Regression Analysis

In statistics, regression analysis consists of techniques for modeling the relationship between a dependent variable (also called response variable) and one or more independent variables (also known as explanatory variables or predictors). In regression, the dependent variable is modeled as a function of independent variables, corresponding regression parameters (coefficients), and a random error term which represents variation in the dependent variable unexplained by the function of the dependent variables and coefficients. In linear regression, the dependent variable is modeled as a linear function of a set of regression parameters and a random error. The parameters need to be estimated so that the model gives the “ best fit ” to the data.

Linear Regression Analysis Theory and Computing

Overview Regression analysis is a process used to estimate a function which predicts the value of the response variable in terms of values of other independent variables. If the regression function is determined only through a set of parameters the type of regression is the parametric regression. Many methods have been developed to determine various parametric relationships between the response variable and independent variables. These methods typically depend on the form of the parametric regression function and the distribution of the error term in a regression model. For example, linear regression, logistic regression, Poisson regression, and probit regression, etc. These particular regression models assume different regression functions and error terms from corresponding underline distributions. A generalization of linear regression models have been formalized in the “ generalized linear model ” and it requires to specify a link function which provides the relationship between the linear predictor and the mean of the distribution function.

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Topics

The topics on regression analysis covered in this book are distributed among 9 chapters. Chapter 1 briefly introduces the basic concept of regression and defines the linear regression model. Chapters 2 and 3 cover the simple linear regression and multiple linear regression. Although the simple linear regression is a special case of the multiple linear regression, we present it without using matrix and give detailed derivations that highlight the fundamental concepts in linear regression. The presentation of multiple regression focus on the concept of vector space, linear projection, and linear hypothesis test. The theory of matrix is used extensively for the proofs of the statistical properties of the linear regression model. Chapters 4 through 6 discuss the diagnosis of a linear regression model. These chapters cover outlier detection, influential observations identification, collinearity, confounding, regression on dummy variables, checking for equal variance assumption, graphical display of residual diagnosis, and variable transformation technique in linear regression analysis. Chapters 7 and 8 provide further discussions on the generalizations of the ordinary least squares estimation in linear regression. In these two chapters, we discuss how to extend the regression model to the situation where the equal variance assumption on the error term fails. To model the regression data with unequal variance the generalized least squares method is introduced. In Chapter 7, two shrinkage estimators, the ridge regression and the LASSO are introduced and discussed. A brief discussion of the least-squares method for nonlinear regression is also included. Chapter 8 briefly introduces generalized linear models. In particular, the Poisson Regression for count data and the logistic regression for binary data are discussed. Chapter 9 briefly discussed the Bayesian linear regression models. The Bayes averaging method is introduced and discussed.

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About the Author:

Ed Neil O. Maratas an instructor of Jose Rizal Memorial State University, Dapitan Campus, Philippines as regular status. He earned his Bachelor of Science in Statistics at Mindanao State University-Tawi-Tawi College of Technology and Oceanography in the year 2003 and finished Master of Arts in Mathematics at Jose Rizal Memorial State University year 2009. He Became a researcher, a data analyst, and engaged to several projects linked to the university as data processor.

Prepared by:ednielmaratas@gmail.com or you can visit the facebook pageStatisticss For Funfor more details about statistics.

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