Comparisons among regression estimators under the generalized mean square error criterion

It is not always prossible to establish a preference ordering among regression estimators in terms of the generalized mean square error criterion. In the paper, we determine when it is feasible to use this criteion to couduct comparisons among ordinary least squares, principal components, ridge regression, and shrunken least squares estimators.     

Component selection norms for principal components regression

Multicollinearity or near exact linear dependence among the vectors of regressor variables in a multiple linear regression analysis can have important effects on the quality of least squares parameter estimates. One frequently suggested approach for these problems is principal components regression. This paper investigates alternative variable selection procedures and their implications for such an analysis.     

Dimensionality reduction approach to multivariate prediction

The authors consider dimensionality reduction methods used for prediction, such as reduced rank regression, principal component regression and partial least squares. They show how it is possible to obtain intermediate solutions by estimating simultaneously the latent variables for the predictors and for the responses. They obtain a continuum of solutions that goes from reduced rank regression to principal component regression via maximum likelihood and least squares estimation. Different solutions are compared using simulated and real data.     

Regression methods for high dimensional multicollinear data

To compare their performance on high dimensional data, several regression methods are applied to data sets in which the number of exploratory variables greatly exceeds the sample sizes. The methods are stepwise regression, principal components regression, two forms of latent root regression, partial least squares, and a new method developed here. The data are four sample sets for which near infrared reflectance spectra have been determined and the regression methods use the spectra to estimate the concentration of various chemical constituents, the latter having been determined by standard chemical analysis. Thirty-two regression equations are estimated using each method and their performances are evaluated using validation data sets. Although it is the most widely used, stepwise regression was decidedly poorer than the other methods considered. Differences between the latter were small with partial least squares performing slightly better than other methods under all criteria examined, albeit not by a statistically significant amount.     

Evaluation of different statistical methods using SAS software: an in silico approach for analysis of real-time PCR data

Real-time polymerase chain reaction (PCR) is reliable quantitative technique in gene expression studies. The statistical analysis of real-time PCR data is quite crucial for results analysis and explanation. The statistical procedures of analyzing real-time PCR data try to determine the slope of regression line and calculate the reaction efficiency. Applications of mathematical functions have been used to calculate the target gene relative to the reference gene(s). Moreover, these statistical techniques compare C_t (threshold cycle) numbers between control and treatments group. There are many different procedures in SAS for real-time PCR data evaluation. In this study, the efficiency of calibrated model and delta delta C_t model have been statistically tested and explained. Several methods were tested to compare control with treatment means of C_t. The methods tested included t-test (parametric test), Wilcoxon test (non-parametric test) and multiple regression. Results showed that applied methods led to similar results and no significant difference was observed between results of gene expression measurement by the relative method.     

An asymptotic theory for semiparametric generalized least squares estimation in partially linear regression models

Consider a partially linear regression model with an unknown vector parameter β, an unknown functiong(·), and unknown heteroscedastic error variances. In this paper we develop an asymptotic semiparametric generalized least squares estimation theory under some weak moment conditions. These moment conditions are satisfied by many of the error distributions encountered in practice, and our theory does not require the number of replications to go to infinity.     

Latent root regression: a biased regression methodology for use with collinear predictor variables

Many different biased regression techniques have been proposed for estimating parameters of a multiple linear regression model when the predictor variables are collinear. One particular alternative, latent root regression analysis, is a technique based on analyzing the latent roots and latent vectors of the correlation matrix of both the response and the predictor variables. It is the purpose of this paper to review the latent root regression estimator and to re-examine some of its properties and applications. It is shown that the latent root estimator is a member of a wider class of estimators for linear models     

P-spline Estimation of Functional Classification Methods for Improving the Quality in the Food Industry

The aim of this article is to improve the quality of cookies production by classifying them as good or bad from the curves of resistance of dough observed during the kneading process. As the predictor variable is functional, functional classification methodologies such as functional logit regression and functional discriminant analysis are considered. A P-spline approximation of the sample curves is proposed to improve the classification ability of these models and to suitably estimate the relationship between the quality of cookies and the resistance of dough. Inference results on the functional parameters and related odds ratios are obtained using the asymptotic normality of the maximum likelihood estimators under the classical regularity conditions. Finally, the classification results are compared with alternative functional data analysis approaches such as componentwise classification on the logit regression model.     

An efficient procedure for computing minque of variance components and generalized least squares estimates of fixed effects

An efficient method for computing minimum norm quadratic unbiased estimates (MINQUE) of variance components and generalized least squares estimates of the fixed effects in the mixed model is developed. The computing algorithm uses a modification of the W transformation.     

Partial Least Squares: A First-order Analysis

We compare the partial least squares (PLS) and the principal component analysis (PCA), in a general case in which the existence of a true linear regression is not assumed. We prove under mild conditions that PLS and PCA are equivalent, to within a first-order approximation, hence providing a theoretical explanation for empirical findings reported by other researchers. Next, we assume the existence of a true linear regression equation and obtain asymptotic formulas for the bias and variance of the PLS parameter estimator     

Compound Regression and Constrained Regression: Nonparametric Regression Frameworks for EIV Models

Abstract

Errors-in-variable (EIV) regression is often used to gauge linear relationship between two variables both suffering from measurement and other errors, such as, the comparison of two measurement platforms (e.g., RNA sequencing vs. microarray). Scientists are often at a loss as to which EIV regression model to use for there are infinite many choices. We provide sound guidelines toward viable solutions to this dilemma by introducing two general nonparametric EIV regression frameworks: the compound regression and the constrained regression. It is shown that these approaches are equivalent to each other and, to the general parametric structural modeling approach. The advantages of these methods lie in their intuitive geometric representations, their distribution free nature, and their ability to offer candidate solutions with various optimal properties when the ratio of the error variances is unknown. Each includes the classic nonparametric regression methods of ordinary least squares, geometric mean regression (GMR), and orthogonal regression as special cases. Under these general frameworks, one can readily uncover some surprising optimal properties of the GMR, and truly comprehend the benefit of data normalization. Supplementary materials for this article are available online.     

Regression for overdetermined systems: A fisheries example

In trying to establish the relationship between a yearly fisheries recruitment series and meteorological or oceanographic variables such as air pressure or sea surface temperature, we are often faced with the situation where the number of regressors exceeds the number of observations. In this paper we use the techniques of penalized least squares and principal-components regression to determine whether air pressure over the North Atlantic can be used to predict two North Atlantic cod recruitment series. The results suggest that penalized least squares can be very effective in these situations.     

Group screening method for the statistical analysis of -optimal mixed-level supersaturated designs

In this paper, we propose the application of group screening methods for analyzing data using E(f_NOD)-optimal mixed-level supersaturated designs possessing the equal occurrence property. Supersaturated designs are a large class of factorial designs which can be used for screening out the important factors from a large set of potentially active variables. The huge advantage of these designs is that they reduce the experimental cost drastically, but their critical disadvantage is the high degree of confounding among factorial effects. Based on the idea of the group screening methods, the f factors are sub-divided into g “group-factors”. The “group-factors” are then studied using the penalized likelihood statistical analysis methods at a factorial design with orthogonal or near-orthogonal columns. All factors in groups found to have a large effect are then studied in a second stage of experiments. A comparison of the Type I and Type II error rates of various estimation methods via simulation experiments is performed. The results are presented in tables and discussion follows.     

Sensitivity analysis and related analyses: A review of some statistical techniques

This paper reviews five related types of analysis, namely (i) sensitivity or what-if analysis, (ii) uncertainty or risk analysis, (iii) screening, (iv) validation, and (v) optimization. The main questions are: when should which type of analysis be applied; which statistical techniques may then be used? This paper claims that the proper sequence to follow in the evaluation of simulation models is as follows. 1) Validation, in which the availability of data on the real system determines which type of statistical technique to use for validation. 2) Screening: in the simulation‘s pilot phase the really important inputs can be identified through a novel technique, called sequential bifurcation, which uses aggregation and sequential experimentation. 3) Sensitivity analysis: the really important inputs should be subjected to a more detailed analysis, which includes interactions between these inputs; relevant statistical techniques are design of experiments (DOE) and regression analysis. 4) Uncertainty analysis: the important environmental inputs may have values that are not precisely known, so the uncertainties of the model outputs that result from the uncertainties in these model inputs should be quantified; relevant techniques are the Monte Carlo method and Latin hypercube sampling. 5) Optimization: the policy variables should be controlled; a relevant technique is Response Surface Methodology (RSM), which combines DOE, regression analysis, and steepest-ascent hill-climbing. The recommended sequence implies that sensitivity analysis procede uncertainty analysis. Several case studies for each phase are briefly discussed in this paper.     

Regression imputation with Q-mode clustering for rounded zero replacement in high-dimensional compositional data

The logratio methodology is not applicable when rounded zeros occur in compositional data. There are many methods to deal with rounded zeros. However, some methods are not suitable for analyzing data sets with high dimensionality. Recently, related methods have been developed, but they cannot balance the calculation time and accuracy. For further improvement, we propose a method based on regression imputation with Q-mode clustering. This method forms the groups of parts and builds partial least squares regression with these groups using centered logratio coordinates. We also prove that using centered logratio coordinates or isometric logratio coordinates in the response of partial least squares regression have the equivalent results for the replacement of rounded zeros. Simulation study and real example are conducted to analyze the performance of the proposed method. The results show that the proposed method can reduce the calculation time in higher dimensions and improve the quality of results.     

ADE-4: a multivariate analysis and graphical display software

We present ADE-4, a multivariate analysis and graphical display software. Multivariate analysis methods available in ADE-4 include usual one-table methods like principal component analysis and correspondence analysis, spatial data analysis methods (using a total variance decomposition into local and global components, analogous to Moran and Geary indices), discriminant analysis and within/between groups analyses, many linear regression methods including lowess and polynomial regression, multiple and PLS (partial least squares) regression and orthogonal regression (principal component regression), projection methods like principal component analysis on instrumental variables, canonical correspondence analysis and many other variants, coinertia analysis and the RLQ method, and several three-way table (k-table) analysis methods. Graphical display techniques include an automatic collection of elementary graphics corresponding to groups of rows or to columns in the data table, thus providing a very efficient way for automatic k-table graphics and geographical mapping options. A dynamic graphic module allows interactive operations like searching, zooming, selection of points, and display of data values on factor maps. The user interface is simple and homogeneous among all the programs; this contributes to making the use of ADE-4 very easy for non- specialists in statistics, data analysis or computer science.     

Model Reduction for Prediction in Regression Models

We look at prediction in regression models under squared loss for the random x case with many explanatory variables. Model reduction is done by conditioning upon only a small number of linear combinations of the original variables. The corresponding reduced model will then essentially be the population model for the chemometricians' partial least squares algorithm. Estimation of the selection matrix under this model is briefly discussed, and analoguous results for the case with multivariate response are formulated. Finally, it is shown that an assumption of multinormality may be weakened to assuming elliptically symmetric distribution, and that some of the results are valid without any distributional assumption at all.     

Embedding latent class regression and latent class distal outcome models into cluster-weighted latent class analysis: a detailed simulation experiment

Usually in latent class (LC) analysis, external predictors are taken to be cluster conditional probability predictors (LC models with external predictors), and/or score conditional probability predictors (LC regression models). In such cases, their distribution is not of interest. Class-specific distribution is of interest in the distal outcome model, when the distribution of the external variables is assumed to depend on LC membership. In this paper, we consider a more general formulation, that embeds both the LC regression and the distal outcome models, as is typically done in cluster-weighted modelling. This allows us to investigate (1) whether the distribution of the external variables differs across classes, (2) whether there are significant direct effects of the external variables on the indicators, by modelling jointly the relationship between the external and the latent variables. We show the advantages of the proposed modelling approach through a set of artificial examples, an extensive simulation study and an empirical application about psychological contracts among employees and employers in Belgium and the Netherlands.     

Model selection criteria for reduced rank multivariate time series: a simulation study

The main focus of our paper is to compare the performance of different model selection criteria used for multivariate reduced rank time series. We consider one of the most commonly used reduced rank model, that is, the reduced rank vector autoregression (RRVAR (p, r)) introduced by Velu et al. [Reduced rank models for multiple time series. Biometrika. 1986;7(31):105–118]. In our study, the most popular model selection criteria are included. The criteria are divided into two groups, that is, simultaneous selection and two-step selection criteria, accordingly. Methods from the former group select both an autoregressive order p and a rank r simultaneously, while in the case of two-step criteria, first an optimal order p is chosen (using model selection criteria intended for the unrestricted VAR model) and then an optimal rank r of coefficient matrices is selected (e.g. by means of sequential testing). Considered model selection criteria include well-known information criteria (such as Akaike information criterion, Schwarz criterion, Hannan–Quinn criterion, etc.) as well as widely used sequential tests (e.g. the Bartlett test) and the bootstrap method. An extensive simulation study is carried out in order to investigate the efficiency of all model selection criteria included in our study. The analysis takes into account 34 methods, including 6 simultaneous methods and 28 two-step approaches, accordingly. In order to carefully analyse how different factors affect performance of model selection criteria, we consider over 150 simulation settings. In particular, we investigate the influence of the following factors: time series dimension, different covariance structure, different level of correlation among components and different level of noise (variance). Moreover, we analyse the prediction accuracy concerned with the application of the RRVAR model and compare it with results obtained for the unrestricted vector autoregression. In this paper, we also present a real data application of model selection criteria for the RRVAR model using the Polish macroeconomic time series data observed in the period 1997–2007.