Variability explained by covariates in linear mixed‐effect models for longitudinal data

Variability explained by covariates or explained variance is a well‐known concept in assessing the importance of covariates for dependent outcomes. In this paper we study R² statistics of explained variance pertinent to longitudinal data under linear mixed‐effect models, where the R² statistics are computed at two different levels to measure, respectively, within‐ and between‐subject variabilities explained by the covariates. By deriving the limits of R² statistics, we find that the interpretation of explained variance for the existing R² statistics is clear only in the case where the covariance matrix of the outcome vector is compound symmetric. Two new R² statistics are proposed to address the effect of time‐dependent covariate means. In the general case where the outcome covariance matrix is not compound symmetric, we introduce the concept of compound symmetry projection and use it to define level‐one and level‐two R² statistics. Numerical results are provided to support the theoretical findings and demonstrate the performance of the R² statistics. The Canadian Journal of Statistics 38: 352–368; 2010 © 2010 Statistical Society of Canada     

A very simple proof of the multivariate Chebyshev's inequality

Abstract

In this short note, a very simple proof of the Chebyshev's inequality for random vectors is given. This inequality provides a lower bound for the percentage of the population of an arbitrary random vector X with finite mean μ = E(X) and a positive definite covariance matrix V = Cov(X) whose Mahalanobis distance with respect to V to the mean μ is less than a fixed value. The main advantage of the proof is that it is a simple exercise for a first year probability course. An alternative proof based on principal components is also provided. This proof can be used to study the case of a singular covariance matrix V.     

Finding maximum G-criterion values for central composite designs on the hypercube

For quadratic regression on the hypercube, G—efficiencies are often used in the selection process of an experimental design. To calculate a design's G—efficiency, it is necessary to maximize the prediction variance over the experimental design region. However, it is common to approximate a G—efficiency. This is achieved by calculating the prediction variances generated from a subset of points in the design space and taking the maximum to estimate the maximum prediction variance. This estimate is then applied to approximate the G—efficiency. In this paper, it will be shown that over the class of central composite designs (CCDs) on the hypercube. the prediction variance can be expressed in a closed-form. An exact value of the maximum prediction variance can then be determined by evaluating this closed-form expression over a finite subset of barycentric points. Tables of exact G—efficiencies will be presented. Design optimality criteria, quadratic regression on the hypercube, and the structures of the design matrix X, X'X, and (X'X)^?1 for any CCD will be discussed.     

Parameter expansion for sampling a correlation matrix: an efficient GPX-RPMH algorithm

Sampling the correlation matrix (R) plays an important role in statistical inference for correlated models. There are two main constraints on a correlation matrix: positive definiteness and fixed diagonal elements. These constraints make sampling R difficult. In this paper, an efficient generalized parameter expanded re-parametrization and Metropolis-Hastings (GPX-RPMH) algorithm for sampling a correlation matrix is proposed. Drawing all components of R simultaneously from its full conditional distribution is realized by first drawing a covariance matrix from the derived parameter expanded candidate density (PXCD), and then translating it back to a correlation matrix and accepting it according to a Metropolis-Hastings (M-H) acceptance rate. The mixing rate in the M-H step can be adjusted through a class of tuning parameters embedded in the generalized candidate prior (GCP), which is chosen for R to derive the PXCD. This algorithm is illustrated using multivariate regression (MVR) models and a simulation study shows that the performance of the GPX-RPMH algorithm is more efficient than that of other methods.     

Effects of the generalized Box–Cox transformation on Type I error rate and power of Hotelling's T 2

Most multivariate statistical techniques rely on the assumption of multivariate normality. The effects of nonnormality on multivariate tests are assumed to be negligible when variance–covariance matrices and sample sizes are equal. Therefore, in practice, investigators usually do not attempt to assess multivariate normality. In this simulation study, the effects of skewed and leptokurtic multivariate data on the Type I error and power of Hotelling's T ² were examined by manipulating distribution, sample size, and variance–covariance matrix. The empirical Type I error rate and power of Hotelling's T ² were calculated before and after the application of generalized Box–Cox transformation. The findings demonstrated that even when variance–covariance matrices and sample sizes are equal, small to moderate changes in power still can be observed.     

Asymptotic calculations of functions of expected values and covariances of order statistics

Suppose m and V are respectively the vector of expected values and the covariance matrix of the order statistics of a sample of size n from a continuous distribution F. A method is presented to calculate asymptotic values of functions of m and V ^–1, for distributions F which are sufficiently regular. Values are given for the normal, logistic, and extreme-value distributions; also, for completeness, for the uniform and exponential distributions, although for these other methods must be used.     

On estimating a function of the mean of a multivariate normal distribution

The unique minimum variance of unbiased estimator is obtained for analysis functions of the mean of a multivariate normal distribution with either unknown covariance matrix or with covariance matrix of the form σ²v where σ² is unknown.     

A bound for the Euclidean distance between restricted and unrestricted estimators of parametric functions in the general linear model

Let [^(\varveck)]{\widehat{\varvec{\kappa}}} and [^(\varveck)]_r{\widehat{\varvec{\kappa}}_r} denote the best linear unbiased estimators of a given vector of parametric functions \varveck = \varvecKb{\varvec{\kappa} = \varvec{K\beta}} in the general linear models M = {\varvecy, \varvecX\varvecb, s²\varvecV}{{\mathcal M} = \{\varvec{y},\, \varvec{X\varvec{\beta}},\, \sigma^2\varvec{V}\}} and M_r = {\varvecy, \varvecX\varvecb | \varvecR \varvecb = \varvecr, s²\varvecV}{{\mathcal M}_r = \{\varvec{y},\, \varvec{X}\varvec{\beta} \mid \varvec{R} \varvec{\beta} = \varvec{r},\, \sigma^2\varvec{V}\}}, respectively. A bound for the Euclidean distance between [^(\varveck)]{\widehat{\varvec{\kappa}}} and [^(\varveck)]_r{\widehat{\varvec{\kappa}}_r} is expressed by the spectral distance between the dispersion matrices of the two estimators, and the difference between sums of squared errors evaluated in the model M{{\mathcal M}} and sub-restricted model M_r^*{{\mathcal M}_r^*} containing an essential part of the restrictions \varvecR\varvecb = \varvecr{\varvec{R}\varvec{\beta} = \varvec{r}} with respect to estimating \varveck{\varvec{\kappa}}.     

Three estimators for the poisson regression model with measurement errors

We consider two consistent estimators for the parameters of the linear predictor in the Poisson regression model, where the covariate is measured with errors. The measurement errors are assumed to be normally distributed with known error variance σ _u ² . The SQS estimator, based on a conditional mean-variance model, takes the distribution of the latent covariate into account, and this is here assumed to be a normal distribution. The CS estimator, based on a corrected score function, does not use the distribution of the latent covariate. Nevertheless, for small σ _u ² , both estimators have identical asymptotic covariance matrices up to the order of σ _u ² . We also compare the consistent estimators to the naive estimator, which is based on replacing the latent covariate with its (erroneously) measured counterpart. The naive estimator is biased, but has a smaller covariance matrix than the consistent estimators (at least up to the order of σ _u ² ).     

Vector random fields with compactly supported covariance matrix functions

The objective of this paper is to construct covariance matrix functions whose entries are compactly supported, and to use them as building blocks to formulate other covariance matrix functions for second-order vector stochastic processes or random fields. In terms of the scale mixture of compactly supported covariance matrix functions, we derive a class of second-order vector stochastic processes on the real line whose direct and cross covariance functions are of Pólya type. Then some second-order vector random fields in R^d${\mathbb{R}}^d$ whose direct and cross covariance functions are compactly supported are constructed by using a convolution approach and a mixture approach.     

On the bias and variance of some proportion estimators

Providing certain parameters are known, almost any linear map from R^P to R¹ can be adjusted to yield a consistent and unbiased estimator in the context of estimating the mixing proportion θ on the basis of an unclassified sample of observations taken from a mixture of two p-dimensional distributions in proportions θ and 1-θ. Attention is focused on an estimator proposed recently, θ, which has minimum variance over all such linear maps. Unfortunately, the form of θ depends on the means of the component distributions and the covariance matrix of the mixture distribution. The effect of using appropriate sample estimates for these unknown parameters in forming θ is investigated by deriving the asymptotic mean and variance of the resulting estimator. The relative efficiency of this estimator under normality is derived. Also, a study is undertaken of the performance of a similar type of estimator appropriate in the context where an observed data vector is not an observation from either one or the other onent distributions, but is recorded as an integrated measurement over a surface area which is a mixture of two categories whose characteristics have different statistical distributions.The asymptotic bias in this case is compared with some available practical results.     

Improved shrinkage estimators for the mean vector of a scale mixture of normals with unknown variance

The problem of estimating the mean θ of a not necessarily normal p-variate (p > 3) distribution with unknown covariance matrix of the form σ²A (A a known diagonal matrix) on the basis of n_i > 2 observations on each coordinate X_t (1 < i < p) is considered. It is argued that the class of scale (or variance) mixtures of normal distributions is a reasonable class to study. Assuming the loss function is quadratic, a large class of improved shrinkage estimators is developed in the case of a balanced design. We generalize results of Berger and Strawderman for one observation in the known-variance case. This methodology also permits the development of a new class of minimax shrinkage estimators of the mean of a p-variate normal distribution for an unbalanced design. Numerical calculations show that the improvements in risk can be substantial.     

Adaptive fitting of linear mixed-effects models with correlated random effects

Linear mixed-effects model has been widely used in longitudinal data analyses. In practice, the fitting algorithm can fail to converge due to boundary issues of the estimated random-effects covariance matrix G, that is, being near-singular, non-positive definite, or both. Current available algorithms are not computationally optimal because the condition number of matrix G is unnecessarily increased when the random-effects correlation estimate is not zero. We propose an adaptive fitting (AF) algorithm using an optimal linear transformation of the random-effects design matrix. It is a data-driven adaptive procedure, aiming at reducing subsequent random-effects correlation estimates down to zero in the optimal transformed estimation space. Simulations show that AF significantly improves the convergent properties, especially under small sample size, relative large noise and high correlation settings. One real data for insulin-like growth factor protein is used to illustrate the application of this algorithm implemented with software package R (nlme).     

A control chart based on sample ranges for monitoring the covariance matrix of the multivariate processes

For the univariate case, the R chart and the S ² chart are the most common charts used for monitoring the process dispersion. With the usual sample size of 4 and 5, the R chart is slightly inferior to the S ² chart in terms of efficiency in detecting process shifts. In this article, we show that for the multivariate case, the chart based on the standardized sample ranges, we call the RMAX chart, is substantially inferior in terms of efficiency in detecting shifts in the covariance matrix than the VMAX chart, which is based on the standardized sample variances. The user's familiarity with sample ranges is a point in favor of the RMAX chart. An example is presented to illustrate the application of the proposed chart.     

Asymptotic normal tests for integration in panels with cross-dependent units

The asymptotically normal, regression-based LM integration test is adapted for panels with correlated units. The N different units may be integrated of different (fractional) orders under the null hypothesis. The paper first reviews conditions under which the test statistic is asymptotically (as T→∞) normal in a single unit. Then we adopt the framework of seemingly unrelated regression [SUR] for cross-correlated panels, and discuss a panel test statistic based on the feasible generalized least squares [GLS] estimator, which follows a χ ²(N) distribution. Third, a more powerful statistic is obtained by working under the assumption of equal deviations from the respective null in all units. Fourth, feasible GLS requires inversion of sample covariance matrices typically imposing T>N; in addition we discuss alternative covariance matrix estimators for T<N. The usefulness of our results is assessed in Monte Carlo experimentation.     

The formal posterior of a standard flat prior in MANOVA is incoherent

Summary A standard improper prior for the parameters of a MANOVA model is shown to yield an inference that is incoherent in the sense of Heath and Sudderth. The proof of incoherence is based on the fact that the formal Bayes estimate, sayδ ₀ , of the covariance matrix based on the improper prior and a certain bounded loss function is uniformly inadmissible in that there is another estimatorδ _l and an ɛ>0 such that the risk functions satisfyR(δ _l ,Σ)⩽R δ ₀ ,Σ)−ε for all values of the covariance matrix Σ. The estimatorδ _I is formal Bayes for an alternative improper prior which leads to a coherent inference. Research supported by National Science Foundation grants DMS-89-22607 (for Eaton) and DMS-9123358 (for Sudderth).     

Failure Data Analysis with Extended Weibull Distribution

A consecutive k-out-of-n: G system consists of n linearly ordered components functions if and only if at least k consecutive components function. In this article we investigate the consecutive k-out-of-n: G system in a setup of multicomponent stress-strength model. Under this setup, a system consists of n components functions if and only if there are at least k consecutive components survive a common random stress. We consider reliability and its estimation of such a system whenever there is a change and no change in strength. We provide minimum variance unbiased estimation of system reliability when the stress and strength distributions are exponential with unknown scale parameters. A nonparametric minimum variance unbiased estimator is also provided.     

A Coefficient of Determination for Generalized Linear Models

The coefficient of determination, a.k.a. R², is well-defined in linear regression models, and measures the proportion of variation in the dependent variable explained by the predictors included in the model. To extend it for generalized linear models, we use the variance function to define the total variation of the dependent variable, as well as the remaining variation of the dependent variable after modeling the predictive effects of the independent variables. Unlike other definitions that demand complete specification of the likelihood function, our definition of R² only needs to know the mean and variance functions, so applicable to more general quasi-models. It is consistent with the classical measure of uncertainty using variance, and reduces to the classical definition of the coefficient of determination when linear regression models are considered.     

Quadratic subspaces and construction of Bayes invariant quadratic estimators of variance components in mixed linear models

Gnot et al. (J Statist Plann Inference 30(1):223–236, 1992) have presented the formulae for computing Bayes invariant quadratic estimators of variance components in normal mixed linear models of the form where the matrices V _i , 1 ≤ i ≤ k − 1, are symmetric and nonnegative definite and V _k is an identity matrix. These formulae involve a basis of a quadratic subspace containing MV ₁ M,...,MV _k-1 M,M, where M is an orthogonal projector on the null space of X′. In the paper we discuss methods of construction of such a basis. We survey Malley’s algorithms for finding the smallest quadratic subspace including a given set of symmetric matrices of the same order and propose some modifications of these algorithms. We also consider a class of matrices sharing some of the symmetries common to MV ₁ M,...,MV _k-1 M,M. We show that the matrices from this class constitute a quadratic subspace and describe its explicit basis, which can be directly used for computing Bayes invariant quadratic estimators of variance components. This basis can be also used for improving the efficiency of Malley’s algorithms when applied to finding a basis of the smallest quadratic subspace containing the matrices MV ₁ M,...,MV _k-1 M,M. Finally, we present the results of a numerical experiment which confirm the potential usefulness of the proposed methods. Dedicated to the memory of Professor Stanisław Gnot.