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Unformatted text preview: 1 Topic 2 – Simple Linear Regression KKMN Chapters 4 – 7 2 Overview Regression Models; Scatter Plots SAS GPLOT Procedure Estimation and Inference in SLR SAS REG Procedure ANOVA Table & Coefficient of Determination (R 2 ) 3 Simple Linear Regression Model We take n pairs of observations The goal is to find a model that best fits with the data. Model will be linear in terms of the parameters (betas). These won’t appear in exponents or anything unusual. Allowed to be nonlinear in terms of predictor variables (we may transform these somewhat freely). We may also transform the response. ( ) ( ) ( ) 1 1 2 2 , , , ,..., , n n X Y X Y X Y 4 Simple Linear Regression Model (2) Some sample models Notice the betas always function in the same way, and the analysis will always proceed in the same way too (after we make whatever transformations we might need). [ ] [ ] 1 1 1 log log i i i i i i i i i Y X Y X Y X β β ε β β ε β β ε = + + = + + = + + 5 Simple Linear Regression Model (3) Key question: How do you decide on the “best” form for the model? Always view a scatter plot (use PROC GPLOT in SAS). Curvature in the plot will help you determine the need for a transformation on either X or Y. Always consider residual plots. Some patterns in these plots will also indicate the need for transformation (more on this later). 6 Scatter Plot Approach If you can look at a scatter plot and the data “look linear”, then likely no transformation is necessary. Try not to look for things that are not there. If you see curvature, then some transformation may be appropriate: Use scientific theory & experience Try transformations you think may work – look at scatter plots of the transformed data to assess whether they do work. 7 Finding the “Best” Model There is no “absolute” strategy. Some common mistakes (why are these bad?): Try several different methods and simply take the one for which you get the best results (e.g. highest R 2 ) Overfit the model by including lots of extra terms (e.g. squares, cubes, etc.) in hopes to get the curve to go through all of the data points (note that this would be MLR) 8 Collaborative Learning Activity CLG #2.12.3 First, make sure you read enough to understand the dataset we will be considering. Then, please try to answer these questions related to scatter plots. 9 Scatter Plot Examples (1) Wi t h Es t i mat ed Regr es s i on Li ne 800 900 1000 1100 1200 St at ewi de Expendi t ur es 3 4 5 6 7 8 9 10 10 Scatter Plot Examples (2) Wi t h Es t i mat ed Regr es s i on Li ne 800 900 1000 1100 1200 Per cent age of El i gi bl e St udent s Taki ng SAT 10 20 30 40 50 60 70 80 90 11 Scatter Plot Examples (3) Wi t h Nonpar amet er i c Smoot h 800 900 1000 1100 1200 Per cent age of El i gi bl e St udent s Taki ng SAT 10 20 30 40 50 60 70 80 90 12 Scatter Plot Examples (4) Log Tr ans f or med Pr edi c t or Wi t h Nonpar amet er i c Smoot h...
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 Fall '08
 Staff
 Linear Regression, Regression Analysis, Scatter Plots, Estimates, Yi, Linear Association

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