Point and Interval Estimation in the Combination of Bioassay Results with Unequal Variances

Symmetric parallel‐line biological assays involve the estimation of (log) relative potencies. The class of p(≥ 2) combination of symmetric parallel line bioassays are considered in this study. A large sample test for the equality of the several potencies is developed. An estimator and a confidence interval are proposed for the common relative potency parameter. The asymptotic distribution of the proposed test‐statistic under the null hypothesis as well as under a contagious hypothesis is derived.     

Combining multivariate bioassays

Linear multivariate theory is applied to the problem of combining several multivariate bioassays. Results are an asymptotic test of the hypothesis of a common log relative potency; the maximum likelihood estimator of the common log relative potency; and an exact and asymptotic confidence interval estimator for log relative potency.     

Shrinkage and Pretest Nonparametric Estimation of Regression Parameters from Censored Data with Multiple Observations at Each Level of Covariate

A simple linear regression model is considered where the independent variable assumes only a finite number of values and the response variable is randomly right censored. However, the censoring distribution may depend on the covariate values. A class of noniterative estimators for the slope parameter, namely, the noniterative unrestricted estimator, noniterative restricted estimator and noniterative improved pretest estimator are proposed. The asymptotic bias and mean squared errors of the proposed estimators are derived and compared. The relative dominance picture of the estimators is investigated. A simulation study is also performed to asses the properties of the various estimators for small samples.     

A Monte Carlo Study of Robustness of Pretest and Shrinkage Estimators in Pooling Coefficients of Variation

Pooling data, when justified, is advantageous for estimating the true parameter. In this paper the problem of estimating the coefficient of variation is considered when it is a priori suspected that two coefficients of variation are the same. Various estimators based on pretest and shrinkage rules are considered. A comparison through the Simulated Mean Squared Error (SMSE) criterion is carried out among various proposed estimators of the target coefficient of variation. The relative simulated efficiencies of the restricted, shrinkage restricted and shrinkage pretest estimators are studied. It is found that the proposed estimators are quite robust when the sample sizes are not too large. The result of Monte Carlo study indicates that the proposed shrinkage pretest estimator is efficient than the usual estimator in a wider range.     

Shrinkage Pre-Test Estimator of the Intercept Parameter for a Regression Model with Multivariate Student-t Errors

In the presence of an uncertain prior information about the value of the slope parameter, the estimation of the intercept parameter of a simple regression model with a multivariate Student-t error distribution is investigated. The unrestricted, restricted and shrinkage preliminary test maximum likelihood estimators are defined. The expressions for the bias and the mean square error of the three estimators are provided and the relative efficiences are analyzed. A maximin criterion is established, and graphs are constructed for an arbitrary number of degrees of freedom (D.F.) as well as sample sizes. A criterion to select optimal significance level is also discussed.     

On pooling across strata when frequency matching has been followed in a cohort study

In a study designed to assess the relationship between a dichotomous exposure and the eventual occurrence of a dichotomous outcome, frequency matching has been proposed as a way to balance the exposure cohorts with respect to the sampling distribution of potential confounding factors. This paper discusses the pooled estimator for the log relative risk, and provides an estimator for its variance which takes into account the dependency in the pooled outcomes induced by frequency matching. The pooled estimator has asymptotic relative efficiency less than but close to 1, relative to the usual, inverse variance weighted, stratified estimator. Simulations suggest, however, that the pooled estimator is likely to outperform the stratified estimator when samples are of moderate size. This estimator carries the added advantage that it consistently estimates a meaningful population parameter under heterogeneity of the relative risk across strata.     

Model‐Assisted Ranked Survey Sampling

The problem of Ranked Set Sampling [RSS] is analyzed and a ratio estimator is proposed. A simple linear regression superpopulation model is proposed as a counterpart to the design approach. A predictor is developed by using shrinkage techniques.     

A class of permutation tests for stratified survival data

We propose a class of permutation tests for stratified survival data. The tests are derived using the framework of Fay and Shih (1998, Journal of the American Statistical Association 93, 387-396), which creates tests by permuting scores based on a functional of estimated distribution functions. Here the estimated distribution function for each possibly right-, left-, or interval-censored observation is based on a shrinkage estimator similar to the nonparametric empirical estimator of Ghosh, Lahiri, and Tiwari (1989, Communications in Statistics--Theory and Methods 18, 121-146), and permutation is carried out within strata. The proposed test with a weighted Mann-Whitney functional is similar to the permutation form of the stratified log-rank test when there is a large strata effect or the sample size in each stratum is large and is similar to the permutation form of the ordinary log-rank test when there is little strata effect. Thus, the proposed test unifies the advantages of both the stratified and ordinary log-rank tests. By changing the functional, we may obtain a stratified Prentice-Wilcoxon test or a difference in means test with similar unifying properties. We show through simulations the advantage of the proposed test over existing tests for uncensored and right-censored data.     

Adaptive design: estimation and inference with censored data in a semiparametric model

In this article, we provide a method of estimation for the treatment effect in the adaptive design for censored survival data with or without adjusting for risk factors other than the treatment indicator. Within the semiparametric Cox proportional hazards model, we propose a bias-adjusted parameter estimator for the treatment coefficient and its asymptotic confidence interval at the end of the trial. The method for obtaining an asymptotic confidence interval and point estimator is based on a general distribution property of the final test statistic from the weighted linear rank statistics at the interims with or without considering the nuisance covariates. The computation of the estimates is straightforward. Extensive simulation studies show that the asymptotic confidence intervals have reasonable nominal probability of coverage, and the proposed point estimators are nearly unbiased with practical sample sizes.     

On the inadmissibility of Watterson's estimator

We consider the estimation of the scaled mutation parameter θ, which is one of the parameters of key interest in population genetics. We provide a general result showing when estimators of θ can be improved using shrinkage when taking the mean squared error as the measure of performance. As a consequence, we show that Watterson’s estimator is inadmissible, and propose an alternative shrinkage-based estimator that is easy to calculate and has a smaller mean squared error than Watterson’s estimator for all possible parameter values 0<θ<∞. This estimator is admissible in the class of all linear estimators. We then derive improved versions for other estimators of θ, including the MLE. We also investigate how an improvement can be obtained both when combining information from several independent loci and when explicitly taking into account recombination. A simulation study provides information about the amount of improvement achieved by our alternative estimators.     

Designs for slope ratio assays

Optimal designs are obtained for the estimation of relative potency in a slope ratio assay. Three criteria are considered based on the asymptotic variance of the standard point estimator. Efficiency functions are derived and numerical examples are given.     

Regression on Quantile Residual Life

Summary A time‐specific log‐linear regression method on quantile residual lifetime is proposed. Under the proposed regression model, any quantile of a time‐to‐event distribution among survivors beyond a certain time point is associated with selected covariates under right censoring. Consistency and asymptotic normality of the regression estimator are established. An asymptotic test statistic is proposed to evaluate the covariate effects on the quantile residual lifetimes at a specific time point. Evaluation of the test statistic does not require estimation of the variance–covariance matrix of the regression estimators, which involves the probability density function of the survival distribution with censoring. Simulation studies are performed to assess finite sample properties of the regression parameter estimator and test statistic. The new regression method is applied to a breast cancer data set with long‐term follow‐up to estimate the patients' median residual lifetimes, adjusting for important prognostic factors.     

Structural inference in transition measurement error models for longitudinal data

We propose a new class of models, transition measurement error models, to study the effects of covariates and the past responses on the current response in longitudinal studies when one of the covariates is measured with error. We show that the response variable conditional on the error-prone covariate follows a complex transition mixed effects model. The naive model obtained by ignoring the measurement error correctly specifies the transition part of the model, but misspecifies the covariate effect structure and ignores the random effects. We next study the asymptotic bias in naive estimator obtained by ignoring the measurement error for both continuous and discrete outcomes. We show that the naive estimator of the regression coefficient of the error-prone covariate is attenuated, while the naive estimators of the regression coefficients of the past responses are generally inflated. We then develop a structural modeling approach for parameter estimation using the maximum likelihood estimation method. In view of the multidimensional integration required by full maximum likelihood estimation, an EM algorithm is developed to calculate maximum likelihood estimators, in which Monte Carlo simulations are used to evaluate the conditional expectations in the E-step. We evaluate the performance of the proposed method through a simulation study and apply it to a longitudinal social support study for elderly women with heart disease. An additional simulation study shows that the Bayesian information criterion (BIC) performs well in choosing the correct transition orders of the models.     

Resampling methods to reduce the selection bias in genetic effect estimation in genome-wide scans

Using the simulated data of Problem 2 for Genetic Analysis Workshop 14 (GAW14), we investigated the ability of three bootstrap-based resampling estimators (a shrinkage, an out-of-sample, and a weighted estimator) to reduce the selection bias for genetic effect estimation in genome-wide linkage scans. For the given marker density in the preliminary genome scans (7 cM for microsatellite and 3 cM for SNP), we found that the two sets of markers produce comparable results in terms of power to detect linkage, localization accuracy, and magnitude of test statistic at the peak location. At the locations detected in the scan, application of the three bootstrap-based estimators substantially reduced the upward selection bias in genetic effect estimation for both true and false positives. The relative effectiveness of the estimators depended on the true genetic effect size and the inherent power to detect it. The shrinkage estimator is recommended when the power to detect the disease locus is low. Otherwise, the weighted estimator is recommended.     

On-line estimation of biodegradation in an unsaturated soil

The objective of this study was to develop a model-based estimator of biodegradation in unsaturated soil. This would allow real-time assessment of the efficiency of treatment bioprocesses, such as bioventilation and biopile, and eventually permit optimization through the implementation of control strategies. Based on a reduced-order model, an asymptotic observer was designed to estimate on-line the contaminant concentration, using carbon dioxide measurement. Two observer-based estimators were built to approximate: (1) the specific microbial growth rate; and (2) the biocontact kinetics representing the soil resistance to contaminant biodegradation. State observers and parameter estimators were confronted with the experimental results of biodegradation in microcosms. Hexadecane was used as the model compound, representing petroleum hydrocarbons. Three water contents, corresponding to 20%, 50% and 80% of the water-holding capacity, were tested. The asymptotic observer is able to predict hexadecane depletion with an error on the overall time trajectories of 13%, 8% and 4% for the dry, intermediate and wet soils, respectively, which is acceptable given that all the biokinetic parameters were identified from a biodegradation experiment in liquid phase. The observer-based estimator of the specific microbial growth rate, based on the CO₂ measurement, was successfully calibrated using the off-line measurements of hexadecane as validation data, and allowed estimation of the time when biodegradation switched from a microbial to a biocontact limitation. The biocontact kinetics was also identified on-line, using an estimator based on the hexadecane not in biocontact. These results are very encouraging with respect to the potential for on-line assessment of the performance of treatment bioprocesses in unsaturated soils.     

Improved statistical tests for differential gene expression by shrinking variance components estimates

Combining information across genes in the statistical analysis of microarray data is desirable because of the relatively small number of data points obtained for each individual gene. Here we develop an estimator of the error variance that can borrow information across genes using the James-Stein shrinkage concept. A new test statistic (FS) is constructed using this estimator. The new statistic is compared with other statistics used to test for differential expression: the gene-specific F test (F1), the pooled-variance F statistic (F3), a hybrid statistic (F2) that uses the average of the individual and pooled variances, the regularized t-statistic, the posterior odds statistic B, and the SAM t-test. The FS-test shows best or nearly best power for detecting differentially expressed genes over a wide range of simulated data in which the variance components associated with individual genes are either homogeneous or heterogeneous. Thus FS provides a powerful and robust approach to test differential expression of genes that utilizes information not available in individual gene testing approaches and does not suffer from biases of the pooled variance approach.     

Stabilization Coefficient of Random Variable

Coefficient of variation, standard deviation divided by mean, has some essential defects. Its density, expectation and variance are too complex to make the statistical inference for such a coefficient. The definition of stabilization coefficient is just the reciprocal of variation coefficient, mean divided by standard deviation. Such a coefficient has a simple expectation and a simple variance, and is an asymptotically unbiased estimator and a consistent estimator of its true value. Furthermore, coefficient of stabilization has an asymptotic normality. Due to its statistical advantages, coefficient of stabilization is easy to be tested statistically. In some applied fields, usually, there is an increasing standard deviation accompanying an increasing mean. Coefficient of stabilization can be practically used for some comparison studies in such fields. Illustrations about comparing microorganism strains are given in this paper. The robustness of stabilization coefficient is satisfactory.     

Sparse sufficient dimension reduction

Existing sufficient dimension reduction methods suffer fromthe fact that each dimension reduction component is a linearcombination of all the original predictors, so that it is difficultto interpret the resulting estimates. We propose a unified estimationstrategy, which combines a regression-type formulation of sufficientdimension reduction methods and shrinkage estimation, to producesparse and accurate solutions. The method can be applied tomost existing sufficient dimension reduction methods such assliced inverse regression, sliced average variance estimationand principal Hessian directions. We demonstrate the effectivenessof the proposed method by both simulations and real data analysis.     

Bayesian shrinkage estimation of the relative abundance of mRNA transcripts using SAGE

Serial analysis of gene expression (SAGE) is a technology for quantifying gene expression in biological tissue that yields count data that can be modeled by a multinomial distribution with two characteristics: skewness in the relative frequencies and small sample size relative to the dimension. As a result of these characteristics, a given SAGE sample may fail to capture a large number of expressed mRNA species present in the tissue. Empirical estimators of mRNA species' relative abundance effectively ignore these missing species, and as a result tend to overestimate the abundance of the scarce observed species comprising a vast majority of the total. We have developed a new Bayesian estimation procedure that quantifies our prior information about these characteristics, yielding a nonlinear shrinkage estimator with efficiency advantages over the MLE. Our prior is mixture of Dirichlets, whereby species are stochastically partitioned into abundant and scarce classes, each with its own multivariate prior. Simulation studies reveal our estimator has lower integrated mean squared error (IMSE) than the MLE for the SAGE scenarios simulated, and yields relative abundance profiles closer in Euclidean distance to the truth for all samples simulated. We apply our method to a SAGE library of normal colon tissue, and discuss its implications for assessing differential expression.     

Interval Estimation of Simple Difference under Independent Negative Binomial Sampling

When the sample size is not large or when the underlying disease is rare, to assure collection of an appropriate number of cases and to control the relative error of estimation, one may employ inverse sampling, in which one continues sampling subjects until one obtains exactly the desired number of cases. This paper focuses discussion on interval estimation of the simple difference between two proportions under independent inverse sampling. This paper develops three asymptotic interval estimators on the basis of the maximum likelihood estimator (MLE), the uniformly minimum variance unbiased estimator (UMVUE), and the asymptotic likelihood ratio test (ALRT). To compare the performance of these three estimators, this paper calculates the coverage probability and the expected length of the resulting confidence intervals on the basis of the exact distribution. This paper finds that when the underlying proportions of cases in both two comparison populations are small or moderate (≤0.20), all three asymptotic interval estimators developed here perform reasonably well even for the pre-determined number of cases as small as 5. When the pre-determined number of cases is moderate or large (≥50), all three estimators are essentially equivalent in all the situations considered here. Because application of the two interval estimators derived from the MLE and the UMVUE does not involve any numerical iterative procedure needed in the ALRT, for simplicity we may use these two estimators without losing efficiency.