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Lec12.MultiDiag - MultipleRegression Diagnostics...

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Click to edit Master subtitle style Multiple Regression  Diagnostics KNNL Chapter 10 (10.1-10.3,10.5) Building the Regression Model II Lec12.MultiDiag.ssc
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Multicolinearity: KNNL Chapters 7.6, 10.5 Comments section also Adding or deleting a predictor variable changes  the regression coefficients Estimated standard deviations of coefficients  become large  Coefficients may not be statistically significant Make sure to read …
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alligator$Length 60 80 100 120 140 100 200 300 400 500 600 alligator$Weight Alligator Polynomial Model
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F-Statistic: Kutner et al. pg 73
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anova(alligator2.lm, alligator3.lm, test = "F") Analysis of Variance Table Response: Weight                          Terms Resid.Df Resid.Dev           1            Length + Length^2        21 5267.637                                                 2 Length + Length^2 + Length^3        20 2684.465  Test  Df Sum of Sq F Value        Pr(F) +I(Length^3) 1 2583.172 19.24534 0.0002847591 0 : hypothesis null Test 3 = β
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anova(alligator3.lm, alligator4.lm, test = "F") Analysis of Variance Table Response: Weight Terms  Resid. Df Resid. Dev          1 Length + Length^2 + Length^3     20   2684.465                                               2 Length + Length^2 + Length^3 + Length^4                                            19   2581.174 Test    Df Sum of Sq F Value     Pr(F) +I(Length^4)  1  103.2905 0.7603201 0.3941139  0 : hypothesis null Test 4 = β
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( 29 - - = = 2 1 2 2 2 2 ) ( exp ) 2 ( 1 ; σ θ πσ θ n i i n x L x The Normal Likelihood Function Obj105
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2 2 2 2 ) ( ) 2 ln( 2 ln σ θ πσ - + = - i x n L The Negative Log Likelihood 2 ) ( ln σ θ θ - - = - i x L d d 2 2 2 ln σ θ n L d d = - Hessian Matrix
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Akaike Information Criteria (AIC) Schwarz’ Bayesian Criteria (SBC,  AIC = n*log(SSE) - n*log(n) + 2*p SBC = n*log(SSE) - n*log(n) + log(n)*p Kutner et al. Notation
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# Splus/R AIC # -2*log-likelihood + 2*npar # where npar includes coefficients and sigma2 sum(resid(alligator3.lm)^2) 2684.465 SSE = sum(resid(alligator3.lm)^2) p = length(coef(alligator3.lm)) n = length(resid(alligator3.lm))
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lm1=lm(Weight~Length,data=alligator) text(1,AIC(lm1),1) lm1=lm(Weight~Length^2,data=alligator) text(1,AIC(lm1),2) lm1=lm(Weight~Length^3,data=alligator) text(1,AIC(lm1),3) lm1=lm(Weight~Length^4,data=alligator) text(1,AIC(lm1),4) One at a time
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Number of Parameters 0 1 2 3 4 5 6 100 150 200 250 300 350 AIC 1 2 3 4 AIC Plot: Single Variable Models
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lm1=lm(Weight~Length+Length^2,data=alligator) AIC.p = rbind(AIC.p,c(3,AIC(lm1),12)) lm1=lm(Weight~Length+Length^3,data=alligator) AIC.p = rbind(AIC.p,c(3,AIC(lm1),13)) lm1=lm(Weight~Length+Length^4,data=alligator) AIC.p = rbind(AIC.p,c(3,AIC(lm1),14)) lm1=lm(Weight~Length^2+Length^3,data=alligator) AIC.p = rbind(AIC.p,c(3,AIC(lm1),23))
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