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Separation or monotone-likelihood can be observed in fitting process of a multinomial logistic model using maximum likelihood estimation (MLE) when sample size is small and/or one of the outcome categories is rare and/or there is one or more influential covariates, resulting in infinite or biased estimate of at least one regression coefficient of the model. This study investigated empirically to identify the optimal data condition to define both ‘separation’ and ‘near-to-separation’ (partial separation) and explored their consequences in MLE and provided a solution by applying a penalized likelihood approach, which has been proposed in the literature, by adding a Jeffreys prior-based penalty term to the original likelihood function to remove the first-order bias in the MLEs of the multinomial logit model via equivalent Poisson regression. Furthermore, the penalized estimating equation (PMLE) is extended to a weighted estimating equation allowing for survey-weight for analyzing data from a complex survey. The simulation study suggests that the PMLE outperforms the MLE by providing smaller amount of bias and mean squared of error and better coverage. The methods are applied to analyze data on choice of health facility for treatment of childhood diseases.

In parametric estimates, the maximum likelihood estimation method is the most popular method widely used in the social sciences and psychology, although it is biased in situations where sample sizes are small or the data are heavily censored. Therefore, the main objective of this research is to improve this estimation method using the Runge-Kutta technique. The improved method was applied to derive the estimators of the shape scale family parameters and compare them with Bayesian estimators based on the informative and kernel priors, via Monte Carlo simulation. The simulation results showed that the improved maximum likelihood estimation method is highly efficient and outperforms the Bayesian method for different sample sizes. Finally, from a future perspective, the proposed model could be important for analyzing real data sets including data on COVID-19 deaths in Egypt, for potential comparative studies with other countries.

Asymptotic approaches are traditionally used to calculate confidence intervals for intraclass correlation coefficient in a clustered binary study. When sample size is small to medium, or correlation or response rate is near the boundary, asymptotic intervals often do not have satisfactory performance with regard to coverage. We propose using the importance sampling method to construct the profile confidence limits for the intraclass correlation coefficient. Importance sampling is a simulation based approach to reduce the variance of the estimated parameter. Four existing asymptotic limits are used as statistical quantities for sample space ordering in the importance sampling method. Simulation studies are performed to evaluate the performance of the proposed accurate intervals with regard to coverage and interval width. Simulation results indicate that the accurate intervals based on the asymptotic limits by Fleiss and Cuzick generally have shorter width than others in many cases, while the accurate intervals based on Zou and Donner asymptotic limits outperform others when correlation and response rate are close to their boundaries.

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Phase II clinical trials investigate whether a new drug or treatment has sufficient evidence of effectiveness against the disease under study. Two-stage designs are popular for phase II since they can stop in the first stage if the drug is ineffective. Investigators often face difficulties in determining the target response rates, and adaptive designs can help to set the target response rate tested in the second stage based on the number of responses observed in the first stage. Popular adaptive designs consider two alternate response rates, and they generally minimise the expected sample size at the maximum uninterested response rate. Moreover, these designs consider only futility as the reason for early stopping and have high expected sample sizes if the provided drug is effective. Motivated by this problem, we propose an adaptive design that enables us to terminate the single-arm trial at the first stage for efficacy and conclude which alternate response rate to choose. Comparing the proposed design with a popular adaptive design from literature reveals that the expected sample size decreases notably if any of the two target response rates are correct. In contrast, the expected sample size remains almost the same under the null hypothesis.

We propose what appears to be the first Bayesian procedure for the analysis of seasonal variation when the sample size and the amplitude are small. Such data occur often in the medical sciences, where seasonality analyses and environmental considerations can help clarify disease etiologies. The method is explained in terms of a simple physico-geometric setting. We present the Bayesian version of a frequentist test that performs well. Two examples of real data illustrate the procedure's application.

Sample size calculation is a critical issue in clinical trials because a small sample size leads to a biased inference and a large sample size increases the cost. With the development of advanced m...

Despite the popularity of high dimension, low sample size data analysis, there has not been enough attention to the sample integrity issue, in particular, a possibility of outliers in the data. A n...

More and more studies have shown that genetic determinants may mediate variability among persons in the response to a drug. In other words, some therapeutics benefit only a subset of treated patien...

© 2017, © 2017 Informa UK Limited, trading as Taylor & Francis Group. The purpose of acceptance sampling is to develop decision rules to accept or reject production lots based on sample data. When testing is destructive or expensive, dependent sampling procedures cumulate results from several preceding lots. This chaining of past lot results reduces the required size of the samples. A large part of these procedures only chain past lot results when defects are found in the current sample. However, such selective use of past lot results only achieves a limited reduction of sample sizes. In this article, a modified approach for chaining past lot results is proposed that is less selective in its use of quality history and, as a result, requires a smaller sample size than the one required for commonly used dependent sampling procedures, such as multiple dependent sampling plans and chain sampling plans of Dodge. The proposed plans are applicable for inspection by attributes and inspection by variables. Several properties of their operating characteristic-curves are derived, and search procedures are given to select such modified chain sampling plans by using the two-point method. ispartof: Journal of Applied Statistics vol:45 issue:8 pages:1447-1464 status: published

The logistic regression model has been widely used in the social and natural sciences and results from studies using this model can have significant policy impacts. Thus, confidence in the reliability of inferences drawn from these models is essential. The robustness of such inferences is dependent on sample size. The purpose of this article is to examine the impact of alternative data sets on the mean estimated bias and efficiency of parameter estimation and inference for the logistic regression model with observational data. A number of simulations are conducted examining the impact of sample size, nonlinear predictors, and multicollinearity on substantive inferences (e.g. odds ratios, marginal effects) when using logistic regression models. Findings suggest that small sample size can negatively affect the quality of parameter estimates and inferences in the presence of rare events, multicollinearity, and nonlinear predictor functions, but marginal effects estimates are relatively more robust to...

Mahalanobis square distances (MSDs) based on robust estimators improves outlier detection performance in multivariate data. However, the unbiasedness of robust estimators are not guaranteed when the sample size is small and this reduces their performance in outlier detection. In this study, we propose a framework that uses MSDs with incorporated small sample correction factor (c) and show its impact on performance when the sample size is small. This is achieved by using two prototypes, minimum covariance determinant estimator and S-estimators with bi-weight and t-biweight functions. The results from simulations show that distribution of MSDs for non-extreme observations are more likely to fit to chi-square with p degrees of freedom and MSDs of the extreme observations fit to F distribution, when c is incorporated into the model. However, without c, the distributions deviate significantly from chi-square and F observed for the case with incorporated c. These results are even more prominent for S-estimators. We present seven distinct comparison methods with robust estimators and various cut-off values and test their outlier detection performance with simulated data. We also present an application of some of these methods to the real data.

The way of investigating a distribution knowing its interesting properties might be often inadequate when the shapes of two distributions are almost similar. In each of these circumstances, the accurate decision about the genesis of a random sample from any of the two parent distributions will be very much ambiguous even with the availability of the existing testing procedure of the circular data. A sequential discrimination procedure has been suggested which is also invariant to the sample size. The performance of the proposed discrimination procedure has been evaluated by checking its capability of detecting the genesis of the known samples from the two identically shaped wrapped distributions.

Sample size determination is one of the most commonly encountered tasks in the design of every applied research. The general guideline suggests that a pilot study can offer plausible planning values for the vital model characteristics. This article examines two viable approaches to taking into account the imprecision of a variance estimate in sample size calculations for linear statistical models. The multiplier procedure employs an adjusted sample variance in the form of a multiple of the observed sample variance. The Bayesian method accommodates the uncertainty of a sample variance through a prior distribution. It is shown that the two seemingly distinct techniques are equivalent for sample size determination under the designated assurance requirements that the actual power exceeds the planned threshold with a given tolerance probability, or the expected power attains the desired level. The selection of optimum pilot sample size for minimizing the expected total cost is also considered.

A discrimination procedure, based on the location model is described and suggested for use in situation where the discriminating variables are mixtures of continuous and binary variables. Some procedures that have been previously employed, in a similar situation, like Fisher's linear discriminant function and the logistic regression were compared with this method using error rate (ER). Optimal ERs for these procedures are reported using real and simulated data for the case of varying sample size and number of continuous and binary variables and were used as a measure for assessing the performance of the various procedures. The suggested procedure performed considerably better in the cases considered and never did produce a result that is poor when compared with other procedures. Hence, the suggested procedure might be considered for such situations.

Interval-censored data are very common in the reliability and lifetime data analysis. This paper investigates the performance of different estimation procedures for a special type of interval-censored data, i.e. grouped data, from three widely used lifetime distributions. The approaches considered here include the maximum likelihood estimation, the minimum distance estimation based on chi-square criterion, the moment estimation based on imputation (IM) method and an ad hoc estimation procedure. Although IM-based techniques are extensively used recently, we show that this method is not always effective. It is found that the ad hoc estimation procedure is equivalent to the minimum distance estimation with another distance metric and more effective in the simulation. The procedures of different approaches are presented and their performances are investigated by Monte Carlo simulation for various combinations of sample sizes and parameter settings. The numerical results provide guidelines to analyse g...

Smoothing methods for curve estimation have received considerable attention in statistics with a wide range of applications. However, to our knowledge, sample size planning for testing significance of curves has not been discussed in the literature. This paper focuses on sample size calculations for nonparametric regression and partially linear models based on local linear estimators. We describe explicit procedures for sample size calculations based on non- and semi-parametric F-tests. Data examples are provided to demonstrate the use of the procedures.

The problem of testing the similarity of two normal populations is reconsidered, in this article, from a nonclassical point of view. We introduce a test statistic based on the maximum likelihood estimate of Weitzman's overlapping coefficient. Simulated critical points are provided for the proposed test for various sample sizes and significance levels. Statistical powers of the proposed test are computed via simulation studies and compared to those of the existing tests. Furthermore, Type-I error robustness of the proposed and the existing tests are studied via simulation studies when the underlying distributions are non-normal. Two data sets are analyzed for illustration purposes. Finally, the proposed test has been implemented to assess the bioequivalence of two drug formulations.

Cut-off sampling has been widely used for business survey which has the right-skewed population with a long tail. Several methods are suggested to obtain the optimal cut-off point. The LH algorithm suggested by Lavallee and Hidiroglou [6] is commonly used to get the optimum boundaries by minimizing the total sample size with a given precision. In this paper, we suggest a new cut-off point determination method which minimizes a cost function. And that leads to reducing the size of take-all stratum. Also we investigate an optimal cut-off point using a typical parametric estimation method under the assumptions of underlying distributions. Small Monte-Carlo simulation studies are performed in order to compare the new cut-off point method to the LH algorithm. The Korea Transportation Origin – Destination data are used for real data analysis.

One of the general problems in clinical trials and mortality rates is the comparison of competing risks. Most of the test statistics used for independent and dependent risks with censored data belong to the class of weighted linear rank tests in its multivariate version. In this paper, we introduce the saddlepoint approximations as accurate and fast approximations for the exact p-values of this class of tests instead of the asymptotic and permutation simulated calculations. Real data examples and extensive simulation studies showed the accuracy and stability performance of the saddlepoint approximations over different scenarios of lifetime distributions, sample sizes and censoring.

A popular choice when analyzing ordinal data is to consider the cumulative proportional odds model to relate the marginal probabilities of the ordinal outcome to a set of covariates. However, application of this model relies on the condition of identical cumulative odds ratios across the cut-offs of the ordinal outcome; the well-known proportional odds assumption. This paper focuses on the assessment of this assumption while accounting for repeated and missing data. In this respect, we develop a statistical method built on multiple imputation (MI)based on generalized estimating equations that allows to test the proportionality assumption under the missing at random setting. The performance of the proposed method is evaluated for two MI algorithms for incomplete longitudinal ordinal data. The impact of both MI methods is compared with respect to the type I error rate and the power for situations covering various numbers of categories of the ordinal outcome, sample sizes, rates of missingness, well-balanced and skewed data. The comparison of both MI methods with the complete-case analysis is also provided. We illustrate the use of the proposed methods on a quality of life data from a cancer clinical trial.

We propose a class of general partially linear additive transformation models (GPLATM) with right-censored survival data in this work. The class of models are flexible enough to cover many commonly used parametric and nonparametric survival analysis models as its special cases. Based on the B spline interpolation technique, we estimate the unknown regression parameters and functions by the maximum marginal likelihood estimation method. One important feature of the estimation procedure is that it does not need the baseline and censoring cumulative density distributions. Some numerical studies illustrate that this procedure can work very well for the moderate sample size.

Five sampling schemes (SS) for price index construction – one cut-off sampling technique and four probability-proportional-to-size (pps) methods – are evaluated by comparing their performance on a homescan market research data set across 21 months for each of the 13 classification of individual consumption by purpose (COICOP) food groups. Classifications are derived for each of the food groups and the population index value is used as a reference to derive performance error measures, such as root mean squared error, bias and standard deviation for each food type. Repeated samples are taken for each of the pps schemes and the resulting performance error measures analysed using regression of three of the pps schemes to assess the overall effect of SS and COICOP group whilst controlling for sample size, month and population index value. Cut-off sampling appears to perform less well than pps methods and multistage pps seems to have no advantage over its single-stage counterpart. The jackknife resampling technique is also explored as a means of estimating the standard error of the index and compared with the actual results from repeated sampling.

We propose penalized minimum φ-divergence estimator for parameter estimation and variable selection in logistic regression. Using an appropriate penalty function, we show that penalized φ-divergence estimator has oracle property. With probability tending to 1, penalized φ-divergence estimator identifies the true model and estimates nonzero coefficients as efficiently as if the sparsity of the true model was known in advance. The advantage of penalized φ-divergence estimator is that it produces estimates of nonzero parameters efficiently than penalized maximum likelihood estimator when sample size is small and is equivalent to it for large one. Numerical simulations confirm our findings.

The multivariate log-normal distribution is a good candidate to describe data that are not only positive and skewed, but also contain many characteristic values. In this study, we apply the generalized variable method to compare the mean vectors of two independent multivariate log-normal populations that display heteroscedasticity. Two generalized pivotal quantities are derived for constructing the generalized confidence region and for testing the difference between two mean vectors. Simulation results indicate that the proposed procedures exhibit satisfactory performance regardless of the sample sizes and heteroscedasticity. The type I error rates obtained are consistent with expectations and the coverage probabilities are close to the nominal level when compared with the other method which is currently available. These features make the proposed method a worthy alternative for inferential analysis of problems involving multivariate log-normal means. The results are illustrated using three examples.

When conducting research with controlled experiments, sample size planning is one of the important decisions that researchers have to make. However, current methods do not adequately address this issue with regard to variance heterogeneity with some cost constraints for comparing several treatment means. This paper proposes a sample size allocation ratio in the fixed-effect heterogeneous analysis of variance when group variances are unequal and in cases where the sampling and/or variable cost has some constraints. The efficient sample size allocation is determined for the purpose of minimizing total cost with a designated power or maximizing the power with a given total cost. Finally, the proposed method is verified by using the index of relative efficiency and the corresponding total cost and the total sample size needed. We also apply our method in a pain management trial to decide an efficient sample size. Simulation studies also show that the proposed sample size formulas are efficient in terms of sta...

Since multi-attribute control charts have received little attention compared with multivariate variable control charts, this research is concerned with developing a new methodology to employ the multivariate exponentially weighted moving average (MEWMA) charts for m-attribute binomial processes; the attributes being the number of nonconforming items. Moreover, since the variable sample size and sampling interval (VSSI) MEWMA charts detect small process mean shifts faster than the traditional MEWMA, an economic design of the VSSI MEWMA chart is proposed to obtain the optimum design parameters of the chart. The sample size, the sampling interval, and the warning/action limit coefficients are obtained using a genetic algorithm such that the expected total cost per hour is minimized. At the end, a sensitivity analysis has been carried out to investigate the effects of the cost and the model parameters on the solution of the economic design of the VSSI MEWMA chart.

In this paper, we study the bioequivalence (BE) inference problem motivated by pharmacokinetic data that were collected using the serial sampling technique. In serial sampling designs, subjects are independently assigned to one of the two drugs; each subject can be sampled only once, and data are collected at K distinct timepoints from multiple subjects. We consider design and hypothesis testing for the parameter of interest: the area under the concentration–time curve (AUC). Decision rules in demonstrating BE were established using an equivalence test for either the ratio or logarithmic difference of two AUCs. The proposed t-test can deal with cases where two AUCs have unequal variances. To control for the type I error rate, the involved degrees-of-freedom were adjusted using Satterthwaite's approximation. A power formula was derived to allow the determination of necessary sample sizes. Simulation results show that, when the two AUCs have unequal variances, the type I error rate is better controlled by t...

In this article we examine sample size calculations for a binomial proportion based on the confidence interval width of the Agresti-Coull, Wald and Wilson Score intervals. We pointed out that the commonly used methods based on known and fixed standard errors cannot guarantee the desired confidence interval width given a hypothesized proportion. Therefore, a new adjusted sample size calculation method was introduced, which is based on the conditional expectation of the width of the confidence interval given the hypothesized proportion. With the reduced sample size, the coverage probability can still maintain at the nominal level and is very competitive to the converge probability for the original sample size.

This study examined the influence of heterogeneity of variance on Type I error rates and power of the independent-samples Student's t-test of equality of means on samples of scores from normal and 10 non-normal distributions. The same test of equality of means was performed on corresponding rank-transformed scores. For many non-normal distributions, both versions produced anomalous power functions, resulting partly from the fact that the hypothesis test was biased, so that under some conditions, the probability of rejecting H 0 decreased as the difference between means increased. In all cases where bias occurred, the t-test on ranks exhibited substantially greater bias than the t-test on scores. This anomalous result was independent of the more familiar changes in Type I error rates and power attributable to unequal sample sizes combined with unequal variances.

The poor performance of the Wald method for constructing confidence intervals (CIs) for a binomial proportion has been demonstrated in a vast literature. The related problem of sample size determination needs to be updated and comparative studies are essential to understanding the performance of alternative methods. In this paper, the sample size is obtained for the Clopper–Pearson, Bayesian (Uniform and Jeffreys priors), Wilson, Agresti–Coull, Anscombe, and Wald methods. Two two-step procedures are used: one based on the expected length (EL) of the CI and another one on its first-order approximation. In the first step, all possible solutions that satisfy the optimal criterion are obtained. In the second step, a single solution is proposed according to a new criterion (e.g. highest coverage probability (CP)). In practice, it is expected a sample size reduction, therefore, we explore the behavior of the methods admitting 30% and 50% of losses. For all the methods, the ELs are inflated, as expected, but the...

Sampling cost is a crucial factor in sample size planning, particularly when the treatment group is more expensive than the control group. To either minimize the total cost or maximize the statistical power of the test, we used the distribution-free Wilcoxon–Mann–Whitney test for two independent samples and the van Elteren test for randomized block design, respectively. We then developed approximate sample size formulas when the distribution of data is abnormal and/or unknown. This study derived the optimal sample size allocation ratio for a given statistical power by considering the cost constraints, so that the resulting sample sizes could minimize either the total cost or the total sample size. Moreover, for a given total cost, the optimal sample size allocation is recommended to maximize the statistical power of the test. The proposed formula is not only innovative, but also quick and easy. We also applied real data from a clinical trial to illustrate how to choose the sample size for a randomized two...

Parametric methods for the calculation of reference intervals in clinical studies often rely on the identification of a suitable transformation so that the transformed data can be assumed to be drawn from a Gaussian distribution. In this paper, the two-stage transformation recommended by the International Federation for Clinical Chemistry is compared with a novel generalised Box–Cox family of transformations. Investigation is also made of sample sizes needed to achieve certain criteria of reliability in the calculated reference interval. Simulations are used to show that the generalised Box–Cox family achieves a lower bias than the two-stage transformation. It was found that there is a possibility that the two-stage transformation will result in percentile estimates that cannot be back-transformed to obtain the required reference intervals, a difficulty not observed when using the generalised Box–Cox family introduced in this paper.

In this article, we are interested in comparing growth curves for the Red Delicious apple in several locations to that of a reference site. Although such multiple comparisons are common for linear models, statistical techniques for nonlinear models are not prolific. We theoretically derive a test statistic, considering the issues of sample size and design points. Under equal sample sizes and same design points, our test statistic is based on the maximum of an equi-correlated multivariate chi-square distribution. Under unequal sample sizes and design points, we derive a general correlation structure, and then utilize the multivariate normal distribution to numerically compute critical points for the maximum of the multivariate chi-square. We apply this statistical technique to compare the growth of Red Delicious apples at six locations to a reference site in the state of Washington in 2009. Finally, we perform simulations to verify the performance of our proposed procedure for Type I error and marginal pow...

In this paper, a test statistic for testing upper outliers with a slippage alternative, in an exponential sample is proposed. No tables for critical values are required as they can be calculated easily for any sample size. A simulation study is also carried out to compare the performance of the test with the maximum likelihood ratio test and other existing tests.

DECORATE (Diverse Ensemble Creation by Oppositional Relabeling of Artificial Training Examples) is a classifier combination technique to construct a set of diverse base classifiers using additional artificially generated training instances. The predictions from the base classifiers are then integrated into one by the mean combination rule. In order to gain more insight about its effectiveness and advantages, this paper utilizes a large experiment to study the bias–variance analysis of DECORATE as well as some other widely used ensemble methods (such as bagging, AdaBoost, random forest) at different training sample sizes. The experimental results yield the following conclusions. For small training sets, DECORATE has a dominant advantage over its rivals and its success is attributed to the larger bias reduction achieved by it than the other algorithms. With increase in training data, AdaBoost benefits most and the bias reduced by it gradually turns to be significant while its variance reduction is also medi...

Principal axis factoring (PAF) and maximum likelihood factor analysis (MLFA) are two of the most popular estimation methods in exploratory factor analysis. It is known that PAF is better able to recover weak factors and that the maximum likelihood estimator is asymptotically efficient. However, there is almost no evidence regarding which method should be preferred for different types of factor patterns and sample sizes. Simulations were conducted to investigate factor recovery by PAF and MLFA for distortions of ideal simple structure and sample sizes between 25 and 5000. Results showed that PAF is preferred for population solutions with few indicators per factor and for overextraction. MLFA outperformed PAF in cases of unequal loadings within factors and for underextraction. It was further shown that PAF and MLFA do not always converge with increasing sample size. The simulation findings were confirmed by an empirical study as well as by a classic plasmode, Thurstone's box problem. The present results are...

Microarray studies are now common for human, agricultural plant and animal studies. False discovery rate (FDR) is widely used in the analysis of large-scale microarray data to account for problems associated with multiple testing. A well-designed microarray study should have adequate statistical power to detect the differentially expressed (DE) genes, while keeping the FDR acceptably low. In this paper, we used a mixture model of expression responses involving DE genes and non-DE genes to analyse theoretical FDR and power for simple scenarios where it is assumed that each gene has equal error variance and the gene effects are independent. A simulation study was used to evaluate the empirical FDR and power for more complex scenarios with unequal error variance and gene dependence. Based on this approach, we present a general guide for sample size requirement at the experimental design stage for prospective microarray studies. This paper presented an approach to explicitly connect the sample size with FDR a...

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In multi-category response models, categories are often ordered. In the case of ordinal response models, the usual likelihood approach becomes unstable with ill-conditioned predictor space or when the number of parameters to be estimated is large relative to the sample size. The likelihood estimates do not exist when the number of observations is less than the number of parameters. The same problem arises if constraint on the order of intercept values is not met during the iterative procedure. Proportional odds models (POMs) are most commonly used for ordinal responses. In this paper, penalized likelihood with quadratic penalty is used to address these issues with a special focus on POMs. To avoid large differences between two parameter values corresponding to the consecutive categories of an ordinal predictor, the differences between the parameters of two adjacent categories should be penalized. The considered penalized-likelihood function penalizes the parameter estimates or differences between the para...

Recent studies have shown that using variable sampling size and control limits (VSSC) schemes result in charts with more statistical power than variable sampling size (VSS) when detecting small to moderate shifts in the process mean vector. This paper presents an economic-statistical design (ESD) of the VSSC T2 control chart using the general model of Lorenzen and Vance [22]. The genetic algorithm approach is then employed to search for the optimal values of the six test parameters of the chart. We then compare the expected cost per unit of time of the optimally designed VSSC chart with optimally designed VSS and FRS (fixed ratio sampling) T2 charts as well as MEWMA charts.

This paper develops a method of estimating micro-level poverty in cases where data are scarce. The method is applied to estimate district-level poverty using the household level Indian national sample survey data for two states, viz., West Bengal and Madhya Pradesh. The method involves estimation of state-level poverty indices from the data formed by pooling data of all the districts (each time excluding one district) and multiplying this poverty vector with a known weight matrix to obtain the unknown district-level poverty vector. The proposed method is expected to yield reliable estimates at the district level, because the district-level estimate is now based on a much larger sample size obtained by pooling data of several districts. This method can be an alternative to the “small area estimation technique” for estimating poverty at sub-state levels in developing countries.

The semiparametric LABROC approach of fitting binormal model for estimating AUC as a global index of accuracy has been justified (except for bimodal forms), while for estimating a local index of accuracy such as TPF, it may lead to a bias in severe departure of data from binormality. We extended parametric ROC analysis for quantitative data when one or both pair members are mixture of Gaussian (MG) in particular for bimodal forms. We analytically showed that AUC and TPF are a mixture of weighting parameters of different components of AUCs and TPFs of a mixture of underlying distributions. In a simulation study of six configurations of MG distributions:{bimodal, normal} and {bimodal, bimodal} pairs, the parameters of MG distributions were estimated using the EM algorithm. The results showed that the estimated AUC from our proposed model was essentially unbiased, and that the bias in the estimated TPF at a clinically relevant range of FPF was roughly 0.01 for a sample size of n=100/100. In practice, with severedepartures from binormality, we recommend an extension of the LABROC and software development for future research to allow for each member of the pair of distributions to be a mixture of Gaussian that is a more flexible parametric form.

Variance estimators for probability sample-based predictions of species richness (S) are typically conditional on the sample (expected variance). In practical applications, sample sizes are typically small, and the variance of input parameters to a richness estimator should not be ignored. We propose a modified bootstrap variance estimator that attempts to capture the sampling variance by generating B replications of the richness prediction from stochastically resampled data of species incidence. The variance estimator is demonstrated for the observed richness (SO), five richness estimators, and with simulated cluster sampling (without replacement) in 11 finite populations of forest tree species. A key feature of the bootstrap procedure is a probabilistic augmentation of a species incidence matrix by the number of species expected to be ‘lost’ in a conventional bootstrap resampling scheme. In Monte-Carlo (MC) simulations, the modified bootstrap procedure performed well in terms of tracking the average MC ...

Three Bayesian methods are considered for the determination of sample sizes for sampling from the Laplace distribution – the distribution of time between rare events – with a normal prior. These methods are applied to the sizing of aircraft mid-air collisions in a navigation system or large flight path deviations of aircraft in air traffic management scenarios. A computer program handles all computations and gives a good insight about the best suggested method.