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ANOVA Handout

# ANOVA Handout - STAT E-50 Introduction to Statistics...

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STAT E-50 - Introduction to Statistics One-Way Analysis of Variance To compare two or more groups: To test two proportions, use a z-test To test more than two proportions, use a χ 2 test To test two means, use a t-test To test more than two means, use Analysis of Variance (ANOVA) In Analysis of Variance, we are testing for the equality of the means of several levels of a variable. The technique is to compare the variation between the levels and the variation within each level. (The levels of the variable are also referred to as groups, or treatments.) If the variation due to the level (variation between levels) is significantly larger than the variation within each level, then we can conclude that the means of the levels are not all the same. We will test the hypothesis H 0 : μ 1 = μ 2 = … = μ k vs. H a : the means are not all equal using the ratio: differences among means variability within the groups When the numerator is large compared to the denominator, we will reject the null hypothesis. The numerator measures the variation between groups; this is called the Treatment Mean Square, or MS T The denominator measures the variation within groups; this is called the Error Mean Square, or MS E The test statistic is T E MS F = MS ; reject H 0 when F is large. MS T has k - 1 degrees of freedom, where k = the number of groups MS E has k(n - 1) degrees of freedom, where n = the number of observations from each group

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Assumptions and Conditions First plot the data in side-by-side boxplots. Look for outliers, similar spreads, similar centers. Independence Assumption The groups must be independent of each other. The data within each group must be independent data drawn at random from a homogeneous population, or generated by a randomized comparative experiment. Randomization Condition: Was the data collected randomly? Equal Variance Assumption We need the variances of the treatment groups to be equal, so we can find a pooled variance Similar Variance Condition: Compare the spreads of the groups in the side-by-side boxplots Are the spreads changing systematically with the centers? Plot the residuals vs. the predicted values to be sure you don’t have larger residuals for larger values Normal Population Assumption Nearly Normal Condition: Compare the boxplots for skewness Pool the residuals and create a histogram or NPP The ANOVA table: Source df SS MS F p Factor Error Total T T T SS MS = df E E E SS MS = df T E MS F = MS Page 2