the differences in the tails of the groups is often of interest. In this study, Harrell–Davis estimator, Sfakianakis–Verginis estimators and default quantile estimator of R are used in conjunction with a percentile bootstrap method with the aim of comparing two independent groups via the quantiles, and the relative efficiencies of Harrell–Davis and Sfakianakis–Verginis estimators are compared. General performance of Sfakianakis–Verginis estimators was much better than Harrell–Davis estimator in terms of both saving actual type I error and relative efficiency.

Comparison of two independent groups by using the lower and upper quantiles and percentile bootstrap

The frequently used way of comparing two independent groups is to compare in terms of some measure of location such as mean. For non-normal and heteroscedastic cases, trimmed mean, median or some other robust measures of location can be used instead. However, determination of the differences in the tails of the groups might be of interest. For this reason, comparing the lower and upper quantiles becomes an important issue. In this study, Harrell-Davis estimator and the default quantile estimator of R are compared in terms of actual Type I error rates. When quantiles close to zero or one are compared with small sample sizes Gumbel's estimator, and when quantiles close to median are compared with large sample sizes Harrell Davis estimator saved actual Type I error rate better.

Comparing two independent groups via the lower and upper quantiles

The most common strategy for comparing two independent groups is in terms of some measure of location intended to reflect the typical observation. However, it can be informative and important to compare the lower and upper quantiles as well, but when there are tied values, extant techniques suffer from practical concerns reviewed in the paper. For the special case where the goal is to compare the medians, a slight generalization of the percentile bootstrap method performs well in terms of controlling Type I errors when there are tied values [Wilcox RR. Comparing medians. Comput. Statist. Data Anal. 2006;51:1934–1943]. But our results indicate that when the goal is to compare the quartiles, or quantiles close to zero or one, this approach is highly unsatisfactory when the quantiles are estimated using a single order statistic or a weighted average of two order statistics. The main result in this paper is that when using the Harrell–Davis estimator, which uses all of the order statistics to estimate a quantile, control over the Type I error probability can be achieved in simulations, even when there are tied values, provided the sample sizes are not too small. It is demonstrated that this method can also have substantially higher power than the distribution free method derived by Doksum and Sievers [Plotting with confidence: graphical comparisons of two populations. Biometrika 1976;63:421–434]. Data from two studies are used to illustrate the practical advantages of the method studied here.

Comparing two dependent groups via quantiles

This paper considers two general ways dependent groups might be compared based on quantiles. The first compares the quantiles of the marginal distributions. The second focuses on the lower and upper quantiles of the usual difference scores. Methods for comparing quantiles have been derived that typically assume that sampling is from a continuous distribution. There are exceptions, but generally, when sampling from a discrete distribution where tied values are likely, extant methods can perform poorly, even with a large sample size. One reason is that extant methods for estimating the standard error can perform poorly. Another is that quantile estimators based on a single-order statistic, or a weighted average of two-order statistics, are not necessarily asymptotically normal. Our main result is that when using the Harrell–Davis estimator, good control over the Type I error probability can be achieved in simulations via a standard percentile bootstrap method, even when there are tied values, provided the sample sizes are not too small. In addition, the two methods considered here can have substantially higher power than alternative procedures. Using real data, we illustrate how quantile comparisons can be used to gain a deeper understanding of how groups differ.

An exact bootstrap approach towards modification of the Harrell–Davis quantile function estimator for censored data

A new kernel quantile estimator is proposed for right-censored data, which takes the form of , where w j(u, c) is based on a beta kernel with bandwidth parameter c. The advantage of this estimator is that exact bootstrap methods may be employed to estimate the mean and variance of [Qcirc](u; c). It follows that a novel solution for finding the optimal bandwidth may be obtained through minimization of the exact bootstrap mean squared error (MSE) estimate of [Qcirc](u; c). We prove the large sample consistency of [Qcirc](u; c) for fixed values of c. A Monte Carlo simulation study shows that our estimator is significantly better than the product-limit quantile estimator [Qcirc] KM(u)=inf{t:[Fcirc] n (t)≥u}, with respect to various MSE criteria. For general simplicity, setting c=1 leads to an extension of classical Harrell–Davis estimator for censored data and performs well in simulations. The procedure is illustrated by an application to lung cancer survival data.

Estimating a population median with a small sample

Various estimators of the population median were compared for small samples (N 50). The competitors included the sample median, Harrell-Davis (HD) estimate, six new estimates (Circle, DblExp, Exp+, Exp−, AltExp, Fibon) that require no a priori information about the population, and one estimate (AllExp) that requires information on the general form of the distribution The two benchmark criteria were the mean square error (MSE) and the rank based error (RBE). When there is no information available on the population, (a) AltExp was the best estimator for N 30 for the MSE criterion, (b) the estimate based on the Fibonacci sequence was the best for N 30 for the RBE criterion, and (c) the Harrell-Davis estimator was superior for N > 30. There was no support for using the sample median as an estimate of the population median.

Some Results on Comparing the Quantiles of Dependent Groups

Recently, Lombard derived an extension of the Doksum–Sievers shift function to dependent groups. This article suggests using a particular numerical method for determining the critical value, reports on the ability of the method to control the probability of a Type I error when sample sizes are small, and it provides comparisons with methods aimed at comparing deciles. It is found that for continuous distributions, Lombard's method performs well and in particular has high power relative to the other two methods considered. But when tied values can occur, now it can have relatively poor power; a method based on the Harrell-Davis estimator is found to give more satisfactory results.

Six Modifications Of The Aligned Rank Transform Test For Interaction

Testing for interactions in multivariate experiments is an important function. Studies indicate that much data from social studies research is not normally distributed, thus violating that assumption of the AN OVA procedure. The aligned rank transformation test (ART), aligning using the means of columns and rows, has been found, in limited situations, to be robust to Type I error rates and to have greater power than the ANOVA. This study explored a variety of alignments, including the median, Winsorized trimmed means (10%) and (20%), the Huber1.28 M-estimator, and the Harrell-Davis estimator of the median. Results are reported for Type I errors and power.

Testing whether independent treatment groups have equal medians

The paper suggests new methods for comparing the medians corresponding to independent treatment groups. The procedures are based on the Harrell-Davis estimator in conjunction with a slight modification and extension of the bootstrap calibration technique suggested by Loh. Alternatives to the Harrell-Davis estimator are briefly discussed. For the special case of two treatment groups, the proposed procedure always had more power than the Fligner-Rust solution, as well as the procedure examined by Wilcox and Charlin. Included is an illustration, using real data, that comparing medians, rather than means, can yield a substantially different conclusion as to whether two distributions differ in terms of some measure of central location.