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Regression Analysis Regression Analysis 1 CHAPTER OUTLINE • INTRODUCTION TO EMPIRICAL MODELS • LEAST SQUARES ESTIMATION OF THE PARAMETERS • PROPERTIES OF THE LEAST SQUARES ESTIMATORS AND ESTIMATION OF s 2 • HYPOTHESIS TESTING IN LINEAR REGRESSION • CONFIDENCE INTERVALS IN LINEAR REGRESSION • PREDICTION OF NEW OBSERVATIONS • ASSESSING THE ADEQUACY OF THE REGRESSION MODEL Regression Analysis 2 Definitions Regress The act of reasoning backward Regression A functional relationship between two or more correlated variables that is often empirically determined from data and is used esp. to predict values of one variable when given values of the others. Regression Analysis 3 Models Abstraction/simplification of the system used as a proxy for the system itself Can try wide-ranging ideas in the model Make your mistakes on the computer where they don’t count, rather for real where they do count Issue of model validity Two types of models Physical (iconic) Logical/Mathematical -- quantitative and logical assumptions, approximations Regression Analysis 4 What Do You Do with a Logical Model? If model is simple enough, use traditional mathematics (queueing theory, differential equations, linear programming) to get “answers” Nice in the sense that you get “exact” answers to the model But might involve many simplifying assumptions to make the model analytically tractable -- validity?? Many complex systems require complex models for validity — simulation needed Regression Analysis 5 INTRODUCTION TO EMPIRICAL MODELS • models • theoretical (mechanical) model • empirical model • scatter diagram Regression Analysis 6 Regression Analysis 7 Regression Analysis 8 • linear model (equation) • probabilistic linear model • simple linear regression model • regression coefficients Regression Analysis 9 • multiple regression model • multiple linear regression model • intercept • partial regression coefficients • contour plot Regression Analysis 10 • dependent variable or response y may be related to k independent or regressor variables • interaction • any regression model that is linear in parameters (the b’s) is a linear regression model, regardless of the shape of the surface that it generates. Regression Analysis 11 Regression Analysis 12 Regression Analysis 13 LEAST SQUARES ESTIMATION OF THE PARAMETERS Simple Linear Regression Y 0 1 x Regression Analysis 14 • method of least squares • least squares normal equations L yi 0 1 xi 2 2 i • fitted or estimated regression line ^ ^ ^ y 0 1 x • residual ^ ei yi y i Regression Analysis 15 Regression Analysis 16 S xx xi x 2 S xy yi xi x then, ^ 1 S xy S xx Example 10-1, pp. 436 Regression Analysis 17 Regression Analysis 18 Regression Analysis 19 Multiple Linear Regression Regression Analysis 20 Regression Analysis 21 Regression Analysis 22 Regression Analysis 23 PROPERTIES OF THE LEAST SQUARES ESTIMATORS AND ESTIMATION OF s2 • unbiased estimators • covariance matrix • estimated standard error • residual mean square (or error mean square) Regression Analysis 24 Hypothesis Testing on 0and 1, pp. 447 Regression Analysis 25 HYPOTHESIS TESTING IN LINEAR REGRESSION Regression Analysis 26 Regression Analysis 27 *k=p-1 Regression Analysis 28 Regression Analysis 29 Tests on Individual Regression Coefficients Regression Analysis 30 Confidence Intervals on Individual Regression Coefficients Regression Analysis 31 Regression Analysis 32 Confidence Interval on the Mean Response Regression Analysis 33 Regression Analysis 34 Regression Analysis 35 PREDICTION OF NEW OBSERVATIONS Regression Analysis 36 Regression Analysis 37 • simple linear regression Regression Analysis 38 Regression Analysis 39 ASSESSING THE ADEQUACY OF THE REGRESSION MODEL • normal probability plot of residuals • standardize • outlier Regression Analysis 40 Regression Analysis 41 Regression Analysis 42 Regression Analysis 43 Regression Analysis 44 Regression Analysis 45 Regression Analysis 46 Regression Analysis 47 Regression Analysis 48 Coefficient of Multiple Determination Regression Analysis 49 Regression Analysis 50 Influential Observations Regression Analysis 51 Regression Analysis 52 Regression Analysis 53 Regression Analysis 54