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Fourth exact moment result for improving MRBP based inferences
Journal
of Statistical
Planning
and Inference
263
28 (1991) 263-270
North-Holland
Fourth exact moment result for improving MRBP based inferences Derrick
S. Tracy and Khushnood
Department
of Mathematics
and Statistics,
Received 21 August 1989; revised Recommended by J.N. Srivastava
Abstract:
Tests similar
(1982) to analyze of this test statistic moment AMS
Subject
Classification:
Key words and phrases:
which
could
Randomized
procedures
distribution.
secondary
block design;
Ontario,
were introduced
block design.
help in obtaining
62GlO;
Windsor,
Canada N9B 3P4
15 May 1990
permutation
its permutation
Primary
of Windsor,
received
data for the randomized
to approximate
of this test statistic
University
manuscript
to multiresponse
multivariate
A. Khan
They obtained
by Mielke and lyer three exact moments
In this paper we obtain a better
62El5,
permutation
approximating
the fourth
exact
distribution.
62KlO. procedure;
Pearson
type distributions.
1. Introduction In order to avoid the assumptions required in the classical approach of hypothesis testing, a test based on a permutation procedure for analyzing multivariate data for the randomized block design was introduced by Mielke and Iyer (1982). This technique is appropriate for any number of responses with the condition that the responses are commensurate (e.g., utilizing rank order transformations) and the data is at the ordinal or higher level. Let X,, be a q-variate response for the r-th block and i-th treatment in a randomized block design with b blocks and g treatments. Then the MRBP (multivariate randomized block permutation procedures) test statistic due to Mielke and Iyer (1982) is given by
where A(x, JJ) is a symmetric
where pz
distance
function
defined
as
1 and v>O.
037%3758/91/$03.50
0
1991-Elsevier
Science Publishers
B.V. (North-Holland)
264
D.S.
Tracy, K.A.
Khan / Fourth
moment
for MRBP
inferences
Under H,: There is no difference in the treatment effects and the permutation distribution of 6 assigns equal probabilities to the (g!)b possible allocations. The asymptotic distribution of 6 is not known except in few cases when 6 is equivalent to well known test statistics. For a p-value 5 a, we reject HO in favour of Hi: The treatments are different. Calculating the exact p-value for an observed value of 6 becomes unreasonable even with moderate values of b and g. Consequently Mielke and Iyer (1982) provided an algorithm to obtain approximate p-values based on the first three exact moments of this test statistic. The distribution of this test statistic is often negatively skewed; see either Brockwell and Mielke (1984) and/or Mielke and Berry (1982) for the special case involving matched pairs. Due to the negative skewness, the distribution of the standardized test statistic is approximated by a Pearson type III distribution; see Iyer, Berry and Mielke (1983), Mielke (1984), Mielke and Iyer (1982). For a good survey of the topic, see Mielke (1986). In this paper we develop the fourth exact moment of this test statistic which may help in obtaining a better approximation than the one based on three exact moments.
2. Expression The MRBP
of d4 test statistic
6= [gb’2)]~1 ;
can be rewritten
as
r;,
i=l
with &= CFzsEi Al where power of 6, for gZ 4, is
n(‘)=n!/(n-t)!
and d~~=d(x,,,x,).
>
Then
the fourth
.
In the following, the running variables used for the g treatments are i, j, k, 1,m, n, o,p. When two or more response measurements occur in the same block for different treatments, the variables used are i, i’, i”, . . . . Similarly, the running variables used for the b blocks are r, s, t, u, u, w,x, y. We use J$ for summation over distinct # blocks. For example,
represents the sum of bc3) terms. In this notation, the expressions
of the above
mentioned
respective
terms are
D.S.
3. Expected
Tracy, K.A. Khan / Fourth moment for MRBP inferences
265
value
Now we need to take the expectations of all the above mentioned terms. For illustration, the expectation of Cf+j=, Cf,,7,,=, (A:,‘)2(A,:‘)2 in the expansion of <,?tJ2 can be written as i i+j=
; I rfs#1=
=
=
E[(d;:)2($321
1
4 f f i: g
!#I'j=l
$c
i
(D;;)2(D;;‘)2
k=l r#s#/=l
; (D;;)2(D;;)2
with 0;; = d(d;,, dj,,), where x,,= and C symbolizes Using below.
g c d/J/, /= I
if xlr=a,,, otherwise,
Cf+,, CT=, C”,=, in this example.
this notation,
we list the expectations
of the above
mentioned
five terms
D.S.
266
Tf’my, K.A.
KhQn
/
Fourth
+ ***+
momenffor MRBP inferences 12(g- 1) D{‘D”D’U D”W JS kt MU ox 2
+ ... + *(g-
1) D(rD;irDh D”w kt
JS
iTI”
OX
g6
i
I
i#j#k=
E(&;&) = c
; ( g(g _ l;cg _2) (D;;)2D;.kD,‘:; + g2c;6 1) (D;;)2D;;D;‘; +
32 &--)2
D”Df;‘D&D;‘; Js
Js
+ ... + %-l)k2)
DirD;rD/u D”” Js kt mu ox 2
+
16k-2)
DfrD;‘rD/u JS kt
D”w
MD
OX
g6
+ (g-
l)(g-2) D;;Dil;‘D;;D;;
>
8’
E(t,t,tit,> =
f i+j#kfl=
I
c; (
g(g
+
_
ljtg~ 2)(g
3)
_
,
D;iD;;D:eD;‘;
48 D !rD‘;rD;;‘D;:;’ g2(g- 1>2 Js Js
t . . . + wg
- wg
-
3) ,t,,i’rDlu JS
D””
kt
mu
OX
2 + (g -
I)&
- 2)(g
-
3, D(rDktDmuDox JS
8’
Now by using
relations
such as
/!A
nw
PY
>
.
D.S.
Tracy,
K.A.
Khan / Fourth
moment
for MRBP
267
inferences
or
and adding
similar
terms,
4. Fourth exact moment The following
expression
we get the fourth
exact moment,
for gz4.
of 6 is obtained
for the fourth
exact moment
of 6.
E(S4) = [g@] -4
+ 24(g - l)(LI;;)2I$D;, + 12(g + l)(D;;)*D;D:,‘+
24g(o;~)2o$I;,’
+ 3(g2 - 3g + l)(D;;y(L$y +4(g-
1)2(D;;)3D;-448DJ,‘D&D;DJ~r
- 24DJ,‘D;D;;D&
- 12D;;D&D,‘;D&
- 12(g - 2)(D;;)*D,r’Dg - 24(g - 1)(D~~)2D~D~~~+ 6D;;D;D;sD; f 6D;;D&D;sD;s + 16D;;D~D;;D,~‘+ - 12Dis’D; D& 0;;
+ 24D;;D;D;;D; 6(g - 2)(D,;)*D;‘D~ f D;;DED;‘D;;
48 + g2(g - 1)2 { g2(DJ,‘)2(D;)2 - 2g(D;;)*D,:‘D; - 2g(Dj,‘)‘D,:‘D;‘+
2g(D;;)*DI:‘Dz’
]
268
D.S. Tracy, K.A. Khan / Fourth moment for MRBP inferences
+ g2(g _
Y*(,_2) {g3(D;;)2D;D;-
4g2D;;DFD;;Df
- g2(D;;)2D;Df
- 2g2(D,f;)2D;D;
+ g(D;;)2D,:‘D,F
+ 6gD;;DED;,D$,
- 4(g - 2)D;;D;D;;D~ + 4(g - l)D;;D;D;;Df, + 2g(g - l)D;;DFD;,D;f + 4(g - l)D;;DjyD/;D;f
JS ks
It
+ 2g(D;;)2D;D;, + 4gD;;D;D;;D,:’
nt
+ (g2 - 3g + l)(D;;)2Dl:‘D;
- 4D;;D$D;, + 2D;;DED;‘D;;
02
- 2(g + l)D;;DjyD/;D,$ - 2(g - S)D;;D;D;,D,:s
+ (g - 3)D;;D;‘DnyrD;;
}
D.S.
Tracy,
+
K.A.
Khan
48 g3(g-
/ Fourth
{ g3D;;D;
for MRBP
moment
D:,“o;
269
inferences
- 4g’D;;D;
D;;Dz’
1)3 + 4gD;;D;
D;;D;’
+ 2gD;;D;
D;, 0,;;
-4D~;D~D~,D~~+(g2-3g+3)D;;D$rD;~D;~} +
192
2 { g’D;;D; D~Df;, g3(g - 1) - 2D;;D,:‘D$D;u + 4(g - 1)D;;D;
+ 2D;;D,:‘DtD;,, - 4(g - l)D;;D;D;D;;, 06 D;;, - gD;;Dl:‘D:;‘D;
+ 2(g - l)D;;D,:rD;tDD,k: - 2(g - l)D;;D,:‘D,‘;sD;u + 2(3g - 5)D;;D,:‘D;tD~
- 6(g-
l)DJ,‘D;D;D,m:
- 2(g - 2)D:,‘DI:‘Dl:‘sD,k: - (2g - S)D,,‘D; D,“, 0,:: + 2(g - 2)D;;DrD;D;;
+ 2D;;D,:‘D,~D,:
+ (g - 2)D,‘;D,:‘D,~D;;} 192 + g3(g_ 1) { - 2Dj,‘D$D;‘D;, + g(D;;)2D,:‘Dn;’ - 4 D;;D;‘D;,
+ 4D;;D;Dz
On”: - Dj,‘DFD;,
- D;;DFD,;‘D,:: + g2(g _ ;;(,
+ 2D;;DcD,‘;D;,
_ 2) {g2(D;;)3D::
0;;’ 0,“:
+ D;;D;rD~rD;;}
- 6g(D;:)‘D&D;,
270
D.S. Tracy, K.A. Khan / Fourth moment for MRBP inferences
Acknowledgements Working details for Mielke and Iyer (1982), very kindly supplied by Prof. Paul W. Mielke, Jr., were immensely useful to the authors for the derivations in Section 3. The partial support of this research from the Natural Sciences and Engineering Research Council of Canada Grant No. A31 11 is gratefully acknowledged.
References Brockwell, statistics Iyer, H.K.,
P.J. based K.J.
proximate
and
P.W.
P.W.
(1984).
Asymptotic
distributions
of matched-pairs
permutation
Austral. J. Sfatist. 26, 30-38.
measure.
Mielke (1983). Computation
values for multi-response
randomized
of finite population block permutation
parameters procedures
and ap(MRBP).
Simulation Comput. 12, 479-499.
(1984).
Meteorological
tions. In: P.R. Krishnaiah Amsterdam, 813-830. Mielke,
Mielke
Berry and P.W.
probability
Comm. Statist. Mielke,
P.W.
on distance
(1986).
applications
and P.K.
Non-metric
of permutation
statistical
techniques
based
on distance
func-
Handbook of Siatistics, Vol. 4. North-Holland,
Sen, Eds., analyses:
some metric
alternatives.
J. Slat&t. Plann. In-
ference 13, 377-387. Mielke,
P.W.
and K.J.
Comm. Stat&. Mielke,
P.W.
randomized
and H.K. block
Berry (1982). An extended
class of permutation
techniques
for matched
pairs.
Theory Methods 11, 1197-1207. Iyer (1982).
experiments.
Permutation
techniques
Comm. Sfatist. -
for analyzing
multi-response
Theory Mefhods 11, 1427-1437.
data
from
of Statistical
Planning
and Inference
263
28 (1991) 263-270
North-Holland
Fourth exact moment result for improving MRBP based inferences Derrick
S. Tracy and Khushnood
Department
of Mathematics
and Statistics,
Received 21 August 1989; revised Recommended by J.N. Srivastava
Abstract:
Tests similar
(1982) to analyze of this test statistic moment AMS
Subject
Classification:
Key words and phrases:
which
could
Randomized
procedures
distribution.
secondary
block design;
Ontario,
were introduced
block design.
help in obtaining
62GlO;
Windsor,
Canada N9B 3P4
15 May 1990
permutation
its permutation
Primary
of Windsor,
received
data for the randomized
to approximate
of this test statistic
University
manuscript
to multiresponse
multivariate
A. Khan
They obtained
by Mielke and lyer three exact moments
In this paper we obtain a better
62El5,
permutation
approximating
the fourth
exact
distribution.
62KlO. procedure;
Pearson
type distributions.
1. Introduction In order to avoid the assumptions required in the classical approach of hypothesis testing, a test based on a permutation procedure for analyzing multivariate data for the randomized block design was introduced by Mielke and Iyer (1982). This technique is appropriate for any number of responses with the condition that the responses are commensurate (e.g., utilizing rank order transformations) and the data is at the ordinal or higher level. Let X,, be a q-variate response for the r-th block and i-th treatment in a randomized block design with b blocks and g treatments. Then the MRBP (multivariate randomized block permutation procedures) test statistic due to Mielke and Iyer (1982) is given by
where A(x, JJ) is a symmetric
where pz
distance
function
defined
as
1 and v>O.
037%3758/91/$03.50
0
1991-Elsevier
Science Publishers
B.V. (North-Holland)
264
D.S.
Tracy, K.A.
Khan / Fourth
moment
for MRBP
inferences
Under H,: There is no difference in the treatment effects and the permutation distribution of 6 assigns equal probabilities to the (g!)b possible allocations. The asymptotic distribution of 6 is not known except in few cases when 6 is equivalent to well known test statistics. For a p-value 5 a, we reject HO in favour of Hi: The treatments are different. Calculating the exact p-value for an observed value of 6 becomes unreasonable even with moderate values of b and g. Consequently Mielke and Iyer (1982) provided an algorithm to obtain approximate p-values based on the first three exact moments of this test statistic. The distribution of this test statistic is often negatively skewed; see either Brockwell and Mielke (1984) and/or Mielke and Berry (1982) for the special case involving matched pairs. Due to the negative skewness, the distribution of the standardized test statistic is approximated by a Pearson type III distribution; see Iyer, Berry and Mielke (1983), Mielke (1984), Mielke and Iyer (1982). For a good survey of the topic, see Mielke (1986). In this paper we develop the fourth exact moment of this test statistic which may help in obtaining a better approximation than the one based on three exact moments.
2. Expression The MRBP
of d4 test statistic
6= [gb’2)]~1 ;
can be rewritten
as
r;,
i=l
with &= CFzsEi Al where power of 6, for gZ 4, is
n(‘)=n!/(n-t)!
and d~~=d(x,,,x,).
>
Then
the fourth
.
In the following, the running variables used for the g treatments are i, j, k, 1,m, n, o,p. When two or more response measurements occur in the same block for different treatments, the variables used are i, i’, i”, . . . . Similarly, the running variables used for the b blocks are r, s, t, u, u, w,x, y. We use J$ for summation over distinct # blocks. For example,
represents the sum of bc3) terms. In this notation, the expressions
of the above
mentioned
respective
terms are
D.S.
3. Expected
Tracy, K.A. Khan / Fourth moment for MRBP inferences
265
value
Now we need to take the expectations of all the above mentioned terms. For illustration, the expectation of Cf+j=, Cf,,7,,=, (A:,‘)2(A,:‘)2 in the expansion of <,?tJ2 can be written as i i+j=
; I rfs#1=
=
=
E[(d;:)2($321
1
4 f f i: g
!#I'j=l
$c
i
(D;;)2(D;;‘)2
k=l r#s#/=l
; (D;;)2(D;;)2
with 0;; = d(d;,, dj,,), where x,,= and C symbolizes Using below.
g c d/J/, /= I
if xlr=a,,, otherwise,
Cf+,, CT=, C”,=, in this example.
this notation,
we list the expectations
of the above
mentioned
five terms
D.S.
266
Tf’my, K.A.
KhQn
/
Fourth
+ ***+
momenffor MRBP inferences 12(g- 1) D{‘D”D’U D”W JS kt MU ox 2
+ ... + *(g-
1) D(rD;irDh D”w kt
JS
iTI”
OX
g6
i
I
i#j#k=
E(&;&) = c
; ( g(g _ l;cg _2) (D;;)2D;.kD,‘:; + g2c;6 1) (D;;)2D;;D;‘; +
32 &--)2
D”Df;‘D&D;‘; Js
Js
+ ... + %-l)k2)
DirD;rD/u D”” Js kt mu ox 2
+
16k-2)
DfrD;‘rD/u JS kt
D”w
MD
OX
g6
+ (g-
l)(g-2) D;;Dil;‘D;;D;;
>
8’
E(t,t,tit,> =
f i+j#kfl=
I
c; (
g(g
+
_
ljtg~ 2)(g
3)
_
,
D;iD;;D:eD;‘;
48 D !rD‘;rD;;‘D;:;’ g2(g- 1>2 Js Js
t . . . + wg
- wg
-
3) ,t,,i’rDlu JS
D””
kt
mu
OX
2 + (g -
I)&
- 2)(g
-
3, D(rDktDmuDox JS
8’
Now by using
relations
such as
/!A
nw
PY
>
.
D.S.
Tracy,
K.A.
Khan / Fourth
moment
for MRBP
267
inferences
or
and adding
similar
terms,
4. Fourth exact moment The following
expression
we get the fourth
exact moment,
for gz4.
of 6 is obtained
for the fourth
exact moment
of 6.
E(S4) = [g@] -4
+ 24(g - l)(LI;;)2I$D;, + 12(g + l)(D;;)*D;D:,‘+
24g(o;~)2o$I;,’
+ 3(g2 - 3g + l)(D;;y(L$y +4(g-
1)2(D;;)3D;-448DJ,‘D&D;DJ~r
- 24DJ,‘D;D;;D&
- 12D;;D&D,‘;D&
- 12(g - 2)(D;;)*D,r’Dg - 24(g - 1)(D~~)2D~D~~~+ 6D;;D;D;sD; f 6D;;D&D;sD;s + 16D;;D~D;;D,~‘+ - 12Dis’D; D& 0;;
+ 24D;;D;D;;D; 6(g - 2)(D,;)*D;‘D~ f D;;DED;‘D;;
48 + g2(g - 1)2 { g2(DJ,‘)2(D;)2 - 2g(D;;)*D,:‘D; - 2g(Dj,‘)‘D,:‘D;‘+
2g(D;;)*DI:‘Dz’
]
268
D.S. Tracy, K.A. Khan / Fourth moment for MRBP inferences
+ g2(g _
Y*(,_2) {g3(D;;)2D;D;-
4g2D;;DFD;;Df
- g2(D;;)2D;Df
- 2g2(D,f;)2D;D;
+ g(D;;)2D,:‘D,F
+ 6gD;;DED;,D$,
- 4(g - 2)D;;D;D;;D~ + 4(g - l)D;;D;D;;Df, + 2g(g - l)D;;DFD;,D;f + 4(g - l)D;;DjyD/;D;f
JS ks
It
+ 2g(D;;)2D;D;, + 4gD;;D;D;;D,:’
nt
+ (g2 - 3g + l)(D;;)2Dl:‘D;
- 4D;;D$D;, + 2D;;DED;‘D;;
02
- 2(g + l)D;;DjyD/;D,$ - 2(g - S)D;;D;D;,D,:s
+ (g - 3)D;;D;‘DnyrD;;
}
D.S.
Tracy,
+
K.A.
Khan
48 g3(g-
/ Fourth
{ g3D;;D;
for MRBP
moment
D:,“o;
269
inferences
- 4g’D;;D;
D;;Dz’
1)3 + 4gD;;D;
D;;D;’
+ 2gD;;D;
D;, 0,;;
-4D~;D~D~,D~~+(g2-3g+3)D;;D$rD;~D;~} +
192
2 { g’D;;D; D~Df;, g3(g - 1) - 2D;;D,:‘D$D;u + 4(g - 1)D;;D;
+ 2D;;D,:‘DtD;,, - 4(g - l)D;;D;D;D;;, 06 D;;, - gD;;Dl:‘D:;‘D;
+ 2(g - l)D;;D,:rD;tDD,k: - 2(g - l)D;;D,:‘D,‘;sD;u + 2(3g - 5)D;;D,:‘D;tD~
- 6(g-
l)DJ,‘D;D;D,m:
- 2(g - 2)D:,‘DI:‘Dl:‘sD,k: - (2g - S)D,,‘D; D,“, 0,:: + 2(g - 2)D;;DrD;D;;
+ 2D;;D,:‘D,~D,:
+ (g - 2)D,‘;D,:‘D,~D;;} 192 + g3(g_ 1) { - 2Dj,‘D$D;‘D;, + g(D;;)2D,:‘Dn;’ - 4 D;;D;‘D;,
+ 4D;;D;Dz
On”: - Dj,‘DFD;,
- D;;DFD,;‘D,:: + g2(g _ ;;(,
+ 2D;;DcD,‘;D;,
_ 2) {g2(D;;)3D::
0;;’ 0,“:
+ D;;D;rD~rD;;}
- 6g(D;:)‘D&D;,
270
D.S. Tracy, K.A. Khan / Fourth moment for MRBP inferences
Acknowledgements Working details for Mielke and Iyer (1982), very kindly supplied by Prof. Paul W. Mielke, Jr., were immensely useful to the authors for the derivations in Section 3. The partial support of this research from the Natural Sciences and Engineering Research Council of Canada Grant No. A31 11 is gratefully acknowledged.
References Brockwell, statistics Iyer, H.K.,
P.J. based K.J.
proximate
and
P.W.
P.W.
(1984).
Asymptotic
distributions
of matched-pairs
permutation
Austral. J. Sfatist. 26, 30-38.
measure.
Mielke (1983). Computation
values for multi-response
randomized
of finite population block permutation
parameters procedures
and ap(MRBP).
Simulation Comput. 12, 479-499.
(1984).
Meteorological
tions. In: P.R. Krishnaiah Amsterdam, 813-830. Mielke,
Mielke
Berry and P.W.
probability
Comm. Statist. Mielke,
P.W.
on distance
(1986).
applications
and P.K.
Non-metric
of permutation
statistical
techniques
based
on distance
func-
Handbook of Siatistics, Vol. 4. North-Holland,
Sen, Eds., analyses:
some metric
alternatives.
J. Slat&t. Plann. In-
ference 13, 377-387. Mielke,
P.W.
and K.J.
Comm. Stat&. Mielke,
P.W.
randomized
and H.K. block
Berry (1982). An extended
class of permutation
techniques
for matched
pairs.
Theory Methods 11, 1197-1207. Iyer (1982).
experiments.
Permutation
techniques
Comm. Sfatist. -
for analyzing
multi-response
Theory Mefhods 11, 1427-1437.
data
from