To simplify the notation all zero elements have been omitted from the matrix. By assumption (5 :28) has pq rows and 1+p+q+pq columns and thus rank (5:28) ~ pq. As the last pq columns form an identity matrix there is equality and rank (5 :28) = pq. There are thus p+q + 1 independent linear relations among the columns. The same two relations from the additive model still hold, namely that the sum of the a-regressors and the sum of the ~-re gressors each equal the Jl-regressor. Furthermore, for each i the sum of the ( a~) .. columns over j equals the a· column and for each j the sum of the (a~) .. lJ 1 lJ columns over all i equals the ~. colun1n. However, this does not give p+q inJ dependent relations, but p+q-l; because one can be obtained froln the others. This is seen from the following vector relations. p
(5:29)
p. = L a· ; i= 1
1
q-l ~
q
= Jl -
q-l
p
L ~.
j=l
L a· - L
J
i=l
~.
j=l J
1
;
(5:30)
q
a· = L (a~) .. ; 1 j=l lJ
i=l, ... ,p
(5:31)
j=l, ... ,(q-l)
(5 :32)
p ~.
J
=L
i=l
«(X~) .. ;
lJ
After substitution of (5 :29) and (5 :32) into (5 :30) and a change of the summation order we obtain q ~
q
= L
p
q
p
L (0:(3) .. - L L lJ
j=l i=l
j=l i=l
p
(a~) ..
lJ
=L
i=l
(0:(3).
lq
;
(5:33)
p+q+ 1 independent linear relations have now been demonstrated. The vector {ln</>..} 1 of a two·factor model with interactions as (5 :13) lj pqx thus belongs to a pq-<limensional space, while a model without interactions is restricted to a p+q-l dimensional space. The restrictions imposed on the additive model are of the following kind. For instance for p=q=2, (5 :13) gives lnep .. = Il lJ
+
(x.
1
+ ~J..'
i=1,2.
j=1,2
(5:34)
which yields the restriction (5 :35)
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