Expected Mean Squares

1
Expected Mean Squares

• Crossed and nested factorial models with fixed
  and random factors
• Results can be derived as we need for the
  two-stage nested model
• Algorithms for SS, df, and EMS (though we will
  cover only df and EMS)

2
Rules for df

• The replication subscript is nested for the error
  term
• There are 3 types of subscripts
• Live Present and not nested
• Dead Present and nested
• Absent Absent from term (only used for SS)
• DfProduct of levels of each dead subscript and
  (levels-1) of each live subscript

3
Example 1

• Hospitals are randomly selected to test surgical
  methods (M). At each hospital (H), doctors (D)
  are randomly selected to treat randomly selected
  patientseach doctor uses all surgical methods
  with n patients.
• YijklmHi Mj HMijDk(i) MDjk(i)el(ijk),
  i1,,a
• j1,,b k1,,c l1,,n

4
Example 1
Source Df
H I-1
M J-1
D(H) I(K-1)
MD(H) I(J-1)(K-1)
Error IJK(n-1)
Total IJKn-1
5
Example 1
I J K n
R F R R
i j k l Component
Hi 1 J K n sH2
Mj I 0 K n SMj2/(J-1)
HMij 1 0 K n sHM2
Dk(i) 1 J 1 n sD2
MDjk(i) 1 0 1 n sMD2
el(ijk) 1 1 1 1 s2
6
Example 1
Source EMS
H s2 Jn sD2JKn sH2
M s2 n sMD2 Kn sHM2IKn SMj2/(J-1)
HM s2 n sMD2Kn sHM2
D(H) s2Jn sD2
MD(H) s2n sMD2
Error s2
7
Example 2.1

• Two groups of 5 randomly selected students each
  are tested on knowledge of 4 sentence types.
  Every student answers the same 3 randomly
  selected sentences of each type.
• YijklmGi Pj(i) TkSl(k) GTjk
• PTjk(i) GSil(k) PSjl(ik)

8
2 5 4 3
F R F R
i j k l Component
Gi 0 5 4 3 SGi2
Pj(i) 1 1 4 3 s2P
Tk 2 5 0 3 STk2/3
Sl(k) 2 5 1 1 s2S
GTik 0 5 0 3 SSGTik2/3
PTjk(i) 1 1 0 3 s2PT
GSil(k) 0 5 1 1 s2GS
PSjl(ik) 1 1 1 1 s2PS
9
Source EMS
G s2PS 5s2GS 12s2P 60 SGi2
P(G) s2PS 12s2P
T s2PS 3s2PT 10s2S 30 STk2/3
S(T) s2PS 10s2S
GxT s2PS 5s2GS 3s2PT 15 SSGTik2/3
P(G)xT s2PS 3s2PT
GxS(T) s2PS 5s2GS
P(G)xS(T) s2PS
10
Example 2.2

• Discard the interactions involving random effects
• YijklmGi Pj(i) TkSl(k) GTjk
• el(ijk)

11
Example 2.2
2 5 4 3
F R F R
i j k l Component
Gi 0 5 4 3 SGi2
Pj(i) 1 1 4 3 s2P
Tk 2 5 0 3 STk2/3
Sl(k) 2 5 1 1 s2S
GTjk 0 5 0 3 SSGTik2/3
el(ijk) 1 1 1 1 s2
12
Example 2.2
Source EMS
G s2 12s2P 60 SGi2
P(G) s2 12s2P
T s210s2S 30 SGi2/3
S(T) s2 10s2S
GxT s2 15 SSGTik2/3
Error s2