Estimation of parameters of a polynomial model under intra class correlation structure for incomplete longitudinal data

The problem of analyzing and modeling incomplete longitudinal data arising from clinical and epidemiological studies are discussed, A method for handling arbitrarily missing observations under the intra class correlation structure and a polynomial model is developed. Explicit expressions for likelihood equations and information matrix for a second degree polynomial model are provided. The method is illustrated through an example.     

A new regression model for rates and proportions data with applications

We propose a new continuous distribution in the interval (0,1) based on the generalized odd log-logistic-G family, whose density function can be symmetrical, asymmetric, unimodal and bimodal. The new model is implemented using the gamlss packages in R. We propose an extended regression based on this distribution which includes as sub-models some important regressions. We employ a frequentist and Bayesian analysis to estimate the parameters and adopt the non-parametric and parametric bootstrap methods to obtain better efficiency of the estimators. Some simulations are conducted to verify the empirical distribution of the maximum likelihood estimators. We compare the empirical distribution of the quantile residuals with the standard normal distribution. The extended regression can give more realistic fits than other regressions in the analysis of proportional data.     

A semiparametric stochastic mixed effects model for bivariate cyclic longitudinal data

We propose a flexible semiparametric stochastic mixed effects model for bivariate cyclic longitudinal data. The model can handle either single cycle or, more generally, multiple consecutive cycle data. The approach models the mean of responses by parametric fixed effects and a smooth nonparametric function for the underlying time effects, and the relationship across the bivariate responses by a bivariate Gaussian random field and a joint distribution of random effects. The proposed model not only can model complicated individual profiles, but also allows for more flexible within-subject and between-response correlations. The fixed effects regression coefficients and the nonparametric time functions are estimated using maximum penalized likelihood, where the resulting estimator for the nonparametric time function is a cubic smoothing spline. The smoothing parameters and variance components are estimated simultaneously using restricted maximum likelihood. Simulation results show that the parameter estimates are close to the true values. The fit of the proposed model on a real bivariate longitudinal dataset of pre-menopausal women also performs well, both for a single cycle analysis and for a multiple consecutive cycle analysis. The Canadian Journal of Statistics 48: 471–498; 2020 © 2020 Statistical Society of Canada     

A sequential discrimination procedure for two almost identically shaped wrapped distributions

The way of investigating a distribution knowing its interesting properties might be often inadequate when the shapes of two distributions are almost similar. In each of these circumstances, the accurate decision about the genesis of a random sample from any of the two parent distributions will be very much ambiguous even with the availability of the existing testing procedure of the circular data. A sequential discrimination procedure has been suggested which is also invariant to the sample size. The performance of the proposed discrimination procedure has been evaluated by checking its capability of detecting the genesis of the known samples from the two identically shaped wrapped distributions.     

Additional comparisons of designs to estimate variance components in a two-way classification model

In this paper, the research of Muse and Anderson is extended to include additional comparisons of designs, featuring planned unbalance, for the estimation of variance components in a two-way cross classification model. Their results are extended to Include the following: (i) a small sample study of the original off-diagonal (OD) design and (ii) an asymptotic maximum likelihood investigation of three modifica-tions of the balanced diagonal rectangles (BD) design and one modification of the 01) design to permit the estimation of row, column, interaction and error variance components. Also a general iterative least.     

A comparison of estimation procedures for the parameters of the star model

A simulation experiment compares the accuracy and precision of three alternate estimation techniques for the parameters of the STARMA model. Maximum likelihood estimation, in most ways the "best" estimation procedure, involves a large amount of computational effort so that two approximate techniques, exact least squares and conditional maximum likelihood, are often proposed for series of moderate lengths. This simulation experiment compares the accuracy of these three estimation procedures for simulated series of various lengths, and discusses the appropriateness of the three procedures as a function of the length of the observed series.     

A statistical model of tracking

A method is proposed to model individual patterns of growth over time by linear combinations of optimally chosen weighted orthogonal vectors. The goal is to distinguish individuals who track from nontrackers. Nontrackers are defined as those who follow different, usually more complex, growth patterns than trackers. Thus, nontrackers require more vectors than do trackers in modeling their longitudinal observations. A method of specifying the class-specific vectors and individual weights is demonstrated. When the proportion of nontrackers in the population is small, a modified form of the Akaike maximum entropy criterion is used to select the number of vectors appopriate for each person and also to classify each person into a tracking category. When the proportion of nontrackers is large, the modified Akaike criterion together with scatterplots of the growth curve weights are needed to distinguish trackers from nontrackers. The apprach is illustrated with longitudinal observations of height measured in an epidemiologic survey of children.     

Resampling approaches for common intraclass correlation coefficients

In this paper, we present several resampling methods for interval estimation for the common intraclass correlation coefficients. Comparisons are made on the coverage probabilities and average lengths with confidence intervals estimated by using the generalized pivots. Most of the methods proposed in this article produce confidence intervals with better probabilities and shorter average lengths than that produced by using generalized pivots.     

Choice of the primary analysis in longitudinal clinical trials

Missing data, and the bias they can cause, are an almost ever‐present concern in clinical trials. The last observation carried forward (LOCF) approach has been frequently utilized to handle missing data in clinical trials, and is often specified in conjunction with analysis of variance (LOCF ANOVA) for the primary analysis. Considerable advances in statistical methodology, and in our ability to implement these methods, have been made in recent years. Likelihood‐based, mixed‐effects model approaches implemented under the missing at random (MAR) framework are now easy to implement, and are commonly used to analyse clinical trial data. Furthermore, such approaches are more robust to the biases from missing data, and provide better control of Type I and Type II errors than LOCF ANOVA. Empirical research and analytic proof have demonstrated that the behaviour of LOCF is uncertain, and in many situations it has not been conservative. Using LOCF as a composite measure of safety, tolerability and efficacy can lead to erroneous conclusions regarding the effectiveness of a drug. This approach also violates the fundamental basis of statistics as it involves testing an outcome that is not a physical parameter of the population, but rather a quantity that can be influenced by investigator behaviour, trial design, etc. Practice should shift away from using LOCF ANOVA as the primary analysis and focus on likelihood‐based, mixed‐effects model approaches developed under the MAR framework, with missing not at random methods used to assess robustness of the primary analysis. Copyright © 2004 John Wiley & Sons, Ltd.     

Marginal Projected Multivariate Linear Models for Clustered Angular Data

Among the diverse frameworks that have been proposed for regression analysis of angular data, the projected multivariate linear model provides a particularly appealing and tractable methodology. In this model, the observed directional responses are assumed to correspond to the angles formed by latent bivariate normal random vectors that are assumed to depend upon covariates through a linear model. This implies an angular normal distribution for the observed angles, and incorporates a regression structure through a familiar and convenient relationship. In this paper we extend this methodology to accommodate clustered data (e.g., longitudinal or repeated measures data) by formulating a marginal version of the model and basing estimation on an EM‐like algorithm in which correlation among within‐cluster responses is taken into account by incorporating a working correlation matrix into the M step. A sandwich estimator is used for the parameter estimates’ covariance matrix. The methodology is motivated and illustrated using an example involving clustered measurements of microbril angle on loblolly pine (Pinus taeda L.) Simulation studies are presented that evaluate the finite sample properties of the proposed fitting method. In addition, the relationship between within‐cluster correlation on the latent Euclidean vectors and the corresponding correlation structure for the observed angles is explored.     

A comparison of stein-like procedures for estimating linear regression models with multicollinear data

This paper compares several Stein-like estimation methods for estimating regression parameters. The criterion function was the mean-squared error of prediction and the parameter of interest was the mean of the response variable at the sampled values of the control variables. Large sample simulation techniques were used to evaluate the mean-squared error of the predictions. The parameters of interest were varied systematically over wide ranges.     

On the Convergence of the Monte Carlo Maximum Likelihood Method for Latent Variable Models

While much used in practice, latent variable models raise challenging estimation problems due to the intractability of their likelihood. Monte Carlo maximum likelihood (MCML), as proposed by Geyer & Thompson (1992 ), is a simulation-based approach to maximum likelihood approximation applicable to general latent variable models. MCML can be described as an importance sampling method in which the likelihood ratio is approximated by Monte Carlo averages of importance ratios simulated from the complete data model corresponding to an arbitrary value of the unknown parameter. This paper studies the asymptotic (in the number of observations) performance of the MCML method in the case of latent variable models with independent observations. This is in contrast with previous works on the same topic which only considered conditional convergence to the maximum likelihood estimator, for a fixed set of observations. A first important result is that when is fixed, the MCML method can only be consistent if the number of simulations grows exponentially fast with the number of observations. If on the other hand, is obtained from a consistent sequence of estimates of the unknown parameter, then the requirements on the number of simulations are shown to be much weaker.     

Likelihood inference for small variance components

The authors explore likelihood‐based methods for making inferences about the components of variance in a general normal mixed linear model. In particular, they use local asymptotic approximations to construct confidence intervals for the components of variance when the components are close to the boundary of the parameter space. In the process, they explore the question of how to profile the restricted likelihood (REML). Also, they show that general REML estimates are less likely to fall on the boundary of the parameter space than maximum‐likelihood estimates and that the likelihood‐ratio test based on the local asymptotic approximation has higher power than the likelihood‐ratio test based on the usual chi‐squared approximation. They examine the finite‐sample properties of the proposed intervals by means of a simulation study.     

Testing random effects in linear mixed models: another look at the F‐test (with discussion)

This article re‐examines the F‐test based on linear combinations of the responses, or FLC test, for testing random effects in linear mixed models. In current statistical practice, the FLC test is underused and we argue that it should be reconsidered as a valuable method for use with linear mixed models. We present a new, more general derivation of the FLC test which applies to a broad class of linear mixed models where the random effects can be correlated. We highlight three advantages of the FLC test that are often overlooked in modern applications of linear mixed models, namely its computation speed, its generality, and its exactness as a test. Empirical studies provide new insight into the finite sample performance of the FLC test, identifying cases where it is competitive or even outperforms modern methods in terms of power, as well as settings in which it performs worse than simulation‐based methods for testing random effects. In all circumstances, the FLC test is faster to compute.     

Comparison of designs for the three-fold nested random model

The quality of estimation of variance components depends on the design used as well as on the unknown values of the variance components. In this article, three designs are compared, namely, the balanced, staggered, and inverted nested designs for the three-fold nested random model. The comparison is based on the so-called quantile dispersion graphs using analysis of variance (ANOVA) and maximum likelihood (ML) estimates of the variance components. It is demonstrated that the staggered nested design gives more stable estimates of the variance component for the highest nesting factor than the balanced design. The reverse, however, is true in case of lower nested factors. A comparison between ANOVA and ML estimation of the variance components is also made using each of the aforementioned designs.     

Missing data techniques for multilevel data: implications of model misspecification

When modeling multilevel data, it is important to accurately represent the interdependence of observations within clusters. Ignoring data clustering may result in parameter misestimation. However, it is not well established to what degree parameter estimates are affected by model misspecification when applying missing data techniques (MDTs) to incomplete multilevel data. We compare the performance of three MDTs with incomplete hierarchical data. We consider the impact of imputation model misspecification on the quality of parameter estimates by employing multiple imputation under assumptions of a normal model (MI/NM) with two-level cross-sectional data when values are missing at random on the dependent variable at rates of 10%, 30%, and 50%. Five criteria are used to compare estimates from MI/NM to estimates from MI assuming a linear mixed model (MI/LMM) and maximum likelihood estimation to the same incomplete data sets. With 10% missing data (MD), techniques performed similarly for fixed-effects estimates, but variance components were biased with MI/NM. Effects of model misspecification worsened at higher rates of MD, with the hierarchical structure of the data markedly underrepresented by biased variance component estimates. MI/LMM and maximum likelihood provided generally accurate and unbiased parameter estimates but performance was negatively affected by increased rates of MD.