Dirichlet Processes in Nonlinear Mixed Effects Models

In this article, we use two efficient approaches to deal with the difficulty in computing the intractable integrals when implementing Gibbs sampling in the nonlinear mixed effects model (NLMM) based on Dirichlet processes (DP). In the first approach, we compute the Laplace's approximation to the integral for its high accuracy, low cost, and ease of implementation. The second approach uses the no-gaps algorithm of MacEachern and Müller (1998 MacEachern , S. , Müller , P. ( 1998 ). Estimating mixtures of Dirichlet process models . Journal of Computational and Graphical Statistics 7 : 223 – 238 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]) to perform Gibbs sampling without evaluating the difficult integral. We apply both approaches to real problems and simulations. Results show that both approaches perform well in density estimation and prediction and are superior to the parametric analysis in that they can detect important model features, such as skewness, long tails, and multimodality, whereas the parametric analysis cannot.     

A comparison of several analysis of variance programs with unequal cell size

Various programs in statistical packages for analysis of variance with unequal cell size give different results to the same data because of nonorthogonality of the main effects and interactions. This paper explains how these programs treat the problem of analysis of variance of unbalanced data.     

Using regression mixture models with non-normal data: examining an ordered polytomous approach

Mild to moderate skew in errors can substantially impact regression mixture model results; one approach for overcoming this includes transforming the outcome into an ordered categorical variable and using a polytomous regression mixture model. This is effective for retaining differential effects in the population; however, bias in parameter estimates and model fit warrant further examination of this approach at higher levels of skew. The current study used Monte Carlo simulations; 3000 observations were drawn from each of two subpopulations differing in the effect of X on Y. Five hundred simulations were performed in each of the 10 scenarios varying in levels of skew in one or both classes. Model comparison criteria supported the accurate two-class model, preserving the differential effects, while parameter estimates were notably biased. The appropriate number of effects can be captured with this approach but we suggest caution when interpreting the magnitude of the effects.     

Maximum Likelihood Estimation for Stochastic Differential Equations with Random Effects

Abstract. We consider N independent stochastic processes (X _i (t), t ∈ [0,T _i ]), i=1,…, N, defined by a stochastic differential equation with drift term depending on a random variable φ _i . The distribution of the random effect φ _i depends on unknown parameters which are to be estimated from the continuous observation of the processes X_i. We give the expression of the exact likelihood. When the drift term depends linearly on the random effect φ _i and φ _i has Gaussian distribution, an explicit formula for the likelihood is obtained. We prove that the maximum likelihood estimator is consistent and asymptotically Gaussian, when T _i =T for all i and N tends to infinity. We discuss the case of discrete observations. Estimators are computed on simulated data for several models and show good performances even when the length time interval of observations is not very large.     

New Developments in Statistical Computing: New Bias Evaluation Features of EXPLOR—A Program for Assessing Experimental Design Properties

Longitudinal investigations play an increasingly prominent role in biomedical research. Much of the literature on specifying and fitting linear models for serial measurements uses methods based on the standard multivariate linear model. This article proposes a more flexible approach that permits specification of the expected response as an arbitrary linear function of fixed and time-varying covariates so that mean-value functions can be derived from subject matter considerations rather than methodological constraints. Three families of models for the covariance function are discussed: multivariate, autoregressive, and random effects. Illustrations demonstrate the flexibility and utility of the proposed approach to longitudinal analysis.     

Spatial Clustering Tests Based on the Domination Number of a New Random Digraph Family

We use the domination number of a parametrized random digraph family called proportional-edge proximity catch digraphs (PCDs) for testing multivariate spatial point patterns. This digraph family is based on relative positions of data points from various classes. We extend the results on the distribution of the domination number of proportional-edge PCDs, and use the domination number as a statistic for testing segregation and association against complete spatial randomness. We demonstrate that the domination number of the PCD has binomial distribution when size of one class is fixed while the size of the other (whose points constitute the vertices of the digraph) tends to infinity and has asymptotic normality when sizes of both classes tend to infinity. We evaluate the finite sample performance of the test by Monte Carlo simulations and prove the consistency of the test under the alternatives. We find the optimal parameters for testing each of the segregation and association alternatives. Furthermore, the methodology discussed in this article is valid for data in higher dimensions also.     

Computer-Generated Change-Over Designs for Correlated Observations

Change-over designs with independently distributed errors in the model have been studied extensively in the literature. Martin and Eccleston (2001 Martin , R. J. , Eccleston , J. A. ( 2001 ). Optimal and near optimal designs for dependent observations . Statist. Applic. 3 : 101 – 116 . [Google Scholar]) gave an algorithm for the generation of efficient change-over designs when the errors are correlated. This article proposes an algorithm for the generation of efficient change-over designs for estimation of direct effects of treatments in the presence of first-order residual effects in the model and when the errors are correlated.     

Diagnostic tests for the distribution of random effects in multivariate mixed effects models

Abstract

Fourier methods are proposed for testing the distribution of random effects in classical and robust multivariate mixed effects models. The test statistics involve estimation of the characteristic function of random effects. Theoretical and computational issues are addressed while Monte Carlo results show that the new procedures compare favorably with other methods.     

Optimal allocation of time points for the random effects model

In this article the problem of the optimal selection and allocation of time points in repeated measures experiments is considered. D‐ optimal designs for linear regression models with a random intercept and first order auto‐regressive serial correlations are computed numerically and compared with designs having equally spaced time points. When the order of the polynomial is known and the serial correlations are not too small, the comparison shows that for any fixed number of repeated measures, a design with equally spaced time points is almost as efficient as the D‐ optimal design. When, however, there is no prior knowledge about the order of the underlying polynomial, the best choice in terms of efficiency is a D‐ optimal design for the highest possible relevant order of the polynomial. A design with equally‐spaced time points is the second best choice     

A boundary kernel for local polynomial regression

We define and compute a boundary kernel for local polynomial regression, We prove that the new kernel provides improvement over the existing kernels, Simulations show the improvement in finite samples.