Random Regression Models Based On The Elliptically Contoured Distribution Assumptions With Applications To Longitudinal Data (original) (raw)
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Permutation tests increasingly are the statistical method of choice for addressing business questions and research hypotheses across a broad range of industries. Their distribution-free nature maintains test validity where many parametric tests (and even other nonparametric tests), encumbered by restrictive and often inappropriate data assumptions, fail miserably. The computational demands of permutation tests, however, have severely limited other vendors' attempts at providing useable permutation test software for anything but highly stylized situations or small datasets and few tests. PermuteIt TM addresses this unmet need by utilizing a combination of algorithms to perform non-parametric permutation tests very quickly -often more than an order of magnitude faster than widely available commercial alternatives when one sample is large and many tests and/or multiple comparisons are being performed (which is when runtimes matter most). PermuteIt TM can make the difference between making deadlines, or missing them, since data inputs often need to be revised, resent, or recleaned, and one hour of runtime quickly can become 10, 20, or 30 hours. SM , at JDOpdyke@DataMineIt.com or www.DataMineIt.com.
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Repeated measurements of surrogate markers are frequently used to track disease progression, but these series are often prematurely terminated due to disease progression or death. Analysing such data through standard likelihood-based approaches can yield severely biased estimates if the censoring mechanism is non-ignorable. Motivated by this problem, we have proposed the bivariate joint multivariate random effects (JMRE) model, which has shown that when correctly specified it performs well in terms of bias reduction and precision. The bivariate JMRE model is fully parametric and belongs to the class of shared parameters joint models where a survival model for the dropouts and a mixed model for the markers' evolution are linked through a multivariate normal distribution of random effects. As in every parametric model, robustness under violations of its distributional assumptions is of great importance. In this study we generated 500 simulated data sets assuming that random effects jointly follow a heavy-tailed distribution, two skewed distributions or a mixture of two normal distributions. Moreover, we generated data where level-1 errors or residuals in the survival part of the model follow a skewed distribution. Further sensitivity analysis on the effects of reduced sample size, increased level-1 variances and altered fixed effects values was also performed.
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In AIDS studies it is quite common to observe viral load measurements collected irregularly over time. Moreover, these measurements can be subjected to some upper and/or lower detection limits depending on the quantification assays. A complication arises when these continuous repeated measures have a heavy-tailed behavior. For such data structures, we propose a robust structure for a censored linear model based on the multivariate Student-t distribution. To compensate for the autocorrelation existing among irregularly observed measures, a damped exponential correlation structure is employed.
Journal of Statistical Planning and Inference, 2006
In this paper statistical inference is developed for the estimation and testing problems of the location and scale parameters of the elliptically contoured family of distributions. The data matrix is of a monotone missing pattern. The analytic form of the maximum likelihood estimators of location and scale are derived, and based on them, the likelihood ratio test statistics are obtained for testing the following: (i) the location and scale parameters are, separately, equal to a specified vector and matrix, (ii) the location and scale parameters are, simultaneously, equal to a specified vector and matrix, and (iii) the hypothesis of lack of correlation between sets of variates that jointly described by the elliptically contoured family of distributions. The test of sphericity is also derived in the particular case of the multivariate normal distribution. The asymptotic null distributions of the resulting test statistics are derived for k = 2, as well as, for k > 2 steps of monotone missing data. The results are illustratively applied in the Appendix A, to specific elliptically contoured models like the multivariate t-distribution. The results are also illustrated using simulated data from a multivariate t-distribution.
Journal of the Royal Statistical Society: Series C (Applied Statistics), 2000
Objectives in many longitudinal studies of individuals infected with the human immuno-de®ciency virus (HIV) include the estimation of population average trajectories of HIV ribonucleic acid (RNA) over time and tests for differences in trajectory across subgroups. Special features that are often inherent in the underlying data include a tendency for some HIV RNA levels to be below an assay detection limit, and for individuals with high initial levels or high rates of change to drop out of the study early because of illness or death. We develop a likelihood for the observed data that incorporates both of these features. Informative drop-outs are handled by means of an approach previously published by Schluchter. Using data from the HIV Epidemiology Research Study, we implement a maximum likelihood procedure to estimate initial HIV RNA levels and slopes within a population, compare these parameters across subgroups of HIV-infected women and illustrate the importance of appropriate treatment of left censoring and informative drop-outs. We also assess model assumptions and consider the prediction of random intercepts and slopes in this setting. The results suggest that marked bias in estimates of ®xed effects, variance components and standard errors in the analysis of HIV RNA data might be avoided by the use of methods like those illustrated.
Journal of Modern Applied Statistical Methods, 2008
Predictive distributions of future response and future regression matrices under multivariate elliptically contoured distributions are discussed. Under the elliptically contoured response assumptions, these are identical to those obtained under matric normal or matric-t errors using structural, Bayesian with improper prior, or classical approaches. This gives inference robustness with respect to departure from the reference case of independent sampling from the matric normal or matric t to multivariate elliptically contoured distributions. The importance of the predictive distribution for skewed elliptical models is indicated; the elliptically contoured distribution, as well as matric t distribution, have significant applications in statistical practices.
Empirical Likelihood Based Longitudinal Data Analysis
Open Journal of Statistics, 2020
In longitudinal data analysis, our primary interest is in the estimation of regression parameters for the marginal expectations of the longitudinal responses, and the longitudinal correlation parameters are of secondary interest. The joint likelihood function for longitudinal data is challenging, particularly due to correlated responses. Marginal models, such as generalized estimating equations (GEEs), have received much attention based on the assumption of the first two moments of the data and a working correlation structure. The confidence regions and hypothesis tests are constructed based on the asymptotic normality. This approach is sensitive to the misspecification of the variance function and the working correlation structure which may yield inefficient and inconsistent estimates leading to wrong conclusions. To overcome this problem, we propose an empirical likelihood (EL) procedure based on a set of estimating equations for the parameter of interest and discuss its characteristics and asymptotic properties. We also provide an algorithm based on EL principles for the estimation of the regression parameters and the construction of its confidence region. We have applied the proposed method in two case examples.
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Joint modeling of longitudinal and time to event data have been widely used for analyzing medical data, where longitudinal measurements is gathered with a time to event or survival data. In most of these studies, distributional assumption for modeling longitudinal response is normal, which leads to vulnerable inference in the presence of outliers in longitudinal measurements and violation of this assumption. Violation of the normality assumption can also make the statistical inference vague. Powerful distributions for robust analyzing and relaxing normality assumption, are skew-elliptical distributions, which include univariate and multivariate versions of the student's t, the Laplace and normal distributions. In this paper, a linear mixed effects model with skew-elliptical distribution for both random effects and residuals and a Cox's model for time to event data are used for joint modeling. This strategy allows for the skewness and the heavy tails of random effect distributions and thus makes inferences robust to the violation. For estimation, a Bayesian parametric approach using Markov chain Monte Carlo is adopted. The method is illustrated in a real Intensive Care Unit (ICU) data set and the best model is selected using some Bayesian criteria for model selection.