A robust biplot

This paper introduces a robust biplot which is related to multivariate M-estimates. The n × p data matrix is first considered as a sample of size n from some p-variate population, and robust M-estimates of the population location vector and scatter matrix are calculated. In the construction of the biplot, each row of the data matrix is assigned a weight determined in the preliminary robust estimation. In a robust biplot, one can plot the variables in order to represent characteristics of the robust variance-covariance matrix: the length of the vector representing a variable is proportional to its robust standard deviation, while the cosine of the angle between two variables is approximately equal to their robust correlation. The proposed biplot also permits a meaningful representation of the variables in a robust principal-component analysis. The discrepancies between least-squares and robust biplots are illustrated in an example.     

Robust multivariate mixture regression models with incomplete data

Multivariate mixture regression models can be used to investigate the relationships between two or more response variables and a set of predictor variables by taking into consideration unobserved population heterogeneity. It is common to take multivariate normal distributions as mixing components, but this mixing model is sensitive to heavy-tailed errors and outliers. Although normal mixture models can approximate any distribution in principle, the number of components needed to account for heavy-tailed distributions can be very large. Mixture regression models based on the multivariate t distributions can be considered as a robust alternative approach. Missing data are inevitable in many situations and parameter estimates could be biased if the missing values are not handled properly. In this paper, we propose a multivariate t mixture regression model with missing information to model heterogeneity in regression function in the presence of outliers and missing values. Along with the robust parameter estimation, our proposed method can be used for (i) visualization of the partial correlation between response variables across latent classes and heterogeneous regressions, and (ii) outlier detection and robust clustering even under the presence of missing values. We also propose a multivariate t mixture regression model using MM-estimation with missing information that is robust to high-leverage outliers. The proposed methodologies are illustrated through simulation studies and real data analysis.     

Compositional biplots including external non-compositional variables

Biplots represent a widely used statistical tool for visualizing the resulting loadings and scores of a dimension reduction technique applied to multivariate data. If the underlying data carry only relative information (i.e. compositional data expressed in proportions, mg/kg, etc.) they have to be pre-processed with a logratio transformation before the dimension reduction is carried out. In the context of principal component analysis, the resulting biplot is called compositional biplot. We introduce an alternative, the ilr biplot, which is based on a special choice of orthonormal coordinates resulting from an isometric logratio (ilr) transformation. This allows to incorporate also external non-compositional variables, and to study the relations to the compositional variables. The methodology is demonstrated on real data sets.     

Dynamic approach to linear statistical calibration with an application in microwave radiometry

The problem of statistical calibration of a measuring instrument can be framed both in a statistical context as well as in an engineering context. In the first, the problem is dealt with by distinguishing between the ‘classical’ approach and the ‘inverse’ regression approach. Both of these models are static models and are used to estimate exact measurements from measurements that are affected by error. In the engineering context, the variables of interest are considered to be taken at the time at which you observe it. The Bayesian time series analysis method of Dynamic Linear Models can be used to monitor the evolution of the measures, thus introducing a dynamic approach to statistical calibration. The research presented employs a new approach to performing statistical calibration. A simulation study in the context of microwave radiometry is conducted that compares the dynamic model to traditional static frequentist and Bayesian approaches. The focus of the study is to understand how well the dynamic statistical calibration method performs under various signal-to-noise ratios, r.     

The construction of a partial least-squares biplot

Biplots are useful tools to explore the relationship among variables. In this paper, the specific regression relationship between a set of predictors X and set of response variables Y by means of partial least-squares (PLS) regression is represented. The PLS biplot provides a single graphical representation of the samples together with the predictor and response variables, as well as their interrelationships in terms of the matrix of regression coefficients.     

Frequency of Selecting Noise Variables in Subset Regression Analysis: A Simulation Study

This article presents the results of a simulation study of variable selection in a multiple regression context that evaluates the frequency of selecting noise variables and the bias of the adjusted R ² of the selected variables when some of the candidate variables are authentic. It is demonstrated that for most samples a large percentage of the selected variables is noise, particularly when the number of candidate variables is large relative to the number of observations. The adjusted R ² of the selected variables is highly inflated.     

Extension of biplot methodology to multivariate regression analysis

At the core of multivariate statistics is the investigation of relationships between different sets of variables. More precisely, the inter-variable relationships and the causal relationships. The latter is a regression problem, where one set of variables is referred to as the response variables and the other set of variables as the predictor variables. In this situation, the effect of the predictors on the response variables is revealed through the regression coefficients. Results from the resulting regression analysis can be viewed graphically using the biplot. The consequential biplot provides a single graphical representation of the samples together with the predictor variables and response variables. In addition, their effect in terms of the regression coefficients can be visualized, although sub-optimally, in the said biplot.KEYWORDS: Biplot, regression analysis, multivariate regression, rank approximation     

Standard and robust orthogonal regression

A fast routine for converting regression algorithms into corresponding orthogonal regression (OR) algorithms was introduced in Ammann and Van Ness (1988). The present paper discusses the properties of various ordinary and robust OR procedures created using this routine. OR minimizes the sum of the orthogonal distances from the regression plane to the data points. OR has three types of applications. First, L ₂ OR is the maximum likelihood solution of the Gaussian errors-in-variables (EV) regression problem. This L ₂ solution is unstable, thus the robust OR algorithms created from robust regression algorithms should prove very useful. Secondly, OR is intimately related to principal components analysis. Therefore, the routine can also be used to create L ₁, robust, etc. principal components algorithms. Thirdly, OR treats the x and y variables symmetrically which is important in many modeling problems. Using Monte Carlo studies this paper compares the performance of standard regression, robust regression, OR, and robust OR on Gaussian EV data, contaminated Gaussian EV data, heavy-tailed EV data, and contaminated heavy-tailed EV data.     

Semiparametric Regression Analysis of Longitudinal Skewed Data

In this paper, we develop a semiparametric regression model for longitudinal skewed data. In the new model, we allow the transformation function and the baseline function to be unknown. The proposed model can provide a much broader class of models than the existing additive and multiplicative models. Our estimators for regression parameters, transformation function and baseline function are asymptotically normal. Particularly, the estimator for the transformation function converges to its true value at the rate n ^{? 1 ∕ 2}, the convergence rate that one could expect for a parametric model. In simulation studies, we demonstrate that the proposed semiparametric method is robust with little loss of efficiency. Finally, we apply the new method to a study on longitudinal health care costs.     

On regression modelling with dummy variables versus separate regressions per group: Comment on Holgersson et al.

In a recent issue of this journal, Holgersson et al. [Dummy variables vs. category-wise models, J. Appl. Stat. 41(2) (2014), pp. 233–241, doi:10.1080/02664763.2013.838665] compared the use of dummy coding in regression analysis to the use of category-wise models (i.e. estimating separate regression models for each group) with respect to estimating and testing group differences in intercept and in slope. They presented three objections against the use of dummy variables in a single regression equation, which could be overcome by the category-wise approach. In this note, I first comment on each of these three objections and next draw attention to some other issues in comparing these two approaches. This commentary further clarifies the differences and similarities between dummy variable and category-wise approaches.     

Multicovariate-adjusted regression models

We introduce multicovariate-adjusted regression (MCAR), an adjustment method for regression analysis, where both the response (Y) and predictors (X ₁, …, X _p ) are not directly observed. The available data have been contaminated by unknown functions of a set of observable distorting covariates, Z ₁, …, Z _s , in a multiplicative fashion. The proposed method substantially extends the current contaminated regression modelling capability, by allowing for multiple distorting covariate effects. MCAR is a flexible generalisation of the recently proposed covariate-adjusted regression method, an effective adjustment method in the presence of a single covariate, Z. For MCAR estimation, we establish a connection between the MCAR models and adaptive varying coefficient models. This connection leads to an adaptation of a hybrid backfitting estimation algorithm. Extensive simulations are used to study the performance and limitations of the proposed iterative estimation algorithm. In particular, the bias and mean square error of the proposed MCAR estimators are examined, relative to a baseline and a consistent benchmark estimator. The method is also illustrated with a Pima Indian diabetes data set, where the response and predictors are potentially contaminated by body mass index and triceps skin fold thickness. Both distorting covariates measure aspects of obesity, an important risk factor in type 2 diabetes.     

Partial logistic relevance vector machines in survival analysis

The use of relevance vector machines to flexibly model hazard rate functions is explored. This technique is adapted to survival analysis problems through the partial logistic approach. The method exploits the Bayesian automatic relevance determination procedure to obtain sparse solutions and it incorporates the flexibility of kernel-based models. Example results are presented on literature data from a head-and-neck cancer survival study using Gaussian and spline kernels. Sensitivity analysis is conducted to assess the influence of hyperprior distribution parameters. The proposed method is then contrasted with other flexible hazard regression methods, in particular the HARE model proposed by Kooperberg et al. [16]. A simulation study is conducted to carry out the comparison. The model developed in this paper exhibited good performance in the prediction of hazard rate. The application of this sparse Bayesian technique to a real cancer data set demonstrated that the proposed method can potentially reveal characteristics of the hazards, associated with the dynamics of the studied diseases, which may be missed by existing modeling approaches based on different perspectives on the bias vs. variance balance.     

A Robust Variable Selection to t-type Joint Generalized Linear Models via Penalized t-type Pseudo-likelihood

Although the t-type estimator is a kind of M-estimator with scale optimization, it has some advantages over the M-estimator. In this article, we first propose a t-type joint generalized linear model as a robust extension to the classical joint generalized linear models for modeling data containing extreme or outlying observations. Next, we develop a t-type pseudo-likelihood (TPL) approach, which can be viewed as a robust version to the existing pseudo-likelihood (PL) approach. To determine which variables significantly affect the variance of the response variable, we then propose a unified penalized maximum TPL method to simultaneously select significant variables for the mean and dispersion models in t-type joint generalized linear models. Thus, the proposed variable selection method can simultaneously perform parameter estimation and variable selection in the mean and dispersion models. With appropriate selection of the tuning parameters, we establish the consistency and the oracle property of the regularized estimators. Simulation studies are conducted to illustrate the proposed methods.     

A semiparametric nonlinear mixed model approach to phase I profile monitoring

When process data follow a particular curve in quality control, profile monitoring is suitable and appropriate for assessing process stability. Previous research in profile monitoring focusing on nonlinear parametric (P) modeling, involving both fixed and random-effects, was made under the assumption of an accurate nonlinear model specification. Lately, nonparametric (NP) methods have been used in the profile monitoring context in the absence of an obvious linear P model. This study introduces a novel technique in profile monitoring for any nonlinear and auto-correlated data. Referred to as the nonlinear mixed robust profile monitoring (NMRPM) method, it proposes a semiparametric (SP) approach that combines nonlinear P and NP profile fits for scenarios in which a nonlinear P model is adequate over part of the data but inadequate of the rest. These three methods (P, NP, and NMRPM) account for the auto-correlation within profiles and treats the collection of profiles as a random sample with a common population. During Phase I analysis, a version of Hotelling’s T² statistic is proposed for each approach to identify abnormal profiles based on the estimated random effects and obtain the corresponding control limits. The performance of the NMRPM method is then evaluated using a real data set. Results reveal that the NMRPM method is robust to model misspecification and performs adequately against a correctly specified nonlinear P model. Control charts with the NMRPM method have excellent capability of detecting changes in Phase I data with control limits that are easily computable.     

Small area estimation of the mean using non-parametric M-quantile regression: a comparison when a linear mixed model does not hold

The demand for reliable statistics in subpopulations, when only reduced sample sizes are available, has promoted the development of small area estimation methods. In particular, an approach that is now widely used is based on the seminal work by Battese et al. [An error-components model for prediction of county crop areas using survey and satellite data, J. Am. Statist. Assoc. 83 (1988), pp. 28–36] that uses linear mixed models (MM). We investigate alternatives when a linear MM does not hold because, on one side, linearity may not be assumed and/or, on the other, normality of the random effects may not be assumed. In particular, Opsomer et al. [Nonparametric small area estimation using penalized spline regression, J. R. Statist. Soc. Ser. B 70 (2008), pp. 265–283] propose an estimator that extends the linear MM approach to the case in which a linear relationship may not be assumed using penalized splines regression. From a very different perspective, Chambers and Tzavidis [M-quantile models for small area estimation, Biometrika 93 (2006), pp. 255–268] have recently proposed an approach for small-area estimation that is based on M-quantile (MQ) regression. This allows for models robust to outliers and to distributional assumptions on the errors and the area effects. However, when the functional form of the relationship between the qth MQ and the covariates is not linear, it can lead to biased estimates of the small area parameters. Pratesi et al. [Semiparametric M-quantile regression for estimating the proportion of acidic lakes in 8-digit HUCs of the Northeastern US, Environmetrics 19(7) (2008), pp. 687–701] apply an extended version of this approach for the estimation of the small area distribution function using a non-parametric specification of the conditional MQ of the response variable given the covariates [M. Pratesi, M.G. Ranalli, and N. Salvati, Nonparametric m-quantile regression using penalized splines, J. Nonparametric Stat. 21 (2009), pp. 287–304]. We will derive the small area estimator of the mean under this model, together with its mean-squared error estimator and compare its performance to the other estimators via simulations on both real and simulated data.     

Minimal Markov Basis for Tests of Main Effect Models for 2p-1 Fractional Factorial Designs of Resolution p

We consider conditional exact tests of factor effects in design of experiments for discrete response variables. Similarly to the analysis of contingency tables, Markov chain Monte Carlo methods can be used to perform exact tests, especially when large-sample approximations of the null distributions are poor and the enumeration of the conditional sample space is infeasible. In order to construct a connected Markov chain over the appropriate sample space, one approach is to compute a Markov basis. Theoretically, a Markov basis can be characterized as a generator of a well-specified toric ideal in a polynomial ring and is computed by computational algebraic software. However, the computation of a Markov basis sometimes becomes infeasible, even for problems of moderate sizes. In the present article, we obtain the closed-form expression of minimal Markov bases for the main effect models of 2^{p ? 1} fractional factorial designs of resolution p.     

Comparing variances and means when distributions have non-identical shapes

The paper compares several methods for computing robust 1-α confidence intervals for σ ₁ ²-σ ₂ ², or σ ₁ ²/σ ₂ ², where σ ₁ ² and σ ₂ ² are the population variances corresponding to two independent treatment groups. The emphasis is on a Box-Scheffe approach when distributions have different shapes, and so the results reported here have implications about comparing means. The main result is that for unequal sample sizes, a Box-Scheffe approach can be considerably less robust than indicated by past investigations. Several other procedures for comparing variances, not based on a Box-Scheffe approach, were also examined and found to be highly unsatisfactory although previously published papers found them to be robust when the distributions have identical shapes. Included is a new result on why the procedures examined here are not robust, and an illustration that increasing σ ₁ ²-σ ₂ ² can reduce power in certain situations. Constants needed to apply Dunnett’s robust comparison of means are included.     

Bayesian mismeasurement t-models for censored responses

This paper aims at introducing a Bayesian robust error-in-variable regression model in which the dependent variable is censored. We extend previous works by assuming a multivariate t distribution for jointly modelling the behaviour of the errors and the latent explanatory variable. Inference is done under the Bayesian paradigm. We use a data augmentation approach and develop a Markov chain Monte Carlo algorithm to sample from the posterior distributions. We run a Monte Carlo study to evaluate the efficiency of the posterior estimators in different settings. We compare the proposed model to three other models previously discussed in the literature. As a by-product we also provide a Bayesian analysis of the t-tobit model. We fit all four models to analyse the 2001 Medical Expenditure Panel Survey data.     

Finite-population prediction under error-in-variables superpopulation models

The prediction problem in finite populations is considered under error-in-variables super population models. The models considered are the usual regression models involving at most two variables, x and y, where both may be measured with error. Properties of some classical predictors are investigated. A Bayesian approach is proposed.     

Analysing truncated data with semiparametric transformation models

Left-truncation often arises when patient information, such as time of diagnosis, is gathered retrospectively. In some cases, the distribution function, say G(x), of left-truncated variables can be parameterized as G(x; θ), where θ∈Θ?R^q and θ is a q-dimensional vector. Under semiparametric transformation models, we demonstrated that the approach of Chen et al. (Semiparametric analysis of transformation models with censored data. Biometrika. 2002;89:659–668) can be used to analyse this type of data. The asymptotic properties of the proposed estimators are derived. A simulation study is conducted to investigate the performance of the proposed estimators.