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Unformatted text preview: PubH 7405: REGRESSION ANALYSIS LR: REVIEWS & COMPUTATIONS CORRELATION & REGRESSION • We have 2 continuous measurements made on each subject, one is the response variable Y, the other predictor X. There are two types of analyses: • Correlation : is concerned with the association between them, measuring the strength of the relationship; the aim is to determine if they are correlated – the roles are exchangeable. • Regression : To predict response from predictor . ou normally like to proceed to performing rediction only if the association is strong nough. However, in practice, “correlation nalysis” only covers association whereas “regression analysis” would cover both ssociation and prediction simultaneously. We start with a statistical/algebraic model, called “Normal Error Regression Model” which can be easily extended into a multivariate model for use in the cases we have more than one predictors. Some of the results can be used to for a Geometric Model/Representation. ) , ( : Model Regression Error Normal 2 1 N x Y (Simple) x β β E(Y) 1 : Response Mean The ) , ( : Model Regression Error Normal 2 N ε x β x β β Y (Multiple) k k 1 1 ε x β x β β E(Y) k k 1 1 : Response Mean The Main Results come as four sets of theorems: Theorems 1A and 1B: About the Slope Theorems 2A and 2B: About the Intercept Theorems 3A and 3B: About the Mean Response Theorems 4A and 4B: About Individual Response The first theorem of each set specifies the sampling distribution (Normal), the second lead to a tdistribution  same tdistribution with df = n2 Plus Theorem 5 on the sampling distribution of MSE/ σ 2 ; Together, the five theorems allow us to draw inferences, in the form of Confidence Intervals – as well as to test for Independence – two ttests and an F test (with identical results). ) ( ) ( ) , ( 2 ) ( ) ( ) ( ^ ^ ^ ^ ^ e Var Y Var e Y Cov e Var Y Var Y Var e Y Y Y Y e Here the regression line is used to predict Y from X; the predicted/fitted value is Ŷ , and the error/residual of this prediction is e. In the above formula, (1) Var(Y) is the total variance of interest, (2) Var( Ŷ ) is the “explained” vaiance, and (3) Var(e) is the “unexplained” variance. FITTED VALUE & RESIDUAL Pythagorean Representation 2 2 2 2 2 2 2 2 1 ^ ) ( y x x y x s r s s s r s b Y Var The Standard Deviation of Y The Standard Deviation of e = √ (1r 2 )(St Deviation of Y) The Standard Deviation of Ŷ = (r)(St Deviation of Y) ) ( ) ( ) ( ^ e Var Y Var Y Var 1 1 : or ; 1 r : is Result ) 1 ( ) ( ) ( ) ( ) ( ) ( ) , ( 2 ) ( ) ( ) ( 2 2 2 2 2 2 ^ ^ ^ ^ ^ ^ r s r s r s Y Var Y Var e Var e Var Y Var e Y Cov e Var Y Var Y Var e Y Y Y Y e y y y Now think of representing the “regression data”...
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 Fall '08
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 Regression Analysis, Descriptive statistics, data SBP

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