Shrinkage Estimators for Covariance Matrices

Estimation of covariance matrices in small samples has been studied by many authors. Standard estimators, like the unstructured maximum likelihood estimator (ML) or restricted maximum likelihood (REML) estimator, can be very unstable with the smallest estimated eigenvalues being too small and the largest too big. A standard approach to more stably estimating the matrix in small samples is to compute the ML or REML estimator under some simple structure that involves estimation of fewer parameters, such as compound symmetry or independence. However, these estimators will not be consistent unless the hypothesized structure is correct. If interest focuses on estimation of regression coefficients with correlated (or longitudinal) data, a sandwich estimator of the covariance matrix may be used to provide standard errors for the estimated coefficients that are robust in the sense that they remain consistent under misspecification of the covariance structure. With large matrices, however, the inefficiency of the sandwich estimator becomes worrisome. We consider here two general shrinkage approaches to estimating the covariance matrix and regression coefficients. The first involves shrinking the eigenvalues of the unstructured ML or REML estimator. The second involves shrinking an unstructured estimator toward a structured estimator. For both cases, the data determine the amount of shrinkage. These estimators are consistent and give consistent and asymptotically efficient estimates for regression coefficients. Simulations show the improved operating characteristics of the shrinkage estimators of the covariance matrix and the regression coefficients in finite samples. The final estimator chosen includes a combination of both shrinkage approaches, i.e., shrinking the eigenvalues and then shrinking toward structure. We illustrate our approach on a sleep EEG study that requires estimation of a 24 x 24 covariance matrix and for which inferences on mean parameters critically depend on the covariance estimator chosen. We recommend making inference using a particular shrinkage estimator that provides a reasonable compromise between structured and unstructured estimators.     

MAXIMUM-LIKELIHOOD APPROACHES APPLIED TO QUANTITATIVE GENETICS OF NATURAL POPULATIONS

Growing interest in adaptive evolution in natural populations has spurred efforts to infer genetic components of variance and covariance of quantitative characters. Here, I review difficulties inherent in the usual least-squares methods of estimation. A useful alternative approach is that of maximum likelihood (ML). Its particular advantage over least squares is that estimation and testing procedures are well defined, regardless of the design of the data. A modified version of ML, REML, eliminates the bias of ML estimates of variance components. Expressions for the expected bias and variance of estimates obtained from balanced, fully hierarchical designs are presented for ML and REML. Analyses of data simulated from balanced, hierarchical designs reveal differences in the properties of ML, REML, and F-ratio tests of significance. A second simulation study compares properties of REML estimates obtained from a balanced, fully hierarchical design (within-generation analysis) with those from a sampling design including phenotypic data on parents and multiple progeny. It also illustrates the effects of imposing nonnegativity constraints on the estimates. Finally, it reveals that predictions of the behavior of significance tests based on asymptotic theory are not accurate when sample size is small and that constraining the estimates seriously affects properties of the tests. Because of their great flexibility, likelihood methods can serve as a useful tool for estimation of quantitative-genetic parameters in natural populations. Difficulties involved in hypothesis testing remain to be solved.     

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.     

Methods for estimating the parameters of a linear model for ordered categorical data.

In many empirical analyses, the response of interest is categorical with an ordinal scale attached. Many investigators prefer to formulate a linear model, assigning scores to each category of the ordinal response and treating it as continuous. When the covariates are categorical, Haber (1985, Computational Statistics and Data Analysis 3, 1-10) has developed a method to obtain maximum likelihood (ML) estimates of the parameters of the linear model using Lagrange multipliers. However, when the covariates are continuous, the only method we found in the literature is ordinary least squares (OLS), performed under the assumption of homogeneous variance. The OLS estimates are unbiased and consistent but, since variance homogeneity is violated, the OLS estimates of variance can be biased and may not be consistent. We discuss a variance estimate (White, 1980, Econometrica 48, 817-838) that is consistent for the true variance of the OLS parameter estimates. The possible bias encountered by using the naive OLS variance estimate is discussed. An estimated generalized least squares (EGLS) estimator is proposed and its efficiency relative to OLS is discussed. Finally, an empirical comparison of OLS, EGLS, and ML estimators is made.     

Nonlinear mixed effects models for repeated measures data

We propose a general, nonlinear mixed effects model for repeated measures data and define estimators for its parameters. The proposed estimators are a natural combination of least squares estimators for nonlinear fixed effects models and maximum likelihood (or restricted maximum likelihood) estimators for linear mixed effects models. We implement Newton-Raphson estimation using previously developed computational methods for nonlinear fixed effects models and for linear mixed effects models. Two examples are presented and the connections between this work and recent work on generalized linear mixed effects models are discussed.     

Complexity of the simplest phylogenetic estimation problem

The maximum-likelihood (ML) solution to a simple phylogenetic estimation problem is obtained analytically The problem is estimation of the rooted tree for three species using binary characters with a symmetrical rate of substitution under the molecular clock. ML estimates of branch lengths and log-likelihood scores are obtained analytically for each of the three rooted binary trees. Estimation of the tree topology is equivalent to partitioning the sample space (space of possible data outcomes) into subspaces, within each of which one of the three binary trees is the ML tree. Distance-based least squares and parsimony-like methods produce essentially the same estimate of the tree topology, although differences exist among methods even under this simple model. This seems to be the simplest case, but has many of the conceptual and statistical complexities involved in phylogeny estimation. The solution to this real phylogeny estimation problem will be useful for studying the problem of significance evaluation.     

Estimation of a common effect parameter from sparse follow-up data

Breslow (1981, Biometrika 68, 73-84) has shown that the Mantel-Haenszel odds ratio is a consistent estimator of a common odds ratio in sparse stratifications. For cohort studies, however, estimation of a common risk ratio or risk difference can be of greater interest. Under a binomial sparse-data model, the Mantel-Haenszel risk ratio and risk difference estimators are consistent in sparse stratifications, while the maximum likelihood and weighted least squares estimators are biased. Under Poisson sparse-data models, the Mantel-Haenszel and maximum likelihood rate ratio estimators have equal asymptotic variances under the null hypothesis and are consistent, while the weighted least squares estimators are again biased; similarly, of the common rate difference estimators the weighted least squares estimators are biased, while the estimator employing "Mantel-Haenszel" weights is consistent in sparse data. Variance estimators that are consistent in both sparse data and large strata can be derived for all the Mantel-Haenszel estimators.     

Estimation of evolutionary parameters with phylogenetic trees

An important issue in the phylogenetic analysis of nucleotide sequence data using the maximum likelihood (ML) method is the underlying evolutionary model employed. We consider the problem of simultaneously estimating the tree topology and the parameters in the underlying substitution model and of obtaining estimates of the standard errors of these parameter estimates. Given a fixed tree topology and corresponding set of branch lengths, the ML estimates of standard evolutionary model parameters are asymptotically efficient, in the sense that their joint distribution is asymptotically normal with the variance–covariance matrix given by the inverse of the Fisher information matrix. We propose a new estimate of this conditional variance based on estimation of the expected information using a Monte Carlo sampling (MCS) method. Simulations are used to compare this conditional variance estimate to the standard technique of using the observed information under a variety of experimental conditions. In the case in which one wishes to estimate simultaneously the tree and parameters, we provide a bootstrapping approach that can be used in conjunction with the MCS method to estimate the unconditional standard error. The methods developed are applied to a real data set consisting of 30 papillomavirus sequences. This overall method is easily incorporated into standard bootstrapping procedures to allow for proper variance estimation.     

Sampling distributions, biases, variances, and confidence intervals for genetic correlations

Genetic correlations are frequently estimated from natural and experimental populations, yet many of the statistical properties of estimators of are not known, and accurate methods have not been described for estimating the precision of estimates of Our objective was to assess the statistical properties of multivariate analysis of variance (MANOVA), restricted maximum likelihood (REML), and maximum likelihood (ML) estimators of by simulating bivariate normal samples for the one-way balanced linear model. We estimated probabilities of non-positive definite MANOVA estimates of genetic variance-covariance matrices and biases and variances of MANOVA, REML, and ML estimators of and assessed the accuracy of parametric, jackknife, and bootstrap variance and confidence interval estimators for MANOVA estimates of were normally distributed. REML and ML estimates were normally distributed for but skewed for and 0.9. All of the estimators were biased. The MANOVA estimator was less biased than REML and ML estimators when heritability (H), the number of genotypes (n), and the number of replications (r) were low. The biases were otherwise nearly equal for different estimators and could not be reduced by jackknifing or bootstrapping. The variance of the MANOVA estimator was greater than the variance of the REML or ML estimator for most H, n, and r. Bootstrapping produced estimates of the variance of close to the known variance, especially for REML and ML. The observed coverages of the REML and ML bootstrap interval estimators were consistently close to stated coverages, whereas the observed coverage of the MANOVA bootstrap interval estimator was unsatisfactory for some H, n, and r. The other interval estimators produced unsatisfactory coverages. REML and ML bootstrap interval estimates were narrower than MANOVA bootstrap interval estimates for most H, and r. Received: 6 July 1995 / Accepted: 8 March 1996     

Weighted regression analysis to correct for informative monitoring times and confounders in longitudinal studies

We address estimation of the marginal effect of a time‐varying binary treatment on a continuous longitudinal outcome in the context of observational studies using electronic health records, when the relationship of interest is confounded, mediated, and further distorted by an informative visit process. We allow the longitudinal outcome to be recorded only sporadically and assume that its monitoring timing is informed by patients' characteristics. We propose two novel estimators based on linear models for the mean outcome that incorporate an adjustment for confounding and informative monitoring process through generalized inverse probability of treatment weights and a proportional intensity model, respectively. We allow for a flexible modeling of the intercept function as a function of time. Our estimators have closed‐form solutions, and their asymptotic distributions can be derived. Extensive simulation studies show that both estimators outperform standard methods such as the ordinary least squares estimator or estimators that only account for informative monitoring or confounders. We illustrate our methods using data from the Add Health study, assessing the effect of depressive mood on weight in adolescents.     

Estimation from Censored Medical Cost Data

This paper applies the inverse probability weighted least‐squares method to predict total medical cost in the presence of censored data. Since survival time and medical costs may be subject to right censoring and therefore are not always observable, the ordinary least‐squares approach cannot be used to assess the effects of explanatory variables. We demonstrate how inverse probability weighted least‐squares estimation provides consistent asymptotic normal coefficients with easily computable standard errors. In addition, to assess the effect of censoring on coefficients, we develop a test comparing ordinary least‐squares and inverse probability weighted least‐squares estimators. We demonstrate the methods developed by applying them to the estimation of cancer costs using Medicare claims data. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Empirical Bayes estimation of gene-specific effects in micro-array research

Micro-array technology allows investigators the opportunity to measure expression levels of thousands of genes simultaneously. However, investigators are also faced with the challenge of simultaneous estimation of gene expression differences for thousands of genes with very small sample sizes. Traditional estimators of differences between treatment means (ordinary least squares estimators or OLS) are not the best estimators if interest is in estimation of gene expression differences for an ensemble of genes. In the case that gene expression differences are regarded as exchangeable samples from a common population, estimators are available that result in much smaller average mean-square error across the population of gene expression difference estimates. We have simulated the application of such an estimator, namely an empirical Bayes (EB) estimator of random effects in a hierarchical linear model (normal-normal). Simulation results revealed mean-square error as low as 0.05 times the mean-square error of OLS estimators (i.e., the difference between treatment means). We applied the analysis to an example dataset as a demonstration of the shrinkage of EB estimators and of the reduction in mean-square error, i.e., increase in precision, associated with EB estimators in this analysis. The method described here is available in software that is available at .     

Species richness estimation of bird communities: how to control for sampling effort?

Since estimates of total species richness increase with sampling effort, methods to control for this sampling effect need to be tested and used. We present seven non-parametric and 12 accumulation curve methods that have been used recently in the ecological literature. To test their performance, we used data from bird communities in the Queen Charlotte Islands, Canada. The performance of each method was evaluated by calculating the bias and precision of its estimates against the known total species richness. For our data set, the two Chao estimators were the overall least biased and most precise estimation methods, followed by the two jackknife estimators, thus supporting results of previous studies. Nonparametric estimators tended to perform better than accumulation curve models. Most estimation methods had the problem that they tended to underestimate species richness for early samples, but slightly overestimated it for late samples. We briefly discuss the practical use of these methods which may greatly increase our ability to answer ecological questions and to guide conservation decisions, especially for species-rich tropical bird communities.     

Semiparametric additive time-varying coefficients model for longitudinal data with censored time origin

Statistical analysis of longitudinal data often involves modeling treatment effects on clinically relevant longitudinal biomarkers since an initial event (the time origin). In some studies including preventive HIV vaccine efficacy trials, some participants have biomarkers measured starting at the time origin, whereas others have biomarkers measured starting later with the time origin unknown. The semiparametric additive time-varying coefficient model is investigated where the effects of some covariates vary nonparametrically with time while the effects of others remain constant. Weighted profile least squares estimators coupled with kernel smoothing are developed. The method uses the expectation maximization approach to deal with the censored time origin. The Kaplan–Meier estimator and other failure time regression models such as the Cox model can be utilized to estimate the distribution and the conditional distribution of left censored event time related to the censored time origin. Asymptotic properties of the parametric and nonparametric estimators and consistent asymptotic variance estimators are derived. A two-stage estimation procedure for choosing weight is proposed to improve estimation efficiency. Numerical simulations are conducted to examine finite sample properties of the proposed estimators. The simulation results show that the theory and methods work well. The efficiency gain of the two-stage estimation procedure depends on the distribution of the longitudinal error processes. The method is applied to analyze data from the Merck 023/HVTN 502 Step HIV vaccine study.     

Monotone spline‐based least squares estimation for panel count data with informative observation times

This article discusses the statistical analysis of panel count data when the underlying recurrent event process and observation process may be correlated. For the recurrent event process, we propose a new class of semiparametric mean models that allows for the interaction between the observation history and covariates. For inference on the model parameters, a monotone spline‐based least squares estimation approach is developed, and the resulting estimators are consistent and asymptotically normal. In particular, our new approach does not rely on the model specification of the observation process. The proposed inference procedure performs well through simulation studies, and it is illustrated by the analysis of bladder tumor data.     

Varying coefficient model with unknown within-subject covariance for analysis of tumor growth curves

SUMMARY: In this article we develop a nonparametric estimation procedure for the varying coefficient model when the within-subject covariance is unknown. Extending the idea of iterative reweighted least squares to the functional setting, we iterate between estimating the coefficients conditional on the covariance and estimating the functional covariance conditional on the coefficients. Smoothing splines for correlated errors are used to estimate the functional coefficients with smoothing parameters selected via the generalized maximum likelihood. The covariance is nonparametrically estimated using a penalized estimator with smoothing parameters chosen via a Kullback-Leibler criterion. Empirical properties of the proposed method are demonstrated in simulations and the method is applied to the data collected from an ovarian tumor study in mice to analyze the effects of different chemotherapy treatments on the volumes of two classes of tumors.     

LSimpute: accurate estimation of missing values in microarray data with least squares methods

Microarray experiments generate data sets with information on the expression levels of thousands of genes in a set of biological samples. Unfortunately, such experiments often produce multiple missing expression values, normally due to various experimental problems. As many algorithms for gene expression analysis require a complete data matrix as input, the missing values have to be estimated in order to analyze the available data. Alternatively, genes and arrays can be removed until no missing values remain. However, for genes or arrays with only a small number of missing values, it is desirable to impute those values. For the subsequent analysis to be as informative as possible, it is essential that the estimates for the missing gene expression values are accurate. A small amount of badly estimated missing values in the data might be enough for clustering methods, such as hierachical clustering or K-means clustering, to produce misleading results. Thus, accurate methods for missing value estimation are needed. We present novel methods for estimation of missing values in microarray data sets that are based on the least squares principle, and that utilize correlations between both genes and arrays. For this set of methods, we use the common reference name LSimpute. We compare the estimation accuracy of our methods with the widely used KNNimpute on three complete data matrices from public data sets by randomly knocking out data (labeling as missing). From these tests, we conclude that our LSimpute methods produce estimates that consistently are more accurate than those obtained using KNNimpute. Additionally, we examine a more classic approach to missing value estimation based on expectation maximization (EM). We refer to our EM implementations as EMimpute, and the estimate errors using the EMimpute methods are compared with those our novel methods produce. The results indicate that on average, the estimates from our best performing LSimpute method are at least as accurate as those from the best EMimpute algorithm.     

Robust Portfolio Optimization Using Pseudodistances

The presence of outliers in financial asset returns is a frequently occurring phenomenon which may lead to unreliable mean-variance optimized portfolios. This fact is due to the unbounded influence that outliers can have on the mean returns and covariance estimators that are inputs in the optimization procedure. In this paper we present robust estimators of mean and covariance matrix obtained by minimizing an empirical version of a pseudodistance between the assumed model and the true model underlying the data. We prove and discuss theoretical properties of these estimators, such as affine equivariance, B-robustness, asymptotic normality and asymptotic relative efficiency. These estimators can be easily used in place of the classical estimators, thereby providing robust optimized portfolios. A Monte Carlo simulation study and applications to real data show the advantages of the proposed approach. We study both in-sample and out-of-sample performance of the proposed robust portfolios comparing them with some other portfolios known in literature.     

Estimating FST and kinship for arbitrary population structures

F_ST and kinship are key parameters often estimated in modern population genetics studies in order to quantitatively characterize structure and relatedness. Kinship matrices have also become a fundamental quantity used in genome-wide association studies and heritability estimation. The most frequently-used estimators of F_ST and kinship are method-of-moments estimators whose accuracies depend strongly on the existence of simple underlying forms of structure, such as the independent subpopulations model of non-overlapping, independently evolving subpopulations. However, modern data sets have revealed that these simple models of structure likely do not hold in many populations, including humans. In this work, we analyze the behavior of these estimators in the presence of arbitrarily-complex population structures, which results in an improved estimation framework specifically designed for arbitrary population structures. After generalizing the definition of F_ST to arbitrary population structures and establishing a framework for assessing bias and consistency of genome-wide estimators, we calculate the accuracy of existing F_ST and kinship estimators under arbitrary population structures, characterizing biases and estimation challenges unobserved under their originally-assumed models of structure. We then present our new approach, which consistently estimates kinship and F_ST when the minimum kinship value in the dataset is estimated consistently. We illustrate our results using simulated genotypes from an admixture model, constructing a one-dimensional geographic scenario that departs nontrivially from the independent subpopulations model. Our simulations reveal the potential for severe biases in estimates of existing approaches that are overcome by our new framework. This work may significantly improve future analyses that rely on accurate kinship and F_ST estimates.     

Improved least squares topology testing and estimation

Generalized least squares (GLS) methods provide a relatively fast means of constructing a confidence set of topologies. Because they utilize information about the covariances between distances, it is reasonable to expect additional efficiency in estimation and confidence set construction relative to other least squares (LS) methods. Difficulties have been found to arise in a number of practical settings due to estimates of covariance matrices being ill conditioned or even noninvertible. We present here new ways of estimating the covariance matrices for distances that are much more likely to be positive definite, as the actual covariance matrices are. A thorough investigation of performance is also conducted. An alternative to GLS that has been proposed for constructing confidence sets of topologies is weighted least squares (WLS). As currently implemented, this approach is equivalent to the use of GLS but with covariances set to zero rather than being estimated. In effect, this approach assumes normality of the estimated distances and zero covariances. As the results here illustrate, this assumption leads to poor performance. A 95% confidence set is almost certain to contain the true topology but will contain many more topologies than are needed. On the other hand, the results here also indicate that, among LS methods, WLS performs quite well at estimating the correct topology. It turns out to be possible to improve the performance of WLS for confidence set construction through a relatively inexpensive normal parametric bootstrap that utilizes the same variances and covariances of GLS. The resulting procedure is shown to perform at least as well as GLS and thus provides a reasonable alternative in cases where covariance matrices are ill conditioned.