STATISTICAL RESEARCH REPORT
5
1971
No
Institute of Mathematics
University of Oslo
JOINT CONFIDENCE INTERVALS FOR AIJL LINEAR FUNCTIONS
OF THE MEANS IN THE ONE-WAY LAYOUT WITH UNKNOWN GROUP
VARIANCES
by
Emil Spj0tvoll
- 1 -
ABSTRACT
The one-way layout in the analysis of variance with
unknown group variances is considered.
A family of joint
confidence intervals for all linear functions in the means
with the property that the probability is
1 - a. that all
confidence intervals covers the true values of the linear
functions is found,
Each confidence interval is natural
in the sense that for a given linear function it is equal
to an estimate of th:· s function plus and minus a constant
times an estimate of the variance of the estimate.
the results are analogous to Scheffe's
multiple comparison.
S-method of
Hence
- 2 -
1.
STATEMENT OF THE PROBLEM AND THE METHOD
Consider the one-way layout with unequal group variances in the
analysis of variance.
i
= 1, ••• ,k
Let the random variables
Yij , j = 1, ••• ,ni
be independent with
E Yij = ~i ' Var Yij ·- cri
2
•
The means and the variances are all unlcnown.
The problem is that
of finding joint confidence intervals for all linear functions of
the
,,,
=
I
'I'
where the
k
L
i=~
ci
c
l'
II
1-"l'
'
are known constants.
the case when all
cri
A
solution to this problem in
are equal was given by Scheffe (1953).
other solutions see, e.gq Miller (1966) and Scheffe (1959).
For
We
shall now derive a solution which takes care of the possibility that
the
cr i
A
may be unequal.
natural estimator of
k
ciyi.
'V = I:
j.=1
'
ni
Y·lJ./n.l •
yi• =I:
j=1
·; is
/:..
where
k
O't. 2 -· I:
111
i=1
2
ci O'i
ni
The variance of
2
An estimate of this variance is
1\
crt.
2
*
where
k
= I:
i=1
. 2
c. 2 s.
l
l
n.l
~
is
- 3 -
'life
shall prove that there exists a constant
bility is
that the values
1-~
A such that the proba-
of all the linear functions
~
satisfy
~
- MJ
~
The constant
< ¢<
f
-
IV
-
+ A~
(1)
~
will depend upon a
A
and the
ni , but not upon
unknown parameters.
It is determined by the following.
z (n 1-1 ) , •• , z (nk -1 )
denote
k
independent
Let
:I?- dis tr i bu ted random
variables all with one degree of freedom in the numerator and
n 1 -1, ••• ,nq-1
Then
is determined by
A
k~
P[
degrees of freedom in the denominator, respectively.
. 1
l=
z(n.-1)
l
0]
<A~
-
= 1-a, •
k
Since the exact distribution of
~
z(n.-1)
l
is difficult to
i=1
calculate, a simple approximation is proposed in Section 3.
The approximate value of
A
~
2
is given by
A
( 2)
aF (k,b)
a
where
2 (n.-2)
(n.-1)
1
l
(k-2)(~
+ 4-k
2 {ni-5)
~
{ni-3)
n.
3
i=1
i=1 l -k
2
k
(ni-1)2(:n.i-2)
ni -1
k
- ( ~i=1 n.-3 )
L: i='1
2
l
(ni -3 ) (ni -5)
n. -.1 2
k
k
_l_.)
b
:::
(3)
and
2
a = ( 1 - b)
and
F
(k 'b)
r.
k
i=1
n.-1
l
n.-3
l
is the upper
(4)
'
a-point of the
('(,
k
and
b
degrees of freedom.
F-distribution with
- 4 -
2•
PROOF OF THE
~1ETHOD
To proof the main result of the previous section we need the
following lemma.
LE~IDU.
numbers.
k
Let
d 1 , ••• ,dk
~
z 1 , ••• 1 zk
and
c (>0) be given real
Then
d.z. 2 < c2
r:
i=1
l.
l.
( 5)
-
if and only if
k
l i=1
r:
c. z.
l.
l.
I
<
c
1/2
d ~~ )
r:
(
i=1
for all real numbers
Proof.
c. 2
k
(6)
l.
c 1 , ••• ,ck.
If (6) holds, it follows by using Schwarz's inequality
that
( r
i=1
c..1 z.)
l.
c.
k
2
k
l.
( r
:;::
i=1
di
1/2
from which we obtain (6).
d.1/2z.)
l.
2
<
-
l.
k
ci
(r
2
r l.
i=1
k
i=1
Using the Lemma with
THEOREM.
p
l
[
di = ni/si
k
+ A ( r.
i=1
i:.:1
s. 2
r.
c.y.
- A(
l. l..
'·· i=1
n . ( y . -!-! . ) 2
k
k
r.
k
i=1
c.l. 2 s.l. 2 1/2
ni
)
l.
2
9
zi = yi• - Ui
l.
J.•
l.
1/2
c.2s.2
l.
l.
ni
l.
c
'i
9
ci = dizi , from which we get (5).
we obtain the following theorem.
r
l.
Conversely, if (6) holds for all
it holds in particular for
P
2
)( L: d.z. ·)
)
k
<
r,
i=1
k
r.
-- i=1
Ci!J.i <
for all c 1 1 • • • 1 ck
J.
0
iyi•
and
c = A
- 5 -
Since the distribution of
is the same as the
k
~
distribution of
z(ni-1) , the statement above (1) in Section 1
i=1
is true.
3.
THE APPROXIMATION
We will approximate the distribution of the random variable
k
~
v =
. 1
l=
z(n.-1)
l
by the distribution of
a F(k,b) , where
buted random variable with
constants
of
v
a
and
and
b
k
and
b
F(k,b)
is an
F-distri-
degrees of freedom.
The
are determined so that the first two cumulants
aF(.k,b)
are equal.
distribution when all
ni
The approximation gives the exact
tend to infinity.
Furthermore, Morrison
(1971) has compared the exact and approximate distribution in the
case
n 1 = n 2 , and shown that the approximation is
k = 2 ,
excellent.
The cumulants of
k
=
=
v
are
~
n.-1
l
n.-3
l
k
I:
2
2(ni-1) (ni-2)
2
(ni-3) (ni-5)
i=1
i=1
while those of
aF(k,b)
are
ab
= b-2
2
2
2a (k+b-2)b
= ----"--::::---'-k(b-2)2(b-4)
Solving
a
and
b
from the equations
get the solutions (3) and (4).
t~i
= :x,i*
i
= 1, 2 , we
- 6 -
4•
AN EXAMPLE
We shall use the data in Pearson and Hartley
Here
(1958~
p. 27).
2
k = 2 , n 1 = 10 , n 2 = 15 , y • = 7 3. 4 , y 2 • = 4 7. 1 , s 1 = 51 ,
1
s 2 2 = 141 •
We shall suppose that we want to find confidence
intervals for the difference
separately.
We find
a
= 2.06
that the values obtained for
Using a = .10
is [ 17.5
~
1
-
, b
a
~
2
, as well as for
= 12.51
and
b
and
A
iJ.
1
and
1
= 2.31
~2
Note
•
seem very reasonable.
we find that the confidence interval for
35.1] , for
~
it is [ 68.2 , 78.6]
~
and for
1 -
~
1-!
2
2
it
is [ 40.0 , 54.2] •
1 - w2 obtained by
Pearson and Hartley using a method. due to Welch (1947) is [ 19.8 ,
The 90 percent confidence interval for
32.8
J.
1.1.
As should be expected, since this method is aimed only at
that difference
1 - iJ. 2 , this interval is smaller than the one
obtained by the method of Section 1.
1.1.
If we actually wanted
3
confidence intervals and did not use
the simultaneous method of Section 1, we could still do this and
still have 90 percent probability that all three intervals were
correct by increasing the confidence coefficient of each interval
to
96.67 •
Doing this we find that the confidence intervals for
J
and
2 , using ordinary t-intervals, are [ 67.8 ' 79.0
[ 39.7 ' 54.5] , respectively. The interval for o. 1 - a 2 becomes
this I had to interpolate in the available
[17.7 ' 34.9] (to find
..,.,
iJ.
1
and
tables).
iJ.
It is seen rthat two intervals are slightly wider while
one is slightly narrower than the intervals obtained by the method
of Section 1.
- 7 -
REFERENCES
MILLER, R.G. Jr. (1966). Simultaneous Statistical Inference.
New York : McGraw-Hill.
MORRISON, D.F. (1971). On the distribution of linear functions of
independent F variates. J. Amer. Statist. Ass. 66,
383-85.
PEARSON, E.S. & HARTLEY, H.O. (1958). Piometrika Tables for
Statisticians, 1. Cambridge University Press.
SCHEFFE, H. (1953). A method for judging all contrasts in the
analysis of variance. Biometrika, 40, 87-104.
SCHEFFE, H. (1959). The Analysis of Variance.
Wiley and Sons.
New York
John
WELCH, B.L. (1947). The generalization of 'Students' problem when
several different population variances are involved.
Biometrika, 34, 28-35.