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COCHRAN-SATTERTHWAITE APPROXIMATION FOR LINEAR COMBINATIONS OF MEAN SQUARES Suppose M1 , ..., Mk are independent mean squares and that di Mi ∼ χ2di E(Mi ) ∀ i = 1, . . . , k. It follows that di M i di Mi E(Mi ) 2 E χdi = di , Var = 2di , and Mi ∼ E(Mi ) E(Mi ) di for all i = 1, . . . , k. c Copyright 2012 (Iowa State University) Statistics 511 1/5 Consider the random variable M = a1 M1 + a2 M2 + · · · + ak Mk , where a1 , a2 , . . . , ak are known constants in IR. Note that M is a linear combination of scaled χ2 random variables. The Cochran-Satterthwaite approximation works by assuming that M is approximately distributed as a scaled χ2 , just like each of the variables in the linear combination. dM · 2 · E(M) 2 ∼ χd ⇐⇒ M ∼ χd . E(M) d What is a good choice for d that will make this approximation reasonable? c Copyright 2012 (Iowa State University) Statistics 511 2/5 If · M∼ E(M) 2 χd , d then E(M) d 2 E(M) d 2 Var(M) ≈ = = ≈ c Copyright 2012 (Iowa State University) Var χ2d (2d) 2 [E(M)]2 d 2 2M . d Statistics 511 3/5 Now note that Var(M) = a21 Var(M1 ) + · · · + a2k Var(MK ) = a21 h E(M1 ) d1 i2 2d1 + · · · + a2k h E(Mk ) dk i2 2dk . = 2 P a2i [E(Mi )]2 ≈ 2 Pk c Copyright 2012 (Iowa State University) di 2 2 i=1 ai Mi /di . Statistics 511 4/5 Equating these two variance approximations yields k X 2M 2 =2 a2i Mi2 /di . d i=1 Solving for d yields d = Pk M2 2 2 i=1 ai Mi /di P = Pk k i=1 ai Mi 2 2 2 i=1 ai Mi /di . This is the Cochran-Satterthwaite formula for the approximate degrees of freedom associated with the linear combination of mean squares M. c Copyright 2012 (Iowa State University) Statistics 511 5/5