Seemingly unrelated regression tree

Nonparametric seemingly unrelated regression provides a powerful alternative to parametric seemingly unrelated regression for relaxing the linearity assumption. The existing methods are limited, particularly with sharp changes in the relationship between the predictor variables and the corresponding response variable. We propose a new nonparametric method for seemingly unrelated regression, which adopts a tree-structured regression framework, has satisfiable prediction accuracy and interpretability, no restriction on the inclusion of categorical variables, and is less vulnerable to the curse of dimensionality. Moreover, an important feature is constructing a unified tree-structured model for multivariate data, even though the predictor variables corresponding to the response variable are entirely different. This unified model can offer revelatory insights such as underlying economic meaning. We propose the key factors of tree-structured regression, which are an impurity function detecting complex nonlinear relationships between the predictor variables and the response variable, split rule selection with negligible selection bias, and tree size determination solving underfitting and overfitting problems. We demonstrate our proposed method using simulated data and illustrate it using data from the Korea stock exchange sector indices.     

Goodness-of-Fit Statistics for General Linear Regression Equations in the Presence of Replicated Responses

Goodness-of-fit statistics for general multiple-linear-regression equations are reviewed for the case of replicated responses. A modification of the coefficient of determination is recommended. This statistic has 1.0 as its achievable upper bound and has the coefficient of determination as a special case. It indicates more effectively how close a general-linear-regression equation is relative to the best possible one and is particularly useful when the purpose is to ascertain whether higher-order terms of a given set of explanatory variables are required. Other goodness-of-fit statistics that take into account the variation within replicated responses are reviewed. An illustration example is presented.     

Variable Selection in Semiparametric Quantile Modeling for Longitudinal Data

We propose a penalized quantile regression for partially linear varying coefficient (VC) model with longitudinal data to select relevant non parametric and parametric components simultaneously. Selection consistency and oracle property are established. Furthermore, if linear part and VC part are unknown, we propose a new unified method, which can do three types of selections: separation of varying and constant effects, selection of relevant variables, and it can be carried out conveniently in one step. Consistency in the three types of selections and oracle property in estimation are established as well. Simulation studies and real data analysis also confirm our method.     

A simulation comparison of estimators for a regression coefficient under differential non-response

We consider the estimation of a regression coefficient in a linear regression when observations are missing due to nonresponse. Response is assumed to be determined by a nonobservable variable which is linearly related to an observable variable. The values of the observable variable are assumed to be available for the whole sample but the variable is not includsd in the regression relationship of interest . Several alternative estimators have been proposed for this situation under various simplifying assumptions. A sampling theory approach provides three alternative estimatrs by considering the observatins as obtained from a sub-sample, selected on the basis of the fully observable variable , as formulated by Nathan and Holt (1980). Under an econometric approach, Heckman (1979) proposed a two-stage (probit and OLS) estimator which is consistent under specificconditions. A simulation comparison of the four estimators and the ordinary least squares estimator , under multivariate normality of all the variables involved, indicates that the econometric approach estimator is not robust to departures from the conditions underlying its derivation, while two of the other estimators exhibit a similar degree of stable performance over a wide range of conditions. Simulations for a non-normal distribution show that gains in performance can be obtained if observations on the independent variable are available for the whole population.     

Quality of Fit Measures in the Framework of Quantile Regression

Abstract. In regression experiments, to learn about the strength of the relationship between a covariate vector and a dependent variable, we propose a ‘coefficient of determination’ based on the quantiles. Such a coefficient is a ‘local’ measure in the sense that the strength is measured at a prespecified quantile level. Once estimated, it can be used, for example, to measure the relative importance of a subset of covariates in the quantile regression context. Related to this coefficient, we also propose a new ‘local’ lack‐of‐fit measure of a given parametric model. We provide some asymptotic results of the proposed measures and carry out a Monte Carlo simulation study to illustrate their use and performance in practice.     

An alternative approach to variable selection for prediction

A procedure for selecting a subset of predictor variables in regression analysis is suggested. The procedure is so designed that it leads to the selection of a subset of variables having an adequate degree of informativeness with a directly specified confidence coefficient. Some examples are considered to illustrate the application of the procedure.     

Variable selection in gamma regression models via artificial bee colony algorithm

Variable selection is an important task in regression analysis. Performance of the statistical model highly depends on the determination of the subset of predictors. There are several methods to select most relevant variables to construct a good model. However in practice, the dependent variable may have positive continuous values and not normally distributed. In such situations, gamma distribution is more suitable than normal for building a regression model. This paper introduces an heuristic approach to perform variable selection using artificial bee colony optimization for gamma regression models. We evaluated the proposed method against with classical selection methods such as backward and stepwise. Both simulation studies and real data set examples proved the accuracy of our selection procedure.     

The Sample Coefficient of Determination in Simple Linear Regression

The behavior of the sample coefficient of determination is examined for some arrangements of independent variable values in a simple linear regression with normally distributed error terms. Numerical values of means and standard deviations are presented that provide some insight into the influence of range and arrangement of independent variable values and sample size on the size of the sample coefficient of determination. Some asymptotic results are given.     

Localized classification

The main problem with localized discriminant techniques is the curse of dimensionality, which seems to restrict their use to the case of few variables. However, if localization is combined with a reduction of dimension the initial number of variables is less restricted. In particular it is shown that localization yields powerful classifiers even in higher dimensions if localization is combined with locally adaptive selection of predictors. A robust localized logistic regression (LLR) method is developed for which all tuning parameters are chosen data-adaptively. In an extended simulation study we evaluate the potential of the proposed procedure for various types of data and compare it to other classification procedures. In addition we demonstrate that automatic choice of localization, predictor selection and penalty parameters based on cross validation is working well. Finally the method is applied to real data sets and its real world performance is compared to alternative procedures.     

Krylov Sequences as a Tool for Analysing Iterated Regression Algorithms

Abstract.  We use Krylov sequences to analyse a class of regression methods based on successive identification of latent factors. Some results already proved for partial least squares regression (PLSR) are shown to hold for other methods also. We prove that the well-known peculiar pattern of alternating shrinkage and inflation of the principal components is not unique for PLSR. We also show that for any method in the class under study, the coefficient of determination is always at least as high as for principal components regression with the same number of factors.     

On variable selection in generalized linear and related regression models

This paper is concerned with selection of explanatory variables in generalized linear models (GLM). The class of GLM's is quite large and contains e.g. the ordinary linear regression, the binary logistic regression, the probit model and Poisson regression with linear or log-linear parameter structure. We show that, through an approximation of the log likelihood and a certain data transformation, the variable selection problem in a GLM can be converted into variable selection in an ordinary (unweighted) linear regression model. As a consequence no specific computer software for variable selection in GLM's is needed. Instead, some suitable variable selection program for linear regression can be used. We also present a simulation study which shows that the log likelihood approximation is very good in many practical situations. Finally, we mention briefly possible extensions to regression models outside the class of GLM's.     

Asymptotic distribution of regression correlation coefficient for Poisson regression model

This study treats an asymptotic distribution for measures of predictive power for generalized linear models (GLMs). We focus on the regression correlation coefficient (RCC) that is one of the measures of predictive power. The RCC, proposed by Zheng and Agresti is a population value and a generalization of the population value for the coefficient of determination. Therefore, the RCC is easy to interpret and familiar. Recently, Takahashi and Kurosawa provided an explicit form of the RCC and proposed a new RCC estimator for a Poisson regression model. They also showed the validity of the new estimator compared with other estimators. This study discusses the new statistical properties of the RCC for the Poisson regression model. Furthermore, we show an asymptotic normality of the RCC estimator.     

Hierarchically penalized quantile regression

In many regression problems, predictors are naturally grouped. For example, when a set of dummy variables is used to represent categorical variables, or a set of basis functions of continuous variables is included in the predictor set, it is important to carry out a feature selection both at the group level and at individual variable levels within the group simultaneously. To incorporate the group and variables within-group information into a regularized model fitting, several regularization methods have been developed, including the Cox regression and the conditional mean regression. Complementary to earlier works, the simultaneous group and within-group variables selection method is examined in quantile regression. We propose a hierarchically penalized quantile regression, and show that the hierarchical penalty possesses the oracle property in quantile regression, as well as in the Cox regression. The proposed method is evaluated through simulation studies and a real data application.     

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.     

Model selection for logistic regression via association rules analysis

Interaction is very common in reality, but has received little attention in logistic regression literature. This is especially true for higher-order interactions. In conventional logistic regression, interactions are typically ignored. We propose a model selection procedure by implementing an association rules analysis. We do this by (1) exploring the combinations of input variables which have significant impacts to response (via association rules analysis); (2) selecting the potential (low- and high-order) interactions; (3) converting these potential interactions into new dummy variables; and (4) performing variable selections among all the input variables and the newly created dummy variables (interactions) to build up the optimal logistic regression model. Our model selection procedure establishes the optimal combination of main effects and potential interactions. The comparisons are made through thorough simulations. It is shown that the proposed method outperforms the existing methods in all cases. A real-life example is discussed in detail to demonstrate the proposed method.     

Adaptive varying-coefficient linear models

Summary. Varying-coefficient linear models arise from multivariate nonparametric regression, non-linear time series modelling and forecasting, functional data analysis, longitudinal data analysis and others. It has been a common practice to assume that the varying coefficients are functions of a given variable, which is often called an index . To enlarge the modelling capacity substantially, this paper explores a class of varying-coefficient linear models in which the index is unknown and is estimated as a linear combination of regressors and/or other variables. We search for the index such that the derived varying-coefficient model provides the least squares approximation to the underlying unknown multidimensional regression function. The search is implemented through a newly proposed hybrid backfitting algorithm. The core of the algorithm is the alternating iteration between estimating the index through a one-step scheme and estimating coefficient functions through one-dimensional local linear smoothing. The locally significant variables are selected in terms of a combined use of the t -statistic and the Akaike information criterion. We further extend the algorithm for models with two indices. Simulation shows that the methodology proposed has appreciable flexibility to model complex multivariate non-linear structure and is practically feasible with average modern computers. The methods are further illustrated through the Canadian mink–muskrat data in 1925–1994 and the pound–dollar exchange rates in 1974–1983.     

Analysis of Poisson varying-coefficient models with autoregression

In the regression analysis of time series of event counts, it is of interest to account for serial dependence that is likely to be present among such data as well as a nonlinear interaction between the expected event counts and predictors as a function of some underlying variables. We thus develop a Poisson autoregressive varying-coefficient model, which introduces autocorrelation through a latent process and allows regression coefficients to nonparametrically vary as a function of the underlying variables. The nonparametric functions for varying regression coefficients are estimated with data-driven basis selection, thereby avoiding overfitting and adapting to curvature variation. An efficient posterior sampling scheme is devised to analyse the proposed model. The proposed methodology is illustrated using simulated data and daily homicide data in Cali, Colombia.     

A Semiparametric Regression Model for Longitudinal Data with Non‐stationary Errors

Motivated by the need to analyze the National Longitudinal Surveys data, we propose a new semiparametric longitudinal mean‐covariance model in which the effects on dependent variable of some explanatory variables are linear and others are non‐linear, while the within‐subject correlations are modelled by a non‐stationary autoregressive error structure. We develop an estimation machinery based on least squares technique by approximating non‐parametric functions via B‐spline expansions and establish the asymptotic normality of parametric estimators as well as the rate of convergence for the non‐parametric estimators. We further advocate a new model selection strategy in the varying‐coefficient model framework, for distinguishing whether a component is significant and subsequently whether it is linear or non‐linear. Besides, the proposed method can also be employed for identifying the true order of lagged terms consistently. Monte Carlo studies are conducted to examine the finite sample performance of our approach, and an application of real data is also illustrated.     

The estimation of R 2 and adjusted R 2 in incomplete data sets using multiple imputation

The coefficient of determination, known also as the R ², is a common measure in regression analysis. Many scientists use the R ² and the adjusted R ² on a regular basis. In most cases, the researchers treat the coefficient of determination as an index of ‘usefulness’ or ‘goodness of fit,’ and in some cases, they even treat it as a model selection tool. In cases in which the data is incomplete, most researchers and common statistical software will use complete case analysis in order to estimate the R ², a procedure that might lead to biased results. In this paper, I introduce the use of multiple imputation for the estimation of R ² and adjusted R ² in incomplete data sets. I illustrate my methodology using a biomedical example.     

Quantile-adaptive variable screening in ultra-high dimensional varying coefficient models

The varying-coefficient model is an important nonparametric statistical model since it allows appreciable flexibility on the structure of fitted model. For ultra-high dimensional heterogeneous data it is very necessary to examine how the effects of covariates vary with exposure variables at different quantile level of interest. In this paper, we extended the marginal screening methods to examine and select variables by ranking a measure of nonparametric marginal contributions of each covariate given the exposure variable. Spline approximations are employed to model marginal effects and select the set of active variables in quantile-adaptive framework. This ensures the sure screening property in quantile-adaptive varying-coefficient model. Numerical studies demonstrate that the proposed procedure works well for heteroscedastic data.