Subset Of Explanatory Variables Research Articles

Prediction by Regression in Multivariate Normal Distributions

In the context of the multivariate Normal regression model, a mean squared error of prediction is developed for making the choice of subset of explanatory variables for predicting the response variable in future samples.

Stem volume of tropical forests from polarimetric radar

In this study, we investigated the potential of polarimetric synthetic aperture radar (PolSAR) data for the estimation of stem volume in tropical forests. We used calibrated L-band, high incidence angle data from the airborne system SAR-R99B, acquired over an experimental area in the Tapajós National Forest, Pará, Brazil. To evaluate the potential of PolSAR data for this application we used regression analysis, in which first-order models were fit to predict stem volume per hectare, as determined from field measurements. Unlike previous studies in tropical forests, the set of potential explanatory variables included a series of PolSAR attributes based on phase information, in addition to power measurements. Model selection techniques based on coefficient of determination (R 2) and mean square error (MSE) identified several useful subsets of explanatory variables for stem volume estimation, including backscattering coefficient in HH polarization, cross-polarized ratio, HH-VV phase difference, polarimetric coherence, and the volume scatter component of the Freeman decomposition. Evaluation of the selected models indicated that PolSAR data can be used to quantify stem volume in the study site with a root mean square error (RMSE) of about 20–29 m3 ha−1, corresponding to 8–12% of the mean stem volume. External validation using independent data showed average prediction errors of less than 14%. Saturation effects in measured versus modelled volume were not observed up to volumes of 308 m3 ha−1 (biomasses of ∼357 Mg ha−1). However, no formal assessment of saturation was possible due to limitations of the volume range of the dataset.

The Regional Settlement Patterns of Immigrants to Sweden 1967-2005

The aim of this study is to analyse the initial allocation of immigrants to Sweden 1967-2005 by age in a regional perspective. Vacancies, unemployment and labour market participation as well as geographical areas with many previous immigrants are included in the analysis as pull-factors. A multivariate cross-section OLS regression model will be used to estimate the relative initial regional distribution of immigrants in Sweden in 1967, 1975, 1990 and 2005 by age. The chosen method enables to control for a subset of explanatory variables and examine the effect of a selected independent variable when estimating the regional pull-factors to immigration. This study uses data collected from Statistics Sweden (SCB) and the National Labour Market Board (AMS). The data used is regional macro data, which does not contain information on single individuals.The overall conclusion is that previous immigrants had an impact on age groups in their initial regional settlement pattern in 1967 - 2005. Few of the other independent variables show any coefficients which are statistically separated from zero.

Selection of Working Correlation Structure and Best Model in GEE Analyses of Longitudinal Data

The Generalized Estimating Equations (GEE) method is one of the most commonly used statistical methods for the analysis of longitudinal data in epidemiological studies. A working correlation structure for the repeated measures of the outcome variable of a subject needs to be specified by this method. However, statistical criteria for selecting the best correlation structure and the best subset of explanatory variables in GEE are only available recently because the GEE method is developed on the basis of quasi-likelihood theory. Maximum likelihood based model selection methods, such as the widely used Akaike Information Criterion (AIC), are not applicable to GEE directly. Pan (2001) proposed a selection method called QIC which can be used to select the best correlation structure and the best subset of explanatory variables. Based on the QIC method, we developed a computing program to calculate the QIC value for a range of different distributions, link functions and correlation structures. This program was written in Stata software. In this article, we introduce this program and demonstrate how to use it to select the most parsimonious model in GEE analyses of longitudinal data through several representative examples.

Information Theory and Mixing Least-Squares Regressions

For Gaussian regression, we develop and analyze methods for combining estimators from various models. For squared-error loss, an unbiased estimator of the risk of the mixture of general estimators is developed. Special attention is given to the case that the component estimators are least-squares projections into arbitrary linear subspaces, such as those spanned by subsets of explanatory variables in a given design. We relate the unbiased estimate of the risk of the mixture estimator to estimates of the risks achieved by the components. This results in simple and accurate bounds on the risk and its estimate, in the form of sharp and exact oracle inequalities. That is, without advance knowledge of which model is best, the resulting performance is comparable to or perhaps even superior to what is achieved by the best of the individual models. Furthermore, in the case that the unknown parameter has a sparse representation, our mixture estimator adapts to the underlying sparsity. Simulations show that the performance of these mixture estimators is better than that of a related model-selection estimator which picks a model with the highest weight. Also, the connection between our mixtures with Bayes procedures is discussed

NONPARAMETRIC SIGNIFICANCE TESTING

A procedure for testing the significance of a subset of explanatory variables in a nonparametric regression is proposed. Our test statistic uses the kernel method. Under the null hypothesis of no effect of the variables under test, we show that our test statistic has an nhp2/2 standard normal limiting distribution, where p2 is the dimension of the complete set of regressors. Our test is one-sided, consistent against all alternatives and detects local alternatives approaching the null at rate slower than n−1/2h−p2/4. Our Monte-Carlo experiments indicate that it outperforms the test proposed by Fan and Li (1996, Econometrica 64, 865–890).

A Multilevel Thresholds of Change Model for Analysis of Stages of Change Data

In this article we describe a model for multilevel ordinal response data that allows for non- proportional odds for a subset of explanatory variables. As applied to stages of change data, which are commonly encountered in health promotion research, this model is termed the multilevel thresholds of change model since it focuses on modeling the K-I thresholds that delineate membership in the K ordered stages. Explanatory variables can have the same effect across thresholds (i.e., proportional odds) or varying effects across thresholds (i.e., non- proportional odds). In addition to the explanatory variables of the model, random effects are included to account for the multilevel structure of the data (e.g., repeated observations within subjects, or subjects observed within clusters). A maximum marginal likelihood (MML) solution is described using Gauss-Hermite quadrature to numerically integrate over the distribution of normally-distributed random effects. Data from a skin cancer prevention study, in which subjects were repeatedly measured across time and clustered within schools, are used to illustrate the multilevel thresholds of change model.

A PREDICTIVE APPROACH TO MODEL SELECTION AND MULTICOLLINEARITY

We argue for the adoption of a predictive approach to model specification. Specifically, we derive the difference between means and the ratio of determinants of covariance matrices when a subset of explanatory variables is included or excluded from a regression. Results for an economic application are presented as an example. © 1997 by John Wiley & Sons, Ltd.

A PREDICTIVE APPROACH TO MODEL SELECTION AND MULTICOLLINEARITY

We argue for the adoption of a predictive approach to model specification. Specifically, we derive the difference between means and the ratio of determinants of covariance matrices when a subset of explanatory variables is included or excluded from a regression. For several special cases these measures are shown to be related to widely used tools for studying model specification. Results for a set of simulated data and for two economic applications are presented as examples.

A Modified Akaike Criterion for Model Choice in Generalized Linear Models

summary. In Bonneu (1988) a prediction criterion for model selection is defined for Generalized Linear Models (GLM). This criterion, similar to AIC (Akaike, H. (1973)), Sakamoto, Y. and Akaike, H. (1978)) is based on the expected Kullback-Leibler discrepancy (Kullback, S. 1959)), especially defined for prediction. The model selection strategy is employed to select a Nelder's link function (1972) and a small subset of explanatory variables. Data are observed from an experimental design with unequal numbers of replications. This paper deals with the asymptotic estimate of this prediction criterion and compares it with a simulated bootstrap estimate. Usually asymptotic criteria for model selection can be written as the sum of a statistic and a bias (see Linhart, H. and Volkers, P. (1984); Linhart, H. and Zucchini, W. (1986)). In the present paper, asymptotic arguments are investigated in a different way, taking into account the prediction objective and GLM framework with unequal numbers of replicates. Our as...

Model choice for prediction in generalized linear models

In generalized linear models, a prediction criterion is defined similar to AKAIKE'S information criterion (AIC), see AKAIKE (1973), based on the expectation of the KULL- BACK-LEIBLER information measure ((cf. KULLBACK ( 1959 )) of an estimated pseudo den¬sity and the true density. This criterion chooses a pseudo prediction model which is determined by a subset of explanatory variables and NELDER'S link function (cf. NELDER and WEDDEBUEN ( 1972 )). The estiniatioii of this criterion leads to the definition of a statistic which has an analvtic expression in the gaussiao linear mooch as in the r>aper of BUNKE and DROGE ( 1984a, 1984b ) while in other cases, a simulated bootstrap estimate can be found, As an example, in the case of models of quanta! response which generalize the logistic model, see PREGIBON ( 1980 ), some data are analysed using the GLIM computer program

Robustness Aspects of Score Tests for Generalized Linear and Partially Linear Regression Models

Robustness aspects of score tests for the null hypothesis of no z effect on the mean response μ for some given subset of explanatory variables, z, are explored in the context of regression analysis. The model considered here is θ = θ(μ) = something else + f(β1z′), where θ is a given transformation of μ via some link function, and f, representing the form of the z effect on θ, is monotonic and twice differentiable. It is shown that the score test for H 0: β1 = 0 remains unchanged whatever the form chosen for f under the alternative, H 1: β1 ≠ 0. This type of robustness is desirable in testing how z might affect the mean response, when precise knowledge of the form for f under H 1 is lacking. Further robust score tests against extreme observations, either in the rank-test sense or in the Hampe–Huber sense, can be achieved for the location family with additive errors, as is pointed out in one of the examples, by replacing in the tests the score vector based on the observations by that based on the ranks or s...

Robustness Aspects of Score Tests for Generalized Linear and Partially Linear Regression Models

Robustness aspects of score tests for the null hypothesis of no z effect on the mean response μ for some given subset of explanatory variables, z, are explored in the context of regression analysis. The model considered here is θ = θ(μ) = something else + f(β1z′), where θ is a given transformation of μ via some link function, and f, representing the form of the z effect on θ, is monotonic and twice differentiable. It is shown that the score test for H 0: β1 = 0 remains unchanged whatever the form chosen for f under the alternative, H 1: β1 ≠ 0. This type of robustness is desirable in testing how z might affect the mean response, when precise knowledge of the form for f under H 1 is lacking. Further robust score tests against extreme observations, either in the rank-test sense or in the Hampe–Huber sense, can be achieved for the location family with additive errors, as is pointed out in one of the examples, by replacing in the tests the score vector based on the observations by that based on the ranks or some suitable ψ function. Finally, the simplicity of score tests in seemingly complicated practical situations, a consequence. of their robustness, is demonstrated via testing host variability in dilution experiments.

Use of Orthogonal Factors for Selection of Variables in a Regression Equation-An Illustration

SUMMARY Selection of explanatory variables in the regression equation has been a prime problem in constructing a prediction equation. This paper describes and gives an illustration of a selection technique which makes use of the orthogonality among factors extracted from the correlation matrix. Using the factors not as new variables, but merely as the reference frame, we can identify a near orthogonal subset of explanatory variables. It is indicated that this approach provides the model builder with the flexibility that is not available in the conventional, purely mechanical, selection methods. SELECTION of explanatory variables in multiple regression analysis has been a prime problem in the analysis of unplanned data. The interdependency among the explanatory variables makes it difficult to determine empirically the contribution of each independent variable to the observed variation of the dependent variable. Various alternative selection techniques have been proposed, but the criterion employed in each technique seems quite arbitrary and is known to provide different solutions for the same problem (Draper and Smith, 1966, p. 163). Attempts to include more variables in the equation are often frustrated by near singularity of the normal equation system, and make estimates of regression coefficients highly sensitive to small changes in the original data. Resulting equations often contain coefficients with theoretically incorrect signs restricting the use of the equation as a functional relationship explaining the system under study. This paper describes an approach to the problem of selection of variables in regression analysis. This method purports to come up with a prediction equation in accordance with the principle of parsimony in terms of minimum interdependency among variables. The technique can be viewed as a use of the principal components regression proposed by Kendall (1957). In the procedure to be described, however, we make use of orthogonality among components, using the components not as new variables but merely as the reference frame to identify a near orthogonal subset of explanatory variables. Selection of such variables minimizes overlapping of information supplied by explanatory variables in the regression.