Slide 1Slide 2Slide 3SMOKERS NONSMOKERSImagine the following “observational” study… X = Survival Time (“Time to Death”) in two independent normally-distributed populations Null Hypothesis H0: μ1 = μ2, i.e., μ1 – μ2 = 0 (“No mean difference") Test at signif level α X2 ~ N(μ2, σ2)1σ12σ2X1 ~ N(μ1, σ1)Sample 1, size n1Sample 2, size n2Suppose a statistically significant difference exists, with evidence that μ1 < μ2. “The reason for the significance was that the smokers started out older than the nonsmokers.” How do we prevent this criticism???Now consider two dependent (“matched,” “paired”) populations… Null Hypothesis H0: μ1 = μ2, i.e., μ1 – μ2 = 0 (“No mean difference") Test at signif level α POPULATION 1X and Y normally distributed.POPULATION 2Classic Examples: Twin studies, Left vs. Right, Pre-Tx (Baseline) vs. Post-Tx, etc.Y ~ N(μ2, σ2)1σ12σ2X ~ N(μ1, σ1) … etc….… etc….By design, every individual in Sample 1 is “paired” or “matched” with an individual in Sample 2, on potential confounding variables.Common in human trials to match on Age, Sex, Race,…Now consider two dependent (“matched,” “paired”) populations… Null Hypothesis H0: μ1 = μ2, i.e., μ1 – μ2 = 0 (“No mean difference") Test at signif level α POPULATION 1X, and Y normally distributed. POPULATION 2Y ~ N(μ2, σ2)1σ12σ2X ~ N(μ1, σ1)NOTE: Sample sizes are equal!… etc….… etc….1 2 3 4{ , , , , , }nx x x x xK1 2 3 4{ , , , , , }ny y y y yKSample 1, size nSample 2, size nClassic Examples: Twin studies, Left vs. Right, Pre-Tx (Baseline) vs. Post Tx, etc.Common in human trials to match on Age, Sex, Race,… Since they are paired, subtract!1 1 2 2 3 3 4 4{ , , , , , }n nx y x y x y x y x y- - - - -K1d2d3d4dLndTreat as one sample of the normally distributed variable D = X – Y.1ind d=�2 211( )ins d d-= -�D
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