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# note13 - Chapter 5 Tests for Trends and Association the...

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Chapter 5: Tests for Trends and Association the relationship between two quantitative variables - correlation and regression/ nonparametric correlation and regression the relationship between qualitative variables - contingency tables 1. A Permutation Test for Correlation and Slope Settings: data ( X i , Y i ) i = 1 , . . . , n bivariate sampling: the pairs are selected at random from a bivariate population fixed- X sampling: the pairs are obtained in an experiment where the values of X are fixed by the experimenter The Correlation coefficient for random pair, the correlation coefficient as a measure of the strength of the linear relationsip between X and Y the population correlation coefficient ρ = E [( X μ X ) ( Y μ Y )] σ X σ Y ∗ − 1 ρ 1 ρ = ± 1: perfect linear relationship Y = a + bX ρ = 0: no linear relationship/ uncorrelated the sample correlation coefficient or the Pearson product-moment cor- relation r = n i =1 ( X i ¯ X ) ( Y i ¯ Y ) radicalBig n i =1 ( X i ¯ X ) 2 n i =1 ( Y i ¯ Y ) 2 testing of correlation coefficient Hypothesis H 0 : ρ = 0 H a : ρ > 0 H a : ρ < 0 H a : ρ negationslash = 0 the test statistic t corr = radicalbigg n 2 1 r 2 r under H 0 , t corr t n 2 1

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Slope of Least Squres Line linear regression model Y i = β 0 + β 1 X i + ǫ i where ǫ i iid ( 0 , σ 2 ǫ ) ( σ 2 ǫ < ) fixed- X sampling: how we believe Y i behaves for the fixed value of X i bivariate sampling: conditional distribution of Y i given X i - how we believe Y i would behave if we could fix X i the least squares estimates ˆ β 0 = ¯ Y ˆ β 1 ¯ X ˆ β 1 = n i =1 ( X i ¯ X ) ( Y i ¯ Y ) n i =1 ( X i ¯ X ) 2 by minimizing the sume of squares: min β 0 1 SSE = min β 0 1 n summationdisplay i =1 parenleftBig Y i ˆ β 0 ˆ β 1 X i parenrightBig 2 test whether or not there is a significant relationship between X and Y Hypothesis H 0 : β 1 = 0 test stat: if ǫ ’s are normally distributed t slope = radicalBigg n i =1 ( X i ¯ X ) 2 MSE ˆ β 1 under H 0 t slope t n 2 for linear regression model β 1 = ρ σ Y σ X ˆ β 1 = r S Y S X and t corr = t slope 2
The Permutation Test under H 0 : β 1 = 0 or ρ = 0: an observed value of Y i is as likely to appear with X j as it is to appear with X i for any value of j negationslash = i X 1 2 3 4 Y 5 7 9 8 X 1 = 1 X 2 = 2 X 3 = 3 X 4 = 4 Y 1 Y 2 Y 3 Y 4 Slope Correlation 5 7 8 9 1.3 0.98 5 7 9 8 1.1 0.83 5 8 7 9 1.1 0.83 5 8 9 7 0.7 0.53 5 9 7 8 0.7 0.53 5 9 8 7 0.5 0.38 7 5 8 9 0.9 0.68 7 5 9 8 0.7 0.53 7 8 5 9 0.3 0.23 7 8 9 5 -0.5 -0.38 7 9 5 8 -0.1 -0.08 7 9 8 5 -0.7 -0.53 8 5 7 9 0.5 0.38 8 5 9 7 0.1 0.08 8 7 5 9 0.1 0.08 8 7 9 5 -0.7 -0.53 8 9 5 7 -0.7 -0.53 8 9 7 5 -1.1 -0.83 9 5 7 8 -0.1 -0.08 9 5 8 7 -0.3 -0.23 9 7 5 8 -0.5 -0.38 9 7 8 5 -1.1 -0.83 9 8 5 7 -0.9 -0.68 9 8 7 5 -1.3 -0.98 3

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steps for a permutation test for slope or correlation (a) ˆ β 1 ,obs from the original data (b) permute Y ’s among X ’s in the n ! possible ways ˆ β 1 ,b b = 1 . . . , n !
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note13 - Chapter 5 Tests for Trends and Association the...

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