2 Simple Linear Regression

2 Simple Linear Regression - Statistics 191 Introduction to...

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Unformatted text preview: Statistics 191: Introduction to Applied Statistics Jonathan Taylor Department of Statistics Stanford University Statistics 191: Introduction to Applied Statistics Simple linear regression Jonathan Taylor Department of Statistics Stanford University January 6, 2010 1 / 1 Statistics 191: Introduction to Applied Statistics Jonathan Taylor Department of Statistics Stanford University Outline Simple Linear Regression Some definitions for regression models. Specifying the model. Fitting the model: least squares. Inference. What is a T-statistic? “Inference” for β 1 . Linear combinations of β ,β 1 . 2 / 1 Statistics 191: Introduction to Applied Statistics Jonathan Taylor Department of Statistics Stanford University Reminder What is a “regression” model? A regression model is a model of the relationships between some covariates (predictors) and an outcome . Specifically, regression is a model of the average outcome given the covariates. Mathematical formulation For height of couples data: a mathematical model, using only Husband ’s height: Wife = f ( Husband ) + ε where f gives the average height of the wife of a man of height Husband and ε is the random error. 3 / 1 Statistics 191: Introduction to Applied Statistics Jonathan Taylor Department of Statistics Stanford University Height data 4 / 1 Statistics 191: Introduction to Applied Statistics Jonathan Taylor Department of Statistics Stanford University Regression models Linear regression models A linear regression model says that the function f is a sum (linear combination) of functions of Husband . Simple linear regression model: f ( Husband ) = β + β 1 · Husband for some unknown parameter vector ( β ,β 1 ). Could also be a sum (linear combination) of known functions of Husband : f ( Husband ) = β + β 1 · Husband + β 2 · Husband 2 5 / 1 Statistics 191: Introduction to Applied Statistics Jonathan Taylor Department of Statistics Stanford University Simple linear regression model Specifying the (statistical) model Simple linear regression is the case when there is only one predictor: f ( Husband ) = β + β 1 · Husband . Let Y i be the height of the i-th wife in the sample, X i be the height of the i-th husband. Model: Y i = β + β 1 X i | {z } regression equation + ε i |{z} error where ε i ∼ N (0 ,σ 2 ) are independent. This specifies a distribution for the Y ’s given the X ’s, i.e. it is a statistical model. 6 / 1 Statistics 191: Introduction to Applied Statistics Jonathan Taylor Department of Statistics Stanford University Fitting the model Least squares We will be using “least squares” regression. This measures the goodness of fit of a line by the sum of squared errors, SSE ....
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2 Simple Linear Regression - Statistics 191 Introduction to...

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