Bias-corrected Maximum Likelihood Estimator Research Articles

Molecular Detection of Rickettsia felis in Fleas of Companion Animals in East Texas.

Flea-borne spotted fever is an emerging insect-borne rickettsial infection caused by Rickettsia felis and has been identified worldwide. This study sought to explore the prevalence of rickettsiae associated with fleas on companion dogs and cats from Walker and Montgomery Counties in East Texas. Fleas were collected from animals entering local veterinary clinics for routine checkups. Collected fleas were identified as Ctenocephalides felis or Pulex irritans and analyzed by polymerase chain reaction for the presence of rickettsiae and subsequent sequencing. An estimation of the bcMLE (bias-corrected maximum likelihood estimation) of pooled samples was calculated. Four hundred eighty-eight fleas (comprising C. felis and P. irritans) were collected from 16 cats and 77 dogs. Our results demonstrate R. felis in 21 pools of fleas from dogs (bcMLE 15.28%) and a bcMLE of 7.25% from flea samples collected from cats. Sequence analysis revealed R. felis as the only Rickettsia that could be amplified in our samples using the rickettsial citrate synthase gene and subsequent sequencing. In this study, the presence of R. felis in fleas from companion cats and dogs suggests a potential risk of flea-borne spotted fever in humans who encounter flea-infested animals.

Bias-corrected maximum-likelihood estimation of multiplicity of infection and lineage frequencies.

BackgroundThe UN’s Sustainable Development Goals are devoted to eradicate a range of infectious diseases to achieve global well-being. These efforts require monitoring disease transmission at a level that differentiates between pathogen variants at the genetic/molecular level. In fact, the advantages of genetic (molecular) measures like multiplicity of infection (MOI) over traditional metrics, e.g., R0, are being increasingly recognized. MOI refers to the presence of multiple pathogen variants within an infection due to multiple infective contacts. Maximum-likelihood (ML) methods have been proposed to derive MOI and pathogen-lineage frequencies from molecular data. However, these methods are biased.Methods and findingsBased on a single molecular marker, we derive a bias-corrected ML estimator for MOI and pathogen-lineage frequencies. We further improve these estimators by heuristical adjustments that compensate shortcomings in the derivation of the bias correction, which implicitly assumes that data lies in the interior of the observational space. The finite sample properties of the different variants of the bias-corrected estimators are investigated by a systematic simulation study. In particular, we investigate the performance of the estimator in terms of bias, variance, and robustness against model violations. The corrections successfully remove bias except for extreme parameters that likely yield uninformative data, which cannot sustain accurate parameter estimation. Heuristic adjustments further improve the bias correction, particularly for small sample sizes. The bias corrections also reduce the estimators’ variances, which coincide with the Cramér-Rao lower bound. The estimators are reasonably robust against model violations.ConclusionsApplying bias corrections can substantially improve the quality of MOI estimates, particularly in areas of low as well as areas of high transmission—in both cases estimates tend to be biased. The bias-corrected estimators are (almost) unbiased and their variance coincides with the Cramér-Rao lower bound, suggesting that no further improvements are possible unless additional information is provided. Additional information can be obtained by combining data from several molecular markers, or by including information that allows stratifying the data into heterogeneous groups.

Improved maximum likelihood estimation of the parameters of the Gamma-Uniform distribution with bias-corrections

A two-parameter Gamma-Uniform distribution was recently introduced as a prominent alternative in modeling bounded phenomena. Unfortunately, however, its maximum likelihood estimators (MLEs) are found to be highly biased in finite samples, a limitation that might effect this model’s application in data modeling. In this article, we construct nearly unbiased estimators for the unknown parameters of this distribution by deriving analytical bias-corrected maximum likelihood estimators applying the Cox and Snell methodology, the Firth’s method and also via the parametric Bootstrap bias correction approach. Our extensive simulation clearly revealed that the three bias reduction methods yield very good estimates which are nearly unbiased and exhibit comparable efficiency. Finally, we consider a real data set where the variable under enquiry is the proportion of unemployed labor force reported across some 158 nations in 2018 to show case the positive gain of incorporating the bias correction in the model fitting.

Molecular epidemiology of mosquito-borne viruses at the China\u2013Myanmar border: discovery of a potential epidemic focus of Japanese encephalitis

BackgroundMosquito-based arbovirus surveillance can serve as an early warning in evaluating the status of mosquito-borne virus prevalence and thus prevent local outbreaks. Although Tengchong County in Yunnan Province—which borders Myanmar—is abundant and diverse in mosquitoes, very few mosquito-based arbovirus investigations have been conducted in the recent decade. Herein, this study aims to evaluate the presence and the diffusion of mosquito-borne pathogens, currently prevalent in this region.MethodsWe collected 9486 mosquitoes, representing eight species, with Culex tritaeniorhynchus and Anopheles sinensis as the dominant species, during high mosquito activity seasons (July–October) in Tengchong, in 2018. Samples collected from 342 pools were tested using reverse-transcription PCR to determine the species, distribution, and infection rates of virus and parasite, and further analyze their genotypes, phylogenetic relationships, infection rate, and potential pathogenicity.ResultsFifteen Japanese encephalitis virus (JEV) strains from Cx. tritaeniorhynchus pools were detected. Seven strains of insect-specific flaviviruses (ISFVs), including two Aedes flavivirus (AeFV) and Yunnan Culex flavivirus strains each, one Culex theileri flavivirus, Yamadai flavivirus (YDFV) and Anopheles-associated flavivirus (AAFV) strains each were detected in Aedes albopictus, Cx. tritaeniorhynchus, Cx. vagans, Cx. pseudovihnui, and An. sinensis pools, respectively. The whole-genome was successfully amplified in one strain of JEV and AeFV each. Phylogenetic analysis using the E gene placed all the newly detected JEV strains into the GI-b genotype. They showed highly nucleotide identities, and were most closely related to the strain detected in Tengchong in 2010. The comparison of the E protein of JEV strains and vaccine-derived strain, showed six amino residue differences. The bias-corrected maximum likelihood estimation values (and 95% confidence interval) for JEV in Cx. tritaeniorhynchus collected in Tengchong in 2018 were 2.4 (1.4–3.9).ConclusionsA potential Japanese encephalitis epidemic focus with the abundance of host mosquitoes and high JEV infection rate was observed in Tengchong. In addition, at least five species of ISFVs co-circulate in this area. This study highlights the importance of widespread and sustained mosquito-based arbovirus surveillance in local areas to prevent the transmission of JEV, and other emerging/re-emerging mosquito-borne pathogens.Graphic

Estimation of the binomial probabilities in a two-stage phase II clinical trial with two co-primary endpoints

In cancer research, two-stage designs are usually used to assess the effect of a new agent in phase II clinical trials. Optimal two-stage designs with two co-primary endpoints have been proposed to assess the effects of new cancer treatments, such as cytostatic or molecularly targeted agents (MTAs), based on both response rate and early progression rate. Accurate estimation of response and early progression rates based on the data from the phase II trials conducted according to the optimal two-stage designs would be very useful for further testing of the agents in phase II trials. In this paper, we derive some estimation procedures, which include both standard and bias-corrected maximum likelihood estimates (MLE) and uniformly minimum variance unbiased estimate (UMVUE), for two binomial probabilities which are used to define the hypotheses for two co-primary endpoints tested in a two-stage phase II clinical trial. Simulation studies were performed to evaluate the performance of these procedures. These procedures are also applied to analyze the data from a phase II trial conducted by the Canadian Cancer Trials Group.

Monitoring Mosquito-Borne Arbovirus in Various Insect Regions in China in 2018.

BackgroundIncreases in global travel and trade are changing arbovirus distributions worldwide. Arboviruses can be introduced by travelers, migratory birds, or vectors transported via international trade. Arbovirus surveillance in field-collected mosquitoes may provide early evidence for mosquito-borne disease transmission.MethodsDuring the seasons of high mosquito activity of 2018, 29,285 mosquitoes were sampled from seven sentinel sites in various insect regions. The mosquitoes were analyzed by RT-PCR for alphaviruses, flaviviruses, and orthobunyaviruses.ResultsWe detected three strains of Japanese encephalitis virus (JEV), five strains of Getah virus (GETV), and 45 strains of insect-specific flaviviruses including Aedes flavivirus (AeFV, 1), Chaoyang virus (CHAOV, 1), Culex flavivirus (CxFV, 17), Hanko virus (HANKV, 2), QuangBinh virus (QBV, 22), and Yunnan Culex flavivirus (YNCxFV, 2). Whole genomes of one strain each of GETV, CxFV, CHAOV, and AeFV were successfully amplified. Phylogenetic analysis revealed that the new JEV strains detected in the Shanghai and Hubei Provinces belong to the GI-b strain and are phylogenetically close to the NX1889 strain (MT134112) isolated from a patient during a JE outbreak in Ningxia in 2018. GETVs were found in Inner Mongolia, Hubei, and Hainan and belonged to Group III. They were closely related to strains isolated from swine. HANKV was recorded for the first time in China and other ISFVs were newly detected at several sentinel sites. The bias-corrected maximum likelihood estimation value for JEV in Jinshan, Shanghai was 4.52/1,000 (range 0.80–14.64). Hence, there is a potential risk of a JEV epidemic in that region.ConclusionGI-b is the dominant circulating JEV genotype in nature and poses a health risk to animals and humans. The potential threat of widespread GETV distribution as a zoonosis is gradually increasing. The present study also disclosed the dispersion and host range of ISFVs. These findings highlight the importance of tracing the movements of the vectors and hosts of mosquito-borne pathogens in order to prevent and control arbovirus outbreaks in China.

Estimation of the generalized process capability index C pyk based on bias-corrected maximum-likelihood estimators for the generalized inverse Lindley distribution and bootstrap confidence intervals

In this paper, we are interested in estimating the generalized process capability index () proposed by Maiti et al. [On generalizing process capability indices. Qual Technol Quant Manag. 2010;7(3):279–300], when the underlying distribution is the generalized inverse Lindley (GIL) distribution. We estimate parameters of the GIL distribution using maximum likelihood (ML), bias-corrected maximum-likelihood (BCML) and bootstrap bias-corrected maximum-likelihood (BBCML) methodologies. are obtained using proposed estimators. Bootstrap confidence intervals called standard bootstrap (SB), percentile bootstrap (PB) and bias-corrected percentile bootstrap (BCPB) are constructed based on the estimators of . We compare efficiencies of the parameter estimators and the performance of ML, BCML and BBCML based Cpyk via an extensive Monte Carlo simulation study. A simulation study is also described to compare the coverage probabilities (CP) and average lengths (AL) of SB, PB and BCPB confidence intervals for proposed . Finally, two real datasets are analysed for illustrative purposes.

Serological and molecular epidemiology of Japanese Encephalitis in Zhejiang, China, 2015-2018.

BackgroundShifts have occurred in the epidemiological characteristics of Japanese encephalitis (JE), extending from the molecular level to the population level. The aim of this study was to investigate the seroprevalence of JE neutralizing antibodies in healthy populations from different age groups in Zhejiang Province, and to conduct mosquito monitoring to evaluate the infection rate of Japanese encephalitis virus (JEV) among vectors, as well as the molecular characteristics of the E gene of isolated JEV strains.Methodology/Principal findingsA total of 1190 sera samples were screened by a microseroneutralization test, including 429 infants (28d-11m) and 761 participants (2y-82y). For those under 1 year old, the geometric mean titers (GMTs) of the JE neutralizing antibody was 9.49 at birth and significantly declined as the age of month increased (r = -0.225, P<0.001). For those above 1-year old, seropositive proportions were higher in subjects aged 1–3 years old as well as ≥25 years old (65%-75%), and relatively lower in subjects aged between 4–25 years old (22%-55%). Four or more years after the 2nd dose of JEV-L (first dose administered at 8 months and the second at 2 years of age), the seropositive proportion decreased to 32.5%, and GMTs decreased to 8.08. A total of 87,201 mosquitoes were collected from livestock sheds in 6 surveillance sites during 2015–2018, from which 139 E gene sequences were successfully amplified. The annual infection rate according to bias-corrected maximum likelihood estimation of JEV in Culex tritaeniorhynchus was 1.56, 2.36, 5.65 and 1.77 per 1000, respectively. JEV strains isolated during 2015–2018 all belonged to Genotype I. The E gene of amplified 139 samples differed from the JEV-L vaccine strain at fourteen amino acid residues, including the eight key residues related to virulence and virus attenuation. No divergence was observed at the sites related to antigenicity.Conclusions/SignificanceZhejiang Province was at a high risk of JE exposure due to relatively lower neutralizing antibody levels among the younger-aged population and higher infection rates of JEV in mosquitoes. Continuous, timely and full coverage of JE vaccination are essential, as well as the separation of human living areas and livestock shed areas. In addition, annual mosquito surveillance and periodic antibody level monitoring are important for providing evidence for improvement in JE vaccines and immunization schedules.

Bias-corrected Maximum-Likelihood Estimator for the Parameter of the Logarithmic Series Distribution and its Characterizations

Bias-corrected Maximum-Likelihood Estimator for the Parameter of the Logarithmic Series Distribution and its Characterizations

Accounting for Wood, Foliage Properties, and Laser Effective Footprint in Estimations of Leaf Area Density from Multiview-LiDAR Data

The spatial distribution of Leaf Area Density (LAD) in a tree canopy has fundamental functions in ecosystems. It can be measured through a variety of methods, including voxel-based methods applied to LiDAR point clouds. A theoretical study recently compared the numerical errors of these methods and showed that the bias-corrected Maximum Likelihood Estimator was the most efficient. However, it ignored (i) wood volumes, (ii) vegetation sub-grid clumping, (iii) the instrument effective footprint, and (iv) was limited to a single viewpoint. In practice, retrieving LAD is not straightforward, because vegetation is not randomly distributed in sub-grids, beams are divergent, and forestry plots are sampled from more than one viewpoint to mitigate occlusion. In the present article, we extend the previous formulation to (i) account for both wood volumes and hits, (ii) rigorously include correction terms for vegetation and instrument characteristics, and (iii) integrate multiview data. Two numerical experiments showed that the new approach entailed reduction of bias and errors, especially in the presence of wood volumes or when multiview data are available for poorly-explored volumes. In addition to its conciseness, completeness, and efficiency, this new formulation can be applied to multiview TLS—and also potentially to UAV LiDAR scanning—to reduce errors in LAD estimation.

Two-Stage Procedure of Fixed-Width Confidence Intervals for the Risk Ratio

A two-stage procedure is considered for obtaining fixed-width confidence intervals and optimal sample sizes for the risk ratio of two independent binomial proportions. We study desirable properties of the proposed estimator based on a bias-corrected maximum likelihood estimator (MLE). The two-stage procedure provides flexible sampling strategies, thus can be more advantageous in decision-making as well as in inference for the risk ratio. As a result, the proposed procedure can be a remedy not only for asymptotic consistency, but also for drawbacks of coverage to the nominal probability of the purely sequential method. To investigate large-sample properties of the proposed procedure, first-order asymptotic expansions are obtained. Through Monte Carlo experiments, we examine finite sample behavior for various scenarios of samples for illustrations.

Improved Estimation for a New Class of Parametric Link Functions in Binary Regression

We develop nearly unbiased maximum likelihood estimators for a new class of asymmetric link functions proposed recently in the statistic literature by Lemonte and Bazan (TEST 27, 597–617 2018). These authors have introduced a broad class of parametric link functions in binary regression that contains as special cases both symmetric as well as asymmetric links. We discuss maximum likelihood estimation for the model parameters and derive a closed-form expression for the second order bias of these estimators. The second order bias can be easily computed as an ordinary weighted least-squares regression and is then used to define bias corrected maximum likelihood estimators. Monte Carlo simulation experiments are conducted in order to investigate the performance of the corrected estimators. The numerical results reveal that the bias correction scheme yields nearly unbiased estimates without increasing the mean squared errors. Empirical applications are considered for illustrative purposes.

Estimators and confidence intervals for plant area density at voxel scale with T-LiDAR

Estimating leaf and plant area density with Terrestrial Laser Scanners (TLS) continues to be more and more popular, as tridimensional point clouds appear as an appealing measurement technique for heterogeneous environments. Some approaches implement a discretization of the point cloud in a grid (referred to as “voxel-based”) to account for this vegetation heterogeneity and significant work has been done in this recent research field, but no general theoretical analysis is available. Although estimators have been proposed and several causes of biases have been identified, their unbiasedness (zero bias) and efficiency (smallest error) have not been evaluated. Also, confidence intervals are almost never provided.In the present paper, we assumed that the vegetation elements were randomly distributed within voxels and that TLS beams were infinitely thin, in order to focus on the remaining sources of biases and errors. In this simplified context, we both solve the transmittance equation and use the Maximum Likelihood Estimation (MLE), to derive some new estimators for the attenuation coefficient, which is proportional to leaf area density at voxel scale in this idealized context. These estimators include bias corrections and confidence intervals, and account for the number of beams crossing the voxel (beam number), the inequality of path lengths in voxel, the size of vegetation elements, as well as for the variability of element positions between vegetation samples. These theoretical derivations are complemented by numerous numerical simulations for the evaluation of estimator bias and efficiency, as well as the assessment of the coverage probabilities of confidence intervals. Our simulations reveal that the usual estimators are biased and exhibit 95% confidence intervals on the order of ±100% of the estimate, when the beam number is smaller than 30. Second, our bias-corrected estimators -especially the bias-corrected MLE- are truly unbiased and efficient in a wider range of validity than the usual ones, even for beam number as low as 5. Third, we found that the confidence intervals can be as high as ≈ ± 50% when the projected area of a single element was on the order of 10% of voxel cross-sectional area and vegetation was dense (optical depth of the voxel equal to 2), even for a beam number larger than 1000. This is explained by the variability of element positions between vegetation samples, which implies that a significant part of residual error is caused by random effects. When LAD estimates are averaged over several small voxels -typically to determine a vertical profile at plot scale or to compute the total leaf area of a single plant-, confidence intervals are typically on the order of ±5 to 10% with bias-corrected estimators, which is twice as small as with usual estimators.Our study provides some new ready-to-use estimators and confidence intervals for attenuation coefficients, which are unbiased and efficient within a fairly large range of parameter values. The unbiasedness is achieved for a fairly low beam number, which is promising for application to airborne LiDAR data. They permit to raise the level of understanding and confidence on LAD estimation. Among other applications, their usage should help determine the most suitable voxel size, for given vegetation types and scanning density, whereas existing guidelines are highly variable among studies, probably because of differences in vegetation, scanning design and estimators. The impact of other sources of biases and errors, such as vegetation heterogeneity inside voxels or TLS specifications are not addressed in the present manuscript and would require further investigations.

Efficient closed-form maximum a posteriori estimators for the gamma distribution

ABSTRACTWe proposed a new class of maximum a posteriori estimators for the parameters of the Gamma distribution. These estimators have simple closed-form expressions and can be rewritten as a bias-corrected maximum likelihood estimators presented by Ye and Chen [Closed-form estimators for the gamma distribution derived from likelihood equations. Am Statist. 2017;71(2):177–181]. A simulation study was carried out to compare different estimation procedures. Numerical results revels that our new estimation scheme outperforms the existing closed-form estimators and produces extremely efficient estimates for both parameters, even for small sample sizes.

On the Bias of the Maximum Likelihood Estimators of Parameters of the Weibull Distribution

Usually, the parameters of a Weibull distribution are estimated by maximum likelihood estimation. To reduce the biases of the maximum likelihood estimators (MLEs) of two-parameter Weibull distributions, we propose analytic bias-corrected MLEs. Two other common estimators of Weibull distributions, least-squares estimators and percentiles estimators, are also introduced. Based on a comparison of their performances in the simulation study, we strongly recommend the analytic bias-corrected MLEs for the parameters of Weibull distributions, especially when the sample size is small.

Bias corrected MLEs under progressive type-II censoring scheme

ABSTRACTCensoring frequently occurs in survival analysis but naturally observed lifetimes are not of a large size. Thus, inferences based on the popular maximum likelihood (ML) estimation which often give biased estimates should be corrected in the sense of bias. Here, we investigate the biases of ML estimates under the progressive type-II censoring scheme (pIIcs). We use a method proposed in Efron and Johnstone [Fisher's information in terms of the hazard rate. Technical Report No. 264, January 1987, Stanford University, Stanford, California; 1987] to derive general expressions for bias corrected ML estimates under the pIIcs. This requires derivation of the Fisher information matrix under the pIIcs. As an application, exact expressions are given for bias corrected ML estimates of the Weibull distribution under the pIIcs. The performance of the bias corrected ML estimates and ML estimates are compared by simulations and a real data application.

A note on comparison and improvement of estimators based on likelihood

The problem of estimation of a parameter of interest in the presence of a nuisance parameter, which is either location or scale, is considered. Three estimators are taken into account: usual maximum likelihood (ML) estimator, maximum integrated likelihood estimator and the bias-corrected ML estimator. General results on comparison of these estimators w.r.t. the second-order risk based on the mean-squared error are obtained. Possible improvements of basic estimators via the notion of admissibility and methodology given in Ghosh and Sinha [A necessary and sufficient condition for second order admissibility with applications to Berkson's bioassay problem. Ann Stat. 1981;9(6):1334–1338] are considered. In the recent paper by Tanaka et al. [On improved estimation of a gamma shape parameter. Statistics. 2014; doi:10.1080/02331888.2014.915842], this problem was considered for estimating the shape parameter of gamma distribution. Here, we perform more accurate comparison of estimators for this case as well as for some other cases.

Minimax Estimation of Functionals of Discrete Distributions.

We propose a general methodology for the construction and analysis of essentially minimax estimators for a wide class of functionals of finite dimensional parameters, and elaborate on the case of discrete distributions, where the support size S is unknown and may be comparable with or even much larger than the number of observations n. We treat the respective regions where the functional is nonsmooth and smooth separately. In the nonsmooth regime, we apply an unbiased estimator for the best polynomial approximation of the functional whereas, in the smooth regime, we apply a bias-corrected version of the maximum likelihood estimator (MLE). We illustrate the merit of this approach by thoroughly analyzing the performance of the resulting schemes for estimating two important information measures: 1) the entropy [Formula: see text] and 2) [Formula: see text], α > 0. We obtain the minimax L2 rates for estimating these functionals. In particular, we demonstrate that our estimator achieves the optimal sample complexity n ≍ S/ln S for entropy estimation. We also demonstrate that the sample complexity for estimating Fα (P), 0 < α < 1, is n ≍ S1/α /ln S, which can be achieved by our estimator but not the MLE. For 1 < α < 3/2, we show the minimax L2 rate for estimating Fα (P) is (n ln n)-2(α-1) for infinite support size, while the maximum L2 rate for the MLE is n-2(α-1). For all the above cases, the behavior of the minimax rate-optimal estimators with n samples is essentially that of the MLE (plug-in rule) with n ln n samples, which we term "effective sample size enlargement." We highlight the practical advantages of our schemes for the estimation of entropy and mutual information. We compare our performance with various existing approaches, and demonstrate that our approach reduces running time and boosts the accuracy. Moreover, we show that the minimax rate-optimal mutual information estimator yielded by our framework leads to significant performance boosts over the Chow-Liu algorithm in learning graphical models. The wide use of information measure estimation suggests that the insights and estimators obtained in this paper could be broadly applicable.

Quantitative T1 estimation from T1-weighted images affected by partial volume effects

Event Abstract Back to Event Quantitative T1 estimation from T1-weighted images affected by partial volume effects Piet Bladt1*, Gabriel Ramos Llordén1, Gwendolyn Van Steenkiste1, Jan Sijbers1 and Arjan Den Dekker1 1 Vision Lab, University of Antwerp, Department of Physics, Belgium Fundamental MRI parameters, such as the spin-lattice relaxation time T1, reflect the local tissue structure and their quantification provides valuable clinical information. Specifically, quantification of T1 has significant value in many brain disorders, such as multiple sclerosis, tumors and stroke. T1 values can be retrieved by fitting a voxel-wise model to a set of T1-weighted MR images, referred to as T1 mapping. Conventionally, a mono-exponential model is chosen to describe the T1 relaxation. However, many voxels consist of two tissue types, in particular at a tissue border, a phenomenon called partial volume effect (PVE). For voxels suffering from PVEs, the mono-exponential estimator will yield an inaccurate T1 value. For such voxels, a bi-exponential model is needed to account for PVEs. Unfortunately, single-voxel bi-exponential estimators (SBEs) are inaccurate and imprecise in clinically realistic circumstances. We propose a joint multi-voxel bi-exponential estimator (JMBE) assuming that a cluster of four neighbouring voxels contains two tissue types with two global T1 values. The JMBE is constructed based on a bias-corrected maximum likelihood (ML) estimation framework. To assess its accuracy and efficiency at certain fixed imaging settings, the JMBE is compared to an ML SBE using numerical brain phantom data and a Cramér-Rao lower bound analysis. We show that the JMBE is accurate and efficient for clinically realistic single-voxel signal-to-noise ratios (SNRs), in contrast to the SBE. This demonstrates the need to incorporate information of neighbouring voxels for accurate bi-exponential T1 estimation.

Testing equality of two negative binomial means in presence of unequal over-dispersion parameters: a Behrens–Fisher problem analog

Two samples count data with unequal over-dispersions arise in many applied statistics problems (biostatistics, epidemiology, etc.). In this situation, it is of interest to test the equality of the means. The traditional Behrens–Fisher problem is to test the equality of the means μ1 and μ2 of two normal populations where the variances and are unknown. The purpose of this paper is to deal with the corresponding problem for over-dispersed count data. We develop six test procedures, namely, a likelihood ratio test LR, a likelihood ratio test based on the bias corrected maximum likelihood estimates of the nuisance parameters LR(bc), a score test T2, a score test based on the bias corrected estimates of the nuisance parameters T2 (bc), a C(α) test based on the method of moments estimates of the nuisance parameters T1 with Welch's [The significance of the difference between two means when the population variances are unequal. Biometrika. 1937;29:350–362] degree of freedom correction, and a test TN using the asymptotic normal distribution of T1. These procedures are then compared in terms of size and power using simulations. Simulations show the best overall performance of the statistic T1 in terms of size and power and it is easy to calculate. For large sample sizes, for example, for n1=n2=50, all six statistics do well in terms of level and their power performances are also similar. So, for large sample sizes, the statistic TN should be used as it is very easy to use in practice.