A Comparison of Analysis of Covariate-Adjusted Residuals and Analysis of Covariance

Various methods to control the influence of a covariate on a response variable are compared. These methods are ANOVA with or without homogeneity of variances (HOV) of errors and Kruskal–Wallis (K–W) tests on (covariate-adjusted) residuals and analysis of covariance (ANCOVA). Covariate-adjusted residuals are obtained from the overall regression line fit to the entire data set ignoring the treatment levels or factors. It is demonstrated that the methods on covariate-adjusted residuals are only appropriate when the regression lines are parallel and covariate means are equal for all treatments. Empirical size and power performance of the methods are compared by extensive Monte Carlo simulations. We manipulated the conditions such as assumptions of normality and HOV, sample size, and clustering of the covariates. The parametric methods on residuals and ANCOVA exhibited similar size and power when error terms have symmetric distributions with variances having the same functional form for each treatment, and covariates have uniform distributions within the same interval for each treatment. In such cases, parametric tests have higher power compared to the K–W test on residuals. When error terms have asymmetric distributions or have variances that are heterogeneous with different functional forms for each treatment, the tests are liberal with K–W test having higher power than others. The methods on covariate-adjusted residuals are severely affected by the clustering of the covariates relative to the treatment factors when covariate means are very different for treatments. For data clusters, ANCOVA method exhibits the appropriate level. However, such a clustering might suggest dependence between the covariates and the treatment factors, so makes ANCOVA less reliable as well.     

Linear Signed-Rank Tests for Paired Survival Data Subject to a Common Censoring Time

In this paper we consider rank-based tests for paired survival data, in which pair members are subject to the same right censoring time. Linear signed-rank tests have already been developed for the two-treatment problem in which pair members receive the opposite treatments. Assuming a bivariate accelerated failure time model, we extend this class of linear signed-rank tests to the case of multiple covariates, making this methodology applicable to more complicated experimental designs. These tests can be reformulated as weighted sums of contigency table measures, giving an alternative method of computation and intuitive view of how these tests work. A simulation study of their small-sample performance relative to other tests demonstrates that the linear signed-rank tests have greater power in cases of moderately to highly correlated data.     

A multivariate zero-inflated binomial model for the analysis of correlated proportional data

In this paper, a new multivariate zero-inflated binomial (MZIB) distribution is proposed to analyse the correlated proportional data with excessive zeros. The distributional properties of purposed model are studied. The Fisher scoring algorithm and EM algorithm are given for the computation of estimates of parameters in the proposed MZIB model with/without covariates. The score tests and the likelihood ratio tests are derived for assessing both the zero-inflation and the equality of multiple binomial probabilities in correlated proportional data. A limited simulation study is performed to evaluate the performance of derived EM algorithms for the estimation of parameters in the model with/without covariates and to compare the nominal levels and powers of both score tests and likelihood ratio tests. The whitefly data is used to illustrate the proposed methodologies.     

Bayesian analysis of covariance under inverse Gaussian model

This paper considers the problem of analysis of covariance (ANCOVA) under the assumption of inverse Gaussian distribution for response variable from the Bayesian point of view. We develop a fully Bayesian model for ANCOVA based on the conjugate prior distributions for parameters contained in the model. The Bayes estimator of parameters, ANCOVA model and adjusted effects for both treatments and covariates along with predictive distribution of future observations are developed. We also provide the essentials for comparing adjusted treatments effects and adjusted factor effects. A simulation study and a real world application are also performed to illustrate and evaluate the proposed Bayesian model.     

Joint GEEs for multivariate correlated data with incomplete binary outcomes

This study considers a fully-parametric but uncongenial multiple imputation (MI) inference to jointly analyze incomplete binary response variables observed in a correlated data settings. Multiple imputation model is specified as a fully-parametric model based on a multivariate extension of mixed-effects models. Dichotomized imputed datasets are then analyzed using joint GEE models where covariates are associated with the marginal mean of responses with response-specific regression coefficients and a Kronecker product is accommodated for cluster-specific correlation structure for a given response variable and correlation structure between multiple response variables. The validity of the proposed MI-based JGEE (MI-JGEE) approach is assessed through a Monte Carlo simulation study under different scenarios. The simulation results, which are evaluated in terms of bias, mean-squared error, and coverage rate, show that MI-JGEE has promising inferential properties even when the underlying multiple imputation is misspecified. Finally, Adolescent Alcohol Prevention Trial data are used for illustration.     

Some comments on a class of simultaneous inference procedures in ancova

Methods for constructing simultaneous confidence intervals for contrasts of treatment effects in analysis of covariance (ANCOVA) models are discussed and compared. A simple procedure is given by which, under a normality assumption made on the covariates, any method appropriate in an analysis of variance (ANOVA) may be applied to a corresponding ANCOVA by means of a simple adjustment of the required critical values. Some properties of this procedure are noted.     

Fast accelerated failure time modeling for case-cohort data

Semiparametric accelerated failure time (AFT) models directly relate the expected failure times to covariates and are a useful alternative to models that work on the hazard function or the survival function. For case-cohort data, much less development has been done with AFT models. In addition to the missing covariates outside of the sub-cohort in controls, challenges from AFT model inferences with full cohort are retained. The regression parameter estimator is hard to compute because the most widely used rank-based estimating equations are not smooth. Further, its variance depends on the unspecified error distribution, and most methods rely on computationally intensive bootstrap to estimate it. We propose fast rank-based inference procedures for AFT models, applying recent methodological advances to the context of case-cohort data. Parameters are estimated with an induced smoothing approach that smooths the estimating functions and facilitates the numerical solution. Variance estimators are obtained through efficient resampling methods for nonsmooth estimating functions that avoids full blown bootstrap. Simulation studies suggest that the recommended procedure provides fast and valid inferences among several competing procedures. Application to a tumor study demonstrates the utility of the proposed method in routine data analysis.     

Model checks for nonparametric regression with missing data: a comparative study

ABSTRACT

This paper analyses the behaviour of the goodness-of-fit tests for regression models. To this end, it uses statistics based on an estimation of the integrated regression function with missing observations either in the response variable or in some of the covariates. It proposes several versions of one empirical process, constructed from a previous estimation, that uses only the complete observations or replaces the missing observations with imputed values. In the case of missing covariates, a link model is used to fill the missing observations with other complete covariates. In all the situations, Bootstrap methodology is used to calibrate the distribution of the test statistics. A broad simulation study compares the different procedures based on empirical regression methodology, with smoothed tests previously studied in the literature. The comparison reflects the effect of the correlation between the covariates in the tests based on the imputed sample for missing covariates. In addition, the paper proposes a computational binning strategy to evaluate the tests based on an empirical process for large data sets. Finally, two applications to real data illustrate the performance of the tests.     

Mixed Correlated Bivariate Ordinal and Negative Binomial Longitudinal Responses with Nonignorable Missing Values

Regression models with random effects are proposed for joint analysis of negative binomial and ordinal longitudinal data with nonignorable missing values under fully parametric framework. The presented model simultaneously considers a multivariate probit regression model for the missing mechanisms, which provides the ability of examining the missing data assumptions and a multivariate mixed model for the responses. Random effects are used to take into account the correlation between longitudinal responses of the same individual. A full likelihood-based approach that allows yielding maximum likelihood estimates of the model parameters is used. The model is applied to a medical data, obtained from an observational study on women, where the correlated responses are the ordinal response of osteoporosis of the spine and negative binomial response is the number of joint damage. A sensitivity of the results to the assumptions is also investigated. The effect of some covariates on all responses are investigated simultaneously.     

Feature Screening for Ultrahigh Dimensional Categorical Data With Applications

Ultrahigh dimensional data with both categorical responses and categorical covariates are frequently encountered in the analysis of big data, for which feature screening has become an indispensable statistical tool. We propose a Pearson chi-square based feature screening procedure for categorical response with ultrahigh dimensional categorical covariates. The proposed procedure can be directly applied for detection of important interaction effects. We further show that the proposed procedure possesses screening consistency property in the terminology of Fan and Lv (2008). We investigate the finite sample performance of the proposed procedure by Monte Carlo simulation studies and illustrate the proposed method by two empirical datasets.     

A class of one sample tests based on the Mann–Whitney–Wilcoxon functional

The main idea behind the proposed class of tests is rooted on an extension of the technique used in the derivation of the Mann–Whitney–Wilcoxon test. Just like the case of two-sample rank-based tests, the new class consists of tests defined through score functions. When properly selected, these score functions lead to consistent and often more powerful tests compared with classical goodness-of-fit tests. Theoretical results are supported by an extensive simulation study.     

Gaussian mixture analysis of covariance

ABSTRACT

In many real-world applications, the traditional theory of analysis of covariance (ANCOVA) leads to inadequate and unreliable results because of violation of the response variable observations from the essential Gaussian assumption that may be due to the heterogeneity of population, the presence of outlier or both of them. In this paper, we develop a Gaussian mixture ANCOVA model for modelling heterogeneous populations with a finite number of subpopulation. We provide the maximum likelihood estimates of the model parameters via an EM algorithm. We also drive the adjusted effects estimators for treatments and covariates. The Fisher information matrix of the model and asymptotic confidence intervals for the parameter are also discussed. We performed a simulation study to assess the performance of the proposed model. A real-world example is also worked out to explained the methodology.     

Weighting for item non-response in attitude scales by using latent variable models with covariates

We discuss the use of latent variable models with observed covariates for computing response propensities for sample respondents. A response propensity score is often used to weight item and unit responders to account for item and unit non-response and to obtain adjusted means and proportions. In the context of attitude scaling, we discuss computing response propensity scores by using latent variable models for binary or nominal polytomous manifest items with covariates. Our models allow the response propensity scores to be found for several different items without refitting. They allow any pattern of missing responses for the items. If one prefers, it is possible to estimate population proportions directly from the latent variable models, so avoiding the use of propensity scores. Artificial data sets and a real data set extracted from the 1996 British Social Attitudes Survey are used to compare the various methods proposed.     

Rank-based group variable selection

A robust rank-based estimator for variable selection in linear models, with grouped predictors, is studied. The proposed estimation procedure extends the existing rank-based variable selection [Johnson, B.A., and Peng, L. (2008), ‘Rank-based Variable Selection’, Journal of Nonparametric Statistics, 20(3):241–252] and the ww-scad [Wang, L., and Li, R. (2009), ‘Weighted Wilcoxon-type Smoothly Clipped Absolute Deviation Method’, Biometrics, 65(2):564–571] to linear regression models with grouped variables. The resulting estimator is robust to contamination or deviations in both the response and the design space.The Oracle property and asymptotic normality of the estimator are established under some regularity conditions. Simulation studies reveal that the proposed method performs better than the existing rank-based methods [Johnson, B.A., and Peng, L. (2008), ‘Rank-based Variable Selection’, Journal of Nonparametric Statistics, 20(3):241–252; Wang, L., and Li, R. (2009), ‘Weighted Wilcoxon-type Smoothly Clipped Absolute Deviation Method’, Biometrics, 65(2):564–571] for grouped variables models. This estimation procedure also outperforms the adaptive hlasso [Zhou, N., and Zhu, J. (2010), ‘Group Variable Selection Via a Hierarchical Lasso and its Oracle Property’, Interface, 3(4):557–574] in the presence of local contamination in the design space or for heavy-tailed error distribution.     

Goodness-of-fit tests for general linear models with covariates missed at random

In this paper, we consider a model checking problem for general linear models with randomly missing covariates. Two types of score type tests with inverse probability weight, which is estimated by parameter and nonparameter methods respectively, are proposed to this goodness of fit problem. The asymptotic properties of the test statistics are developed under the null and local alternative hypothesis. Simulation study is carried out to present the performance of the sizes and powers of the tests. We illustrate the proposed method with a data set on monozygotic twins.     

Improving precision and power in randomized trials with a two-stage study design: Stratification using clustering method

In a randomized controlled trial (RCT), it is possible to improve precision and power and reduce sample size by appropriately adjusting for baseline covariates. There are multiple statistical methods to adjust for prognostic baseline covariates, such as an ANCOVA method. In this paper, we propose a clustering-based stratification method for adjusting for the prognostic baseline covariates. Clusters (strata) are formed only based on prognostic baseline covariates, not outcome data nor treatment assignment. Therefore, the clustering procedure can be completed prior to the availability of outcome data. The treatment effect is estimated in each cluster, and the overall treatment effect is derived by combining all cluster-specific treatment effect estimates. The proposed implementation of the procedure is described. Simulations studies and an example are presented.     

Robust surface estimation in multi-response multistage statistical optimization problems

As the ordinary least squares (OLS) method is very sensitive to outliers as well as to correlated responses, a robust coefficient estimation method is proposed in this paper for multi-response surfaces in multistage processes based on M-estimators. In this approach, experimental designs are used in which the intermediate response variables may act as covariates in the next stages. The performances of both the ordinary multivariate OLS and the proposed robust multi-response surface approach are analyzed and compared through extensive simulation experiments. Sum of the squared errors in estimating the regression coefficients reveals the efficiency of the proposed robust approach.     

Nonparametric Inference for Time-Varying Coefficient Quantile Regression

The article considers nonparametric inference for quantile regression models with time-varying coefficients. The errors and covariates of the regression are assumed to belong to a general class of locally stationary processes and are allowed to be cross-dependent. Simultaneous confidence tubes (SCTs) and integrated squared difference tests (ISDTs) are proposed for simultaneous nonparametric inference of the latter models with asymptotically correct coverage probabilities and Type I error rates. Our methodologies are shown to possess certain asymptotically optimal properties. Furthermore, we propose an information criterion that performs consistent model selection for nonparametric quantile regression models of nonstationary time series. For implementation, a wild bootstrap procedure is proposed, which is shown to be robust to the dependent and nonstationary data structure. Our method is applied to studying the asymmetric and time-varying dynamic structures of the U.S. unemployment rate since the 1940s. Supplementary materials for this article are available online.     

A Bayesian random effects model for analyzing mixed negative binomial and continuous longitudinal responses

ABSTRACT

A general Bayesian random effects model for analyzing longitudinal mixed correlated continuous and negative binomial responses with and without missing data is presented. This Bayesian model, given some random effects, uses a normal distribution for the continuous response and a negative binomial distribution for the count response. A Markov Chain Monte Carlo sampling algorithm is described for estimating the posterior distribution of the parameters. This Bayesian model is illustrated by a simulation study. For sensitivity analysis to investigate the change of parameter estimates with respect to the perturbation from missing at random to not missing at random assumption, the use of posterior curvature is proposed. The model is applied to a medical data, obtained from an observational study on women, where the correlated responses are the negative binomial response of joint damage and continuous response of body mass index. The simultaneous effects of some covariates on both responses are also investigated.