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Unformatted text preview: STA 100 Lecture 26 Paul Baines Department of Statistics University of California, Davis March 9th, 2011 Admin for the Day I Final project due Friday, 3pm I Office Hours – project questions please! I Office hours Friday – 9.3011.30am 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.) Testing Model Parameters Recall that our linear model is: Y i = α + β X i + i , i iid ∼ N ( ,σ 2 ) . Two simpler models would be: Y i = β X i + i , i iid ∼ N ( ,σ 2 ) , [ α = ] (1) Y i = α + i , i iid ∼ N ( ,σ 2 ) , [ β = ] (2) We can test the hypothesis that α = 0 or the hypothesis that β = 0, and see if a simpler model would suffice. Testing the Slope 1. H : 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 , 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 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$... > cor(log.brain,log.body) [1] 0.7056812 > cor.test(log.brain,log.body) Pearson’s productmoment correlation data: animals$log.brain and animals$log.body t = 4.9799, df = 25, pvalue = 3.926e05 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. Take Home Messages We have shown that: I You can conduct correlation test of: H : ρ = 0 vs. H : ρ 6 = 0 by performing a linear regression and testing: H : β = 0 vs. H : β 6 = 0. I For a linear regression, you can conduct a test of: H : β = 0 vs. H : β 6 = 0 by performing a correlation test of H : ρ = 0 vs. H : 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 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!...
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This note was uploaded on 01/17/2012 for the course STAT 100 taught by Professor drake during the Fall '10 term at UC Davis.
 Fall '10
 DRAKE
 Statistics

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