IOE366-Ch13-Multiple+Regression

IOE366-Ch13-Multiple+Regression - Ch13: Nonlinear and...

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Ch13: Nonlinear and Multiple Regression 13.1 Aptness of Model (Checking Assumptions) 13.2 Regression with Transformed Variables. 3.3 Polynomial Regression 13.3 Polynomial Regression 13.4 Multiple Regression 3 5 Issues in Multiple Regression 13.5 Issues in Multiple Regression (… the Kitchen Sink) 13.1 Aptness of the SLR Model L i n e a r i t y 1. Linearity 2. Homogeneity 3. Normality 4. Independence ε i ~ NID (0, σ 2 ) 5. Model Omissions 6. Outliers 2 Standardized Residuals i ˆ Recall e , is called the ith residual. i s(rv) there spon sea tx ii yy =− i Y is ( r.v.), the response at x , ˆ is (r.v.), the estimated response at x i Y then, e i is a random variable with mean 0 and variance aac e 2 2 ˆ () 1 ˆ 1 xx V YY σ ⎡⎤ −= 3 nS ⎢⎥ ⎣⎦ Standardized Residuals ± The standardized residuals are * 2 ˆ 1,2,. .., ˆ 1 y ei n == 2 1 −− 4
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MINITAB OUTPUT (regular residuals) Residuals Versus the Fitted Values (response is Won) 4 3 2 1 0 -1 Residual 2 -2 -3 5 12 7 2 Fitted Value MINITAB OUTPUT (standardized residuals) Residuals Versus the Fitted Values (response is Won) 2 1 sidual 0 dardized Re -1 -2 Stan 6 12 7 2 Fitted Value Graph: versus ˆ i y 14 0 Scatterplot of Y (Games Won) vs Y hat 12 10 n) 8 6 Y (Games Wo 4 2 0 0 7 12 10 8 6 4 2 0 Y hat Residuals versus ˆ Residuals Versus the Fitted Values (response is Games Wo) 2 1 esidual 0 ndardized R -1 -2 Sta 8 12 7 2 Fitted Value
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Residuals versus Observation # Residuals Versus the Order of the Data (response is Games Wo) 2 1 esidual 0 andardized R 25 20 15 10 5 -1 -2 St 9 Observation Order … Four Pack 99 5.0 Normal Probability Plot Versus Fits Residual Plots for Y (Games Won) 90 50 10 Percent 2.5 0.0 -2.5 Residual 5.0 2.5 0.0 -2.5 -5.0 1 Residual 12 9 6 3 -5.0 Fitted Value Histogram Versus Order 8 6 4 2 Frequency 5.0 2.5 0.0 -2.5 10 4 2 0 -2 -4 0 Residual 28 26 24 22 20 18 16 14 12 10 8 6 4 2 -5.0 Observation Order 1. Linearity (…Would a better function fit the data?) To check for “linearity”, we simply need more than one observation at one or more levels of x. Compute 2 j s and y for each level of x with repeated bserva ons observations. 11 We have two estimates of σ 2 : () 2 11 n c PE ij ji SS == =− ∑∑ ( ) 1 1 df N = = / MS = 12 2 EMS σ =
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From the ANOVA Table we obtained: () 2 1 ˆ j n c Ei ij PE LF SS y =− = + ∑∑ 11 ji == 2 E df N / EE MS = 13 The pure error estimate of σ 2 , , should be smaller than since: SS E = SS PE + SS LF Sum of Squares ck of it Lack of Fit 14 2 ˆ 1 = whats left = (N-2) -(N-c) c-2 / = ( ) 2 01 1 2 jj nEy x EMS ββ σ = ⎡⎤ −+ ⎣⎦ =+ 15 2 ( ) ( ) ( ) L F If E MS Ey = If another model is more apt ()( ) LF →≥ 16
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To test the linearity hypothesis ( ) ( ) 00 1 10 1 H:±± H: jj Ey x ββ =+ ≠+ 2 , ) ~ LF MS F −− (2 cNc PE Reject H 0 if . .. MINITAB automatically conducts this test when plicates are available for some levels of X 17 replicates are available for some levels of X. Example: Strength vs Additive Fitted Line Plot Strength = 10.94 + 0.1640 % Additive 25 20 S 5.12174 R-Sq 5.3% R-Sq(adj) 1.2% 15 Strength 10 5 18 35 30 25 20 15 % Additive Example: F-Test for LOF Results for: Additive.MTW Regression Analysis: Strength versus % Additive he regression equation is The regression equation is Strength = 10.9 + 0.164 % Additive Predictor Coef
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This note was uploaded on 03/17/2010 for the course IOE 366 taught by Professor Garyherrin during the Winter '10 term at University of Michigan-Dearborn.

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IOE366-Ch13-Multiple+Regression - Ch13: Nonlinear and...

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