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PDB_Stat_100_Lecture_26_Printable

PDB_Stat_100_Lecture_26_Printable - STA 100 Lecture 26 Paul...

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STA 100 Lecture 26 Paul Baines Department of Statistics University of California, Davis March 9th, 2011
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Admin for the Day I Final project due Friday, 3pm I Office Hours – project questions please! I Office hours Friday – 9.30-11.30am I Coming Soon: Practice Final Exam Questions References for Today: Rosner, Ch 11, Ch 12 (7th Ed.) References for Friday: Rosner, Ch 11, Ch 12 (7th Ed.)
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Testing Model Parameters Recall that our linear model is: Y i = α + β X i + i , i iid N ( 0 , σ 2 ) . Two simpler models would be: Y i = β X i + i , i iid N ( 0 , σ 2 ) , [ α = 0 ] (1) Y i = α + i , i iid N ( 0 , σ 2 ) , [ β = 0 ] (2) We can test the hypothesis that α = 0 or the hypothesis that β = 0, and see if a simpler model would suffice.
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Testing the Slope 1. H 0 : β = 0 vs . H 1 : β 6 = 0 2. Test statistic: t = ˆ β SE ( ˆ β ) = ˆ β ˆ σ/ s xx where s xx = n i =1 ( x i - ¯ x ) 2 . 3. Reference distribution: under H 0 , the test statistic t follows a t - distribution with n - 2 degrees of freedom. 4. The p - value as usual is p = P ( | t n - 2 | > t ) 5. Decide to reject or not depending on the value of p 6. Interpret the meaning for your example
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Testing Correlations The sample correlation coefficent r is an estimate of the unknown population correlation ρ . We can test the hypothesis that ρ = 0: > attach(animals) # Avoid the need to type animals$... > cor(log.brain,log.body) [1] 0.7056812 > cor.test(log.brain,log.body) Pearson’s product-moment correlation data: animals$log.brain and animals$log.body t = 4.9799, df = 25, p-value = 3.926e-05 alternative hypothesis: true correlation is not equal to 0 95 percent confidence interval: 0.4450032 0.8561126 sample estimates: cor 0.7056812 > detach(animals) This is the same p - value as the test of β = 0.
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Take Home Messages We have shown that: I You can conduct correlation test of: H 0 : ρ = 0 vs. H 0 : ρ 6 = 0 by performing a linear regression and testing: H 0 : β = 0 vs. H 0 : β 6 = 0. I For a linear regression, you can conduct a test of: H 0 : β = 0 vs. H 0 : β 6 = 0 by performing a correlation test of H 0 : ρ = 0 vs. H 0 : ρ 6 = 0 I There is no point in doing both – they are equivalent! I The linear regression approach gives you more information I Which you use depends on if you are interested in β or ρ
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Principles of Statistical Modeling 1. Simpler is better (Parsimony) I Given the choice between two models that do a similar job in describing the data, we prefer the simpler model I More complex models should do better than simpler models – but do they improve things enough to make the additional complexity worthwhile? 2. Models should be physically sensible I Never use linear regression for response variables that are not continuous! I Ensure parameters are restricted to physically possible values! 3. Avoid overfitting! I We usually want to apply the conclusions from analyzing a dataset more broadly I While general trends may hold more widely, the specifics usually will not I This is also true for prediction: fitting your data too well often leads to poor predictive performance!
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Goodness-of-Fit Previously we used the χ 2 goodness of fit test as a way to test if a model was a good fit to the data.
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