Alternative modeling techniques for the quantal response data in mixture experiments

Mixture experiments are commonly encountered in many fields including chemical, pharmaceutical and consumer product industries. Due to their wide applications, mixture experiments, a special study of response surface methodology, have been given greater attention in both model building and determination of designs compared with other experimental studies. In this paper, some new approaches are suggested on model building and selection for the analysis of the data in mixture experiments by using a special generalized linear models, logistic regression model, proposed by Chen et al. [7]. Generally, the special mixture models, which do not have a constant term, are highly affected by collinearity in modeling the mixture experiments. For this reason, in order to alleviate the undesired effects of collinearity in the analysis of mixture experiments with logistic regression, a new mixture model is defined with an alternative ratio variable. The deviance analysis table is given for standard mixture polynomial models defined by transformations and special mixture models used as linear predictors. The effects of components on the response in the restricted experimental region are given by using an alternative representation of Cox's direction approach. In addition, odds ratio and the confidence intervals of odds ratio are identified according to the chosen reference and control groups. To compare the suggested models, some model selection criteria, graphical odds ratio and the confidence intervals of the odds ratio are used. The advantage of the suggested approaches is illustrated on tumor incidence data set.     

Beta Regression for Modelling Rates and Proportions

This paper proposes a regression model where the response is beta distributed using a parameterization of the beta law that is indexed by mean and dispersion parameters. The proposed model is useful for situations where the variable of interest is continuous and restricted to the interval (0, 1) and is related to other variables through a regression structure. The regression parameters of the beta regression model are interpretable in terms of the mean of the response and, when the logit link is used, of an odds ratio, unlike the parameters of a linear regression that employs a transformed response. Estimation is performed by maximum likelihood. We provide closed-form expressions for the score function, for Fisher's information matrix and its inverse. Hypothesis testing is performed using approximations obtained from the asymptotic normality of the maximum likelihood estimator. Some diagnostic measures are introduced. Finally, practical applications that employ real data are presented and discussed.     

Logistic regression analysis of non‐randomized response data collected by the parallel model in sensitive surveys

To study the relationship between a sensitive binary response variable and a set of non‐sensitive covariates, this paper develops a hidden logistic regression to analyse non‐randomized response data collected via the parallel model originally proposed by Tian (2014). This is the first paper to employ the logistic regression analysis in the field of non‐randomized response techniques. Both the Newton–Raphson algorithm and a monotone quadratic lower bound algorithm are developed to derive the maximum likelihood estimates of the parameters of interest. In particular, the proposed logistic parallel model can be used to study the association between a sensitive binary variable and another non‐sensitive binary variable via the measure of odds ratio. Simulations are performed and a study on people's sexual practice data in the United States is used to illustrate the proposed methods.     

Improved approximation for transformation diagnostics

The suitability of a normal linear regression model may require transformation of the original response, and transformation diagnostics are designed to detect the need for such transformation. A common approach to transformation diagnostics is to construct an artificial explanatory variable, which is then tested in the augmented linear regression model for the original response. This paper describes corresponding diagnostics based directly on score statistics with accurate approximations for their standard errors. Several transformation models are covered. Some numerical illustrations are given.     

Diagnostics in elliptical regression models with stochastic restrictions applied to econometrics

We propose an influence diagnostic methodology for linear regression models with stochastic restrictions and errors following elliptically contoured distributions. We study how a perturbation may impact on the mixed estimation procedure of parameters in the model. Normal curvatures and slopes for assessing influence under usual schemes are derived, including perturbations of case-weight, response variable, and explanatory variable. Simulations are conducted to evaluate the performance of the proposed methodology. An example with real-world economy data is presented as an illustration.     

Empirical Likelihood Based Synthetic Data Method for Censored Regression Analysis

In this article, we propose a new empirical likelihood method for linear regression analysis with a right censored response variable. The method is based on the synthetic data approach for censored linear regression analysis. A log-empirical likelihood ratio test statistic for the entire regression coefficients vector is developed and we show that it converges to a standard chi-squared distribution. The proposed method can also be used to make inferences about linear combinations of the regression coefficients. Moreover, the proposed empirical likelihood ratio provides a way to combine different normal equations derived from various synthetic response variables. Maximizing this empirical likelihood ratio yields a maximum empirical likelihood estimator which is asymptotically equivalent to the solution of the estimating equation that are optimal linear combination of the original normal equations. It improves the estimation efficiency. The method is illustrated by some Monte Carlo simulation studies as well as a real example.     

Least circular distance regression for directional data

Least-squares regression is not appropriate when the response variable is circular, and can lead to erroneous results. The reason for this is that the squared difference is not an appropriate measure of distance on the circle. In this paper, a circular analog to least-squares regression is presented for predicting a circular response variable by another circular variable and a set of linear covariates. An alternative maximum-likelihood formulation yields the same regression parameter estimates. Under the maximum-likelihood model, asymptotic standard errors of the parameter estimates are obtained. As an example, the regression model is used to model data from a marine biology study.     

A note on variable selection in functional regression via random subspace method

Variable selection problem is one of the most important tasks in regression analysis, especially in a high-dimensional setting. In this paper, we study this problem in the context of scalar response functional regression model, which is a linear model with scalar response and functional regressors. The functional model can be represented by certain multiple linear regression model via basis expansions of functional variables. Based on this model and random subspace method of Mielniczuk and Teisseyre (Comput Stat Data Anal 71:725–742, 2014), two simple variable selection procedures for scalar response functional regression model are proposed. The final functional model is selected by using generalized information criteria. Monte Carlo simulation studies conducted and a real data example show very satisfactory performance of new variable selection methods under finite samples. Moreover, they suggest that considered procedures outperform solutions found in the literature in terms of correctly selected model, false discovery rate control and prediction error.     

More efficient logistic analysis using moving extreme ranked set sampling

Logistic regression is the most popular technique available for modeling dichotomous-dependent variables. It has intensive application in the field of social, medical, behavioral and public health sciences. In this paper we propose a more efficient logistic regression analysis based on moving extreme ranked set sampling (MERSS_min) scheme with ranking based on an easy-to-available auxiliary variable known to be associated with the variable of interest (response variable). The paper demonstrates that this approach will provide more powerful testing procedure as well as more efficient odds ratio and parameter estimation than using simple random sample (SRS). Theoretical derivation and simulation studies will be provided. Real data from 2011 Youth Risk Behavior Surveillance System (YRBSS) data are used to illustrate the procedures developed in this paper.     

Omitted variables in longitudinal data models

The omission of important variables is a well‐known model specification issue in regression analysis and mixed linear models. The author considers longitudinal data models that are special cases of the mixed linear models; in particular, they are linear models of repeated observations on a subject. Models of omitted variables have origins in both the econometrics and biostatistics literatures. The author describes regression coefficient estimators that are robust to and that provide the basis for detecting the influence of certain types of omitted variables. New robust estimators and omitted variable tests are introduced and illustrated with a case study that investigates the determinants of tax liability.     

Efficiently weighted estimating equations with application to proportional excess hazards

A general approach to estimation, that can lead to efficient estimation in two stages, is presented. The method will not always be available, but sufficient conditions for efficiency are provided together with four examples of its use: (1) estimation of the odds ratio in 1:M matched case-control studies with a dichotomous exposure variable; (2) estimation of the relative hazard in a two-sample survival setting; (3) estimation of the regression parameters in the proportional excess hazards model; and (4) estimation in a partly linear parametric additive hazards model. The method depends upon finding a family of weighted estimating equations, which includes a simple initial equation yielding a consistent estimate and also an equation that yields an efficient estimate, provided the optiomal weights are used.     

Fast and approximate exhaustive variable selection for generalised linear models with APES

We present APproximated Exhaustive Search (APES), which enables fast and approximated exhaustive variable selection in Generalised Linear Models (GLMs). While exhaustive variable selection remains as the gold standard in many model selection contexts, traditional exhaustive variable selection suffers from computational feasibility issues. More precisely, there is often a high cost associated with computing maximum likelihood estimates (MLE) for all subsets of GLMs. Efficient algorithms for exhaustive searches exist for linear models, most notably the leaps‐and‐bound algorithm and, more recently, the mixed integer optimisation (MIO) algorithm. The APES method learns from observational weights in a generalised linear regression super‐model and reformulates the GLM problem as a linear regression problem. In this way, APES can approximate a true exhaustive search in the original GLM space. Where exhaustive variable selection is not computationally feasible, we propose a best‐subset search, which also closely approximates a true exhaustive search. APES is made available in both as a standalone R package as well as part of the already existing mplot package.     

Direction dependence in a regression line

In this paper, we derive some simple formulae to express the association between two random variables in the case of a linear relationship, One of these representations, the cube of the correlation coefficient, is given as the ratio of the skewness of the response variable to that of the explanatory variable. This result, along with other expressions of the correlation coefficient presented in this paper, has implications for choosing the response variable in a linear regression modelling.     

Self‐modelling regression for longitudinal data with time‐invariant covariates

The authors propose the use of self‐modelling regression to analyze longitudinal data with time invariant covariates. They model the population time curve with a penalized regression spline and use a linear mixed model for transformation of the time and response scales to fit the individual curves. Fitting is done by an iterative algorithm using off‐the‐shelf linear and nonlinear mixed model software. Their method is demonstrated in a simulation study and in the analysis of tree swallow nestling growth from an experiment that includes an experimentally controlled treatment, an observational covariate and multi‐level sampling.     

A Monte Carlo EM algorithm for random-coefficient-based dropout models

Longitudinal studies of neurological disorders suffer almost inevitably from non-compliance, which is likely to be non-ignorable. It is important in these cases to model the response variable and the dropout mechanism jointly. In this article we propose a Monte Carlo version of the EM algorithm that can be used to fit random-coefficient-based dropout models. A linear mixed model is assumed for the response variable and a discrete-time proportional hazards model for the dropout mechanism; these share a common set of random coefficients. The ideas are illustrated using data from a five-year trial assessing the efficacy of two drugs in the treatment of patients in the early stages of Parkinson's disease.     

Variable Selection in Linear Mixed Models Using an Extended Class of Penalties

There is an emerging need to advance linear mixed model technology to include variable selection methods that can simultaneously choose and estimate important effects from a potentially large number of covariates. However, the complex nature of variable selection has made it difficult for it to be incorporated into mixed models. In this paper we extend the well known class of penalties and show that they can be integrated succinctly into a linear mixed model setting. Under mild conditions, the estimator obtained from this mixed model penalised likelihood is shown to be consistent and asymptotically normally distributed. A simulation study reveals that the extended family of penalties achieves varying degrees of estimator shrinkage depending on the value of one of its parameters. The simulation study also shows there is a link between the number of false positives detected and the number of true coefficients when using the same penalty. This new mixed model variable selection (MMVS) technology was applied to a complex wheat quality data set to determine significant quantitative trait loci (QTL).     

Bayesian analysis of a linear model involving structural changes in either regression parameters or disturbances precision

ABSTRACT

The present paper considers the Bayesian analysis of a linear regression model involving structural change, which may occur either due to shift in disturbances precision or due to shift in regression parameters. The posterior density for the regression parameter has been derived and posterior odds ratio for testing the hypothesis that structural change is due to shift in disturbances precision against the alternative that the change is due to shift in regression parameters has been obtained. The findings of a numerical simulation have been presented. The proposed model has been applied to RBI data set on corporate sector.     

Robust group-Lasso for functional regression model

In this article, we consider the problem of selecting functional variables using the L1 regularization in a functional linear regression model with a scalar response and functional predictors, in the presence of outliers. Since the LASSO is a special case of the penalized least-square regression with L1 penalty function, it suffers from the heavy-tailed errors and/or outliers in data. Recently, Least Absolute Deviation (LAD) and the LASSO methods have been combined (the LAD-LASSO regression method) to carry out robust parameter estimation and variable selection simultaneously for a multiple linear regression model. However, variable selection of the functional predictors based on LASSO fails since multiple parameters exist for a functional predictor. Therefore, group LASSO is used for selecting functional predictors since group LASSO selects grouped variables rather than individual variables. In this study, we propose a robust functional predictor selection method, the LAD-group LASSO, for a functional linear regression model with a scalar response and functional predictors. We illustrate the performance of the LAD-group LASSO on both simulated and real data.     

An extended sweep operator for the cross validation of variable selection in linear regression

In its application to variable selection in the linear model, cross-validation is traditionally applied to an individual model contained in a set of potential models. Each model in the set is cross-validated independently of the rest and the model with the smallest cross-validated sum of squares is selected. In such settings, an efficient algorithm for cross-validation must be able to add and to delete single points quickly from a mixed model. Recent work in variable selection has applied cross-validation to an entire process of variable selection, such as Backward Elimination or Stepwise regression (Thall, Simon and Grier, 1992). The cross-validated version of Backward Elimination, for example, divides the data into an estimation and validation set and performs a complete Backward Elimination on the estimation set, while computing the cross-validated sum of squares at each step with the validation set. After doing this process once, a different validation set is selected and the process is repeated. The final model selection is based on the cross-validated sum of squares for all Backward Eliminations. An optimal algorithm for this application of cross-validation need not be efficient in adding and deleting observations from a single model but must be efficient in computing the cross-validation sum of squares from a series of models using a common validation set. This paper explores such an algorithm based on the sweep operator.