Detecting disease gene in DNA haplotype sequences by nonparametric dissimilarity test

Abstract

Association studies for complex diseases based on haplotype data have received increasing attention in the last few years. A commonly used nonparametric method, which takes haplotype structure into consideration, is to use the U-statistic to compare the similarities between genetic compositions in the case and control populations. Although the method and its variants are convenient to use in practice, there are some areas where the tests cannot detect even large differences between cases and controls. To overcome this problem and enhance the power, we propose a new form of the weighted U-statistic, which directly compares the dissimilarity between the haplotype structures in the case and control populations. We show that this test statistic is asymptotically a linear combination of the absolute values of normal random variables under the null hypothesis, and shifts strictly toward the right under the alternative, and therefore has no blind areas of detection. Simulation studies indicate that our test statistic overcomes the weakness of the existing ones and is robust and powerful as well.

Appendix

Proof of the proposition

(i) Note under H ₀, $U_{m,n} = \mu_{\hat{p}, \hat{q}} = \hat{p}^{\prime} D\hat{q}$ and D = D _p,p , so we have

$$ \hat{U}_{m,n} - \mu_{\check{p},\check{p}} = \left( \hat{U}_{m,n} - U_{m,n} \right) + (U_{m,n} - \mu_{p,p}) + \left( \mu_{p,p} - \mu_{\check{p},\check{p}} \right). $$

(6)

Let D _p,q ^(1,0)(·,·) = ∂D _p,q (·,·)/∂p and D _p,q ^(0,1)(·,·) = ∂D _p,q (·, ·)/∂q be the column vectors of first partial derivatives, and D _p,q ^(0,1) and D _p,q ^(0,1) be the corresponding matrices of column arrays. The first term in (6) is

$$\begin{aligned} \hat{U}_{m,n} - U_{m,n} &= \hat{p}^{\prime} \left(D_{\hat{p},\hat{q}} -D_{p,q} \right) \hat{q} = \hat{p}^{\prime} \left[D^{(1,0)}_{p,q} \left(\hat{p} - p \right) + D^{(0,1)}_{p,q} \left( \hat{q} - p \right) + O \left( \left\|\hat{p} - p \right\|^2 + \left\| \hat{q} - q \right\|^2 \right) \right] \hat{q}\\ &= \hat{p}^{\prime} \left[D^{(1,0)}_{p,q} \left( \hat{p} - p \right) + D^{(0,1)}_{p,q} \left(\hat{q} - p \right) \right] \hat{q} + O_P(1/N). \end{aligned}$$

Note the (i, j)th component of D _p,q ^(1,0) is the vector with lth entry

$$ {\frac{\partial}{\partial p_l}} D_{p,q} \left( {\bf h}_i, {\bf h}_j \right) = \left\{\begin{array}{*{20}l} {\frac{{\rm e}^{p_i-q_i} - {\rm e}^{q_i-p_i}}{2}}\;{\frac{{\rm e}^{p_j-q_j} + {\rm e}^{q_j-p_j}}{2}} D \left({\bf h}_i, {\bf h}_j \right) &\hbox{if}\;l=i\\ 0 & \hbox{else}\end{array}\right. \quad(l=1, \ldots, k). $$

Thus D _p,q ^(1,0) = 0 under H ₀, similarly D _p,q ^(0,1) = 0 under H ₀, and so under H ₀,

$$ \hat{U}_{m,n} - U_{m,n} = O_P(1/N). $$

Note ∂(p′Dq)/∂p = q′ D, ∂(p′Dq)/∂q = p′D, so under H ₀ the second term in (6) is

$$ \hat{p}^{\prime} D\hat{q} - p^{\prime} Dp = p^{\prime} D \left( \hat{p} - p \right) + p^{\prime} D\left( \hat{q} - p \right) + O \left( \left\| \hat{p} - p \right\|^2 + \left\| \hat{q} - p \right\|^2 \right). $$

Also, ∂μ _p,p /∂p = ∂(p′Dp)/∂p = 2p′D, so the third term in (6) under H ₀ is

$$ -2p^{\prime} D\left( \check{p} - p \right) + O \left( \left\| \check{p} - p \right\|^2 \right) = -2p^{\prime} D\left( \check{p} - p \right) + O_P(1/N). $$

Now collect the above relationships, we have

$$ \sqrt{N} \left( \hat{U}_{m,n} - \mu_{\check{p},\check{p}} \right) = \sqrt{N}p^{\prime} D\left[ \left(\hat{p} - p \right) + \left( \hat{q} - p \right) - 2 \left(\check{p} - p \right) \right] + O_P \left(1/\sqrt{N} \right). $$

Note that under H ₀,

$$ \sqrt{N} \left[ \left( \hat{p} - p \right) - \left( \hat{q} - p \right) \right] \mathop{\rightarrow}\limits^{D} N \left(0, \left(\gamma_1+\gamma_2 \right)R \right), $$

and for each i, the i-th entry of $\left( \hat{p} - p \right) + \left( \hat{q} - p \right) - 2\left(\check{p}-p \right)$ is

$$\left\{\begin{array}{*{20}l} \left( \hat{p}_i - p_i \right) - \left(\hat{q}_i - p_i \right) & \hbox{if}\;\hat{p}_i - p_i >= \hat{q}_i - p_i \\ \left(\hat{q}_i - p_i \right) - \left(\hat{p}_i - p_i \right) & \hbox{else}. \end{array} \right.$$

Thus, for a vector a = (a ₁, ..., a _k )′, denote |a| = (|a ₁|, ..., |a _k |)′, we have

$$ \sqrt{N} \left(\hat{U}_{m,n} - \mu_{\check{p},\check{p}} \right) = p^{\prime} D \left| \sqrt{N} \left[ \left( \hat{p} - p \right) - \left( \hat{q} - p \right) \right] \right| + O_P \left(1/\sqrt{N} \right) \mathop{\rightarrow}\limits^{D} \left(\gamma_1 + \gamma_2 \right)^{1/2} p^{\prime} D|W|, $$

where W ∼ N(0,R).

(ii) In this case

$$\begin{aligned} \hat{U}_{m,n} - \check{p}^{\prime} D\check{p} &= \left( \hat{p}^{\prime} D_{\hat{p},\hat{q}} \hat{q} - \hat{p}^{\prime} D_{p,q} \hat{q} \right) + \left( \hat{p}^{\prime} D_{p,q} \hat{q} - p^{\prime} D_{p,q}q \right) + \left( p^{\prime} D_{p,q}q - \check{p}^{\prime} D\check{p} \right)\\ &= \hat{p}^{\prime} \left[ D_{p,q}^{(1,0)} \left( \hat{p} - p \right) + D_{p,q}^{(0,1)} \left( \hat{q} - q \right) \right] \hat{q} + q^{\prime} D_{p,q} \left(\hat{p} - p \right)\\ &\quad + p^{\prime} D_{p,q} \left(\hat{q} - q \right) + \left( p^{\prime} D_{p,q}q - \tilde{p}^{\prime} D\tilde{p} \right) + O_P(1/N). \end{aligned}$$

Since $\hat{p} \rightarrow p$ (a.s.) and $\hat{q} \rightarrow q$ (a.s), by Slutsky’s theorem, $\sqrt{N} \hat{p}^{\prime} \left[ D_{p,q}^{(1,0)} \left( \hat{p} - p \right)+ D_{p,q}^{(0,1)} \left( \hat{q} - q \right) \right] \hat{q}\;\hbox{and}\;\sqrt{N} p^{\prime} \left[ D_{p,q}^{(1,0)} \left( \hat{p} - p \right) + D_{p,q}^{(0,1)} \left( \hat{q} - q \right) \right] q$ has the same asymptotic distribution. With the components of D ^(1,0) _p,q given in (i), it is easy to check that

$$ p^{\prime} D_{p,q}^{(1,0)} \left( \hat{p} - p \right) q = a \left( \hat{p} - p \right), \quad p^{\prime} D_{p,q}^{(0,1)} \left( \hat{q} - q \right)q = b \left( \hat{q} - q \right) $$

where a = (a ₁, ..., a _k )′ and b = (b ₁, ..., b _k )′ with

$$\begin{aligned} a_i &= p_i {\frac{{\rm e}^{p_i-q_i} - {\rm e}^{q_i-p_i}}{2}} \sum\limits_{j=1}^k {\frac{{\rm e}^{p_j-q_j} + {\rm e}^{q_j-p_j}}{2}} q_jD \left( {\bf h}_i, {\bf h}_j \right)\quad (i=1, \ldots, k)\\ b_i &= q_i {\frac{{\rm e}^{q_i-p_i} - {\rm e}^{p_i-q_i}}{2}} \sum\limits_{j=1}^k {\frac{{\rm e}^{q_j-p_j} + {\rm e}^{p_j-q_j}}{2}} p_jD \left({\bf h}_i, {\bf h}_j \right) \quad (i=1,\ldots,k). \end{aligned}$$

This gives

$$\begin{aligned} \, &\sqrt{N} \left( \hat{U}_{m,n} - \check{p}^{\prime} D\check{p} \right) - \sqrt{N} \left( p^{\prime} D_{p,q}q - \tilde{p}^{\prime} D\tilde{p} \right)\\ \, &\quad = \sqrt{N} \left[ \left( a^{\prime} + q^{\prime} D_{p,q} \right) \left( \hat{p} - p \right) + \left( b^{\prime} + p^{\prime} D_{p,q} \left( \hat{q} - q \right) \right) \right] + O_P \left(1/\sqrt{N} \right) \mathop{\rightarrow}\limits^{D} N(0, \sigma^2) \end{aligned}$$

with

$$ \sigma^2 = \gamma_1 \left( a^{\prime} + q^{\prime} D_{p,q} \right) R \left(a + D_{p,q}q \right) + \gamma_2 \left( b^{\prime} + p^{\prime} D_{p,q} \right) Q \left( b + D_{p,q}p \right). $$

Similarly, we obtained asymptotic versions of Tzeng et al.’s results. let σ _T ² and $\tilde{\sigma}^2_T$ be their asymptotic variance under H ₀ and H ₁, respectively, assume $0 \lim N/m = \gamma_1 < \infty,\;0 \lim N/n = \gamma_2 < \infty$ and non-degeneracy of the kernel, then

$$ \sqrt{N} \left( \hat{p}^{\prime} A \hat{p} - \hat{q}^{\prime} A\hat{q} \right)/\sigma_T \mathop{\rightarrow}\limits^{D} N(0,1), $$

where σ _T ² = 4(γ₁ + γ₂) p′ARAp. σ _T ² is consistently estimated by $\hat{\sigma}^2_T$ which is σ _T ² with p replaced by $\hat{p},$ the estimate of p from the pooled data. Under H ₁,

$$ \sqrt{N} \tilde{\sigma}^{-1}_T \left( \hat{p}^{\prime} A\hat{p} - \hat{q}^{\prime} A\hat{q} - p^{\prime} Ap + q^{\prime} Aq \right) \mathop{\rightarrow}\limits^{D} N(0,1), $$

where $\tilde{\sigma}^2_T =4 \left( \gamma_1 p^{\prime}ARAp + \gamma_2 q^{\prime}AQAq \right),\;\hbox{and}\;\tilde{\sigma}^{2}_T$ is consistently estimated by its empirical version.