Estimable Functions of β: An Example
customer
1
2
3
4
movie
1 2 3
4 1 ?
? 3 5
? ? 3
3 1 ?
yij
= customer i’s rating
of movie j
yij
= µ + ci + mj + ij
c
Copyright 2010
(Iowa State University)
Can we guess ratings
for customer/movie
combinations not
in the dataset?
Which movie is
best?
Statistics 511
1 / 11
The Linear Model in Vector/Matrix Form










4
1
3
5
3
3
1


 
 
 
 
=
 
 
 
 
1
1
1
1
1
1
1
y=
c
Copyright 2010
(Iowa State University)
1
1
0
0
0
0
0
0
0
1
1
0
0
0
0
0
0
0
1
0
0
X
0
0
0
0
0
1
1
1
0
0
0
0
1
0
0
1
1
0
0
0
1
0
0
0
1
1
0
0













β
µ
c1
c2
c3
c4
m1
m2
m3



 
 
 
 
+
 
 
 
 

+
11
12
22
23
33
41
42










Statistics 511
2 / 11
Can We Estimate the Means that Underly the Missing
Table Entries?





Xβ = 



µ + c1 + m1
µ + c1 + m2
µ + c2 + m2
µ + c2 + m3
µ + c3 + m3
µ + c4 + m1
µ + c4 + m2









Can we estimate
µ + c1 + m3 ?
µ + c2 + m1 ?
µ + c3 + m1 ?
µ + c3 + m2 ?
µ + c4 + m3 ?
m1 − m2 is estimable because
[1, −1, 0, 0, 0.0.0] Xβ = m1 − m2 .
c
Copyright 2010
(Iowa State University)
Statistics 511
3 / 11
Can We Estimate the Means that Underly the Missing
Table Entries?





Xβ = 



µ + c1 + m1
µ + c1 + m2
µ + c2 + m2
µ + c2 + m3
µ + c3 + m3
µ + c4 + m1
µ + c4 + m2









Can we estimate
µ + c1 + m3 ?
µ + c2 + m1 ?
µ + c3 + m1 ?
µ + c3 + m2 ?
µ + c4 + m3 ?
Likewise, m2 − m3 is estimable because
[0, 0, 1, −1, 0, 0, 0] Xβ = m2 − m3 .
c
Copyright 2010
(Iowa State University)
Statistics 511
4 / 11
We can estimate all pairwise differences between
movie effects.
Because m1 − m2 and m2 − m3 are estimable, we can also
estimate
(m1 − m2 ) + (m2 − m3 ) = m1 − m3 .
This follows because any linear combination of estimable
functions is also estimable.
c
Copyright 2010
(Iowa State University)
Statistics 511
5 / 11
We can estimate the mean underlying the rating for
any combination of customer and movie.
It follows that any linear combination of the form
µ + ci + mj
can be estimated ∀ i = 1, 2, 3, 4 and j = 1, 2, 3 because
µ + ci + mj = (µ + ci + mj0 ) + (mj − mj0 )
∀ i = 1, 2, 3, 4 and j, j0 = 1, 2, 3.
c
Copyright 2010
(Iowa State University)
Statistics 511
6 / 11
Movie lsmeans
If our goal is to compare movies to see which is most highly
rated, we can accomplish that by estimating the pairwise
differences between movie effects.
However, if we want to retain information about the mean rating
rather than the difference between mean ratings, it is natural to
consider estimating the average (across all customers) of the
mean rating for each movie.
c
Copyright 2010
(Iowa State University)
Statistics 511
7 / 11
Movie lsmeans
This average for the jth movie is
4
4
1X
1X
(µ + ci + mj ) = µ +
ci + mj = µ + c̄· + mj .
4 i=1
4 i=1
This average is estimable for each movie in our example
because it is a linear combination of estimable functions.
SAS calls estimates of µ + c̄· + mj ∀ j = 1, 2, 3 movie lsmeans,
where ls presumably denotes least squares to indicate that
these estimates are based on OLS estimators.
c
Copyright 2010
(Iowa State University)
Statistics 511
8 / 11
Suppose we consider a different model.
customer
1
2
3
4
movie
1 2 3
4 1 ?
? 3 5
? ? 3
3 1 ?
yij
= customer i’s rating
of movie j
yij
= µij + ij
c
Copyright 2010
(Iowa State University)
Can we guess ratings
for customer/movie
combinations not
in the dataset?
Which movie is
best?
Statistics 511
9 / 11
The Linear Model in Vector/Matrix Form











4
1
3
5
3
3
1


 
 
 
 
=
 
 
 
 
1
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
c
Copyright 2010
(Iowa State University)
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
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0
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0
0
1
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0
0
0
0
0
1
0
0
0
0
0
0
0




















µ11
µ12
µ13
µ21
µ22
µ23
µ31
µ32
µ33
µ41
µ42
µ43



 


 
 
 
 
 
+
 
 
 






11
12
22
23
33
41
42
Statistics 511










10 / 11
Can we estimate the means that underly the missing
table entries? No.





Xβ = 



µ11
µ12
µ22
µ23
µ33
µ41
µ42









Can we estimate
µ13 ?
µ21 ?
µ31 ?
µ32 ?
µ43 ?
None of the means underlying missing table entries are
estimable under this cell-means model.
c
Copyright 2010
(Iowa State University)
Statistics 511
11 / 11