Bayesian Estimation in a Generalized Negative Binomial Distribution

A generalized negative binomial (GNB) distribution was introduced by JAIN and CONSUL (1971) and was modified by NELSON (1975). The probability function of the distribution is defined by the function p(x; m, β, θ)= θ^x (1 - θ)^m+βx—x for x=0, 1, …, and zero otherwise, where m>0, 0<θ<1 and β=0 or 1≦β<θ^?1. The Bayes estimators for a number of parametric functions of θ when m and β are known are derived. The prior information on θ may be given by a beta distribution, B(a, b), to which no subjective significance is attached. It has been illustrated that the parameters in the prior distribution can be assigned by a computer. Comparisons are made of the Bayes estimate of P(X=k) to the corresponding ML estimate and the MVU estimate for any given sample to the order n^?1 for different values of k..     

Pairwise Comparisons Using Trimmed Means or M-Estimators when Working with Dependent Groups

For J dependent groups, let θ_j, j = 1, …, J, be some measure of location associated with the jth group. A common goal is computing confidence intervals for the pairwise differences, θ_j — θ_k, j < k, such that the simultaneous probability coverage is 1 — α. If means are used, it is well known that slight departures from normality (as measured by the Kolmogorov distance) toward a heavy-tailed distribution can substantially inflate the standard error of the sample mean, which in turn can result in relatively low power. Also, when distributions differ in shape, or when sampling from skewed distributions with relatively light tails, practical problems arise when the goal is to obtain confidence intervals with simultaneous probability coverage reasonably close to the nominal level. Extant theoretical and simulation results suggest replacing means with trimmed means. The Tukey-McLaughlin method is easily adapted to the problem at hand via the Bonferroni inequality, but this paper illustrates that practical concerns remain. Here, the main result is that the percentile t bootstrap method, used in conjunction with trimmed means, gives improved probability coverage and substantially better power. A method based on a one-step M-estimator is also considered but found to be less satisfactory.     

Estimating ancestral distributions of lineages with uncertain sister groups: a statistical approach to Dispersal–Vicariance Analysis and a case using Aesculus L. (Sapindaceae) including fossils

Abstract We propose a simple statistical approach for using Dispersal–Vicariance Analysis (DIVA) software to infer biogeographic histories without fully bifurcating trees. In this approach, ancestral ranges are first optimized for a sample of Bayesian trees. The probability P of an ancestral range r at a node is then calculated as where Y is a node, and F(r_Y ) is the frequency of range r among all the optimal solutions resulting from DIVA optimization at node Y, t is one of n topologies optimized, and Pt is the probability of topology t. Node Y is a hypothesized ancestor shared by a specific crown lineage and the sister of that lineage “x”, where x may vary due to phylogenetic uncertainty (polytomies and nodes with posterior probability <100%). Using this method, the ancestral distribution at Y can be estimated to provide inference of the geographic origins of the specific crown group of interest. This approach takes into account phylogenetic uncertainty as well as uncertainty from DIVA optimization. It is an extension of the previously described method called Bayes‐DIVA, which pairs Bayesian phylogenetic analysis with biogeographic analysis using DIVA. Further, we show that the probability P of an ancestral range at Y calculated using this method does not equate to pp*F(r_Y ) on the Bayesian consensus tree when both variables are <100%, where pp is the posterior probability and F(r_Y ) is the frequency of range r for the node containing the specific crown group. We tested our DIVA‐Bayes approach using Aesculus L., which has major lineages unresolved as a polytomy. We inferred the most probable geographic origins of the five traditional sections of Aesculus and of Aesculus californica Nutt. and examined range subdivisions at parental nodes of these lineages. Additionally, we used the DIVA‐Bayes data from Aesculus to quantify the effects on biogeographic inference of including two wildcard fossil taxa in phylogenetic analysis. Our analysis resolved the geographic ranges of the parental nodes of the lineages of Aesculus with moderate to high probabilities. The probabilities were greater than those estimated using the simple calculation of pp*F(r_y) at a statistically significant level for two of the six lineages. We also found that adding fossil wildcard taxa in phylogenetic analysis generally increased P for ancestral ranges including the fossil's distribution area. The ΔP was more dramatic for ranges that include the area of a wildcard fossil with a distribution area underrepresented among extant taxa. This indicates the importance of including fossils in biogeographic analysis. Exmination of range subdivision at the parental nodes revealed potential range evolution (extinction and dispersal events) along the stems of A. californica and sect. Parryana.     

Empirical Bayes Estimation and Prediction Using Summary-Level Information From External Big Data Sources Adjusting for Violations of Transportability

Large external data sources may be available to augment studies that collect data to address a specific research objective. In this article we consider the problem of building regression models for prediction based on individual-level data from an “internal” study while incorporating summary information from an “external” big data source. We extend the work of Chatterjee et al. (J Am Stat Assoc 111(513):107–117, 2006) by introducing an adaptive empirical Bayes shrinkage estimator that uses the external summary-level information and the internal data to trade bias with variance for protection against departures in the conditional probability distribution of the outcome given a set of covariates between the two populations. We use simulation studies and a real data application using external summary information from the Prostate Cancer Prevention Trial to assess the performance of the proposed methods in contrast to maximum likelihood estimation and the constrained maximum likelihood (CML) method developed by Chatterjee et al. (J Am Stat Assoc 111(513):107–117, 2006). Our simulation studies show that the CML method can be biased and inefficient when the assumption of a transportable covariate distribution between the external and internal populations is violated, and our empirical Bayes estimator provides protection against bias and loss of efficiency.     

Maximum likelihood estimation of ordered multinomial parameters

The pool adjacent violator algorithm Ayer et al. (1955, The Annals of Mathematical Statistics, 26, 641-647) has long been known to give the maximum likelihood estimator of a series of ordered binomial parameters, based on an independent observation from each distribution (see Barlow et al., 1972, Statistical Inference under Order Restrictions, Wiley, New York). This result has immediate application to estimation of a survival distribution based on current survival status at a set of monitoring times. This paper considers an extended problem of maximum likelihood estimation of a series of 'ordered' multinomial parameters p(i)= (p(1i),p(2i),.,p(mi)) for 1 相似文献   

A Finite Stage Decision Procedure in the Analysis of Variance

In this paper we have developed a finite stage Bayes test in the analysis of variance. Determining such a decision rule the losses of erroneously accepting the hypotheses as well as the observation costs and the a priori probabilities were considered. We have given a method of constructing the continuation intervals (a_t, b_t) (t = 1, …, r) of a r-stage test and have demonstrated how the decision process operates. Furthermore it has been investigated in which way the Bayes risk depends on the several parameters. Especially we have shown that the Bayes risk of an r-stage test is much smaller than the Bayes risk of a corresponding test with fixed sample size as a rule.     

Convergence to Normality of the Asymptotic Quasi‐Score Function on a Linear Model

The asymptotic quasi‐likelihood method is considered for the model y_t = f_t(θ) + M_t, t = 0,1, …,T where f_tθ) is a linear predictable process of the parameter of interest θ, M_t is a martingale difference, and the nature of E(M_t² | ℱ_t–1) is unknown. This paper is concerned with the limiting distribution of the asymptotic quasi‐score function of such a model. Confidence intervals and hypothesis testing of θ is derived from the limiting distribution. Comparison is made between the estimates obtained through this method and those obtained through the least squares method.     

Concordance indices with left-truncated and right-censored data

In the context of time-to-event analysis, a primary objective is to model the risk of experiencing a particular event in relation to a set of observed predictors. The Concordance Index (C-Index) is a statistic frequently used in practice to assess how well such models discriminate between various risk levels in a population. However, the properties of conventional C-Index estimators when applied to left-truncated time-to-event data have not been well studied, despite the fact that left-truncation is commonly encountered in observational studies. We show that the limiting values of the conventional C-Index estimators depend on the underlying distribution of truncation times, which is similar to the situation with right-censoring as discussed in Uno et al. (2011) [On the C-statistics for evaluating overall adequacy of risk prediction procedures with censored survival data. Statistics in Medicine 30(10), 1105–1117]. We develop a new C-Index estimator based on inverse probability weighting (IPW) that corrects for this limitation, and we generalize this estimator to settings with left-truncated and right-censored data. The proposed IPW estimators are highly robust to the underlying truncation distribution and often outperform the conventional methods in terms of bias, mean squared error, and coverage probability. We apply these estimators to evaluate a predictive survival model for mortality among patients with end-stage renal disease.     

A system for predicting and preventing work-related musculoskeletal disorders among dentists

Work-related musculoskeletal disorders (WMSDs) have become increasingly common among dentists and initiate a series of events that could result in a career ending. This study aims to construct a system for predicting and preventing WMSD among dentists. We used Bayesian network (BN) that describes the mutual relationships among multiple variables contributing to WMSDs. The data-sets were prepared from direct measurements of dentist's movements and a questionnaire survey. We applied BN learning algorithms to the training data-sets to develop WMSD prediction model using 10-fold cross-validation. To evaluate the system performance, 16 dentists were randomly assigned into a 2 × 2 crossover trial scheduled to each of two sequences of dental working: receiving feedback or no feedback including the probability of WMSD and related risk factors from the system. The group that received feedback decreased significantly (t-test, p < 0.05) the extensions of neck and upper back in the y-axis as well as the WMSD probability on the post-test. In conclusion, the system for predicting and preventing WMSD aids the correction of neck and upper back extensions and reduction in WMSD probability, which may potentially contribute to reduce the risk of WMSD among dentists.     

Interaction of estrogen receptor β5 and interleukin 6 receptor in the progression of non–small cell lung cancer

Numerous studies have shown that the estrogen receptor beta (ERβ) and interleukin 6 receptor (IL-6R) had interaction in many tumors, including lung cancer. Previous studies found that ERβ5 exhibits a different biological function compared with the other subtypes of ERβ. Therefore, this study mainly explores the interaction between ERβ5 and IL-6R in the progression of lung cancer. We found that the expression of ERβ5, IL-6 and glycoprotein 130 (GP130) were significantly increased (P < 0.001) and the 5-year survival rate with the co-expression of ERβ5 and GP130 is significantly lower (P = 0.0315) in non-small cell lung cancer (NSCLC) patients. The cell proliferation, invasion, and cell cycle were markedly increased, and the cell apoptotic was markedly inhibited with the concurrent action of ERβ5 and IL-6 in A549 cells (P < 0.05). In addition, the expression of ERβ5, GP130, p-AKT, and p-44/42 MAPK was also significantly increased in A549 cells (P < 0.05). These results indicate that ERβ5 and GP130 can synergistically promote the progression of NSCLC and maybe combined as an independent prognostic factor in patients. In addition, these results also provide a theoretical basis for the combined targeting therapy of ERβ5 and GP130 in NSCLC.     

Prediction of Survival by Neutropenia According To Delivery Schedule of Oxaliplatin–5-Fluorouracil–Leucovorin for Metastatic Colorectal Cancer in a Randomized International Trial (EORTC 05963)

Circadian clocks control cellular proliferation and drug metabolism over the 24?h. However, circadian chronomodulated chemotherapy with 5-fluorouracil, leucovorin, and oxaliplatin (chronoFLO4) offered no survival benefit as compared with the non–time-stipulated FOLFOX2, in an international randomized trial involving patients with previously untreated metastatic colorectal cancer (EORTC 05963). The authors hypothesized that treatment near maximum tolerated dose could disrupt circadian clocks thus impairing the efficacy of chronoFLO4 but not of FOLFOX2. Patients with available data (N?=?556) were categorized into three subgroups according to the worst grade (G) of neutropenia experienced during treatment. Distinct multivariate models with time-dependent covariates were constructed for each treatment schedule. Neutropenia incidence (all grades) was 33% on chronoFLO4 and 61% on FOLFOX2 (p?<?.0001), and G3–4 were 7% and 25%, respectively (p < .0001). Neutropenia was significantly more frequent in women than men on either schedule (FOLFOX2, p = .003; chronoFLO4, p = .04). Median survival was 20.7 mo in patients with G3–4 neutropenia versus 12.5 mo in neutropenia-free patients on FOLFOX2 (p < .0001). Corresponding figures were 13.7 and 19.4 mo, respectively, on chronoFLO4 (p?=?.36). Multivariate analysis confirmed occurrence of severe neutropenia independently predicted for better overall survival on FOLFOX2 (HR?=?0.56; p = .015), and worse survival on chronoFLO4 (HR?=?1.77, p = .06), with a significant interaction test (p < .0001). Prediction of better survival in neutropenic patients on FOLFOX2 supports the administration of conventional chemotherapy near maximum tolerated dose. The opposite trend shown here for chronoFLO4 supports the novel concept of jointly optimized hematologic tolerability and efficacy through personalized circadian-timed therapy. (Author correspondence: francis.levi@inserm.fr)     

Estimation of a Joint Distribution Function with Multivariate Interval‐Censored Data when the Nonparametric MLE is not Unique

A nonparametric estimator of a joint distribution function F₀ of a d‐dimensional random vector with interval‐censored (IC) data is the generalized maximum likelihood estimator (GMLE), where d ≥ 2. The GMLE of F₀ with univariate IC data is uniquely defined at each follow‐up time. However, this is no longer true in general with multivariate IC data as demonstrated by a data set from an eye study. How to estimate the survival function and the covariance matrix of the estimator in such a case is a new practical issue in analyzing IC data. We propose a procedure in such a situation and apply it to the data set from the eye study. Our method always results in a GMLE with a nonsingular sample information matrix. We also give a theoretical justification for such a procedure. Extension of our procedure to Cox's regression model is also mentioned.     

Modelling the distribution of the invasive Ziziphus mauritiana along road corridors in Zimbabwe

We investigate how human fruit consumption affects the spread of the alien invasive Ziziphus mauritiana along road corridors in northern Zimbabwe. A field survey was conducted to identify and map Z. Mauritiana stems at 25 randomly located 6‐ha plots placed along two main roads connecting the Zambezi Valley to the Highveld region. The L‐function was used to test for evidence of significant spatial clustering of the stems. The inhomogeneous point model fitted by maximum likelihood was also applied to check whether distance from the road explains variation in the number of stems per unit area. Finally, a t test was executed on log‐transformed abundance data to test for significant differences in the mean number of saplings and adults between the Zambezi Valley and the Highveld. Results of the L‐function indicated that Z. mauritiana had a clustered and linear distribution along roads both in the Zambezi Valley and Highveld regions. Results of the t test showed that the mean number of saplings per plot in the Zambezi Valley (μ₁ = 275) was three times higher than in the Highveld (μ₂ = 78), with p < 0.01. The mean number of adult trees was also significantly higher in the Zambezi Valley than in the Highveld.     

Consistency of the Asymptotic Quasi-Likelihood Estimate on Linear Models

Consider a model y_t = f_t(θ) + M_t, 0 ⩽ t ⩽ T where θ∈ Θ in an unknown parameter, f_t(θ) is a linear predictable process, M_t is a martingale difference, and the nature of E(M²_t/ℱ_t—1) is unknown. This paper presents an estimating procedure for θ based on the asymptotic quasi-likelihood methodology. Conditions under which the asymptotic quasi-likelihood estimate converges to the true parameter θ₀ are discussed. This method is applied to several simulated examples, and estimates of the unknown parameter are obtained by means of a two-stage technique. Comparison is made between the estimates obtained via this method and those obtained via the ordinary least squares method. Discussion is provided on the application of the model.     

On the empirical choice of the time window for restricted mean survival time

The t-year mean survival or restricted mean survival time (RMST) has been used as an appealing summary of the survival distribution within a time window [0, t]. RMST is the patient's life expectancy until time t and can be estimated nonparametrically by the area under the Kaplan-Meier curve up to t. In a comparative study, the difference or ratio of two RMSTs has been utilized to quantify the between-group-difference as a clinically interpretable alternative summary to the hazard ratio. The choice of the time window [0, t] may be prespecified at the design stage of the study based on clinical considerations. On the other hand, after the survival data have been collected, the choice of time point t could be data-dependent. The standard inferential procedures for the corresponding RMST, which is also data-dependent, ignore this subtle yet important issue. In this paper, we clarify how to make inference about a random “parameter.” Moreover, we demonstrate that under a rather mild condition on the censoring distribution, one can make inference about the RMST up to t, where t is less than or even equal to the largest follow-up time (either observed or censored) in the study. This finding reduces the subjectivity of the choice of t empirically. The proposal is illustrated with the survival data from a primary biliary cirrhosis study, and its finite sample properties are investigated via an extensive simulation study.     

The Negative Binomial Distribution of Order k and Some of Its Properties

The negative binomial distribution of order k is introduced and briefly studied. First it is shown that it is a proper probability distribution. Then its probability generating function, mean and variance are derived. Finally it is shown that the number of trials until the rth kth consecutive success (r ≧ 1, k ≧ 1) in independent trials with constant success probability p (0 < p < 1) is distributed as negative binomial distribution of order k. The present paper generalizes results of SHANE (1973), PHILIPPOU and MUWAFI (1982), and PHILIPPOU, GEORGHIOU and PHILIPPOU (1982).     

Effect of maize stunting mollicutes on survival and fecundity of Dalbulus leafhopper vectors

The effect of corn stunt spiroplasma (CSS) on survival and fecundity of three Dalbulus leafhopper species was determined. CSS significantly reduced the survival, as measured by the time to 50% (t₅₀) and 25% (t₂₅) survival, and by the scale parameter (b) of the Weibull model, for D. elimatus and D. gelbus. Fecundity of these two species, as measured by the net and gross reproductive rates, was also reduced by CSS. CSS did not significantly affect the corn leafhopper, D. maidis. In a separate experiment, maize bushy stunt mycoplasma (MBSM) reduced the survival and fecundity of D. maidis at temperatures from 20 to 29 °C. The effect of MBSM on D. maidis survival was less severe than CSS on D. elimatus and D. gelbus; t₂₅, but not t₅₀, was reduced by MBSM. Survival times and the cohort generation time generally declined with increasing temperature. Fecundity, however, generally increased with increasing temperature.     

Semiparametric estimation of the transformation model by leveraging external aggregate data in the presence of population heterogeneity

Leveraging information in aggregate data from external sources to improve estimation efficiency and prediction accuracy with smaller scale studies has drawn a great deal of attention in recent years. Yet, conventional methods often either ignore uncertainty in the external information or fail to account for the heterogeneity between internal and external studies. This article proposes an empirical likelihood-based framework to improve the estimation of the semiparametric transformation models by incorporating information about the t-year subgroup survival probability from external sources. The proposed estimation procedure incorporates an additional likelihood component to account for uncertainty in the external information and employs a density ratio model to characterize population heterogeneity. We establish the consistency and asymptotic normality of the proposed estimator and show that it is more efficient than the conventional pseudopartial likelihood estimator without combining information. Simulation studies show that the proposed estimator yields little bias and outperforms the conventional approach even in the presence of information uncertainty and heterogeneity. The proposed methodologies are illustrated with an analysis of a pancreatic cancer study.     

Bayesian Reliability Estimation of a Two-parameter Cauchy Distribution

This paper gives an approximate Bayes procedure for the estimation of the reliability function of a two-parameter Cauchy distribution using Jeffreys' non-informative prior with a squared-error loss function, and with a log-odds ratio squared-error loss function. Based on a Monte Carlo simulation study, two such Bayes estimators of the reliability are compared with the maximum likelihood estimator.     

Daily Rhythms in Blood and Milk Lead Toxicokinetics Following Intravenous Administration of Lead Acetate to Dairy Cows in Winter

The objective was to investigate circadian variations of blood and milk lead toxicokinetics in dairy cows in winter. Twenty lactating Holstein animals were randomly assigned to 4 treatments, corresponding to different hours after onset of light (HALO): 2, 8, 14 and 20. Cows received a single intravenous administration of 2.5 mg/kg lead as lead acetate. Blood and milk samples were taken and analyzed by atomic absorption spectrophotometry. For each toxicokinetic parameter, a one-way analysis of variance was performed to outline the existence of daily variations. Significant differences as a function of HALO were detected in blood for the hybrid constant of elimination (β), half-life of elimination (t_1/2β), area under the curve (AUC) and clearance (Cl) (p &lt; 0.01) and volume of distribution at steady state (V_ss) (p < 0.05). Half-life of elimination was highest when lead acetatae was injected at 2 HALO, and lowest following the 14 HALO administration. Milk data showed significant differences for maximum concentration, AUC and Cl (p &lt; 0.001), volume of distribution and V_ss (p < 0.005). The ratio AUC_milk/ABC_blood was utilized to estimate penetration of lead in milk. It differed significantly along the day (p < 0.01). Blood daily variations could be fitted a circadian rhythms. No circadian rhythms were detected in milk parameters or in the ratio AUC_milk/ABC_blood. It was concluded that blood and milk lead toxicokinetics are distinctly affected by the time of the day.