# COCHRAN-SATTERTHWAITE APPROXIMATION FOR LINEAR COMBINATIONS OF MEAN SQUARES

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
(Iowa State University)
Statistics 511
1/5
Consider the random variable
M = a1 M1 + a2 M2 + &middot; &middot; &middot; + 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 &middot; 2
&middot; E(M) 2
∼ χd ⇐⇒ M ∼
χd .
E(M)
d
What is a good choice for d that will make this approximation
reasonable?
c
(Iowa State University)
Statistics 511
2/5
If
&middot;
M∼
E(M) 2
χd ,
d
then
E(M)
d
2
E(M)
d
2
Var(M) ≈
=
=
≈
c
(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 ) + &middot; &middot; &middot; + a2k Var(MK )
= a21
h
E(M1 )
d1
i2
2d1 + &middot; &middot; &middot; + a2k
h
E(Mk )
dk
i2
2dk
.
= 2
P a2i [E(Mi )]2
≈ 2
Pk
c
(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