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lect16-12_ANOVA

# lect16-12_ANOVA - Lecture 16 One Way ANOVA Stat 102 2012...

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1 Lecture 16 – One Way ANOVA Stat 102 2012 • One Way Analysis of Variance (ANOVA) Read Ch 9.1 • Comparison of means among I groups • Individual t tests vs. multiple comparison: Two methods: Bonferroni and Tukey-Kramer

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2 One-way An alysis o f Va riance One-way ANOVA: Compares the means of two or more groups. Extends the equal-variance two sample test discussed in Lecture 2 Uses an F-test to determine whether there are – overall – any significant differences among the means Then (if there are any overall differences) uses special “multiple comparison” tests to determine which differences between pairs of means are significant. We’ll discuss the theory in the context of an example. This example uses data printed in USA Today (~ 8 years ago) that reports the returns for prior years of a sample of Mutual Funds.
3 Stock Returns Example USA Today stock fund data has 5 yr. Returns from various stock funds, classified according to th Type of mutual fund. Here there are four main Types [ aka “Broad Objectives”] and we will concentrate on these: B = Balanced, GI = Growth and Income, G = Growth GL = Global Here are side-by-side plots of the returns for the four major groups. This plot shows means diamonds and quantile box plots for each group. (The means diamonds are computed from the standard “assuming equal variance” analysis discussed below.) 5 yr Return (%) By Broad Objective There are clearly noticeable differences among the returns. Overall, are these statistically significant? If so, which differences are significant?

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4 Here are the means and standard deviations for each group, and the SEs for the mean of each group as computed from the SD of that group. Means and Std Deviations Level Number Mean Std Dev S.E. Mean B 6 106.2 26.23 10.71 G 31 192.6 51.07 9.17 GI 26 150.5 40.25 7.89 GL 9 98.44 38.94 12.98 Individual means
5 One Way ANOVA (Theory) Groups labeled i = 1,…, I . Observations Y ij in the i th group, with j = 1,…, n i . n = n i observations in all. Model : Y ij = i + ij , where ij indep. normal with mean=0 & var = 2 () i ij EY An alternate form of the model: Y ij = i + ij = + i + ij with 1 and i i i i I . (This implies that 0 i  .) Basic test is of H 0 : All the group means are the same - ie 1 = 2=…. .= I vs H a : They’re not all the same.

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6 Analysis of Variance (Explanation of Calculations, Formulas, & Relation to Regression) The Model has   ij i EY . So, we’ll estimate each i by the mean of the corresponding : 1,. ., ij i Y j n ; denoted by 1 i i ij j Y n Y . As with all our previous regression estimators, this is a Least Squares Estimator, ie it minimizes the total SSError:   2 22 , 1 ˆˆ ( )s =s where ij ij e ij i i ij i SSE Y Y n Y Y    As in other types of regression settings, this is compared to 2 () ij SST y Y  where Y denotes the grand mean.
7 Test of 01 : ..

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lect16-12_ANOVA - Lecture 16 One Way ANOVA Stat 102 2012...

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