11formulas - Math/Stat 423/523 Section 11.1 Chapter 11...

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Math/Stat 423/523 Chapter 11 Formulas Set Section 11.1 Two-Factor ANOVA with No Replications Notation A = 1 st factor, I = number of levels of A B = 2 nd factor, J = number of levels of B X ij = the measurement from the combination of the i th level of A and j th level of B x ij = actual (observed) value of X ij Two-Way Additive Fixed Model Model equation and assumptions are X ij = μ + α i + β j + ij where 0 , 0 J 1 j j I 1 i i = = = β = α , and ij 's are iid N(0, σ 2 ). The average response at the level i of A and level j of B is μ ij = E(X ij ) = μ + α i + β j . -------------------------------------------------------------------------------------- Parameter Estimates Factor A, level i total and average : J x x , x x . i . i J 1 j ij . i = = = Factor B, level j total and average : I x x , x x j . j . I 1 i ij j . = = = , Grand Average : IJ x x I 1 i J 1 j ij .. ∑ ∑ = = = Parameter Estimate μ .. x ˆ = μ α i .. . i i x x ˆ - = α β j .. j . j x x ˆ - = β ij = μ ˆ + i ˆ α + j ˆ β is the predicted or fitted value . e ij = x ij - ij is a residual which estimates ij . Chapter 11 Formulas 1 A B
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Hypothesis Tests Factor A: H 0 : α 1 = α 2 = . .. = α I = 0 vs. H a : at least one α i 0 Factor B: H 0 : β 1 = β 2 = . .. = β J = 0 vs. H a : at least one β j 0 Sums of Squares df ( 29 ∑ ∑ ∑ ∑ = = = = - = - = I 1 i 2 .. J 1 j 2 ij I 1 i J 1 j 2 .. ij IJ x x x x SST IJ-1 ( 29 IJ x x J 1 x x J SSA 2 .. I 1 i 2 . i I 1 i 2 .. . i - = - = = = I-1 ( 29 IJ x x I 1 x x I SSB 2 .. J 1 j 2 j . J 1 j 2 .. j . - = - = = = J-1 ( 29 ∑ ∑ = = + - - = I 1 i J 1 j 2 .. j . . i ij x x x x SSE (I-1)(J-1) ANOVA Table Source df SS MS F P-value Factor A I-1 SSA 1 I SSA MSA - = MSE MSA F = ( 29 F F P ) 1 J )( 1 I ( , 1 I - - - Factor B J-1 SSB 1 J SSB MSB - = MSE MSB F = ( 29 F F P ) 1 J )( 1 I ( , 1 J - - - Error (I-1)(J-1) SSE ) 1 J )( 1 I ( SSE MSE - - = Total IJ-1 SST ∑ ∑ = = = I 1 i J 1 j 2 ij e SSE , SSE = SST - SSA – SSB = = β - + σ = α - + σ = σ = J 1 j 2 j 2 I 1 i 2 i 2 2 1 J I ) MSB ( E , 1 I J ) MSA ( E , ) MSE ( E Factor A: RR = {F=MSA/MSE > F α ,I-1,(I-1)(J-1) } Factor B: RR = {F=MSB/MSE > F α ,J-1,(I-1)(J-1) } ------------------------------------------------------------------------------------- T Method for Significant Differences Compute s comparison B factor for I MSE Q w s comparison A factor for J MSE Q w ) 1 J )( 1 I ( , J , B ) 1 J )( 1 I ( , I , A - - α - - α = = . Apply w A to . I . 2 . 1 x ,..., x , x or w B to J . 2 . 1 . x ,..., x , x Block designs : ANOVA, T Method the same as above with Factor A = Blocks. Chapter 11 Formulas 2 C D
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Two-Way Additive Random Model: X ij = μ + A i + B j + ij where the A i 's are iid N(0, σ A 2 ), the B j 's are iid N(0, σ B 2 ), and ij 's are iid N(0, σ 2 ). 2
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11formulas - Math/Stat 423/523 Section 11.1 Chapter 11...

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