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# 33121 - Review of oneway ANOVA Kristin Sainani Ph.D...

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Review of one-way ANOVA Kristin Sainani Ph.D. http://www.stanford.edu/~kcobb Stanford University Department of Health Research and Policy

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ANOVA for comparing means between  more than 2 groups
The F-distribution A ratio of variances follows an F-distribution: 2 2 2 2 0 : : within between a within between H H σ σ σ σ = The F-test tests the hypothesis that two variances are equal. F will be close to 1 if sample variances are equal. m n within between F , 2 2 ~ σ σ

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How to calculate ANOVA’s by  hand… Treatment 1 Treatment 2 Treatment 3 Treatment 4 y 11 y 21 y 31 y 41 y 12 y 22 y 32 y 42 y 13 y 23 y 33 y 43 y 14 y 24 y 34 y 44 y 15 y 25 y 35 y 45 y 16 y 26 y 36 y 46 y 17 y 27 y 37 y 47 y 18 y 28 y 38 y 48 y 19 y 29 y 39 y 49 y 110 y 210 y 310 y 410 n =10 obs./group k =4 groups The group means 10 10 1 1 1 = = j j y y 10 10 1 2 2 = = j j y y 10 10 1 3 3 = = j j y y 10 10 1 4 4 = = j j y y The (within) group variances 1 10 ) ( 10 1 2 1 1 - - = j j y y 1 10 ) ( 10 1 2 2 2 - - = j j y y 1 10 ) ( 10 1 2 3 3 - - = j j y y 1 10 ) ( 10 1 2 4 4 - - = j j y y
Sum of Squares Within  (SSW), or Sum of Squares  Error (SSE) The (within) group variances 1 10 ) ( 10 1 2 1 1 - - = j j y y 1 10 ) ( 10 1 2 2 2 - - = j j y y 1 10 ) ( 10 1 2 3 3 - - = j j y y 1 10 ) ( 10 1 2 4 4 - - = j j y y ∑∑ = = - = 4 1 10 1 2 ) ( i j i ij y y + = - 10 1 2 1 1 ) ( j j y y = - 10 1 2 2 2 ) ( j j y y = - 10 3 2 3 3 ) ( j j y y = - 10 1 2 4 4 ) ( j j y y + + Sum of Squares Within (SSW) (or SSE , for chance e rror)

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Sum of Squares Between (SSB), or  Sum of Squares Regression (SSR) Sum of Squares Between (SSB). Variability of the group means compared to the grand mean (the variability due to the treatment).
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