How to measure premature mortality? A proposal combining “relative” and “absolute” approaches

Background The concept of “premature mortality” is at the heart of many national and global health measurement and benchmarking efforts. However, despite the intuitive appeal of its underlying concept, it is far from obvious how to best operationalise it. The previous work offers at least two basic approaches: an absolute and a relative one. The former—and far more widely used— approach sets a unique age threshold (e.g. 65 years), below which deaths are defined as premature. The relative approach derives the share of premature deaths from the country-specific age distribution of deaths in the country of interest. The biggest disadvantage of the absolute approach is that of using a unique, arbitrary threshold for different mortality patterns, while the main disadvantage of the relative approach is that its estimate of premature mortality strongly depends on how the senescent deaths distribution is defined in each country. Method We propose to overcome some of the downsides of the existing approaches, by combining features of both, using a hierarchical model, in which senescent deaths distribution is held constant for each country as a pivotal quantity and the premature mortality distribution is allowed to vary across countries. In this way, premature mortality estimates become more comparable across countries with similar characteristics. Results The proposed hierarchical models provide results, which appear to align with related evidence from specific countries. In particular, we find a relatively high premature mortality for the United States and Denmark. Conclusions While our hybrid approach overcomes some of the problems of previous measures, some issues require further research, in particular the choice of the group of countries that a given country is assigned to and the choice of the benchmarks within the groups. Hence, our proposed method, combined with further study addressing these issues, could provide a valid alternative way to measure and compare premature mortality across countries.

Quantile Jensen’s inequalities

Quantiles of random variable are crucial quantities that give more delicate information about distribution than mean and median and so on. We establish Jensen’s inequality for q-quantile ($q\geq 0.5$ q ≥ 0.5 ) of a random variable, which includes as a special case Merkle (Stat. Probab. Lett. 71(3):277–281, 2005) where Jensen’s inequality about median (i.e. $q= 0.5$ q = 0.5 ) was given. We also refine this inequality in the case where $q<0.5$ q < 0.5 . An application to the confidence interval of parameters in pivotal quantity is also considered by virtue of the rigorous description on the relationship between quantiles and intervals that have required probability.

How to measure premature mortality? A proposal combining “relative” and “absolute” approaches

Background : the concept of "premature mortality" is at the heart of many national and global health measurement and benchmarking efforts. However, despite the intuitive appeal of its underlying concept, it is far from obvious how to best operationalise it. The previous work offers at least two basic approaches: an absolute and a relative one. The former -- and far more widely used -- sets a unique age threshold (e.g. 65 years), below which deaths are defined as premature. The relative approach derives the share of premature deaths from the country--specific age distribution of deaths in the country of interest. The biggest disadvantage of the absolute approach is that of using a unique, arbitrary threshold for different mortality patterns, while the main disadvantage of the relative approach is that its estimate of premature mortality strongly depends on how the senescent deaths distribution is defined in each country.\\ Method : we propose to overcome some of the downsides of the existing approaches, by combining features of both, using a hierarchical model, in which senescent deaths distribution is held constant for each country as a pivotal quantity and the premature mortality distribution is allowed to vary across countries. In this way, premature mortality estimates become more comparable across countries with similar characteristics.\\ Results : the proposed hierarchical models provides results which lead to shared conclusions researchers developed analyzing the specific countries. In particular, we find a relatively high premature mortality for United States and Denmark. Conclusions : while our hybrid approach overcomes some of the problems of the previous measures, some issues needs further work. In particular the choice of the group of countries a given country is assigned to and the choice of the benchmarks within the groups. Hence our proposed method, combined with further study addressing the clustering issue, could provide a valid alternative way to measure and compare premature mortality across countries.

Hierarchical Missing Data and Multivariate Behrens–Fisher Problem

This article firstly defines hierarchical data missing pattern, which is a generalization of monotone data missing pattern. Then multivariate Behrens–Fisher problem with hierarchical missing data is considered to illustrate that how ideas in dealing with monotone missing data can be extended to deal with hierarchical missing pattern. A pivotal quantity similar to the Hotelling T 2 is presented, and the moment matching method is used to derive its approximate distribution which is for testing and interval estimation. The precision of the approximation is illustrated through Monte Carlo data simulation. The results indicate that the approximate method is very satisfactory even for moderately small samples.

Constructing a confidence interval for the ratio of normal distribution quantiles

In this paper, to construct a confidence interval (general and shortest) for quantiles of normal distribution in one population, we present a pivotal quantity that has non-central t distribution. In the case of two independent normal populations, we propose a confidence interval for the ratio of quantiles based on the generalized pivotal quantity, and we introduce a simple method for extracting its percentiles, based on which a shorter confidence interval can be created. Also, we provide general and shorter confidence intervals using the method of variance estimate recovery. The performance of five proposed methods will be examined by using simulation and examples.

New Approach for a Weibull Distribution under the Progressive Type-II Censoring Scheme

This paper proposes a new approach based on the regression framework employing a pivotal quantity to estimate unknown parameters of a Weibull distribution under the progressive Type-II censoring scheme, which provides a closed form solution for the shape parameter, unlike its maximum likelihood estimator counterpart. To resolve serious rounding errors for the exact mean and variance of the pivotal quantity, two different types of Taylor series expansion are applied, and the resulting performance is enhanced in terms of the mean square error and bias obtained through the Monte Carlo simulation. Finally, an actual application example, including a simple goodness-of-fit analysis of the actual test data based on the pivotal quantity, proves the feasibility and applicability of the proposed approach.

Estimation and Prediction of Record Values Using Pivotal Quantities and Copulas

Recently, the area of sea ice is rapidly decreasing due to global warming, and since the Arctic sea ice has a great impact on climate change, interest in this is increasing very much all over the world. In fact, the area of sea ice reached a record low in September 2012 after satellite observations began in late 1979. In addition, in early 2018, the glacier on the northern coast of Greenland began to collapse. If we are interested in record values of sea ice area, modeling relationships of these values and predicting future record values can be a very important issue because the record values that consist of larger or smaller values than the preceding observations are very closely related to each other. The relationship between the record values can be modeled based on the pivotal quantity and canonical and drawable vine copulas, and the relationship is called a dependence structure. In addition, predictions for future record values can be solved in a very concise way based on the pivotal quantity. To accomplish that, this article proposes an approach to model the dependence structure between record values based on the canonical and drawable vine. To do this, unknown parameters of a probability distribution need to be estimated first, and the pivotal-based method is provided. In the pivotal-based estimation, a new algorithm to deal with a nuisance parameter is proposed. This method allows one to reduce computational complexity when constructing exact confidence intervals of functions with unknown parameters. This method not only reduces computational complexity when constructing exact confidence intervals of functions with unknown parameters, but is also very useful for obtaining the replicated data needed to model the dependence structure based on canonical and drawable vine. In addition, prediction methods for future record values are proposed with the pivotal quantity, and we compared them with a time series forecasting method in real data analysis. The validity of the proposed methods was examined through Monte Carlo simulations and analysis for Arctic sea ice data.

Generalized Performance Measures of X- Control Charts Based on Different Sampling Schemes

Different versions of X- control chart structure are available under various ranked set strategies. In these control charts, computation of performance measures was carried out through Monte Carlo simulation method (MCSM). In this article, we have defined a generalized structure of X- control charts under variant sampling strategies followed by derivation of their different performance measures. For the derivation of different performance measures, we have proposed pivotal quantity. For comparative analysis, we have presented results of generalized performance measures by involving numerical method (NM) as computation. We found that values of generalized performance measures based on NM are almost similar to values of performance measures based on MCSM. Also, NM is time efficient and can be considered as an alternative of MCSM.

Validation and Measurement Uncertainty Assessment of a Microbiological Method Using Generalized Pivotal Quantity Procedure and Monte-Carlo Simulation

Recently, a novel and effective statistical tool called the uncertainty profile has been developed with the purpose of graphically assessing the validity and estimating the measurement uncertainty of analytical procedures. One way to construct the uncertainty profile is to compute the β-content, γ-confidence tolerance interval. In this study, we propose a tolerance interval based on the combination of the generalized pivotal quantity procedure and Monte-Carlo simulation. The uncertainty profile has been applied successfully in several fields. However, in order to further confirm its universality, this newer approach has been applied to assess the performance of an alternative procedure versus a reference procedure for counting of Escherichia coli bacteria in drinking water. Hence, the aims of this research were to expose how the uncertainty profile can be powerfully applied pursuant to ISO 16140 standards in the frame of interlaboratory study and how to easily make a decision concerning the validity of the procedure. The analysis of the results shows that after the introduction of a correction factor, the alternative procedure is deemed valid over the studied range because the uncertainty limits lie within the acceptability limits set at ±−0.3 log unit/100 ml for a β = 66.7% and γ = 90%.