22ANOVA_Intro

# 22ANOVA_Intro - Statistical Techniques I EXST7005...

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Statistical Techniques I EXST7005 Conceptual Intro to ANOVA

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Analysis of Variance (ANOVA) R. A. Fisher - resolved a problem that had existed for some time. H0: μ 1 = μ 2 = μ 3 = . .. = μ k H1: some μ i is different Conceptually, we have separate (and independent) samples, each giving a mean, an we want to know if they could have come from the same population or if is more likely they come from different populations.
The Problem (continued) One way to do this is a series of t-tests. If we want to test among 3 means we do 3 tests: 1 versus 2, 1 versus 3, 2 versus 3 For 4 means there are 6 tests. 1-2, 1-3, 1-4, 2-3, 4, and 3-4 For 5 means, 10 tests, etc.

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The Problem (continued) This technique is unwieldy, and worse. When we do the first test, there is an α chance error, and for each additional test another α chance of error. So if you do 3 or 6 or 10 tests, the chance of error on each and every test is α . Overall, for the experiment, the chance of error for all tests together is much higher than α .
The Problem (continued) Bonferroni gave a formula that showed that the chance of error would be NO MORE than Σα i. S if we do 3 tests, each with a 5% chance of error the overall probability of error is no greater than 15%, 30 percent for 6 tests, 50% for 10 tests, etc. Of course this is a lower bound. A better calculation comes from a the value. α '=1-(1- α )k

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No. of means pairwise tests Bonferroni's lower bound Duncan's 1-(1- α )k (1- α ) 2 1 0.05 0.0500 0.950 3 3 0.15 0.1426 0.857 4 6 0.30 0.2649 0.735 5 10 0.50 0.4013 0.599 6 15 0.75 0.5367 0.463 7 21 1.05 0.6594 0.341 10 45 2.20 0.9006 0.099 50 1225 61.20 0.9999 0.000 100 4950 247.45 1.0000 0.000 for α = 0.05
(continued) The bottom line: Splitting an experiment into a number of smaller tests is generally a poor idea This applies at higher levels as well (i.e. splitting big ANOVAs into little ones). The solution: We need ONE test that will give u

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## 22ANOVA_Intro - Statistical Techniques I EXST7005...

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