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B AD 6243: Applied Univariate Statistics

B AD 6243: Applied Univariate Statistics. One-way Analysis of Variance Professor Laku Chidambaram Price College of Business University of Oklahoma. Comparing Two Groups vs. Many. Two levels of a single factor --> t-test Multiple levels of a single factor --> One-way ANOVA

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B AD 6243: Applied Univariate Statistics

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  1. B AD 6243: Applied Univariate Statistics One-way Analysis of Variance Professor Laku Chidambaram Price College of Business University of Oklahoma

  2. Comparing Two Groups vs. Many • Two levels of a single factor --> t-test • Multiple levels of a single factor --> One-way ANOVA • They are both part of the general linear model • So, why not use multiple t-tests for multiple comparisons? • Let’s: (a) assume a single factor with three levels, and (b) set the Type I error level to .05 • Now, we will need three t-tests to compare the groups • However, these tests will inflate the Type I error level to .14 or 1 - (1 - )c [where c is # of levels being compared] • So, the probability of incorrectly rejecting the null hypothesis increases to unacceptably high levels • The one-way ANOVA uses a family-wise error rate and enables multiple simultaneous comparisons BAD 6243: Applied Univariate Statistics

  3. Assumptions of One-way ANOVA • Data are normally distributed in each group • Variances are not significantly different across groups (especially where sample sizes are unequal) • Observations are independent • Dependent variable is continuous and independent variable is categorical BAD 6243: Applied Univariate Statistics

  4. Hypothesis Testing H0: 1 = 2 = 3 … = k H1: At least one pair of means is different. F test = SSM/dfM/SSR/dfR = MSM / MSR = Systematic variation / Unsystematic variation = Between group variation / Within group variation If Fcalc > Fcrit, then reject H0 BAD 6243: Applied Univariate Statistics

  5. An Example • A program to help smokers quit smoking wants to test the effectiveness of three types of interventions • a placebo pill (1) • a nicotine patch (2) and • a nicotine gum (3) • At the end of the study, blood oxygen levels are measured for all subjects and converted to an OxLevel index (a measure of effectiveness) • Assume there are three subjects (!) in each group BAD 6243: Applied Univariate Statistics

  6. Error Plots of OxLevels BAD 6243: Applied Univariate Statistics

  7. Viewing ANOVA as Regression BAD 6243: Applied Univariate Statistics

  8. Regression Results BAD 6243: Applied Univariate Statistics

  9. One-way ANOVA & Regression • Regression coefficients: • B0 (Constant) = 7 = Mean of placebo • B1 = 5 = Mean of patch – Mean of placebo • B2 = 2 = Mean of gum – Mean of placebo BAD 6243: Applied Univariate Statistics

  10. One-way ANOVA (contd.) BAD 6243: Applied Univariate Statistics

  11. A Selection of Post-hoc Tests • LSD • Bonferroni • SNK • REGWQ • Tukey’s HSD • Scheffe • Different sample sizes: • Hochberg (equal variances) • Gabriel • Unequal variances: • Tamahane’s T2 (conservative) • Dunnett’s T3 and C • Games-Howell (also when sample sizes unequal) BAD 6243: Applied Univariate Statistics

  12. Pair-wise Comparisons

  13. Homogeneous Subsets BAD 6243: Applied Univariate Statistics

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