In the appendix we prove that *T*
_{
p
}, the Hotelling's *T*
^{2} test, *T*
_{
b
}, and *T*
_{
l
}are not affected by permuting SNPs. Let *G* = (*g*
_{
ij
}) be an *n* × *m* genotype matrix, where *g*
_{
ij
}is the genotype of the *i*th individual at the *j*th marker. Subtract the mean of the *j*th column from *g*
_{
ij
}such that the mean of each column of *G* is 0. The sample covariance matrix of the genotypes is *A* = *G*
^{
T
}
*G*/(*n* - 1). Suppose that *λ*
_{1} ≥ *λ*
_{2}≥...≥*λ*
_{
m
}are the eigenvalues of *A*, and *v*
_{1}, *v*
_{2},..., *v*
_{
m
}are the corresponding eigenvectors. Let *D* be a diagonal matrix with *λ*
_{1} ≥ *λ*
_{2}≥...≥*λ*
_{
m
}as diagonal entries, and let *V* be an *m* × *m* matrix with *v*
_{1}, *v*
_{2},..., *v*
_{
m
}as columns. Then *A* = *VDV*
^{-1}. Note that *v*
_{
i
}is the *i*th principal component, *i* = 1, 2,..., *m*. Write *V* = [*V*
_{1}
*V*
_{2}], where *V*
_{1} contains the used principal components, and *V*
_{2} contains the discarded principal components. In PCReg, the regression model is *y* = *GV*
_{1}
*b* + ϵ.

Suppose the spatial order of SNPs is permuted, then the columns of *G* are also permuted accordingly. Suppose the new genotype matrix is
, then
= *GP* where *P* is obtained from applying the same permutation on the columns of the identity matrix. Assume that
is also centered so that the mean of each column is 0. Let
. Since *P* is an orthogonal matrix, *P*
^{
T
}= *P*
^{-1}. Therefore,
= *P*
^{
T
}
*G*
^{
T
}
*GP* = *P*
^{
T
}
*VDV*
^{-1}
*P* = (*P*
^{
T
}
*V*)*D*(*P*
^{
T
}
*V*)^{-1}. Note that the diagonal matrix of the eigenvalues of
is still *D*, and the matrix of eigenvectors of
is *P*
^{
T
}
*V*. Since *A* and
have the same eigenvalues, the used principal components are the columns of
and the discarded principal components are the columns of
. In PCReg, the regression model after permutation is
which is the same as before permuting SNPs.

Next, we prove that the Hotelling's

*T*
^{2} statistic is not affected by permuting SNPs. Following the notations used in [

3], let

*X* be an

*n*
_{1} ×

*m* matrix of genotypes of cases and let

*Y* be an

*n*
_{2} ×

*m* matrix of genotypes of controls, where

*n*
_{1} is the number of cases,

*n*
_{2} is the number of controls, and

*m* is the number of SNPs. Let

and

be the column mean of

*X* and

*Y*, respectively, written as column vectors. Let

*X*
_{
i
}and

*Y*
_{
i
}denote the

*i*th row of

*X* and

*Y*, respectively, written as column vectors. The pooled-sample variance-covariance matrix of genotypes is

The Hotelling's

*T*
^{2} statistic is

Write

, where

**e**
_{
i
}is the

*i*th column of the identity matrix, and

**1** is a column vector with every entry being 1. Thus,

where

*I* is the identity matrix and

*E* is a square matrix with every entry being 1. Therefore,

Suppose the spatial order of SNPs are permuted, the corresponding columns of the genotype matrices

*X* and

*Y* are permuted accordingly. After permutation, let

and

be the genotype matrix of cases and controls, respectively. Then

=

*XP* and

=

*YP*, where

*P* is a permutation matrix. The new pooled-sample variance-covariance matrix is

After permutation,

and

becomes

and

, respectively. Recall that

*P*
^{-1} =

*P*
^{
T
}. The Hotelling's

*T*
^{2} statistic after permuting the spatial order of SNPs is

The same arguments can be applied to prove that the haplotype *T*
^{2} statistic defined in [3] is not affected by permuting SNPs either. It was proved in [3] that both the multilocus *T*
^{2} and the haplotype *T*
^{2} statistics have the same power. Usually a haplotype-based test will have a higher, or at least a different, power than a genotype-based test. It is a interesting fact that both the multilocus *T*
^{2} and the haplotype *T*
^{2} have the same power and neither have used the information contained in the spatial order of SNPs.

Recall that *T*
_{
b
}is obtained by fitting a regression function with one SNP, followed by Bonferroni correction to find the global *p*-value. Permuting SNPs does not change its results.

The likelihood-ratio test based on logistic regression *T*
_{
l
}is also not affected by permuting SNPs.