# 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 = 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 ~

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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 Regression (SSR) Sum of Squares Between (SSB). Variability of the group means compared to the grand mean (the variability due to the treatment). Overall mean
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33121 - Review of oneway ANOVA Kristin Sainani Ph.D...

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