Confidence Intervals of Gini Coefficient Under Unequal Probability Sampling

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.

The mean residual life (MRL) function for a given random variable T is the expected remaining lifetime of T after a fixed time point t. It is of great interest in survival analysis, reliability, actuarial applications, duration modelling, etc. Liang, Shen, and He [‘Likelihood Ratio Inference for Mean Residual Life of Length-biased Random Variable’, Acta Mathematicae Applicatae Sinica, English Series, 32, 269–282] proposed empirical likelihood (EL) confidence intervals for the MRL based on length-biased right-censored data. However, their -2log(EL ratio) has a scaled chi-squared distribution. To avoid the estimation of the scale parameter in constructing confidence intervals, we propose a new empirical likelihood (NEL) based on i.i.d. representation of Kaplan–Meier weights involved in the estimating equation. We also develop the adjusted new empirical likelihood (ANEL) to improve the coverage probability for small samples. The performance of the NEL and the ANEL compared to the existing EL is demonstrated via simulations: the NEL-based and ANEL-based confidence intervals have better coverage accuracy than the EL-based confidence intervals. Finally, our methods are illustrated with a real data set.

We study statistical inference for the derivative of the nonparametric regression function using local linear model based empirical likelihood. We first derive a normal equation for the derivative through the local linear model and use this equation to construct an empirical likelihood for the derivative. We show that the limiting distribution of the empirical likelihood ratio is a scaled χ 1 2 distribution rather than the usual (unscaled) χ 1 2 distribution. We use this limiting distribution to construct pointwise confidence intervals for the derivative. Such empirical likelihood ratio confidence intervals are easier to obtain than the normal approximation based confidence intervals. A small simulation study also suggests that they are more accurate.

In this paper, we consider the empirical likelihood confidence intervals for mean difference under the assumption of missing at random (MAR). We prove that the empirical log-likelihood ratio of mean difference converges to a standard chi-squared distribution. We conduct simulation study to compare the proposed empirical likelihood with the normal approximation method in terms of the coverage accuracies and average lengths of confidence intervals. By Monte Carlo simulations, we verify that the proposed method performs well.

For an AR(1) process with ARCH(1) errors, we propose empirical likelihood tests for testing whether the sequence is strictly stationary but has infinite variance, or the sequence is an ARCH(1) sequence or the sequence is an iid sequence. Moreover, an empirical likelihood based confidence interval for the parameter in the AR part is proposed. All of these results do not require more than a finite second moment of the innovations. This includes the case of t-innovations for any degree of freedom larger than 2, which serves as a prominent model for real data.

Pretest-posttest studies are commonly used for assessing the effect of a treatment or an intervention. We propose an empirical likelihood based approach to both testing and estimation of the treatment effect in non-randomized pretest-posttest studies where the posttest outcomes are subject to missingness. The proposed empirical likelihood ratio test and the estimation procedure are multiply robust in the sense that multiple working models are allowed for the propensity score of treatment assignment, the missingness probability and the outcome regression, and the validity of the test and the estimation requires only a certain combination of those multiple working models to be correctly specified. An empirical likelihood ratio confidence interval can be constructed for the treatment effect and has better coverage probabilities than confidence intervals based on the Wald statistic. Simulations are conducted to demonstrate the finite-sample performances of the proposed methods.

Recent literature sites ethnic origin as a major factor in developing pulmonary function reference values. Extensive studies established reference values for European and African Americans, but not for Hispanic Americans. The Third National Health and Nutrition Examination Survey defines Hispanic as individuals of Spanish speaking cultures. While no group was excluded from the target population, sample size requirements only allowed inclusion of individuals who identified themselves as Mexican Americans. This research constructs nonparametric reference value confidence intervals for Hispanic American pulmonary function. The method is applicable to all ethnicities. We use empirical likelihood confidence intervals to establish normal ranges for reference values. Its major advantage: it is model free, but shares asymptotic properties of model based methods. Statistical comparisons indicate that empirical likelihood interval lengths are comparable to normal theory intervals. Power and efficiency studies agree with previously published theoretical results.

Nontraumatic abdominal pain is a common and challenging presentation in the emergency department (ED). Point-of-care ultrasound (PoCUS) offers rapid, bedside imaging, but its association with ED length of stay (LOS), resource use, and patient safety, particularly concerning unscheduled return visits, remains unclear. To evaluate the association of early PoCUS (within 1 h) with ED LOS and safety in adults with nontraumatic abdominal pain. A propensity score-weighted retrospective cohort analysis in a tertiary medical center in Taiwan (2021-2023). This study included 39 443 adult patients with nontraumatic abdominal pain (12 399 PoCUS < 1 h and 27 044 no PoCUS), excluding transfers. PoCUS performed by an emergency physician within 1 h of ED arrival. Primary: ED LOS; secondary: ED costs, computed tomography (CT) use, outpatient disposition, and rate of admission after an unscheduled return visit. A key subanalysis evaluated outcomes (costs, hospital LOS, and mortality) for admissions after an unscheduled return visit. Analysis used inverse probability of treatment weighting-adjusted regression. PoCUS performed within 1 h was associated with a 16% reduction in first ED LOS [ratio of means 0.84, 95% confidence interval (CI): 0.83-0.86] and 50% increased odds of outpatient disposition [odds ratio (OR): 1.50, 95% CI: 1.42-1.58]. 1st ED costs were slightly increased by 4% (ratio of means: 1.04, 95% CI: 1.03-1.06), and CT use was not reduced (OR: 0.98, 95% CI: 0.93-1.03). While associated with a 25% increased odds of admission after an unscheduled return visit (OR: 1.25, 95% CI: 1.02-1.53), this subgroup had 33% lower admission costs (ratio of means: 0.67, 95% CI: 0.57-0.80), 26% shorter hospital LOS (ratio of means: 0.74, 95% CI: 0.64-0.85), and 94% lower odds of ED death on return (ratio of means: 0.06, 95% CI: 0.01-0.54). PoCUS performed within 1 h of ED arrival was associated with shorter ED LOS and more outpatient dispositions, without reducing CT use. However, it was also associated with an increased probability of admission following an unscheduled return visit. Future studies are needed to elucidate the causal impact of this practice on patient safety and downstream resource use.

For regression analysis of doubly truncated data, we propose two empirical likelihood (EL) inference approaches, called non-smooth EL and non-smooth Jackknife EL (JEL), to make inference about regression parameters based on the generalized estimating equations of existing weighted rank estimators. The limiting distributions of non-smooth log-EL and log-JEL ratios statistics are derived and non-smooth EL, and JEL confidence intervals for any specified component of regression parameters are obtained. We carry out extensive simulation studies to compare the proposed approaches with the random weighting (RW) approach. The simulation results demonstrate that the non-smooth EL and JEL confidence intervals have better performances than the RW confidence intervals based on coverage probability and average length of confidence intervals of regression parameters when the dependent variable is subject to the double truncation. A real data example is provided to illustrate the proposed approaches.

BackgroundThe Comprehensive Geriatric Assessment (CGA) is a multidisciplinary process that determines a frail older person’s medical, functional, psychological and social capability to ensure that they have a co-ordinated plan for treatment and follow-up.ObjectivesTo improve our understanding of the effectiveness, cost-effectiveness and implementation of the CGA across hospital and hospital-at-home settings.MethodsWe used a variety of methods. We updated a Cochrane review of randomised trials of the CGA in hospital for older people aged ≥ 65 years, conducted a national survey of community CGA, analysed data from three health boards using propensity score matching (PSM) and regression analysis, conducted a qualitative study and used a modified Delphi method.ResultsWe included 29 trials recruiting 13,766 participants in the Cochrane review of the CGA. Older people admitted to hospital who receive the CGA are more likely to be living at home at 3–12 months’ follow-up [relative risk (RR) 1.06, 95% confidence interval (CI) 1.01 to 1.10] (high certainty). The probability that the CGA would be cost-effective at a £20,000 ceiling ratio for quality-adjusted life-years (QALYs), life-years (LYs) and LYs living at home was 0.50, 0.89, and 0.47, respectively (low-certainty evidence). After PSM and regression analysis comparing CGA hospital with CGA hospital at home, we found that the health-care cost (from admission to 6 months after discharge) in site 1 was lower in hospital at home (ratio of means 0.82, 95% CI 0.76 to 0.89), in site 2 there was little difference (ratio of means 1.00, 95% CI 0.92 to 1.09) and in site 3 it was higher (ratio of means 1.15, 95% CI 0.99 to 1.33). Six months after discharge (excluding the index admission), the ratio of means cost in site 1 was 1.27 (95% CI 1.14 to 1.41), in site 2 was 1.09 (95% CI 0.95 to 1.24) and in site 3 was 1.70 (95% CI 1.40 to 2.07). At 6 months’ follow-up (excluding the index admission), there may be an increased risk of mortality (adjusted) in the three hospital-at-home cohorts (site 1: RR 1.09, 95% CI 1.00 to 1.19; site 2: RR 1.29, 95% CI 1.15 to 1.44; site 3: RR 1.27, 95% CI 1.06 to 1.54). The qualitative research indicates the importance of relational aspects of health care, incorporating caregivers’ knowledge in care planning, and a lack of clarity about the end of an episode of health care. Core components that should be included in CGA focus on functional, physical and mental well-being, medication review and a caregiver’s ability to care.LimitationsThe risk of residual confounding limits the certainty of the findings from the PSM analysis; a second major limitation is that the research plan did not include an investigation of social care or primary care.ConclusionsThe CGA is an effective way to organise health care for older people in hospital and may lead to a small increase in costs. There may be an increase in cost and the risk of mortality in the population who received the CGA hospital at home compared with those who received the CGA in hospital; randomised evidence is required to confirm or refute this. Caregiver involvement in the CGA process could be strengthened.FundingThe National Institute for Health Research Health Services and Delivery Research programme.

Empirical likelihood ratio confidence interval estimation of best linear combinations of biomarkers

ABSTRACTIn this article, we apply the empirical likelihood method to make inference on the bivariate survival function of paired failure times by estimating the survival function of censored time with the Kaplan–Meier estimator. Adjusted empirical likelihood (AEL) confidence intervals for the bivariate survival function are developed. We conduct a simulation study to compare the proposed AEL method with other methods. The simulation study shows the proposed AEL method has better performance than other existing methods. We illustrate the proposed method by analyzing the skin graft data.

For some time, so-called empirical likelihoods have been used heuristically for purposes of nonparametric estimation. Owen showed that empirical likelihood ratio statistics for various parameters $\theta(F)$ of an unknown distribution $F$ have limiting chi-square distributions and may be used to obtain tests or confidence intervals in a way that is completely analogous to that used with parametric likelihoods. Our objective in this paper is twofold: first, to link estimating functions or equations and empirical likelihood; second, to develop methods of combining information about parameters. We do this by assuming that information about $F$ and $\theta$ is available in the form of unbiased estimating functions. Empirical likelihoods for parameters are developed and shown to have properties similar to those for parametric likelihood. Efficiency results for estimates of both $\theta$ and $F$ are obtained. The methods are illustrated on several problems, and areas for future investigation are noted.

For the first-order autoregressive model, we establish the asymptotic theory of the weighted least squares estimations whether the underlying autoregressive process is stationary, unit root, near integrated or even explosive under a weaker moment condition of innovations. The asymptotic limit of this estimator is always normal. It is shown that the empirical log-likelihood ratio at the true parameter converges to the standard chi-square distribution. An empirical likelihood confidence interval is proposed for interval estimations of the autoregressive coefficient. The results improve the corresponding ones of Chan et al. (Econ Theory 28:705–717, 2012). Some simulations are conducted to illustrate the proposed method.

Aim. To study the comparative pharmacokinetics and confirmation of bioequivalence of the generic (T) and original (R) dabigatran etexilate in healthy volunteers after a single oral dose under fasted conditions.Material and methods. To confirm bioequivalence, an open-label, randomized, replication, crossover, four-step study was conducted to compare the pharmacokinetics and bioequivalence of generic and original dabigatran with a single oral dose 150 mg dabigatran etexilate under fasted conditions in adult healthy male and female volunteers. Sixty-eight subjects participated in the study. During the study, blood plasma samples were taken from volunteers, in which the concentration of total and free dabigatran was determined. Based on the data obtained, pharmacokinetic and statistical analysis was carried out and 90% confidence intervals were calculated for the ratio of mean pharmacokinetic parameters (Сmax, Tmax, AUC0-t, AUC0-∞, AUCt-∞, T1/2, AUCt-∞/AUC0-∞) for total and free dabigatran.Results. The established 90% confidence intervals for the ratio of AUC0-t, AUC0-∞ and Cmax values for total dabigatran were 82,56-96,36% (mean ratio, 89,19%), 82,39-95,90% (mean ratio, 88,89%) and 85,98-99,17% (mean ratio, 92,34%), respectively. The established 90% confidence intervals for the ratio of AUC0-t, AUC0-∞ and Сmax values for free dabigatran were 83,37-98,29% (mean ratio, 90,53%), 82,98-97,33% (mean ratio, 89,87%) and 85,04-99,28% (mean ratio, 91,88%), respectively. For the estimated pharmacokinetic parameters of dabigatran etexilate, 90% confidence intervals ranged from 80-125% for AUC0-t, AUC0-∞ and Cmax. Additional safety analysis was carried out. Generic and original dabigatran were well tolerated by the volunteers. There were no significant differences in vital signs, paraclinical characteristics throughout the study compared with the initial data, as well as significant differences between the drugs in all studied parameters of adverse events.Conclusion. The study showed that generic and original dabigatran are bio­equivalent. In addition, the data obtained indicate that the agents have similar safety profiles.