Volume 2, Issue 9, September 2013
Page 81
ABSTRACT
This research a review on testing for equality of means against ordered means. Statistical hypothesis testing is a key technique of frequents statistical inference, and it is widely used, but also much criticized. The critical region of a hypothesis test is the set of all possible outcomes which, if they occur, well lead us to reject the null hypothesis in favour of alternative hypothesis.
Keywords: testing, equality , means, ordered ,hypothesis testing
1. INTRODUCTION
Hypothesis testing is a method of making statistical decision using experimental data. One use of hypothesis testing is in deciding whether experimental result contains enough information to cast doubt on conventional wisdom. Hypothesis testing are performed by many researcher in various fields of inquiry, usually to discover something about particular process. Literally, hypothesis testing is a method of testing a claim about a parameter in a population, using data measured in a sample[3,4]. In this method, we test some claim by determine a likelihood that a sample statistic could have been selected if the hypothesis regarding the population parameter were true.
The null hypothesis is a conservative statement about a population parameter, and it is so termed because it is most in variably states that the given sample comes from a population. The purpose of hypothesis testing is to test the variability of the null hypothesis in the light of experimental data.
2. METHODOLOGY
Depending on the data, the null hypothesis either will or will not be rejected as a variable possibility.[6] worked on hypothesis testing of homogeneity of the form
0
:
1 2...
g,
0:
1 2..
gH
against the ordered alternative H
Let
0
,
,
1
1
,
1 11
1
i
i
i
g
then
for
each
pair
of
means
iand
i
i
iif
only
i
Hence the test for hypothesis set above becomes
is
of
estimate
unbiased
an
g
i
H
against
H
i i
0
:
min
0
1
1
:
1 1 10
1
1
1
2 2
1
ˆ ~ , ( )
( ) ( )
( ) ( )
hom var
i i i
i i i
i i i
i i
i i
Y Y
then
N v and
v v Y Y
v Y v Y
under the assumption or ogenity of iances
n n
A REVIEW ON TESTING FOR EQUALITY
OF MEANS AGAINST ORDERED MEANS
1
SAADU,Y. O, 2ADEOYE, A.O AND 3YUSUF, G. A
1
DEPARTMENT OF LIBRARY INFORMATION KWARA STATE POLYTECHNIC ILORIN 2DEPARTMENT OF MATHEMATICS/STATISTICS FEDERAL POLYTECHNIC OFFA
3
Volume 2, Issue 9, September 2013
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Thus
1 2
1
( )
i i ii i
n
n
v
n n
(1) Now consider the quantity mi such that
1 1
1 1
2
ˆ
ˆ ( )
ˆ ~ ( , )
i i i
i i
i i i i i
i i
n n
m and let
n n
m m Y Y
then
N
1 1
i i mi Where
The result of the equation (2.3) above is so because:
) ((
) ( (
) ˆ ( ) ˆ (
1 1
i i
i i i
Y Y E mi
Y Y mi E
i mi E E
Thus,
1
ˆi i i
i i
E m
(2) Also,
22 1
1 1
1 2 1
1 1
1 2 1
ˆ
ˆ
ˆ
(
)
( )
1
ˆ
( ,
ˆ
ˆ
,...,
ˆ
)
ˆ
( ,
,..,
)
i i i i i
i i
i i
g
g
v
v m
m v
n
n
n
n
Now let
and
1 1 2 2 1 1
'
1 1 2 2 1 1
ˆ ( ), ( ),..., ( )
( ), ( ),..., ( )
g g g g g
g g g g g
Then
m Y Y m Y Y m Y Y
and
V m m m
Volume 2, Issue 9, September 2013
Page 83
2 1
1 2
2 1
2
1
) ( 1
) ( 1 1
i ij ni
j g
i
i ij ni
j i i
i i
i o
Y Y g
n S and
Y Y n
S let Now
2 1
2
)
1
(
1
i g
i
ni
S
g
n
S
Then
Where
2
i
S
is the sample variance for the ith population,1
i
g
1
and
S
2is the pooled sample variance for all the g population.Define '
1 2 1
( ,
, ...
g)
T
T T
T
1
(
2 1),
2(
3 2),...
g1(
g g 1)
T
m Y
Y
m Y
Y
m
Y
Y
and U as 1
1 1
T
T
1,
T
2, ...
T
gU
S
S
S
S
Where Ti is defined as
)
(
i 1 ii
i
m
Y
Y
T
Then, we have
1
1 2 1 3 2
1 2 1
1 2 1
1
(
)
(
)
(
)
,
,...
( ,
,...
)
(
)
, 1
1
g g
g
g
i i
Y
Y
Y
Y
Y
Y
U
m
m
m
S
S
S
u u
u
Where
Y
Y
Ui
mi
i
g
S
Then, statistic U has a multivariate t – distribution with mean
and n-g degree of freedom.)
,
(
1
n
g
MT
g
Independent of statistics [1]1 1
min
g i
i
u
v
Let
Volume 2, Issue 9, September 2013
Page 84
0 1 2
1 1 2
:
...
:
...
g
g
H
against
H
[1] proposed the test statistic
(
u
)
given by , ,1,
min
( )
0,
i g n
if
u
t
u
otherwise
(3)
Where min (ui) accepts the preconceived order
1
i1
i i
,
1, 2, ...,
g
1
and t
g n, , is determine such that)]
(
[
u
Eo
= , є(0,1) and predetermined to satisfy sufficiency criterion.However, the critical values
t
g,n, developed using ordered statistic and its distribution as contained in [1] may be use. These critical values are generally smaller than the usual critical values of the t-distribution at a specified degree of freedom. The implication of this is that the critical value of the t-distribution when in use tend to be more conservative than the critical valuest
g,ni, developed for the test. Thus, if test
rejects the null hypothesis using the critical values of the t–distribution it will surely reject it using the critical valuest
g,ni,.In term of
Y Y
1,
2, ...,
Y
g,
and S
2lest
becomes 1, , , 1,..., 1
1 1
2 1 2
(
)
1,
min
( ,
, ...,
,
)
,
i i
g n i g
i g
g
mi Y
Y
if
t
S
Y Y
Y S
o otherwise
That is Ho is rejected if,1
, , , 1,..., 1
1 1
(
)
ˆ
min
i ig n i g
i i
i i
Y
Y
T
mi
t
S
n n
Where mi
n
n
For equal sample sizes
ni
m
i
,
the
test
statistic
T
becomes
1(
)
ˆ
min
1, ...,
1
2
i i
Y
Y
m
T
i
g
S
It should be noted that test
above reduces to the two-sample one tail t-test when g = 2 and it also inherits the three optimality properties of Uniformly Most Powerful Unbiased (UMPU), UniformlyMost Powerful Invariant (UMPI) and Asymptotical Consistency of the two sample one tailed t-test,[1]However, since the
reduces to the two samples one tailed t-test when g = 2, it then shows that statistic T of 5 and 6 follow the multivariate T- distribution. Therefore, the decision rule adopted with unequal sample sizes taken from all the g population is to reject the null hypothesis H0, if1 1
,1 1
(
)
ˆ
min
i i i in g
i i
Y
Y
n n
T
t
n
n
S
(4)With equal sample sizes, this reduce to
.
)
(
2
ˆ
, , 1
n g i i
t
S
Y
Y
m
T
Volume 2, Issue 9, September 2013
Page 85
provides a more conservative test then the actual [2]. Furthermore, if the test statistic rejects using the percentage points of the t-distribution it would be rejected using the actual critical values developed.
However, the rejection or acceptance of the null hypothesis Ho by the proposed test procedure is on the strength of the alternative set H1. In this connection, if Ho is rejected at a specified
level, it shows that the data being analyzed supports the ordered alternative set of H1.[6] developed the test procedure for testing Ho against ordered means with one of them being the control, assuming homogenous variances. For the general case, we can let
i
i1if and only if
i
max
(
g)
where
i
1
,
2
,
3
,
....
g
but for the controlg i
where g is fixed andg
i
i
g
max
(
),
1
,
2
,
...,
in particular, if
1represents the mean of populationi
,
i
1
,
2
,
...
g
,
the problem is to test the hypothesis.1 2 1
1
1 2 1
:
...
:
...
o
g
g
H
against
H
Where
i g i g
Max
1 is control mean from the g th
population. By the term control, we mean that all other g–1 treatment means are compared against
g
and serves as the basis of comparison other.' , 1
1
0
,
1,...
1.
i
i i g
i
i g
s
i
g
Define
Then
if and only if
i i
g
Therefore, the hypothesis of equation (2.9) becomes
1 1 1
:
0
,
1,
1
: max
0
o i i
i i g
H
i
g
against
H
The unbiased estimate of
iis
ˆ
i
Y
iY
g
(5) Therefore,
g i
g i
g i i
i i i
n
n
X
v
X
v
Y
Y
v
v
Where
v
N
2 2
)
(
)
(
)
(
)
ˆ
(
)
(
,
~
ˆ
Volume 2, Issue 9, September 2013
Page 86
2ˆ
( )
i i gi g
n
n
v
n n
Consider the quantity miwhich is defined as
)
(
ˆ
ˆ
1
1
2 1 g i i i i i g i g i iX
X
m
m
let
and
g
i
n
n
n
n
m
Then, we have that, 2
ˆ ~
( ,
)
(
)
i i
i i i g
N
Where
m
2 2] ˆ [ ] ˆ [ ˆ ) ( ) ˆ ( ˆ i i i i i g i i i i r v m m v v and m E m E Since Now, let ) . ... , , ( ˆ ) ˆ .. ,... ˆ , ˆ ( ˆ 1 1 1 1 2 1 2 1 g g i and Then 1 1
1 2 2 1 1
1
1 2 1
1 2 1
ˆ ( ), ( ),... ( ),
ˆ , , ...,
g g g g g
g
g g g g
m m m
and
m m m
Therefore, equation (2.11) becomes
1
1 1
:
0
,
1,...,
1
: max
0
o i i
i i g
H
i
g
against
H
Suppose we define statistic T as
1(
2 1),
2(
3 2),...
...
1(
1)
m
Y
Y
m
Y
Y
m
gY
gY
gT
= (T1, T2,……,Tg-1) and
S2, the pooled sample variance for all the g populations as 2 1 1 2
)
(
1
i ij ni j gi
X
X
g
n
S
i g i i i g i n n Where S n g n S is That 1 2 1 2 , ) 1 ( 1 2Volume 2, Issue 9, September 2013
Page 87
1
1
1 2 1 3 2
1 2 1
1 2 1
( )
( )
( )
, ,...,
( , , ... )
g g
g
g
U S
Y Y Y Y
Y Y
U m m m
S S S
U U U
Therefore, statistics U is distributed
MT
g1(
,
n
g
)
multivariate T – distribution. Define statistic H as, H = h(u) = ik
i 1
U
1
min
. Then H is non-decreasing in T and distributed independent of S2
. Thus, we propose a test procedure for testing hypothesis set up as
0
1 1
1,
max
( )
0,
i i k
if
u
t
u
otherwise
Where max ui shows that the preconceived order
i
i1
i
,
i
1
,
2
,
...
...,
g
and
t
0 is determine such thatE
H0[
(
u
)]
,
(
0
,
1
)
is predetermined.In term of
X
1,
X
2, ...,
X
g,
and S
2 becomes2
0
1 1
(
)
1,
max
( ,
)
,
i g
i g i g
i g
Y
Y
n n
if
t
n
n
S
Y S
o otherwise
(6)
That is in testing (2.9) the rule is to reject the null hypothesis Ho at a specified type I error if
0
1 1
(
)
ˆ
max
i g i gi g
i g
Y
Y
n n
T
t
n
n
S
(7)For equal sample sizes
ni
ng
m
i
,
then the rule is to reject Ho if0
1 1
(
)
ˆ
max
2
i g
i g
Y
Y
m
T
t
S
(8)For
i(
2
.
10
)
define as1
,...
.
1
1
i
g
g
i
Test
reduces to the two sample one side t-test with g = 2and inherits the three optimality properties of Uniformly Most Powerful Unbiased (UMPU), Uniformly Most Powerful Invariant (UMPI) and Asymptotical Consistency of the two sample one tailed test, [1]
In this connection, test
reduces to the two samples one tail t-test when g – 1= 1 and it possesses the three optimality properties of UMPU, UMPI and consistency using Ho against H1Now, if
1of (2.10) is redefined asg i
i
then, for i= 1, …..g – 1,g i
, if and only if
i
0
i,
and hypothesis set of (2.11) will become0
min
:
1
...,
,...
1
,
0
:
1 1
1
i g i
i i
o
H
against
g
i
H
Volume 2, Issue 9, September 2013
Page 88
2
0
1 1
(
)
1,
min
( ,
,
i g
i g i g
i g
Y
Y
n n
if
t
n
n
S
Y S
o otherwise
For some t0. Hence, (2.19) and (2.20) respectively become
0
1 1
0
1 1
(
)
ˆ
min
(
)
ˆ
min
2
i g
i g i g
i g
g i
i g
Y
Y
n n
T
t
n
n
S
Y
Y
m
T
t
S
REFERENCES
[1] Adegboye, O.S. (1981): “On Testing Against Restricted Alternatives In Ganssian Models” Ph.D Dissertation,BGSU. [2] Anne, P.S. and Deborah, I.S. (2002): “Combination of Experimental Design” Clarendon Press. Oxford
[3] Ott, L. (1984): “An Introduction to Statistical Methods and Data Analysis” Second Edition. PWS Publisher [4] Roussas, G.G. (1973): “A First Course in Mathematical Statistics” Addison
– Wesley Publishing Company Inc.
[5] Yahya, W. B. (2003): “Testing Hypothesis Against Ordered Alternative” M.Sc Thesis
Saadu, Yusuf olanrewaju was born on 23rd of December, 1984. He finished his National Diploma and Higher National Diploma in Statistics from kwara state polytechnic, Ilorin in 2005 and 2008 respectively and his just concluded his Post Graduate Diploma in Statistics from University of Ilorin July 2013. Currently am working at Kwara State polytechnic, Ilorin as system Analyst in Library Department.
Adeoye Akeem O was born on October 7th 1976 in owu isin, kwara State of Nigeria. He obtained his Higher National Diploma (H.N.D) in Statistics from the Kwara state polytechnic Ilorin 1n 1999,PGD Statistics in 2008 from University of Ilorin,PGDE Education in 2007 from University of Ado -Ekiti, Teacher training Certificate(TTC) in 2010 from Federal college of Education Akoka Lagos State, Masters degree (M.Sc) in Statistics in 2010 from university of Ilorin, Nigeria and currently on his Ph.D in statistics at the university of Ilorin Nigeria . He has been lecturing Statistics in kwara state polytechnic as an Instructor from April 2005 to March 2013 .He joined Federal Polytechnic offa from April 2013 to date as a lecturer .He attended many workshop and conferences. He has thirteen journals for both national and international.
