Stat664
About Expected Mean Squares
The discussions below are for Balanced Multi-Way Mixed-Effects Models.
Unrestricted Models
Any random effect (including any interaction that involves fixed effects) has its elements sampled
from normal population with zero mean. For instance, assuming that βj ’s are random, even if αi ’s
2
).
are fixed, the random variables (αβ)ij ’s are i.i.d. N (0, σAB
The general formula given on page F.9 in classnotes
Variance Components Model
is for unrestricted models (click the link above and then click on § 4.1 on the navigation tabs on
the left). That is, no side conditions were imposed on summing over the fixed-effect margins of
random effects. The two examples on page F.10, assuming unrestricted model, follow through
this formula.
Three-Way Unrestricted Mixed-Effects Models
Fixed: A; Random: B, C.
u
A
B
C
AB
AC
BC
ABC
Error
Vu
{B,C,BC}
{A,C,AC}
{A,B,AB}
{C}
{B}
{A}
—
—
E[M Su ]
2
2
2
]
+ nσABC
+ nbσAC
σ 2 + nbckA2 + [ncσAB
2
2
2
2
2
σ + nacσB + [ncσAB + naσBC + nσABC ]
2
2
2
σ 2 + nabσC2 + [nbσAC
+ naσBC
+ nσABC
]
2
2
2
σ + ncσAB + [nσABC ]
2
2
]
+ [nσABC
σ 2 + nbσAC
2
2
2
σ + naσBC + [nσABC
]
2
2
σ + nσABC
σ2
Fixed: A, B; Random: C.
u
A
B
C
AB
AC
BC
ABC
Error
Vu
{C,BC}
{C,AC}
{A,B,AB}
{C}
{B}
{A}
—
—
E[M Su ]
2
2
σ 2 + nbckA2 + [nbσAC
+ nσABC
]
2
2
2
2
σ + nackB + [naσBC + nσABC
]
2
2
2
2
2
σ + nabσC + [nbσAC + naσBC + nσABC
]
2
2
2
σ + nckAB + [nσABC ]
2
2
σ 2 + nbσAC
+ [nσABC
]
2
2
2
σ + naσBC + [nσABC ]
2
σ 2 + nσABC
σ2
1
Restricted Models
A restricted model has side conditions imposed on the fixed-effect margins. For the example above,
the random variables (αβ)ij ’s are no longer independent, though still are normally distributed.
Additional side conditions are imposed on these random variables. One key restriction, among
others (about covariance structure), is that
a
X
(αβ)ij = 0, for j = 1, · · · , b.
i
See the discussion on the two-way restricted model on page F.11 of the abovementioned classnotes
link.
Expected Mean Squares for Restricted Models
The general formula on page F.9 still holds except for the replacement of Vu by Vur :
Vur = {v : v is a pure random effect not involving u}
Note that multi-way random effects model is unaffected by this definition.
Three-Way Restricted Mixed-Effects Models
Fixed: A; Random: B, C.
u
A
B
C
AB
AC
BC
ABC
Error
Vur
{B,C,BC}
{C}
{B}
{C}
{B}
—
—
—
E[M Su ]
2
2
2
]
+ nσABC
+ nbσAC
σ 2 + nbckA2 + [ncσAB
2
2
2
σ + nacσB + [naσBC ]
2
σ 2 + nabσC2 + [naσBC
]
2
2
2
σ + ncσAB + [nσABC ]
2
2
]
+ [nσABC
σ 2 + nbσAC
2
2
σ + naσBC
2
σ 2 + nσABC
σ2
Fixed: A, B; Random: C.
u
A
B
C
AB
AC
BC
ABC
Error
Vur
{C}
{C}
—
{C}
—
—
—
—
E[M Su ]
2
σ 2 + nbckA2 + [nbσAC
]
2
2
2
σ + nackB + [naσBC
]
2
2
σ + nabσC
2
2
σ 2 + nckAB
+ [nσABC
]
2
2
σ + nbσAC
2
σ 2 + naσBC
2
2
σ + nσABC
σ2
2
MINITAB
By default, MINITAB assumes unrestricted models in both GLM and AOV (Balanced ANOVA)
unless otherwise specified in Options sub-dialog box. The example follows through
Multi-Way ANOVA Model
using bottler example. The original data were obtained from fixed-effects experiment. Its use
here is for illustration only. For convenience, the verbal names of the factors were changed using
more convenient names A, B, and C. For each case of unrestricted/restricted × (A fixed, B&C
random)/(A&B fixed, C random), only the table of expected mean squares were retained (note that
MINITAB produced ANOVA table using appropriate error terms, and using Satterthwaite pseudo
F when appropriate).
Fixed: A; Random: B, C
1
2
3
4
5
6
7
8
Source
A
B
C
A*B
A*C
B*C
A*B*C
Error
1
2
3
4
5
6
7
8
Source
A
B
C
A*B
A*C
B*C
A*B*C
Error
Variance
component
3.52083
1.77083
0.52083
-0.06250
0.08333
-0.08333
0.70833
Variance
component
3.69444
1.75000
0.52083
-0.06250
0.05556
-0.08333
0.70833
Error
term
*
*
*
7
7
7
8
Expected Mean Square for Each
(using unrestricted model)
(8) + 2 (7) + 4 (5) + 4 (4) +
(8) + 2 (7) + 6 (6) + 4 (4) +
(8) + 2 (7) + 6 (6) + 4 (5) +
(8) + 2 (7) + 4 (4)
(8) + 2 (7) + 4 (5)
(8) + 2 (7) + 6 (6)
(8) + 2 (7)
(8)
Term
Error
term
*
6
6
7
7
8
8
Expected Mean Square for Each Term
(using restricted model)
(8) + 2 (7) + 4 (5) + 4 (4) + 8 Q[1]
(8) + 6 (6) + 12 (2)
(8) + 6 (6) + 12 (3)
(8) + 2 (7) + 4 (4)
(8) + 2 (7) + 4 (5)
(8) + 6 (6)
(8) + 2 (7)
(8)
Q[1]
12 (2)
12 (3)
Fixed: A, B; Random: C
1
2
3
Source
A
B
C
Variance
component
1.77083
Error
term
5
6
*
Expected Mean Square for Each Term
(using unrestricted model)
(8) + 2 (7) + 4 (5) + Q[1,4]
(8) + 2 (7) + 6 (6) + Q[2,4]
(8) + 2 (7) + 6 (6) + 4 (5) + 12 (3)
3
4
5
6
7
8
1
2
3
4
5
6
7
8
A*B
A*C
B*C
A*B*C
Error
Source
A
B
C
A*B
A*C
B*C
A*B*C
Error
-0.06250
0.08333
-0.08333
0.70833
Variance
component
1.77778
-0.10417
0.05556
-0.08333
0.70833
7
7
7
8
Error
term
5
6
8
7
8
8
8
(8)
(8)
(8)
(8)
(8)
+
+
+
+
2
2
2
2
(7) + Q[4]
(7) + 4 (5)
(7) + 6 (6)
(7)
Expected Mean Square
for Each Term (using
restricted model)
(8) + 4 (5) + 8 Q[1]
(8) + 6 (6) + 12 Q[2]
(8) + 12 (3)
(8) + 2 (7) + 4 Q[4]
(8) + 4 (5)
(8) + 6 (6)
(8) + 2 (7)
(8)
SAS
SAS basically uses unrestricted models in PROCs GLM, MIXED, and VARCOMP. However, users
still have the flexibility to declare appropriate error term in a TEST statement. Note that SAS
sort of advocates unrestricted model form (see “Sum-To-Zero Assumption” section in the help
documentation of the RANDOM statement in PROC GLM).
Assume that it’s desired to use unrestricted model. When using RANDOM statement with
TEST option in PROC GLM, SAS does all the tests using appropriate error (pseudo error) terms
and produces the correct output. See
Multi-Way Mixed-Effect Model
for example.
For restricted model, use E = error-term in the TEST statement in PROC GLM. The
error-term must be an effect (one only).
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