anova - ANOVA Suppose X = {xij } is an n × m matrix with...

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Unformatted text preview: ANOVA Suppose X = {xij } is an n × m matrix with numbers. In ANOVA we study the decomposition xij = µ + αi + βj + γij into a main effect, a row effect, a column effect, and an interaction. We identify the decomposition by requiring n αi = 0, i=1 m β j = 0, j =1 n γij = 0 for all j, i=1 m γij = 0 for all i. j =1 Now write 1 xij , nm i=1 j =1 1 xij , m j =1 n m n m x•• = xi• = x•j 1 = xij . n i=1 Date: April 23, 2007. 1 2 ANOVA Then µ = x•• , αi = xi• − x•• , βj = x•j − x•• , γij = xij − xi• − x•j + x•• . Moreover n m n m (xij − µ) = i=1 j =1 n 2 (αi + βj + γij )2 = i=1 j =1 m n m 2 γij , i=1 j =1 =m i=1 α2 i +n j =1 β2 j + which we can write as SSQtotal = SSQr ow + SSQcolumn + SSQinter action , and also as 2 2 2 2 σ 2 χnm−1 = σ 2 {χn−1 + χm−1 + χ(n−1)(m−1) }. Source Rows Columns Interaction Total Sum of Squares Degrees of Freedom n SSQR = m i=1 α2 i m SSQC = n j =1 β2 j n m 2 SSQI = i=1 j =1 γij n m 2 i=1 j =1 (xij − µ) Mean Square 1 n−1 SSQR 1 m−1 SSQC n−1 m−1 (n − 1)(m − 1) nm − 1 1 (n−1)(m−1) SSQI ANOVA anova<−function ( x ) { n<−nrow ( x ) ; m −ncol ( x ) < mu) ^2) mu −mean ( x ) ; s t o t<−sum ( ( x− < alpha<−apply ( x , 1 , mean )−mu; srow<− *sum ( alpha ^2) m 5 beta<−apply ( x , 2 , mean )−mu; scol<−n *sum ( beta ^2) gamma<−x−outer ( alpha , beta , " + " )−mu; s i n t<−sum ( gamma^2) cat ( "−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−\n" ) cat ( " 10 cat ( " Source | Sum of Squares | Degrees of Freedom | Mean Square\n" ) Rows" , formatC ( srow , d i g i t s =6 , width=16 , format= " f " ) , formatC ( n−1, cat ( "−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−\n" ) d i g i t s =0 , width=20 , format= " f " ) , formatC ( srow / ( n−1) , d i g i t s =6 , width=13 , format= " f " ) , " \n" ) cat ( " Columns" , formatC ( scol , d i g i t s =6 , width=16 , format= " f " ) , formatC (m 1, − − d i g i t s =0 , width=20 , format= " f " ) , formatC ( scol / (m 1) , d i g i t s =6 , width=13 , format= " f " ) , " \n" ) 3 cat ( " I n t e r a c t i o n " , formatC ( sint , d i g i t s =6 , width=16 , format= " f " ) , formatC ( ( n−1) * (m −1) , d i g i t s =0 , width=20 , format= " f " ) , formatC ( s i n t / ( ( n−1) * (m 1) ) , d i g i t s =6 , − width=13 , format= " f " ) , " \n" ) cat ( "−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−\n" ) cat ( " Total " , formatC ( stot , d i g i t s =6 , width=16 , format= " f " ) , formatC ( n *m 1, − d i g i t s =0 , width=20 , format= " f " ) , formatC ( s t o t / ( n *m 1) , d i g i t s =6 , width=13 , − format= " f " ) , " \n" ) 15 cat ( "−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−\n" ) } > anova ( x ) −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− Source | Sum of Squares | Degrees of Freedom | Mean Square −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− 5 Rows Columns Interaction Total 10 1.472128 0.496731 16.153535 18.122395 3 4 12 19 0.490709 0.124183 1.346128 0.953810 −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− ...
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