Influence Analysis in Multivariate Restricted and Weighted Linear Regression
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The influence of least squares estimates is discussed for multivariate weighted linear regression under the restricted condition,and three diagnostic statistics as measures of influence are proposed.Several previous results are further extended and developed.Keywords:
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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.
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Part I: Foundations of Multiple Regression Analysis. Overview. Simple Linear Regression and Correlation. Regression Diagnostics. Computers and Computer Programs. Elements of Multiple Regression Analysis: Two Independent Variables. General Method of Multiple Regression Analysis: Matrix Operations. Statistical Control: Partial and Semi-Partial Correlation. Prediction. Part II: Multiple Regression Analysis. Variance Partitioning. Analysis of Effects. A Categorical Independent Variable: Dummy, Effect, And Orthogonal Coding. Multiple Categorical Independent Variables and Factorial Designs. Curvilinear Regression Analysis. Continuous and Categorical Independent Variables I: Attribute-Treatment Interaction, Comparing Regression Equations. Continuous and Categorical Independent Variables II: Analysis of Covariance. Elements of Multilevel Analysis. Categorical Dependent Variable: Logistic Regression. Part III: Structural Equation Models. Structural Equation Models with Observed Variables: Path Analysis. Structural Equation Models with Latent Variables. Part IV: Multivariate Analysis. Regression, Discriminant, And Multivariate Analysis of Variance: Two Groups. Canonical, Discriminant, And Multivariate Analysis of Variance: Extensions. Appendices.
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Regression diagnostics for the multivariate linear model are developed along the lines of the theory for the linear model with univariate response. Internally and externally studentised forms of residuals are given and their distributions found. Distance measures suitable for the assessment of the influence of particular cases on the estimated regression coefficients are considered. The examination of residuals and influence statistics is of great importance in assessing a regression model. Cook & Weisberg (1982) provide an extensive discussion of relevant methods for the linear model with a single response variable. The purpose of the present note is to apply these ideas to the multivariate linear regression problem. Although ordinary least squares estimates of regression coefficients are the same in the multivariate and univariate analyses, there are obvious reasons for carrying out the multivariate analysis to consider simultaneously the different re- sponse variables. One is the possibility that the residual for one response variable in a particular case may not seem to be out of the ordinary in relation to other residuals for that response, but only in relation to the residuals for other responses on the same case. Another is that we may be interested in specifically multivariate aspects of the data. For example, in the problem that prompted this investigation, the main item of interest was the matrix of inter-correlations between five indicators of pollution from sampling stations in the Aegean Sea. This was calculated as the matrix of correlations between the residuals from the regressions of the indicators on covariates including temperature and pH of the seawater. Correlations are particularly vulnerable to distortion by outlying values (Gnanadesikan & Kettenring, 1972), so examination of the multivariate residuals to protect against this was essential. In the following two sections, multivariate residuals and influence measures are presented. Section 4 outlines an application to illustrate the usefulness of the method-
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Abstract Multivariate regression is used to explain the relationship between p > 1 quantitative dependent variables and q quantitative explanatory variables. When a correlation structure among the dependent variables is present, a single multivariate regression is more efficient than regressions analyses for each dependent variable separately.
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Simple Linear Regression Multiple Linear Regression Regression Diagnostics: Detection of Model Violations Qualitative Variables as Predictors Transformation of Variables Weighted Least Squares The Problem of Correlated Errors Analysis of Collinear Data Biased Estimation of Regression Coefficients Variable Selection Procedures Logistic Regression Appendix References Index.
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This study investigates the geographically weighted multivariate logistic regression (GWMLR) model, parameter estimation, and hypothesis testing procedures. The GWMLR model is an extension to the multivariate logistic regression (MLR) model, which has dependent variables that follow a multinomial distribution along with parameters associated with the spatial weighting at each location in the study area. The parameter estimation was done using the maximum likelihood estimation and Newton-Raphson methods, and the maximum likelihood ratio test was used for hypothesis testing of the parameters. The performance of the GWMLR model was evaluated using a real dataset and it was found to perform better than the MLR model.
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