International Journal of Application or Innovation in Engineering & Management (IJAIEM)
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Volume 2, Issue 9, September 2013
ISSN 2319 - 4847
A REVIEW ON TESTING FOR EQUALITY
OF MEANS AGAINST ORDERED MEANS
1
1
SAADU,Y. O, 2ADEOYE, A.O AND 3YUSUF, G. A
DEPARTMENT OF LIBRARY INFORMATION KWARA STATE POLYTECHNIC ILORIN
2
DEPARTMENT OF MATHEMATICS/STATISTICS FEDERAL POLYTECHNIC OFFA
3
DEPARTMENT OF STATISTICS KWARA STATE POLYTECHNIC ILORIN
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
H 0 : 1  2  ...   g against the ordered alternative , H 0 : 1   2  ..   g
Let
 1   i 1   i , 1  i  g  1, then for each pair of means  i and  i 1 ,  i   i 1 if only  i  0
Hence the test for hypothesis set above becomes
H 0 : i  0 against
H 1 : min  1  0
1 i 1 g  1
an unbiased estimate of  i is
 i  Yi 1  Yi
then

ˆi ~ N  i , v ( i )  and

v ( i )  v ( Yi 1  Yi )


v ( Yi 1 )  v ( Yi )
2  2

under the assumption or hom ogenity of var iances
ni 1 ni
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Thus

 ni  ni 1  2
v ( i )  

 ni ni 1 
(1)
Now consider the quantity mi such that
mi 
ni ni 1
and let
ni ni 1
ˆ1  mi ˆi  mi ( Yi 1  Yi )
then
ˆi ~ N ( i ,  2 )
Where  i  mi     
 i 1 1 
The result of the equation (2.3) above is so because:
E (ˆi )  E ( miˆi )
 E ( mi (Yi 1  Yi )
 mi E ((Yi 1  Yi )
Thus, E  ˆ   m       
i
i
i
 i 1
i
(2)

Also,
v ˆi   v (miˆi )  mi2 v (ˆi )

ni  ni  1 2

ni  ni 1
Now let
ˆ1  (ˆ1 , ˆ2 ,..., ˆg 1 )1
and
ˆ1  ( 1 ,  2 ,..,  g 1 )1
Then
ˆ   m1 (Y1  Yg ), m2 (Y2  Yg ),..., mg 1 (Yg 1  Yg ) 
and
V  m1 ( 1   g ), m2 ( 2   g ),..., mg 1 (  g 1   g ) 
Therefore, the hypothesis set can be written in terms of
Volume 2, Issue 9, September 2013
'
 i as
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Volume 2, Issue 9, September 2013
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H o :  i  0  i against H 1 : min  i  0, 1  i  g  1
Now let
S i2 
ni
1
 (Yij  Yi ) 2
j
ni  1  1
and
S2 
g
ni
1
  (Yij  Yi ) 2
n  g i 1 j 1
Then
S2 
1 g
2
 (ni  1) Si
i

1
ng
Where
2
2
Si is the sample variance for the ith population, 1  i  g  1 and S is the pooled sample variance for all the g
population.
Define
T '  (T1 , T2 , ...Tg 1 )
T   m1 (Y2  Y1 ), m2 (Y3  Y2 ),... mg 1 (Yg  Yg 1 ) 
and U as
U1 
Tg  1 
T 1  T1 T2
  , , ...

S
S 
S S
Where Ti is defined as
Ti  mi (Yi 1  Yi )
Then, we have
 (Y  Y1 )
(Yg  Yg 1 ) 
(Y  Y2 )
U 1   m1 2
, m2 3
,...mg 1

S
S
S


 (u1 , u2 ,... u g 1 )
Where
Ui  mi
(Yi  1  Yi )
S
, 1  i  g 1
Then, statistic U has a multivariate t – distribution with mean  and n-g degree of freedom.
MTg 1 ( , n  g ) Independent of statistics [1]
Let v  min u i
1i  g1
Where v is non- decreasing in T and distributed independently of S.
Therefore, in testing the hypothesis,
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Volume 2, Issue 9, September 2013
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H 0 : 1   2  ...   g
against
H1 : 1   2  ...   g
[1] proposed the test statistic  (u ) given by
1, if min ui  t g , n ,

(3)
 (u )  
0, otherwise

Where min (ui) accepts the preconceived order
1  i  1 i, i  1, 2, ..., g  1 and t g , n , is determine such that
Eo[ (u )] = , 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 values t 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 values t g , ni , .
In term of
Y1 , Y2 , ..., Yg , and S 2 lest  becomes

mi (Yi  1  Yi )
t g , n , , i 1,..., g 1
1, if 1min
i  g1
S

 (Y1 , Y2 , ..., Yg , S 2 )  
o, otherwise


That is Ho is rejected if,
Tˆ  min mi
Where mi 
(Yi  1  Yi )
S
t g , n , , i 1,..., g 1
ni ni 1
ni  ni 1
For equal sample sizes ni  m i, the test statistic T becomes
(Yi  1  Yi )
m
min
i 1, ..., g  1
2
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
Tˆ 
optimality properties of Uniformly Most Powerful Unbiased (UMPU), Uniformly Most 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 , if
(Yi  1  Yi )
ni ni 1
 tn  g ,1
ni  ni 1
S
Tˆ  min
(4)
With equal sample sizes, this reduce to
Tˆ 
m
2
(Yi  1  Yi )
 t g , n, .
S
Where ni = m for  i and t g , n , is obtained from table of critical values developed to test  by [1] which is obtained
using ordered statistics and its distribution. However, these critical values are generally smaller than those obtained using
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Volume 2, Issue 9, September 2013
ISSN 2319 - 4847
the t-distributions. The implication of this is that the students t-distribution when in use for the ordered alternative
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   i  1 if and only
 i  max (  g ) where i  1, 2, 3, ....g but for the control
if
 
i
where
g
is
fixed
and
g
 g  max ( i ), i  1, 2, ......., g in particular, if 1 represents the mean of population i, i  1, 2, ......g , the
problem is to test the hypothesis.
H o :     ... 
1
g 1
2
against
H1 :     ...  
1
g 1
2
Where
  Max  is control mean from the gth population. By the term control, we mean that all other g–1
g
1 i  g
i
treatment means are compared against
 and serves as the basis of comparison other.
g
i ' s, 1  i  g  1
Define     
i
i
g
Then
   if and only if  i  0 i, i  1,...g  1.
i
g
Therefore, the hypothesis of equation (2.9) becomes
H o :  i  0 i , i  1, g  1
against
H1 : max  i  0
1 i  g 1
The unbiased estimate of
ˆi  Yi  Yg
 i is
(5)
Therefore,
ˆi ~ N  i , v ( i )
Where
v(ˆi )  v (Yi  Yg )
 v( X i )  v ( X g )

2 2

ni
ng
This occurs under equality of variance assumption and independent sampling.
Thus,
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 ni  ng  2
v(ˆi )  

 ni ng 
Consider the quantity mi which is defined as
1
 ni  n g 
mi  

 ni n g 
2
1 i  g 1
and let
ˆi  mi ˆi  mi ( X i  X g )
Then, we have that,
ˆi ~ N ( i ,  2 )
Where  i  mi ( i   g )
Since E ˆr   mi E ( ˆi )  mi ( 
i
 )
g
and vˆi   v [ miˆi ]  mi2 v [ˆi ]   2
Now, let
ˆ1  (ˆ1 , ˆ2 ,...........ˆ g  1 )
and
ˆi1  ( 1 ,  2 , ...........  g  1 )
Then
ˆ1   m1 ( 1   g ), m2 ( 2   g ),...mg 1 ( g 1   g ), 
and
 




 
ˆ   m1      , m2      , ..., mg 1      
2 g
 g 1 g  
 1 g
Therefore, equation (2.11) becomes
1
H o :  i  0 i , i  1,..., g  1
against
H1 : max  i  0
1  i  g 1
Suppose we define statistic T as

T  m1 (Y 2  Y 1 ), m2 (Y 3  Y 2 ),...........mg 1 (Y g  Y g 1 )

= (T1, T2,……,Tg-1) and
S2, the pooled sample variance for all the g populations as
S2 
g
ni
1
  ( X ij  X i ) 2
n  g i 1 j 1
That is
S2 
g
1
2
 (ni  1)S i
i

n g 1
g
Where, n   ni
i 1
S 2 is the sample variance for the i population, independent of T.
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Volume 2, Issue 9, September 2013
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U1 
T1
S
 (Y  Y1 )
(Y  Yg 1 ) 
(Y  Y2 )
U 1   m1 2
, m2 3
,..., mg 1 g

S
S
S


 (U1 , U 2 , ...U g 1 )
Therefore, statistics U is distributed MT g 1 ( , n  g ) multivariate T – distribution.
Define statistic H as, H = h(u) = min U i . Then H is non-decreasing in T and distributed independent of S2. Thus, we
1  i  k 1
propose a test procedure for testing hypothesis set up as
1, if max ui  t0
1  i  k 1

 (u )  
0, otherwise

Where max ui shows that the preconceived order
 i   i  1 i, i  1, 2, ............., g and t 0 is determine
such that E H 0 [ (u )]   ,  (0,1) is predetermined.
In term of
X 1 , X 2 , ..., X g , and S 2 becomes

1, if max
1 i  g  1


 (Y , S 2 )  
o, otherwise



ni ng
(Yi  Yg )
ni  ng
S
 t0
(6)
That is in testing (2.9) the rule is to reject the null hypothesis Ho at a specified type I error if
Tˆ  max
1 i  g  1
ni ng
(Yi  Yg )
ni  ng
S
 t0
(7)
For equal sample sizes ni  ng  m i, then the rule is to reject Ho if
m (Yi  Yg )
 t0
(8)
1 i  g  1
2
S
For  i ( 2.10) define as  i     i  1,.......... g  1 Test  reduces to the two sample one side t-test with g = 2
Tˆ  max
1
g
and 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 H1
Now, if  1 of (2.10) is redefined as  i     then, for i = 1, …..g – 1,    , if and only if  i  0 i , and
i
g
i
g
hypothesis set of (2.11) will become
H o :  i  0  i , i  1,.................., g  1
against
H 1 : min  i  0
1  i  g 1
Following other procedure all through, test
Volume 2, Issue 9, September 2013
 of (18) becomes
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
1, if min
1 i  g  1


 (Y , S 2  
o, otherwise



ni ng
(Yi  Yg )
ni  ng
S
 t0
For some t0. Hence, (2.19) and (2.20) respectively become
Tˆ  min
1 i  g  1
Tˆ  min
1 i  g  1
ni ng
(Yi  Yg )
ni  ng
S
 t0
m (Yg  Yi )
 t0
2
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.
Yusuf Gbemisola .A (Mrs) was born on 1984 from Ajase in ifelodun Local Government Area of kwara State of Nigeria.
She finished his National Diploma and Higher National Diploma in Statistics from Federal Polytechnic, Offa in 2004 and
2007 respectively and she is just concluded her Post Graduate Diploma in Education from Kwara State College of
Education Ilorin August 2013.
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