# 10formulas - Math/Stat 423/523 Chapter 10 Formulas Section...

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Unformatted text preview: Math/Stat 423/523 Chapter 10 Formulas Section 10.1 Single-Factor ANOVA (Equal Sample Sizes) • I = total number of treatments • J = common number of replications of each treatment • μ i = mean of treatment i (for i = 1, 2, ..., I) • X ij = random variable that represents the measurement from the j th EU under treatment i (for i = 1, ..., I and j = 1, ..., J) The One-Way Fixed Model : X ij = μ i + ∈ ij where ∈ i1 , ∈ i2 , ..., ∈ iJ are iid N(0, σ 2 ). Definition Sums of Squares Treatment i average : J x x J 1 j ij . i ∑ = = Grand Average : IJ x x I 1 i J 1 j ij .. ∑ ∑ = = = • ( 29 ∑ ∑ = =- = I 1 i J 1 j 2 .. ij x x SST • ( 29 ∑ =- = I 1 i 2 .. . i x x J SSTr Treatment i standard deviation = s i • ( 29 ∑ ∑ = =- = I 1 i J 1 j 2 . i ij x x SSE ∑ =- = I 1 i 2 i s ) 1 J ( SSE I / s MSE I 1 i 2 i = ∑ =-------------------------------------------------------------------------------------- Alternative (Working) Formulas Let ∑ ∑ ∑ = = = = = J 1 j ij . i I 1 i J 1 j ij .. x x , x x , IJ x CF factor correction 2 .. = = . • CF x SST I 1 i J 1 j 2 ij- = ∑ ∑ = = • CF J x SSTr I 1 i 2 . i- = ∑ = • SSE = SST – SSTr Remarks: • . i ij ij x x e- = is called a residual and e ij estimates ∈ ij . • SST = SSTr + SSE. SSE = SST – SSTr ....
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10formulas - Math/Stat 423/523 Chapter 10 Formulas Section...

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