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Unformatted text preview: Public Health 6450 Fall 2011 Lynn Eberly and Andy Mugglin Division of Biostatistics School of Public Health University of Minnesota [email protected] Part 20 Review The SLR Model Inference for the SLR Model Predictions from the SLR Model Diagnostics for the SLR Model Where are we going? Previously: I Correlation (Ch. 2) I Introduction to Simple Linear Regression (Ch. 2) Current topic: I Details on Simple Linear Regression (Ch. 10) I SLR Model Inference I SLR Model Predictions I SLR Model Diagnostics Next topic: I Multiple Linear Regression (Ch. 11) Eberly and Mugglin PubH 6450 Fall 2011 Part 20 2 / 72 Review The SLR Model Inference for the SLR Model Predictions from the SLR Model Diagnostics for the SLR Model SLR Model Notation SLR Model Assumptions Estimation and Interpretation of the SLR Model Detailed Example Revisiting the Least Squares Regression Line In Part 19 we learned about the least squares regression line ˆ Y = a + bX where a and b are estimated by minimizing the sum of squared vertical distances between the data points and the line: a = ¯ Y b ¯ X b = r s Y s X . The simple linear regression (SLR) model formalizes this estimation and adds some assumptions about Y so that we can, for example, construct CIs for a , b , and ˆ Y . Eberly and Mugglin PubH 6450 Fall 2011 Part 20 3 / 72 Review The SLR Model Inference for the SLR Model Predictions from the SLR Model Diagnostics for the SLR Model SLR Model Notation SLR Model Assumptions Estimation and Interpretation of the SLR Model Detailed Example Changing notation for intercept and slope What we used to write as ˆ Y = a + bX we will now write as ˆ Y i = b + b 1 X i . (Why? It makes Chapter 11, Multiple Linear Regression, easier.) b is our sample estimate of the population intercept β : β is the intercept we would get if we fit a regression line to the entire population. b 1 is our sample estimate of the population slope β 1 : β 1 is the slope we would get if we fit a regression line to the entire population. Eberly and Mugglin PubH 6450 Fall 2011 Part 20 4 / 72 Review The SLR Model Inference for the SLR Model Predictions from the SLR Model Diagnostics for the SLR Model SLR Model Notation SLR Model Assumptions Estimation and Interpretation of the SLR Model Detailed Example Data, predictions, and residuals It is important to keep the distinction between Y values and ˆ Y values in mind. I ( X i , Y i ) is the original data point. i indexes the observations 1 , 2 , . . . , n . I Thus, each X i came with a Y i in the sample, but each X i can also be used to predict the ˆ Y i value that falls on the regression line. I ( X i , ˆ Y i ) is the corresponding point on the regression line....
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 Fall '10
 AndyMugglin
 Linear Regression, Regression Analysis, Errors and residuals in statistics, SLR model, Lynn Eberly, SLR Model Diagnostics

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