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and above treatment 1 and
t
i
's correspond to the individual physician

effects.

This problem belongs to the class of problems considered by

Neyman and Scott (1948). Suppose we use invariance to eliminate the

nuisance parameters
t
i
's corresponding to the physician effect. Notice

that
X
i,
2

X
i
,1
is a maximal invariant under the group of translations

and
X
i,
2

X
i
,1
~
N
((

1
),2)
. Similarly
Y
j,
2

Y
j
,1
~
N
((

1
)

(

1
),2
.

2

2

2

Let us denote

d
. Now consider the partition

d
,(

1
)

2

1

2

R
2
,

R
2
}
induced by

of the original parameter space

{(

,

,

):

this maximal invariant:

such that

Orbit
(
d
,
d
)

{(

,

)

d
}.

Now consider the likelihood for two sets of parameters,
(

d
,(

1
)

2

1

2

*
)
and

where , that is, they

belong to the same orbit induced by the maximal invariant. Since the

distribution of the maximal invariant is a function of
(
d
,d
)
, it is clear

that the two combinations of the parameters have identical likelihood.

An immediate consequence of this is that
1
and
2
, the change in the

diastolic blood pressure and the change in the systolic blood pressure,

are not identifiable. We can only find out if the change in the diastolic

blood pressure is larger or smaller than the change in the systolic

blood pressure and the magnitude of that difference. But the question:

How much does the treatment 2 reduce systolic blood pressure or dias-

tolic blood pressure, is unanswerable in this situation.

Instead of accepting this limitation of the data, we decide to follow

the superimposition logic and conduct the following analysis.

*
,

1.

Least squares approach:
Since there are nuisance parame-

ters related to the translation group, we first superimpose, that

is translate, the observations in Sample 1 in such a manner

that is minimized. This corre-

sponds to transforming the observations

to Similarly, we translate the observations

in Sample 2 to Now, we average these trans-

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