Two Estimators of Mean Frequency in Products of Multinormal Probability Models

Two estimators, one additive the other multiplicative, are considered for mean frequencies in a complete three-way table. Using the mean square error criterion it is shown that preference for the additive estimator can be as high 7/8 in tables with row-column independence and in homogeneous tables. Extension to other estimators are discussed.     

The dilemma of negative analysis of variance estimators of intraclass correlation

Summary At least two common practices exist when a negative variance component estimate is obtained, either setting it to zero or not reporting the estimate. The consequences of these practices are investigated in the context of the intraclass correlation estimation in terms of bias, variance and mean squared error (MSE). For the one-way analysis of variance random effects model and its extension to the common correlation model, we compare five estimators: analysis of variance (ANOVA), concentrated ANOVA, truncated ANOVA and two maximum likelihood-like (ML) estimators. For the balanced case, the exact bias and MSE are calculated via numerical integration of the exact sample distributions, while a Monte Carlo simulation study is conducted for the unbalanced case. The results indicate that the ANOVA estimator performs well except for designs with family size n = 2. The two ML estimators are generally poor, and the concentrated and truncated ANOVA estimators have some advantages over the ANOVA in terms of MSE. However, the large biases may make the concentrated and truncated ANOVA estimators objectionable when intraclass correlation () is small. Bias should be a concern when a pooled estimate is obtained from the literature since <0.05 in many genetic studies.     

Evaluation of algorithms for ratio imaging in fluorescence microscopy

Ratio imaging in fluorescence microscopy is used in measuring parameters such as pH, pCa, cytoplasmic porosity, and the relative concentration of fluorescent analogs within single cells. The fastest method for ratio imaging is to use lookup tables on special-purpose image processors. Since lookup tables store integers in integer addresses, using a lookup table will generate rounding errors. The magnitude of the error will depend on the transformation performed and on the number of levels used in the lookup table. We examined ratio imaging by lookup table and computed the errors generated by both inversion and log subtraction methods. Both uniformly fluorescing fields and fluorescing cell images were employed to provide data for use in confirming our calculations and illustrating both the magnitude and spatial incidence of errors. It is shown that, through proper design of lookup tables, a significant reduction can be made in the errors generated in comparison with common methods available in most image processors.     

On optimal Bayesian classification and risk estimation under multiple classes

A recently proposed optimal Bayesian classification paradigm addresses optimal error rate analysis for small-sample discrimination, including optimal classifiers, optimal error estimators, and error estimation analysis tools with respect to the probability of misclassification under binary classes. Here, we address multi-class problems and optimal expected risk with respect to a given risk function, which are common settings in bioinformatics. We present Bayesian risk estimators (BRE) under arbitrary classifiers, the mean-square error (MSE) of arbitrary risk estimators under arbitrary classifiers, and optimal Bayesian risk classifiers (OBRC). We provide analytic expressions for these tools under several discrete and Gaussian models and present a new methodology to approximate the BRE and MSE when analytic expressions are not available. Of particular note, we present analytic forms for the MSE under Gaussian models with homoscedastic covariances, which are new even in binary classification.     

Estimation of Heritability in Heteroscedastic One‐Way Unbalanced Random Model

A modified estimator of heritability is proposed under heteroscedastic one way unbalanced random model. The distribution, moments and probability of permissible values (PPV) for conventional and modified estimators are derived. The behaviour of two estimators has been investigated, numerically, to devise a suitable estimator of heritability under variance heterogeneity. The numerical results reveal that under balanced case the heteroscedasticity affects the bias, MSE and PPV of conventional estimator, marginally. In case of unbalanced situations, the conventional estimator underestimates the parameter when more variable group has more observations and overestimates when more variable group has less observations, MSE of the conventional estimator decreases when more variable group has more observations and increases when more variable group has less observations and PPV is marginally decreased. The MSE and PPV are comparable for two estimators while the bias of modified estimator is less than the conventional estimator particularly for small and medium values of the parameter. These results suggest the use of modified estimator with equal or more observations for more variable group in presence of variance heterogeneity.     

Penalized partial likelihood regression for right-censored data with bootstrap selection of the penalty parameter

The Cox proportional hazards model is often used for estimating the association between covariates and a potentially censored failure time, and the corresponding partial likelihood estimators are used for the estimation and prediction of relative risk of failure. However, partial likelihood estimators are unstable and have large variance when collinearity exists among the explanatory variables or when the number of failures is not much greater than the number of covariates of interest. A penalized (log) partial likelihood is proposed to give more accurate relative risk estimators. We show that asymptotically there always exists a penalty parameter for the penalized partial likelihood that reduces mean squared estimation error for log relative risk, and we propose a resampling method to choose the penalty parameter. Simulations and an example show that the bootstrap-selected penalized partial likelihood estimators can, in some instances, have smaller bias than the partial likelihood estimators and have smaller mean squared estimation and prediction errors of log relative risk. These methods are illustrated with a data set in multiple myeloma from the Eastern Cooperative Oncology Group.     

Estimators and tests in the analysis of multiple nonindependent 2 x 2 tables with partially missing observations

We present methods for the analysis of a K-variate binary measure for two independent groups where some observations may be incomplete, as in the case of K repeated measures in a comparative trial. For the K 2 X 2 tables, let theta = (theta 1,..., theta K) be a vector of association parameters where theta k is a measure of association that is a continuous function of the probabilities pi ik in each group (i = 1, 2; k = 1,..., K), such as the log odds ratio or log relative risk. The asymptotic distribution of the estimates theta = (theta 1,..., theta K) is derived. Under the assumption that theta k = theta for all k, we describe the maximally efficient linear estimator theta of the common parameter theta. Tests of contrasts on the theta are presented which provide a test of homogeneity Ha: theta k = theta l for all k not equal to l. We then present maximally efficient tests of aggregate association Hb: theta = theta 0, where theta 0 is a given value. It is shown that the test of aggregate association Hb is asymptotically independent of the preliminary test of homogeneity Ha. These methods generalize the efficient estimators of Gart (1962, Biometrics 18, 601-610), and the Cochran (1954, Biometrics 10, 417-451), Mantel-Haenszel (1959, Journal of the National Cancer Institute 22, 719-748), and Radhakrishna (1965, Biometrics 21, 86-98) tests to nonindependent tables. The methods are illustrated with an analysis of repeated morphologic evaluations of liver biopsies obtained in the National Cooperative Gallstone Study.     

Performance of the Peto odds ratio compared to the usual odds ratio estimator in the case of rare events

For the calculation of relative measures such as risk ratio (RR) and odds ratio (OR) in a single study, additional approaches are required for the case of zero events. In the case of zero events in one treatment arm, the Peto odds ratio (POR) can be calculated without continuity correction, and is currently the relative effect estimation method of choice for binary data with rare events. The aim of this simulation study is a variegated comparison of the estimated OR and estimated POR with the true OR in a single study with two parallel groups without confounders in data situations where the POR is currently recommended. This comparison was performed by means of several performance measures, that is the coverage, confidence interval (CI) width, mean squared error (MSE), and mean percentage error (MPE). We demonstrated that the estimator for the POR does not outperform the estimator for the OR for all the performance measures investigated. In the case of rare events, small treatment effects and similar group sizes, we demonstrated that the estimator for the POR performed better than the estimator for the OR only regarding the coverage and MPE, but not the CI width and MSE. For larger effects and unbalanced group size ratios, the coverage and MPE of the estimator for the POR were inappropriate. As in practice the true effect is unknown, the POR method should be applied only with the utmost caution.     

Comparison of estimators for measures of linkage disequilibrium

The measurement of biallelic pair-wise association called linkage disequilibrium (LD) is an important issue in order to understand the genomic architecture. A plethora of measures of association in two by two tables have been proposed in the literature. Beside the problem of choosing an appropriate measure, the problem of their estimation has been neglected in the literature. It needs to be emphasized that the definition of a measure and the choice of an estimator function for it are conceptually unrelated tasks. In this paper, we compare the performance of various estimators for the three popular LD measures D', r and Y in a simulation study for small to moderate samples sizes (N<=500). The usual frequency-plug-in estimators can lead to unreliable or undefined estimates. Estimators based on the computationally expensive volume measures have been proposed recently as a remedy to this well-known problem. We confirm that volume estimators have better expected mean square error than the naive plug-in estimators. But they are outperformed by estimators plugging-in easy to calculate non-informative Bayesian probability estimates into the theoretical formulae for the measures. Fully Bayesian estimators with non-informative Dirichlet priors have comparable accuracy but are computationally more expensive. We recommend the use of non-informative Bayesian plug-in estimators based on Jeffreys' prior, in particular when dealing with SNP array data where the occurrence of small table entries and table margins is likely.     

Asymptotic Relative Efficiency of Some Jackknife Estimators of a Common Odds Ratio

In the situation of several 2 × 2 tables the asymptotic relative efficiencies of certain jackknife estimators of a common odds ratio are investigated in the case that the number of tables is fixed while the sample sizes within each table tend to infinity. The estimators show very good results over a wide range of parameters. Some situations in which the estimators have low asymptotic relative efficiency are pointed out:.     

Maximum-likelihood estimation of demographic parameters using the frequency spectrum of unlinked single-nucleotide polymorphisms

A maximum-likelihood method for demographic inference is applied to data sets consisting of the frequency spectrum of unlinked single-nucleotide polymorphisms (SNPs). We use simulation analyses to explore the effect of sample size and number of polymorphic sites on both the power to reject the null hypothesis of constant population size and the properties of two- and three-dimensional maximum-likelihood estimators (MLEs). Large amounts of data are required to produce accurate demographic inferences, particularly for scenarios of recent growth. Properties of the MLEs are highly dependent upon the demographic scenario, as estimates improve with a more ancient time of growth onset and smaller degree of growth. Severe episodes of growth lead to an upward bias in the estimates of the current population size, and that bias increases with the magnitude of growth. One data set of African origin supports a model of mild, ancient growth, and another is compatible with both constant population size and a variety of growth scenarios, rejecting greater than fivefold growth beginning >36,000 years ago. Analysis of a data set of European origin indicates a bottlenecked population history, with an 85% population reduction occurring approximately 30,000 years ago.     

Generalized Mantel-Haenszel estimators for K 2 x J tables

Mickey and Elashoff (1985, Biometrics 41, 623-635) gave an extension of Mantel-Haenszel estimation to log-linear models for 2 x J x K tables. Their extension yields two generalizations of the Mantel-Haenszel odds ratio estimator to K 2 x J tables. This paper provides variance and covariance estimators for these generalized Mantel-Haenszel estimators that are dually consistent (i.e., consistent in both large strata and sparse data), and presents comparisons of the efficiency of the generalized Mantel-Haenszel estimators.     

The Chain Ratio Estimator and Regression Estimator with Linear Combination of Two Auxiliary Variables

In sample surveys, it is usual to make use of auxiliary information to increase the precision of the estimators. We propose a new chain ratio estimator and regression estimator of a finite population mean using linear combination of two auxiliary variables and obtain the mean squared error (MSE) equations for the proposed estimators. We find theoretical conditions that make proposed estimators more efficient than the traditional multivariate ratio estimator and the regression estimator using information of two auxiliary variables.     

On the mean square error associated with adaptive generalized ridge regression

Some numerical results are presented for generalized ridge regression where the additive constants are based on the data. The adaptive estimator so obtained is compared with the least-squares estimator on the basis of mean square error (MSE). It is shown that the MSE of each component of the vector of ridge estimators may be as low as 47.1% of the variance of the corresponding component of the least squares vector or as high as 125.2%.     

The ties problem resulting from counting-based error estimators and its impact on gene selection algorithms

MOTIVATION: Feature selection approaches, such as filter and wrapper, have been applied to address the gene selection problem in the literature of microarray data analysis. In wrapper methods, the classification error is usually used as the evaluation criterion of feature subsets. Due to the nature of high dimensionality and small sample size of microarray data, however, counting-based error estimation may not necessarily be an ideal criterion for gene selection problem. RESULTS: Our study reveals that evaluating genes in terms of counting-based error estimators such as resubstitution error, leave-one-out error, cross-validation error and bootstrap error may encounter severe ties problem, i.e. two or more gene subsets score equally, and this in turn results in uncertainty in gene selection. Our analysis finds that the ties problem is caused by the discrete nature of counting-based error estimators and could be avoided by using continuous evaluation criteria instead. Experiment results show that continuous evaluation criteria such as generalised the absolute value of w2 measure for support vector machines and modified Relief's measure for k-nearest neighbors produce improved gene selection compared with counting-based error estimators. AVAILABILITY: The companion website is at http://www.ntu.edu.sg/home5/pg02776030/wrappers/ The website contains (1) the source code of all the gene selection algorithms and (2) the complete set of tables and figures of experiments.     

Least Squares and Minimum MSE Estimators of Variance Components in Mixed Linear Models

The paper deals with the quadratic invariant estimators of the linear functions of variance components in mixed linear model. The estimator with locally minimal mean square error with respect to a parameter ? is derived. Under the condition of normality of the vector Y the theoretical values of MSE of several types of estimators are compared in two different mixed models; under a different types of distributions a simulation study is carried out for the behaviour of derived estimators.     

Statistical analysis of binary data generated from multilocus dominant DNA markers

The use of methodologies such as RAPD and AFLP for studying genetic variation in natural populations is widespread in the ecology community. Because data generated using these methods exhibit dominance, their statistical treatment is less straightforward. Several estimators have been proposed for estimating population genetic parameters, assuming simple random sampling and the Hardy-Weinberg (HW) law. The merits of these estimators remain unclear because no comparative studies of their theoretical properties have been carried out. Furthermore, ascertainment bias has not been explicitly modelled. Here, we present a comparison of a set of candidate estimators of null allele frequency (q), locus-specific heterozygosity (h) and average heterozygosity () in terms of their bias, standard error, and root mean square error (RMSE). For estimating q and h, we show that none of the estimators considered has the least RMSE over the parameter space. Our proposed zero-correction procedure, however, generally leads to estimators with improved RMSE. Assuming a beta model for the distribution of null homozygote proportions, we show how correction for ascertainment bias can be carried out using a linear transform of the sample average of h and the truncated beta-binomial likelihood. Simulation results indicate that the maximum likelihood and empirical Bayes estimator of have negligible bias and similar RMSE. Ascertainment bias in estimators of is most pronounced when the beta distribution is J-shaped and negligible when the latter is inverse J-shaped. The validity of the current findings depends importantly on the HW assumption-a point that we illustrate using data from two published studies.     

Bias-reduced estimators and confidence intervals for odds ratios in genome-wide association studies

Genome-wide association studies (GWAS) provide an important approach to identifying common genetic variants that predispose to human disease. A typical GWAS may genotype hundreds of thousands of single nucleotide polymorphisms (SNPs) located throughout the human genome in a set of cases and controls. Logistic regression is often used to test for association between a SNP genotype and case versus control status, with corresponding odds ratios (ORs) typically reported only for those SNPs meeting selection criteria. However, when these estimates are based on the original data used to detect the variant, the results are affected by a selection bias sometimes referred to the "winner's curse" (Capen and others, 1971). The actual genetic association is typically overestimated. We show that such selection bias may be severe in the sense that the conditional expectation of the standard OR estimator may be quite far away from the underlying parameter. Also standard confidence intervals (CIs) may have far from the desired coverage rate for the selected ORs. We propose and evaluate 3 bias-reduced estimators, and also corresponding weighted estimators that combine corrected and uncorrected estimators, to reduce selection bias. Their corresponding CIs are also proposed. We study the performance of these estimators using simulated data sets and show that they reduce the bias and give CI coverage close to the desired level under various scenarios, even for associations having only small statistical power.     

A restricted maximum likelihood estimator for truncated height samples

A restricted maximum likelihood (ML) estimator is presented and evaluated for use with truncated height samples. In the common situation of a small sample truncated at a point not far below the mean, the ordinary ML estimator suffers from high sampling variability. The restricted estimator imposes an a priori value on the standard deviation and freely estimates the mean, exploiting the known empirical stability of the former to obtain less variable estimates of the latter. Simulation results validate the conjecture that restricted ML behaves like restricted ordinary least squares (OLS), whose properties are well established on theoretical grounds. Both estimators display smaller sampling variability when constrained, whether the restrictions are correct or not. The bias induced by incorrect restrictions sets up a decision problem involving a bias-precision tradeoff, which can be evaluated using the mean squared error (MSE) criterion. Simulated MSEs suggest that restricted ML estimation offers important advantages when samples are small and truncation points are high, so long as the true standard deviation is within roughly 0.5 cm of the chosen value.