Truncated Maximum Likelihood Research Articles

Comparison of age- and sex-dependent reference limits derived from distinct sources for metabolic measurands in basic liver diagnostics.

Reference intervals for basic liver laboratory diagnostic rely on manufacturers' information, remaining unchanged for more than 20 years. This ignores known age and sex dependencies. We performed a retrospective cross-sectional study to compare the age-dependent distribution of flagged and non-flagged laboratory findings between reference limits from 3 distinct sources: manufacturer, published reference study, and the truncated maximum likelihood method applied on a cohort of inpatients aged 18-100 years. Discordance rates adjusted for the permissible analytical uncertainty are reported for serum levels of albumin (n= 150,550), alkaline phosphatase (n= 433,721), gamma-GT (n=580,012), AST (n= 510,620), and ALT (n= 704,546). The number of flagged findings differed notably between reference intervals compared, except for alkaline phosphatase. AST and alkaline phosphatase increased with age in women. Overall discordance for AP, AST, and ALT remained below 10%, respectively, in both sexes. Albumin decreased with age which led to discordant flags in up to 22% in patients ≥70 years. GGT and ALT peaked in 50-59-year-old men with up to 23.5% and 22.8% discordant flags, respectively. We assessed the impact of different reference limits on liver related laboratory results and found up to 25 % discordant flags. We suggest to further analyse the diagnostic and economic effects of reference limits adapted to the population of interest even for well-established basic liver diagnostics.

RIbench: A Proposed Benchmark for the Standardized Evaluation of Indirect Methods for Reference Interval Estimation.

Indirect methods leverage real-world data for the estimation of reference intervals. These constitute an active field of research, and several methods have been developed recently. So far, no standardized tool for evaluation and comparison of indirect methods exists. We provide RIbench, a benchmarking suite for quantitative evaluation of any existing or novel indirect method. The benchmark contains simulated test sets for 10 biomarkers mimicking routine measurements of a mixed distribution of non-pathological (reference) values and pathological values. The non-pathological distributions represent 4 common distribution types: normal, skewed, heavily skewed, and skewed-and-shifted. To identify strengths and weaknesses of indirect methods, test sets have varying sample sizes and pathological distributions differ in location, extent of overlap, and fraction. For performance evaluation, we use an overall benchmark score and sub-scores derived from absolute z-score deviations between estimated and true reference limits. We illustrate the application of RIbench by evaluating and comparing the Hoffmann method and 4 modern indirect methods -TML (Truncated-Maximum-Likelihood), kosmic, TMC (Truncated-Minimum-Chi-Square), and refineR- against one another and against a nonparametric direct method (n = 120). For the modern indirect methods, pathological fraction and sample size had a strong influence on the results: With a pathological fraction up to 20% and a minimum sample size of 5000, most methods achieved results comparable or superior to the direct method. We present RIbench, an open-source R-package, for the systematic evaluation of existing and novel indirect methods. RIbench can serve as a tool for enhancement of indirect methods, improving the estimation of reference intervals.

Outlier-Robust Truncated Maximum Likelihood Parameter Estimation of Compound-Gaussian Clutter with Inverse Gaussian Texture

Compound-Gaussian distributions with inverse Gaussian textures, referred to as the IGCG distributions, are often used to model moderate/high-resolution sea clutter in amplitude. In moderate/high-resolution maritime radars, parameter estimation of the IGCG distributions from radar returns data plays an important role in adaptive target detection. Due to the inevitable existence of outliers of high amplitude in radar returns data from targets and reefs, parameter estimation must be outlier robust. In this paper, an outlier-robust truncated maximum likelihood (TML) estimation method is proposed to mitigate the effect of outliers of high amplitude in data. The data are first transferred into the truncated data by removing a given percentage of the largest samples in amplitude. From the truncated data, the truncated likelihood function is constructed, and its maximum corresponds to the TML estimates of the scale and inverse shape parameters. Further, an iterative algorithm is presented to obtain the TML estimates from data with outliers, which is an extension of the ML estimation method in the case that data contain outliers. In comparison with outlier-sensitive estimation methods and outlier-robust bipercentile estimation methods, the performance of the TML estimation method is close to that of the best ML estimation method in the case that data are without outlier, and it is better in the case that data are with outliers.

Research on Maximum Likelihood b Value and Confidence Limits Estimation in Doubly Truncated Apparent Frequency–Amplitude Distribution in Rock Acoustic Emission Tests

The b value deduced from the Gutenberg–Richter law is an important parameter for sequence and precursory analyses, both in laboratory acoustic emission tests and seismology. As the b value is a statistical value, the maximum likelihood estimation is mostly used to estimate the b value. However, traditional singly truncated maximum likelihood estimation in seismology only considers the minimum magnitude, while the acquisition device in rock acoustic emission tests will set the threshold value and maximum value of the amplitude; therefore, maximum likelihood estimation will estimate the b value in a doubly truncated size distribution, and its confidence limits need to be discussed. Here, in this study, we derive the calculation equations of the b value and the corresponding confidence limits for the maximum likelihood estimation with a narrow amplitude span in a doubly truncated frequency–amplitude distribution. The maximum likelihood b values estimated by the scheme of a singly and doubly truncated frequency–amplitude distribution are compared through acoustic emission data with the known underlying distribution. The results show that the maximum likelihood b value and confidence limits estimation scheme derived for rock acoustic emission tests with a narrow amplitude span is more reasonable. Then, the derived estimation scheme is applied to the rock dilation rupturing test; the results confirm its applicability.

Outlier-robust truncated maximum likelihood parameter estimators of generalized Pareto distributions

Compound-Gaussian model with inverse Gamma-distributed texture is widely used to characterize high-resolution sea clutter and is also referred to as the generalized Pareto (GP) distributions. In real oceanic environments, sea clutter data are often corrupted by a small number of outliers of very high intensity, which are probably from radar returns of ships or reefs. Thus, it is necessary to develop outlier-robust parameter estimators of the amplitude/intensity models of sea clutter for maritime radars. This paper proposes outlier-robust truncated maximum likelihood estimators of the GP distributions. To mitigate the influence of outliers on parameter estimation, the first k largest samples are deleted from the data to generate the right truncated data. The truncated likelihood function (TLF) of the two parameters is derived and the truncated maximum likelihood (TML) equations are given. An iterative algorithm is presented to fast attain the solution of the TML equations and outlier-robust TML estimators are developed, which are a generalization of the ML estimator in the case with outlier in data. The TML estimators are compared with outlier-sensitive estimators and existing outlier-robust estimators. Without outliers in data, the TML estimators have comparable performance with the best ML estimator. With outliers in data, the TML estimators attain higher precision than the existing outlier-robust estimators.

Reference intervals of automated reticulocyte count and immature reticulocyte fraction in a pediatric population.

Reticulocytes are erythroid precursors that develop into mature erythrocytes, and they are an important tool to assess erythropoietic activity, as their count indicates the balance between the cells released from the bone marrow, their stage of maturity, and their rate of development into mature erythrocytes. Considering the described biological variability of the absolute reticulocyte count (ARC) and the immature reticulocyte fraction (IRF) and the limited information available on these hematological parameters in children, this study determined the reference intervals (RIs) of these parameters in a healthy pediatric population. A retrospective, observational, and analytical study was designed to establish RIs for the ARC and the IRF according to age and sex. An indirect sampling method was applied to a mixed database of complete blood counts from children aged 2months to 18years, using the truncated maximum likelihood indirect method for reference interval estimation. Percentiles were calculated to obtain bimodal RIs. From a total of 190,812samples, 6,814 were selected. Gender stratification was not necessary for the ARC and the IRF but they required partitioning into six and two age groups, respectively. This study determined, by an indirect sampling method, RIs for the ARC and the IRF in a pediatric population according to age and sex.

Optimizing Measurement and Interpretation of the G6PD/Hb Ratio.

Glucose-6-phosphate dehydrogenase (G6PD)/hemoglobin (Hb) ratio helps detect G6PD deficiency, an X-linked disorder that can be asymptomatic or cause acute hemolytic anemia and chronic hemolysis. We investigated preanalytical, analytical, and postanalytical aspects to optimize G6PD/Hb measurement and interpretation. G6PD was measured with the Pointe Scientific assay and Hb with Drabkin's reagent on Alinity c® (Abbott Diagnostics). Stability of G6PD/Hb was assessed after 7 and 14 days while stored at 2-8 °C. Stability of hemolysate prepared for G6PD analysis was assessed using QC and patient samples up to 4 h at room temperature or 2-8 °C. Analytical performance specifications including precision, method comparison, linearity, LOQ, and carry-over were established for the enzymatic reaction of G6PD and spectrophotometric reading of Hb. G6PD/Hb reference interval and cut-offs were established indirectly using truncated maximum likelihood method (TML) using retrospective data (n = 4715 patient data points). Samples were stable after 7 days at 2-8°C, unless grossly hemolyzed. Hemolysate prepared for G6PD measurement remained stable for up to 4 h for QC at room temperature and 2-8°C, but up to 30 min-1 h at room temperature and 1-2 h at 2-8 °C for patient samples. Precision, linearity, LOQ, and carryover were acceptable. G6PD/Hb cut-offs were <3.3, ≥3.3, 3.3-8.9, and ≥8.9 U/g Hb for deficient males/females, normal males, intermediate females, and normal females, respectively. In vitro hemolysis and delayed hemolysate analysis significantly reduce G6PD/Hb stability. QC material cannot detect the impact of delayed hemolysate analysis. These findings were foundational for optimizing G6PD/Hb protocols for a new platform and establishing laboratory-specific G6PD/Hb cut-offs.

Application of the TML method to big data analytics and reference interval harmonization

Abstract Significant variation in reported reference intervals across healthcare centers and networks for many well-standardized laboratory tests continues to exist, negatively impacting patient outcomes by increasing the risk of inappropriate and inconsistent test result interpretation. Reference interval harmonization has been limited by challenges associated with direct reference interval establishment as well as hesitancies to apply currently available indirect methodologies. The Truncated Maximum Likelihood (TML) method for indirect reference interval establishment developed by the German Society of Clinical Chemistry and Laboratory Medicine (DGKL) presents unique clinical and statistical advantages compared to traditional indirect methods (Hoffmann and Bhattacharya), increasing the feasibility of developing indirect reference intervals that are comparable to those determined using a direct a priori approach based on healthy reference populations. Here, we review the application of indirect methods, particularly the TML method, to reference interval harmonization and discuss their associated advantages and disadvantages. We also describe the CSCC Reference Interval Harmonization Working Group’s experience with the application of the TML method in harmonization of adult reference intervals in Canada.

Seismic fragility analysis of AP1000 SB considering fluid-structure interaction effects

AP1000 nuclear power plant has an advanced passive cooling system with a vast water tank located above the shield building, which may affect the seismic performance of the structure. This paper conducts a fragility assessment of shield building and considers the fluid-structure interaction (FSI) effect on the damage of the shield building. Arbitrary Lagrangian Eulerian algorithm is applied to calculate the FSI and eight cases of water levels are considered. Three different fragility analysis methods, namely linear regression, quadratic regression and truncated maximum likelihood estimation, are adopted for assessing the seismic vulnerability of shield building. It is indicated from the results that the regression and the truncated maximum likelihood estimation are close to each other. The maximum likelihood estimation not only can effectively save the computational cost, but also can assure the reasonable calculated results. It is observed that different water levels have the individual probabilities of failure, and case 6 has the smallest probability of failure with a more considerable seismic reduction, whereas the case 1 and case 2 have the higher failure probabilities. The fragility analysis framework can thus be applied for evaluating the vulnerability of shield building under earthquakes considering the FSI effect.

COMPARATIVE EFFICIENCY OF AGRICULTURAL COOPERATIVES AND CONVENTIONAL FIRMS IN A SAMPLE OF QUASI‐TWIN COMPANIES

ABSTRACTThe resilience of cooperatives and their positive contribution to employment in times of crisis is well established. However, their overall economic performance relative to conventional firms is still controversial, casting doubt on the ability of this alternative organizational form to govern the fundamental drivers of productivity. To shed new light on the issue, we study the comparative technical efficiency of agricultural cooperatives (ACs) and conventional firms (CFs), drawing on a unique data set comprising all wine‐producing companies in Sardinia (Italy) from 2004 to 2009. Due to the similarity of the habitats in which the firms operate and the careful measurement of several key inputs, the observed units are ‘twins’ in all non‐organizational respects, providing an ideal setting for comparison. Having generated efficiency scores through Data Envelopment Analysis (DEA), we regressed the scores on external covariates and ownership type using a pooled truncated maximum likelihood formulation. Our findings, which survive correction for spatial correlations, indicate that cooperatives are less technically efficient than their capitalist counterparts and struggle more to adapt to extreme weather fluctuations. Both results are particularly worrying in light of the main challenges facing the wine industry in the near future: liberalization of EU planting rights and climate change.

The economics of quality in the specialty coffee industry: insights from the Cup of Excellence auction programs

AbstractThis study estimates price determinants for specialty green coffee auctions using records from the 2004–2010 Cup of Excellence programs hosted by the Alliance for Coffee Excellence. While some literature on coffee has focused on certifications and sustainability labels, the discussion of price determinants has been limited. This article replicates one of the first publications on price determinants and formulates a new model to more accurately describe the market. We include the necessary additional variables and estimate the model using a truncated maximum likelihood estimation technique. While sensory quality has a strong effect on price, the highest premiums stem from obtaining a top rank compared to other coffees from the same country, and North American buyers are more responsive to sensory quality than buyers in Asian and European markets.

Fitting operational risk data using limited information below the threshold

This paper presents a methodology to calibrate the distribution of losses observed in operational risk events. The method is specifically designed to handle the situation where individual event information is only available above an approved threshold and there is a limited set of below-threshold information (BTI). The method requires the use of the number of events and the sum of the losses below the threshold. the authors demonstrate the improved stability of the fitted distribution parameters as the number of events above the threshold is reduced. This is compared with the conventional truncated maximum likelihood estimator (MLE). To demonstrate the improved stability of the parameter estimation using the BTI, a series of fits are performed to m samples of N events drawn from a lognormal distribution, and these are compared with estimates of the truncated MLE. The BTI fitting methodology produces a better estimator of the population distribution parameters, with reduced bias and dispersion. the authors estimate the 99.9th percentile of the severity distribution and show that the BTI provides a better estimator of the percentile with reduced uncertainty.

Length of stay as a performance indicator: robust statistical methodology

Length of stay (LOS) is an important performance indicator for costing and hospital management and a key measure of efficiency of NHS. However, LOS is difficult to analyse because its statistical distribution is non-normal and LOS data habitually have many outliers. Furthermore, the usefulness of LOS for improving NHS performance is undermined because no adjustments are made for some key factors. This paper addresses both these problems. Health episodes statistics data from the UK NHS for 1997/98, and 1998/99 are analysed to investigate the effects of five key variables: admission method, discharge destination, provider (hospital) type, speciality and NHS region. All are found to influence LOS. The effects of some factors are substantial, and were not previously known, and so are not included in planned future NHS performance measures, e.g. LOS is at least 25% longer for patients transferred from other hospitals rather than admitted as an emergency; and LOS for patients discharged to private institutions is more than twice that for patients discharged to NHS institutions or their own home. The problem of finding the most appropriate statistical analysis for data of the LOS type is addressed by comparing standard general linear model methods with an advanced robust method called truncated maximum likelihood (TML). The TML methods are shown to have several advantages over standard methods, in terms of model fit and accuracy of parameter estimation. Implications of these findings for future use of LOS are considered.

Improvements in Maximum Likelihood Estimators of Truncated Normal Samples with Prior Knowledge of O: A Simulation Based Study with Application to Historical Height Samples

Researchers analyzing historical data on human stature have long sought an estimator that performs well in truncated-normal samples. This paper reviews that search, focusing on two currently widespread procedures: truncated least squares (TLS) and truncated maximum likelihood (TML). The first suffers from bias. The second suffers in practical application from excessive variability. A simple procedure is developed to convert TLS truncated means into estimates of the underlying population means, assuming the contemporary population standard deviation. This procedure is shown to be equivalent to restricted TML estimation. Simulation methods are used to establish the mean squared error performance characteristics of the restricted and unconstrained TML estimators in relation to several population and sample parameters. The results provide general insight into the bias-precision tradeoff in restricted estimation and a specific practical guide to optimal estimator choice for researchers in anthropometrics.

Adaptively truncated maximum likelihood regression with asymmetric errors

We consider robust estimators for the linear regression model with asymmetric (or symmetric) error distribution. We assume that the error model belongs to a location-scale family of distributions. Since in the asymmetric case the mean response is very often the parameter of interest and scale is a main component of mean, we do not assume that scale is a nuisance parameter. First, we show how to convert an ordinary robust estimate for the usual model with symmetric errors to an estimate for the more general model with asymmetric errors. Then, in order to improve efficiency, we introduce the truncated maximum likelihood or TML-estimator. A TML-estimate is computed in three steps: first, an initial high breakdown point estimate is computed; then, observations that are unlikely under the estimated model are rejected; finally, the maximum likelihood estimate is computed with the retained observations. The rejection rule used in the second step is based on a cut-off parameter that can be tuned to attain the desired efficiency while maintaining the breakdown point of the initial estimator (e.g., 50%). Optionally, one can use a new adaptive cut-off that, asymptotically, does not reject any observation when the data are generated according to the model. Under the model, the influence function of this adaptive TML-estimator (or ATML-estimator) coincides with the influence function of the maximum likelihood estimator. The ATML-estimator is, therefore, fully efficient at the model; nevertheless, its breakdown point is not smaller than the breakdown point of the initial estimator. We evaluate the TML- and ATML-estimators for finite sample sizes with the help of simulations and discuss an example with real data.

Truncated Maximum Likelihood, Goodness of Fit Tests and Tail Analysis

We propose a new method of tail analysis for data featuring a high degree of leptokurtosis. Heavy tails can typically be found in financial series, like for example, the stock returns or durations between arrivals of trades. In our framework, the shape of tails can be assessed by fitting some selected pseudo-models to extremely valued observations in the sample. A global examination of the density tails is performed using a sequence of truncated pseudo-maximum likelihood estimators, called the tail parameter function (t.p.f.). In practice, data often exhibit local irregularities in the tail behaviour. To detect and approximate these patterns we introduce the local parameter function (l.p.f.), a method of tail analysis involving a selected interval of extreme observations. An immediate extension of the pseudo-value based approach to density analysis yields a new procedure for testing the goodness of fit. We also develop a new nonparametric estimator of the density function. The method is applied to unequally spaced high frequency data. We study an intradaily series of returns on the Alcatel stock, one among the most frequently traded stocks on the Paris Stock Exchange.

On progressively truncated maximum likelihood estimators

The basic regularity conditions pertaining to the asymptotic theory of progressively truncated likelihood functions and maximum likelihood estimators are considered, and the uniform strong consistency and weak convergence of progressively truncated maximum likelihood estimators are studied systematically.

On identification and adaptive estimation for systems with interrupted observations

A linear discrete-time system with interrupted observations is considered. The interrupted observation mechanism is expressed in terms of a finite-state ergodic Markov chain. The transition probability matrix of the Markov chain and some system parameters may be unknown, but constant, and are assumed to belong to a compact set. A novel scheme, called truncated maximum likelihood estimation, is proposed for consistent estimation of the unknown parameters. Sufficient conditions for strong consistency are investigated. The truncated maximum likelihood procedure is computationally feasible whereas the standard maximum likelihood procedure is not, given large observation records. Finally, using the truncated ML algorithm, a suboptimal adaptive state estimator is proposed and its asymptotic behavior is analyzed.

Adaptive estimation and identification for discrete systems with Markov jump parameters

Linear discrete-time stochastic dynamical systems with parameters which may switch among a finite set of values are considered. The switchings are modeled by a finite state ergodic Markov chain whose transition probability matrix is unknown and is assumed to belong to a compact set. A novel scheme, called truncated maximum likelihood estimation, is proposed for consistent estimation of the transition probabilities given noisy observations of the system output variables. Conditions for strong consistency are investigated assuming that the measurements are taken after the system has achieved a statistical steady state. The case when the true transition matrix does not belong to the unknown transition matrix set is also considered. The truncated maximum likelihood procedure is computationally feasible, whereas the standard maximum likelihood procedure is not, given large observation records. Finally, using the truncated ML algorithm, a suboptimal adaptive state estimator is proposed and its asymptotic behavior is analyzed.