INTRODUCTION
The first descriptions of permutation tests for linear statistical models,
including analysis of variance and regression, can be traced back to the work
of Fisher (1935).
Permutation tests were first introduced by Fisher (1935)
and Bizhannia et al. (2010). But they were not
pleased so much because they needed time consuming calculations. But nowadays
these calculations by production of fast and powerful computers that calculating
of a pvalue is faster than finding an amount of charts for a parametric test.
Parametrical tests take place in the area of free distribution. As we know
for doing the question of testing of hypothesis, in addition to hypothesizes
like independency of error term, stability of their variance and random sampling,
we need to the hypothesis of normal distribution of error term, while for doing
permutation tests we don’t need to basic hypothesizes. In fact we for doing
a non parametrical test, we use series of simple hypothesizes that leaves the
researcher. One of these hypothesizes which in fact is the base for permutation
tests is hypothesis of exchange ability of observations which defined like this.
Definition 1: Let's realize exchange ability of random vector (n) dimensional,
x = (x_{1}, x_{2},…, x_{n}) with joint distribution
of f x_{1}, x_{2},…, x_{n} (x_{1}, x_{2},…,
x_{n}) X is called exchangeable if joint density of observations per
each permutation vector is suitable. So for each permutation vector which is
shown by X we have:
Various permutational strategies have been proposed for testing nullity of
a partial regression coefficient in a multiple regression model (ALSalihi
et al., 2010; Bughio et al., 2002;
Edriss et al., 2008; Kandhro
et al., 2002; Laghari et al., 2003;
Rashid et al., 2002; Serhat
Odabas et al., 2007; Tariq et al., 2003;
Alam, 2004). The proposed permutation methods for such
tests have different bases in term of their philosophies and have been proposed
in different contexts.
In Anderson and Legendre (1999) point of view just
four cases of these strategies are suitable. These four methods are: Manly method
(Manly, 1991), Kennedy method (Kennedy
and Cade, 1996) FreedmanLane method (Freedman and Lane,
1983) and Ter braak (TerBraak, 1992) but in this
article we just compare Kennedy’s method and Freedman and Lane’s method.
KENNEDY’S METHOD
Suppose variable answer Y and variable x_{1}, x_{2},…, x_{p} and assume we have n observation for regression. Then we have this equation:
where, ε is (sentence wrong) error term and has an indeterminate distribution F with by a mean zero variance σ^{2}. We can write the Eq. 1 as the following vectormatrix:
Where:
We want to test hypothesis H_{0}: β_{p} = 0 in comparison with, H_{2}: β_{p}≠ 0. For doing this test by permutation method we follow Kennedy algorithm like this: we define matrix:
and then multiple correspondents of Eq. 2 in (I_{n} H_{i}) to solve equation shown here:
Where:
Step 1: 
Then by using least error squares, we estimate β_{p}
like this: 
After that we estimate β_{p}, amount of statistic test
calculate and call it referential t.
Step 2: 
We do permutation for
amounts and show it as 
Step 3: 
We make a regression equation on
, model of Eq. 3, from found vector. Then we estimate
β_{p} by least sum of square error, like one given here: 
Step 4: 
By repeating second and third step and finding t^{*}'s
we also find permutation distribution t's, next, by using that we find pvalue
which is the relation of amount of permutation statistics that their absolute
value is larger than their referential E absolute value and at least we
admit or reject zero hypothesis. 
PROBLEM WITH KENNEDY METHOD
In permutation methods, permutating amounts of y, causes amounts of ε permutate. Thus a question comes to mind is that whether this obligatory permutation changes its parametrical distributions or not? So we first give this definition.
Permutative matrix: Definition 2 permutative matrix is a square matrix that there are numbers of zero and only a one each row and the position of number one in each row is different from other rows. This matrix is shown by sample P and has a special feature of P^{t} P = P P^{t}.
Now, with respect to the above definition, we analyze the mentioned problem: We know that:
so, considering the above definition, we can put the permutated vector ε which is known by the notation of ε^{*} this way ε^{*} pε. As a result:
It is clear that by the permutation of values of y does not change the distribution of ε. Now, we study this in case of the reduced equation of which Kennedy has made use Eq. 3.
It can easily be proved that:
Regarding the definition 2, we can easily prove that:
Which shows the variance of
is dependent on
thus by any permutation of
, is multiplied by a new number that changes the variance. Therefore, by any
permutation of ,
the parameters of distribution of
changes. So due to the null hypothesis H_{0}: β_{P} = 0.
The distribution of
also changes by any permutation.
MODIFIED KENNEDY’S METHOD
Huh and Jhun (2001) improve this problem, with regard
to this point that the matrix (I_{n} H_{i}) is the one to a
power of its own value and with the rank of n’ = np, added the following
steps to kennedy’s algorithm:
Step 1: 
First, we work the eigenvectors and rank of the matrix (I_{n}
H_{i}) out 
Step 2: 
We save the eigenvectors equivalent to the particular measures of one
which equals the rank of in the form of (I_{n} H_{i}) the
matrix 
Step 3: 
Through the orthogonalizing process of Gram Schmith, we change/turn this
matrix to an orthogonal one 
Step 4: 
Divide each column of this matrix by the norm of that column so that an
orthogonal unit vector is resulted. We come to realize this new matrix by
V_{1} the dimensions of which are (nxn’) 
Step 5: 
Using the spectrographic analysis and knowing the point that particular
measures of matrix V_{1} is one, we reduce the matrix (I_{n}
H_{i}) this way: 
Step 6: 
After reduction of this matrix, we multiply two sides of the
Eq. 3 by the matrix
So that a new equation is resulted in this form: 
Where:
Now, we turn to the distribution of the permutated vector of
. To do so, we again refer to the definition 2:
So it is pinned down that
and this gives clue to the fact that this new method keeps the distribution
of the error expressions constant and stable even after the permutation.
After the modification/improvement of the Eq. 3 and turning it into the form (4), we apply the steps contributed by Kennedy to the new resulted equation.
SIMULATION
Anderson and Legendre (1999), having an extent simulation
done (ignoring the error in Kennedy’s method), showed that in FreedmanLawn
method, a Type I error probability is less than that the Kennedy’s methods.
Shadrokh and d'Aubigny (2010), Shadrokh
(2011) analytically show that the Type I error of Freedman and Lane method
is lower than that of Kennedy’s approach.
Here, with the aid of a simulation, we intend to check out whether the claim
Anderson and Legendre (1999) and Shadrokh
(2011) have made remains valid after modifying the error in the method of
Kennedy or not? To do so, we calculate the empirical probability of type I error
in all the three permutating methods on the basis of a double regression model
y = β_{0}+β_{1}x_{1}+β_{2}x_{2}+ε
considering the four following factors:
• 
The sample size {10, 20, 30} 
• 
The correlation between the two variances x_{1}, x_{2},
ρ = {0.1, 0.9} 
• 
The quantity/value of the coefficient of regression which is not tested
β = {0.5, 1.5} 
• 
Thes distribution of error expressions: the exponential with the parameter
2 
The focused simulation is done employing the software Splus and the results are drawn as the following charts.
In all four case (showed in four chart) the probability of type one error of Freedman and Lane method is lower than that of Kennedy’s method and modified Kennedy’s method (Huh and Jhun’s method).
The certain considerable point is that, considering Fig. 1ab,
once β equals a small quantity, ignoring the fact that the correlation
is high or low and when sample size reaches 30, FreedmanLane’s method
and modified Kennedy’s method which is depicted in the charts as the Huh
and Jhun method, both, show a convex quantity.

Fig. 1: 
The consistent views in one view chart; (a) Beta 1 = 0.5,
r = 0.1, (b) Beta 1 = 1.5, r = 0.1, (c) Beta 1 = 0.5, r = 0.9 and (d) Beta
1 = 1.5, r = 0.9 
CONCLUSION
The objective of this article was to select the best test of significance of a single partial regression coefficient in a multiple regression model. Hence the Kennedy’s method, modified Kennedy’s method and Freedman and Lane’s methods compared by simulation. Then, with accordance to the results of the simulation the best method was selected. This led us to the fact that the results Shadrokh’s analytical and results Anderson’s simulation still remain unquestionable after the modification of the problem with Kennedy’s method and it can be asserted that when selecting a method for testing one of the coefficients of multiple regressions, Freedman and Lane’s method is to be the first choice.