SMOKERS NONSMOKERS Sample 1, size n1 Sample 2, size n2

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Presentation transcript:

SMOKERS NONSMOKERS Sample 1, size n1 Sample 2, size n2 Imagine the following “observational” study… X = Survival Time (“Time to Death”) in two independent normally-distributed populations SMOKERS NONSMOKERS X1 ~ N(μ1, σ1) 1 σ1 X2 ~ N(μ2, σ2) Null Hypothesis H0: μ1 = μ2, i.e., μ1 – μ2 = 0 (“No mean difference") Test at signif level α 2 σ2 “The reason for the significance was that the smokers started out older than the nonsmokers.” Sample 1, size n1 Sample 2, size n2 Suppose a statistically significant difference exists, with evidence that μ1 < μ2. How do we prevent this criticism???

        POPULATION 1 POPULATION 2 X ~ N(μ1, σ1) 1 σ1 Now consider two dependent (“matched,” “paired”) populations… X and Y normally distributed. POPULATION 1 POPULATION 2 X ~ N(μ1, σ1) 1 σ1 Y ~ N(μ2, σ2) Null Hypothesis H0: μ1 = μ2, i.e., μ1 – μ2 = 0 (“No mean difference") Test at signif level α 2 σ2 Classic Examples: Twin studies, Left vs. Right, Pre-Tx (Baseline) vs. Post-Tx, etc. Common in human trials to match on Age, Sex, Race,…         By design, every individual in Sample 1 is “paired” or “matched” with an individual in Sample 2, on potential confounding variables. … etc…. … etc….

    POPULATION 1 POPULATION 2 X ~ N(μ1, σ1) 1 σ1 Y ~ N(μ2, σ2) Now consider two dependent (“matched,” “paired”) populations… X, and Y normally distributed. POPULATION 1 POPULATION 2 X ~ N(μ1, σ1) 1 σ1 Y ~ N(μ2, σ2) Null Hypothesis H0: μ1 = μ2, i.e., μ1 – μ2 = 0 (“No mean difference") Test at signif level α 2 σ2 D = Classic Examples: Twin studies, Left vs. Right, Pre-Tx (Baseline) vs. Post Tx, etc. Common in human trials to match on Age, Sex, Race,…     … etc…. Sample 1, size n Sample 2, size n NOTE: Sample sizes are equal! Since they are paired, subtract! Treat as one sample of the normally distributed variable D = X – Y.

http://pages. stat. wisc http://pages.stat.wisc.edu/~ifischer/Intro_Stat/Lecture_Notes/6_-_Statistical_Inference/HYPOTHESIS_TESTING_SUMMARY.pdf

http://pages. stat. wisc http://pages.stat.wisc.edu/~ifischer/Intro_Stat/Lecture_Notes/6_-_Statistical_Inference/HYPOTHESIS_TESTING_SUMMARY.pdf