Estimation of Individual Growth Curves by Empirically Weighted Least Squares

The simultaneous estimation of individual growth curves and a mean growth curve is accomplished by weighted least squares. A polynomial curve is fitted for each individual and the polynomial parameters are linear functions of parameters corresponding to covariates. A simple, computationally efficient variance-covariance estimator is derived. The resultant estimate is used in the weighted least squares estimation. The results are compared to empirical Bayes estimation.     

Testing Hypotheses about Regression Parameters,When the Error Term Is Heteroscedastic

The paper considers methods for testing H₀: β₁ = … = β_p = 0, where β₁, … ,β_p are the slope parameters in a linear regression model with an emphasis on p = 2. It is known that even when the usual error term is normal, but heteroscedastic, control over the probability of a type I error can be poor when using the conventional F test in conjunction with the least squares estimator. When the error term is nonnormal, the situation gets worse. Another practical problem is that power can be poor under even slight departures from normality. Liu and Singh (1997) describe a general bootstrap method for making inferences about parameters in a multivariate setting that is based on the general notion of depth. This paper studies the small-sample properties of their method when applied to the problem at hand. It is found that there is a practical advantage to using Tukey's depth versus the Mahalanobis depth when using a particular robust estimator. When using the ordinary least squares estimator, the method improves upon the conventional F test, but practical problems remain when the sample size is less than 60. In simulations, using Tukey's depth with the robust estimator gave the best results, in terms of type I errors, among the five methods studied.     

Independent contrasts and PGLS regression estimators are equivalent

We prove that the slope parameter of the ordinary least squares regression of phylogenetically independent contrasts (PICs) conducted through the origin is identical to the slope parameter of the method of generalized least squares (GLSs) regression under a Brownian motion model of evolution. This equivalence has several implications: 1. Understanding the structure of the linear model for GLS regression provides insight into when and why phylogeny is important in comparative studies. 2. The limitations of the PIC regression analysis are the same as the limitations of the GLS model. In particular, phylogenetic covariance applies only to the response variable in the regression and the explanatory variable should be regarded as fixed. Calculation of PICs for explanatory variables should be treated as a mathematical idiosyncrasy of the PIC regression algorithm. 3. Since the GLS estimator is the best linear unbiased estimator (BLUE), the slope parameter estimated using PICs is also BLUE. 4. If the slope is estimated using different branch lengths for the explanatory and response variables in the PIC algorithm, the estimator is no longer the BLUE, so this is not recommended. Finally, we discuss whether or not and how to accommodate phylogenetic covariance in regression analyses, particularly in relation to the problem of phylogenetic uncertainty. This discussion is from both frequentist and Bayesian perspectives.     

Closed Form Estimators of Latent Cell Frequencies

One-stage and two-stage closed form estimators of latent cell frequencies in multidimensional contingency tables are derived from the weighted least squares criterion. The first stage estimator is asymptotically equivalent to the conditional maximum likelihood estimator and does not necessarily have minimum asymptotic variance. The second stage estimator does have minimum asymptotic variance relative to any other existing estimator. The closed form estimators are defined for any number of latent cells in contingency tables of any order under exact general linear constraints on the logarithms of the nonlatent and latent cell frequencies.     

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.     

Estimation and comparison of changes in the presence of informative right censoring: conditional linear model

A general linear regression model for the usual least squares estimated rate of change (slope) on censoring time is described as an approximation to account for informative right censoring in estimating and comparing changes of a continuous variable in two groups. Two noniterative estimators for the group slope means, the linear minimum variance unbiased (LMVUB) estimator and the linear minimum mean squared error (LMMSE) estimator, are proposed under this conditional model. In realistic situations, we illustrate that the LMVUB and LMMSE estimators, derived under a simple linear regression model, are quite competitive compared to the pseudo maximum likelihood estimator (PMLE) derived by modeling the censoring probabilities. Generalizations to polynomial response curves and general linear models are also described.     

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 Growth Norms from Incomplete Longitudinal Data

Longitudinal studies are rarely complete due to attrition, mistimed visits and observations missing at random. When the data are missing at random it is possible to estimate the primary location parameters of interest by constructing a modification of Zellner's (1962) seemingly unrelated regression estimator. Such a procedure is developed in this paper and is applied to a longitudinal study of coronary risk factors in children. The method consists of two stages in which the covariance matrix is estimated at the first stage. Using the estimated covariance matrix a generalized least squares estimator of the regression parameter vector is then determined at the second stage. Limitations of the procedure are also discussed.     

The Routine Fitting of Kinetic Data to Models: A Mathematical Formalism for Digital Computers

A mathematical formalism is presented for use with digital computers to permit the routine fitting of data to physical and mathematical models. Given a set of data, the mathematical equations describing a model, initial conditions for an experiment, and initial estimates for the values of model parameters, the computer program automatically proceeds to obtain a least squares fit of the data by an iterative adjustment of the values of the parameters. When the experimental measures are linear combinations of functions, the linear coefficients for a least squares fit may also be calculated. The values of both the parameters of the model and the coefficients for the sum of functions may be unknown independent variables, unknown dependent variables, or known constants. In the case of dependence, only linear dependencies are provided for in routine use. The computer program includes a number of subroutines, each one of which performs a special task. This permits flexibility in choosing various types of solutions and procedures. One subroutine, for example, handles linear differential equations, another, special non-linear functions, etc. The use of analytic or numerical solutions of equations is possible.     

Regularized estimation in the accelerated failure time model with high-dimensional covariates

We consider two regularization approaches, the LASSO and the threshold-gradient-directed regularization, for estimation and variable selection in the accelerated failure time model with multiple covariates based on Stute's weighted least squares method. The Stute estimator uses Kaplan-Meier weights to account for censoring in the least squares criterion. The weighted least squares objective function makes the adaptation of this approach to multiple covariate settings computationally feasible. We use V-fold cross-validation and a modified Akaike's Information Criterion for tuning parameter selection, and a bootstrap approach for variance estimation. The proposed method is evaluated using simulations and demonstrated on a real data example.     

删失数据下非线性半参数回归模型中参数的经验似然推断

考察了响应变量在随机删失情形下的非线性半参数回归模型,构造了未知参数的经验对数似然比统计量和调整经验对数似然比统计量,证明在一定条件下,所构造的经验似然比统计量渐近于X~2分布,并由此构造出未知参数的置信域.此外,又构造了未知参数的最小二乘估计量,证明了它的渐近性质.通过模拟研究表明,经验似然方法在置信域的覆盖概率以及精度方面要优于最小二乘法.     

The Asymptotic Covariance Matrix of the Maximum Likelihood Parameter Estimator in Conditional Poisson Log-linear Models

The asymptotic covariance matrix of the maximum likelihood estimator for the log-linear model is given for a general class of conditional Poisson distributions which include the unconditional Poisson, multinomial and product-multinomial, as special cases. The general conditions are given under which the maximum likelihood covariance matrix is equal to the covariance matrix of an equivalent closed-form weighted least squares estimator.     

Quantitative description of cell cycle kinetics under chemotherapy utilizing flow cytometry.

A discrete time cell cycle kinetics model is developed to account for the effects of cytotoxic chemotherapy, particularly including the existence of cells destined to die. A model structure is determined from related experiments, leaving key parameter values undetermined. These values are found by determining the best least squares fit of the predicted to the observed DNA distribution data at a series of time intervals. The numerical methods include separable least squares, linear inequality constrained least squares and the Gauss--Newton method. This approach is applied to an experiment in which the Ehrlich ascites tumour was given a single dose of bleomycin. The results include several different parameters, including the age response function and a time series of cell age and DNA distributions, which can be used as a basis for further treatment.     

A Modification of the Edgeworth Approximation in the AR (1) Model

This paper considers the sampling distribution problem of the least squares estimator for the parameter of some special autoregressive models. The Edgeworth approximation has been derived and a modification is proposed to improve its accuracy. Comparisons with the exact distribution and the so called Edgeworth-B approximation have been discussed. The results show that the proposed approximation performs more accurately than the Edgeworth-B approximation, especially, when models are close to the non-stationary boundary.     

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%.     

Missing value estimation for DNA microarray gene expression data: local least squares imputation

MOTIVATION: Gene expression data often contain missing expression values. Effective missing value estimation methods are needed since many algorithms for gene expression data analysis require a complete matrix of gene array values. In this paper, imputation methods based on the least squares formulation are proposed to estimate missing values in the gene expression data, which exploit local similarity structures in the data as well as least squares optimization process. RESULTS: The proposed local least squares imputation method (LLSimpute) represents a target gene that has missing values as a linear combination of similar genes. The similar genes are chosen by k-nearest neighbors or k coherent genes that have large absolute values of Pearson correlation coefficients. Non-parametric missing values estimation method of LLSimpute are designed by introducing an automatic k-value estimator. In our experiments, the proposed LLSimpute method shows competitive results when compared with other imputation methods for missing value estimation on various datasets and percentages of missing values in the data. AVAILABILITY: The software is available at http://www.cs.umn.edu/~hskim/tools.html CONTACT: hpark@cs.umn.edu     

Unbiasedness of the Estimator of the Function of Expected Value in the Mixed Linear Model

The traditional method for estimating the linear function of fixed parameters in mixed linear model is a two-stage procedure. In the first stage of this procedure the variance components estimators are calculated and next in the second stage these estimators are taken as true values of variance components to estimating the linear function of fixed parameters according to generalized least squares method. In this paper the general mixed linear model is considered in which a matrix related to fixed parameters and or/a dispersion matrix of observation vector may be deficient in rank. It is shown that the estimators of a set of functions of fixed parameters obtained in second stage are unbiased if only the observation vector is symmetrically distributed about its expected value and the estimators of variance components from first stage are translation-invariant and are even functions of the observation vector.     

Quantitative Description of Cell Cycle Kinetics Under Chemotherapy Utilizing Flow Cytometry

A discrete time cell cycle kinetics model is developed to account for the effects of cytotoxic chemotherapy, particularly including the existence of cells destined to die. A model structure is determined from related experiments, leaving key parameter values undetermined. These values are found by determining the best least squares fit of the predicted to the observed DNA distribution data at a series of time intervals. the numerical methods include separable least squares, linear inequality constrained least squares and the Gauss-Newton method. This approach is applied to an experiment in which the Ehrlich ascites tumour was given a single dose of bleomycin. the results include several different parameters, including the age response function and a time series of cell age and DNA distributions, which can be used as a basis for further treatment.     

Statistical considerations in the estimation of enzyme kinetic parameters by the direct linear plot and other methods

The statistical implications of the direct linear plot for enzyme kinetic data, described in the preceding paper (Eisenthal & Cornish-Bowden, 1974), are discussed for the case of the Michaelis-Menten equation. The plot is shown to lead directly to non-parametric confidence limits for the kinetic parameters, V and K(m), which depend on far less sweeping assumptions about the nature of experimental error than those implicit in the method of least squares. Median estimates of V and K(m) can also be defined, which are shown to be more robust than the least-squares estimates in a wide variety of experimental situations.     

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.