An entropy-based genome-wide transmission/disequilibrium test

Abstract

Availability of a large collection of single nucleotide polymorphisms (SNPs) and efficient genotyping methods enable the extension of linkage and association studies for complex diseases from small genomic regions to the whole genome. Establishing global significance for linkage or association requires small P-values of the test. The original TDT statistic compares the difference in linear functions of the number of transmitted and nontransmitted alleles or haplotypes. In this report, we introduce a novel TDT statistic, which uses Shannon entropy as a nonlinear transformation of the frequencies of the transmitted or nontransmitted alleles (or haplotypes), to amplify the difference in the number of transmitted and nontransmitted alleles or haplotypes in order to increase statistical power with large number of marker loci. The null distribution of the entropy-based TDT statistic and the type I error rates in both homogeneous and admixture populations are validated using a series of simulation studies. By analytical methods, we show that the power of the entropy-based TDT statistic is higher than the original TDT, and this difference increases with the number of marker loci. Finally, the new entropy-based TDT statistic is applied to two real data sets to test the association of the RET gene with Hirschsprung disease and the Fcγ receptor genes with systemic lupus erythematosus. Results show that the entropy-based TDT statistic can reach p-values that are small enough to establish genome-wide linkage or association analyses.

Appendices

Appendix 1

Let ${X = - \hat{p}\log \hat{p}}$ and ${Y = - \hat{q}\log \hat{q}}.$ Both X and Y are nonlinear functions of the allele frequencies. The distribution of (X − Y) is asymptotically normal with mean zero and variance Var(X − Y), where

$$ \begin{aligned} \hbox{Var}(X - Y) & = [{X}\ifmmode{'}\else$'$\fi(p)]^{2} \frac{{pq}} {n} + [{Y}\ifmmode{'}\else$'$\fi(q)]^{2} \frac{{pq}} {n} + 2{X}\ifmmode{'}\else$'$\fi(p){Y}\ifmmode{'}\else$'$\fi(q)\frac{{pq}} {n} \\ & = \frac{{pq}} {n}[{X}\ifmmode{'}\else$'$\fi(p) + {Y}\ifmmode{'}\else$'$\fi(q)]^{2} \\ & = \frac{{pq}} {n}(2 + \log p + \log q)^{2}. \\ \end{aligned} $$

Under the null hypothesis of no linkage or no association, the frequency of the transmitted allele M ₁ is equal to the frequency of the transmitted allele M ₂, thus, we have X(p) = Y(q). Therefore, the distribution of (X−Y) is normal distribution with mean zero and variance ${\frac{{pq}}{n}(2 + \log p + \log q)^{2}}.$ Under the null hypothesis, ${\hbox{TDT}_{e} = \frac{{(X - Y)^{2}}}{{{\rm Var}(X - Y)}} = \frac{{n(\hat{p}\log \hat{p} - \hat{q}\log \hat{q})^{2}}}{{\hat{p}\hat{q}(2 + \log \hat{p} + \log \hat{q})^{2}}}}$ is asymptotically distributed as a central χ² ₍₁₎ distribution.

Appendix 2

First, we calculate ${\hbox{var} (\hat{p}_{{i \cdot}})}.$ By definition, we have

$$ \begin{aligned} \operatorname{var} (\hat p_{i \cdot }) &= \operatorname{var} \left(\frac{1} {n}\sum\limits_{j = 1}^k {n_{ij} } \right) \\ &= \frac{1} {{n^2 }}\left\{ {n\sum\limits_{j = 1}^k {p_{ij} (1 - p_{ij}) + (- n)\sum\limits_{\begin{array}{*{20}c} {j = 1} \\ {j \ne l} \\ \end{array} }^k {\sum\limits_{l = 1}^k {p_{ij} p_{il} } } } } \right\} & \\ &= \frac{1} {n}p_{i \cdot } (1 - p_{i \cdot }) & \\ \end{aligned} $$

where p _ii = 0.

Similarly, we have

$$\hbox{var} (\hat{q}_{{i.}}) = \frac{1}{n}q_{{i.}} (1 - q_{{i.}}).$$

Next we calculate covariance between ${\hat{p}_{{i \cdot}}}$ and ${\hat{p}_{{j \cdot}}(i \ne j}).$ Again, by definition, we obtain

$$ \begin{aligned} \operatorname{cov} (\ifmmode\expandafter\hat\else\expandafter\^\fi{p}_{{i \cdot }},\ifmmode\expandafter\hat\else\expandafter\^\fi{p}_{{j \cdot }}) & = \operatorname{cov} \left(\frac{1} {n}{\sum\limits_{l = 1}^k {n_{{il}} } },\frac{1} {n}{\sum\limits_{m = 1}^k {n_{{jm}} } }\right) \\ & = - \frac{1} {n}{\sum\limits_{l = 1}^k {{\sum\limits_{m = 1}^k {p_{{il}} p_{{jm}} } }} } \\ & = - \frac{1} {n}p_{{i \cdot }} p_{{j \cdot }} \\ \end{aligned} $$

Similarly, we have

$$\hbox{cov} (\hat{q}_{{i.}}, \hat{q}_{{j.}}) = - \frac{1}{n}q_{{i.}} q_{{j.}}$$

Now we calculate ${\hbox{cov} (\hat{p}_{{i \cdot}}, \hat{q}_{{j.}}).}$ First, we consider i ≠ j. In these cases, we have

$$ \begin{aligned} \operatorname{cov} (\ifmmode\expandafter\hat\else\expandafter\^\fi{p}_{{i \cdot }} ,\ifmmode\expandafter\hat\else\expandafter\^\fi{q}_{{j.}}) & = \operatorname{cov} \left(\frac{1} {n}{\sum\limits_{l = 1}^k {n_{{il}} } },\frac{1} {n}{\sum\limits_{m = 1}^k {n_{{mj}} } }\right) \\ & = - \frac{1} {n}{\sum\limits_{\begin{array}{*{20}c} {{l = 1}} \\ {{l \ne j}} \\ \end{array} }^k {{\sum\limits_{\begin{array}{*{20}c} {{m = 1}} \\ {{m \ne i}} \\ \end{array} }^k {p_{{il}} p_{{mj}} } }} } + \frac{1} {n}p_{{ij}} (1 - p_{{ij}}) \\ & = \frac{1} {n}(p_{{ij}} - p_{{i \cdot }} q_{{j.}}) \\ \end{aligned} $$

Similarly, we have ${\hbox{cov} (\hat{q}_{{i.}}, \hat{p}_{{j \cdot}}) = \frac{1}{n}(p_{{ji}} - q_{{i.}} p_{{j \cdot}})}$ when i ≠ j.

Then, we consider i = j. For i = j, we obtain

$$ \begin{aligned} \hbox{cov} (\hat{p}_{{i \cdot }},\hat{q}_{{j.}}) & = \operatorname{cov} \left(\frac{1} {n}{\sum\limits_{m = 1}^k {n_{{im}} } },\frac{1} {n}{\sum\limits_{l = 1}^k {n_{{li}} } }\right) \\ & = - \frac{1} {n}{\sum\limits_{m = 1}^k {{\sum\limits_{l = 1}^k {p_{{im}} p_{{li}} } }} } + \frac{1} {n}p_{{ii}} \\ & = - \frac{1} {n}p_{{i \cdot }} q_{{j.}} \\ \end{aligned} $$

Thus, we have proven Eq. 6.

Let ${h(p) = [h(p_{{1 \cdot}}), \ldots, h(p_{{k \cdot}})]^{T}}$ and ${h(q) = [h(q_{{1 \cdot}}), \ldots, h(q_{{k \cdot}})]^{\rm {T}}},$ where ${h(p_{{i \cdot}}) = - p_{{i \cdot}} \log p_{{i \cdot}}}$ and ${h(q_{{i.}}) = - q_{{i.}} \log q_{{i.}}}.$ Then ${h(\hat{p}) - h(p)}$ and ${h(\hat{q}) - h(q)}$ are asymptotically distributed as normal distribution with mean zero and variance ${\frac{1}{n}B\Sigma_{p} B^{\rm {T}}}$ and ${\frac{1}{n}C\Sigma_{q} C^{\rm {T}}},$ respectively, where ${B = (b_{{ij}})_{{k \times k}}}$ and ${C = (c_{{ij}})_{{k \times k}}, b_{{ii}} = \frac{{\partial h(p_{{i \cdot}})}}{{\partial p_{{i \cdot}}}} = - 1 - \log p_{{i \cdot}}, b_{{ij}} = \frac{{\partial h(p_{{i \cdot}})}}{{\partial p_{{j \cdot}}}} = 0\begin{array}{*{20}c} & {{(j \ne i)}}, \\ \end{array} c_{{ii}} = \frac{{\partial h(q_{{i.}})}}{{\partial q_{{i.}}}} = - 1 - \log q_{{i.}}}$ and ${c_{{ij}} = \frac{{\partial h(q_{{i.}})}}{{\partial q_{{j.}}}} = 0\begin{array}{*{20}c} & {{(j \ne i)}} \\ \end{array}}$ Under the null hypothesis of no linkage or no association, we have h(p) = h(q), thus ${h(\hat{p}) - h(\hat{q}) = h(\hat{p}) - h(p) - [h(\hat{q}) - h(q)]}$ is asymptotically distributed as normal distribution:

$$h(\hat{p}) - h(\hat{q}) = h(\hat{p}) - h(p) - [h(\hat{q}) - h(q)]\sim N\left(0,\frac{1}{n}\Lambda\right),$$

where

$$\Lambda = B\Sigma_{p} B^{\rm {T}} + C\Sigma_{q} C^{\rm {T}} - B\Sigma_{{pq}} C^{\rm {T}} - C\Sigma^{\rm {T}}_{{pq}} B^{\rm {T}}.$$

(11)

Applying Theorem 4.4.3 (Graybill 1976), we obtain ${n[h(\hat{p}) - h(\hat{q})]^{T} \Lambda^{-}_{e} [h(\hat{p}) - h(\hat{q})]}$ is asymptotically distributed as a central χ² _(r) distribution under the null hypothesis of no linkage or no association, where r = rank(Λ _e ), and Λ _e is the estimator of matrix Λ by substituting the estimators of p _i· and p _·i into Eq. 11.

Appendix 3

Following the approach of Sham and Curtis, we can obtain the joint probability that a heterozygous parent with genotype M _i M _j transmits the allele M _i to an affected child. Let TM denote the transmitted marker allele, NM denote the nontransmitted marker allele, TD denote the transmitted disease allele, OTD denote disease allele transmitted by another parent and TH denote the transmitted haplotype. Let P _i be the frequency of the allele M _i at the marker locus, $P_{D_{k}}$ be the frequency of the disease allele D _k , P _ki be the frequency of the haplotype $H_{D_{k}M_{i}}$ and θ be the recombination fraction between the marker and disease loci. Define the measure of LD between the marker and disease loci as

$$\delta_{{kj}} = P_{{kj}} - P_{{{\rm D}_{k}}} P_{j}.$$

Then, the probability that the haplotype ${H_{{{\rm D}_{k} M_{i}}}}$ is transmitted and the marker allele M _j is not transmitted is given by

$$P(\hbox{TH} = \hbox{D}_{k} M_{i}, NM = M_{j}) = (1 - \theta)P_{{ki}} P_{j} + \theta P_{{kj}} P_{i}$$

(12)

The joint probability that a heterozygous parent with genotype M _i M _j transmits the allele M _i to an affected child is given by

$$ \begin{aligned} P(\hbox{TM} & = M_{i}, \hbox{NM} = M_{j} |A) \\ & = \frac{1} {{P(A)}}P(\hbox{TM} = M_{i}, \hbox{NM} = M_{j},A) \\ & = \frac{1} {{P(A)}}{\sum\limits_k {{\sum\limits_l {P(\hbox{TH} = \hbox{D}_{k} M_{i}, \hbox{NM} = M_{j} )P(\hbox{D}_{l} |\hbox{TH} = \hbox{D}_{k} M_{i}, \hbox{NM} = M_{j})P(A|\hbox{D}_{k} \hbox{D}_{l}).} }} } \\ \end{aligned} $$

Using Eq. 12, we have

$$ \begin{aligned} P(\hbox{TM} & = M_{i}, \hbox{NM} = M_{j} |A) \\ & = \frac{1} {{P(A)}}{\sum\limits_k {{\sum\limits_l {f_{{kl}} } }} }P_{{{\rm D}_{l} }} [(1 - \theta)P_{{ki}} P_{j} + \theta P_{{kj}} P_{i} ] \\ & = P_{i} P_{j} + \frac{1} {{P(A)}}[P_{j} {\sum\limits_k {{\sum\limits_l {f_{{kl}} } }} }P_{{{\rm D}_{l} }} \delta _{{ki}} + \theta {\sum\limits_k {{\sum\limits_l {f_{{kl}} P_{{{\rm D}_{l} }} } }} }(P_{i} \delta _{{kj}} - P_{j} \delta _{{ki}})] \\ \end{aligned} $$

where notations are given in the text.

Summarizing over all j in the above equation, we obtain the probability of transmitting the marker allele M _i to an affected child:

$$ \begin{aligned} P_{{i.}} & = P(\hbox{TM} = M_{i} |\hbox{Affected}) = P_{i} + \frac{1} {{P(A)}}\left({\sum\limits_k {{\sum\limits_l {f_{{kl}} P_{{\hbox{D}l}} \delta _{{ki}} - \theta {\sum\limits_k {{\sum\limits_l {f_{{kl}} P_{{\hbox{D}l}} \delta _{{ki}} } }} }} }} }\right) \\ & = P_{i} + \frac{{1 - \theta }} {{P(A)}}[(f_{{11}} - f_{{12}})P_{D} + (f_{{12}} - f_{{22}})P_{d} ]\delta _{{1i}} \\ \end{aligned} $$

Similarly, we have

$$p_{{.j}} = P_{j} + \frac{\theta}{{P(A)}}[(f_{{11}} - f_{{12}})P_{{\rm D}} + (f_{{12}} - f_{{22}})P_{{\rm d}}]\delta_{{1j}}$$

Note that

$$ \begin{aligned} &\delta_{{1j}} + \delta_{{2j}} = P_{{1j}} - P_{{\rm D}} P_{j} + P_{{2j}} - P_{{\rm d}} P_{j} = P_{j} - P_{j} = 0, \min (P_{{\rm D}}, P_{j}) \geqslant P_{{1j}} = \delta_{{1j}} + P_{{\rm D}} P_{j} \geqslant 0,\\ &\min (P_{{\rm d}}, P_{j}) \geqslant P_{{2j}} = - \delta_{{1j}} + (1 - P_{{\rm D}})P_{j} \geqslant 0. \end{aligned} $$

Thus, the measure δ_1j of the LD between the disease allele D and the marker allele M _j should satisfy the following constraints:

$$\min (P_{j} P_{{\rm d}}, (1 - P_{j})P_{{\rm D}}) \geqslant \delta_{{1j}} \geqslant \max (- P_{{\rm D}} P_{j}, - P_{{\rm d}} (1 - P_{j}))$$

(13)

The measure δ_1j of the LD between the disease allele and the marker allele M _j can be calculated by

$$E[\delta_{{1j}} (t)] = \delta_{{1j}} (0)(1 - \theta)^{t}$$

(14)

where t is the time since the generations of the LD between the marker and disease loci, and δ_1j (0) is the measure of the initial LD when the LD was created. The initial measure δ_1j (0) of the LD should satisfy the constraints Eq. 13.

Now we study how to calculate the frequencies of the transmitted and nontransmitted haplotypes. Recall that TH and NH denote the transmitted and nontransmitted haplotypes, respectively. The transmitted three-locus haplotype will experience a non-recombinant, a single recombinant and a double recombinant event (Wilson 1997). Thus, we have

$$ \begin{aligned} P(\hbox{TH} & = H_{{i_{1} si_{2} }}, \hbox{NH} = H_{{j_{1} j_{2} }} ) \\ & = (1 - \theta _{1})(1 - \theta _{2})P_{{i_{1} si_{2} }} P_{{j_{1} j_{2} }} + \theta _{1} (1 - \theta _{2})P_{{j_{1} si_{2} }} P_{{i_{1} j_{2} }} + (1 - \theta _{1})\theta _{2} P_{{i_{1} sj_{2} }} P_{{j_{1} i_{2} }} + \theta _{1} \theta _{2} P_{{j_{1} sj_{2} }} P_{{i_{1} i_{2} }} \\ \end{aligned} $$

$$ \begin{aligned} P(\hbox{TH} & = H_{{i_{1} i_{2} }}, \hbox{NH} = H_{{j_{1} j_{2} }} ,{\rm Affected}) \\ & = {\sum\limits_s {{\sum\limits_u {f_{{\hbox{su}}} } }} }P_{{\hbox{Du}}} [(1 - \theta _{1})(1 - \theta _{2})P_{{i_{1} si_{2} }} P_{{j_{1} j_{2} }} + \theta _{1} (1 - \theta _{2})P_{{j_{1} si_{2} }} P_{{i_{1} j_{2} }} \\ & \quad + (1 - \theta _{1})\theta _{2} P_{{i_{1} sj_{2} }} P_{{j_{1} i_{2} }} + \theta _{1} \theta _{2} P_{{j_{1} sj_{2} }} P_{{i_{1} i_{2} }} ] \\ \end{aligned} $$

(15)

Let ${b = \frac{{(f_{{11}} - f_{{12}})P_{{\rm D}} + (f_{{12}} - f_{{22}})P_{{\rm d}}}}{{P(A)}}}$ and ${a = \frac{{f_{{12}} P_{{\rm D}} + f_{{22}} P_{{\rm d}}}}{{P(A)}}},$ Then, after some algebra on Eq. 15, we can obtain

$$ \begin{aligned} P(\hbox{TH} & = H_{{i_{1} i_{2} }}, \hbox{NH} = H_{{j_{1} j_{2} }} |\hbox{Affected}) \\ & = (1 - \theta _{1})(1 - \theta _{2})P_{{j_{1} j_{2} }} (bP_{{i_{1} \hbox{D}i_{2} }} + aP_{{i_{1} i_{2} }}) + \theta _{1} (1 - \theta _{2})P_{{i_{1} j_{2} }} (bP_{{j_{1} \hbox{D}i_{2} }} + aP_{{j_{1} i_{2} }}) \\ & \quad + (1 - \theta _{1})\theta _{2} P_{{j_{1} i_{2} }} (bP_{{i_{1} \hbox{D}j_{2} }} + aP_{{i_{1} j_{2} }}) + \theta _{1} \theta _{2} P_{{i_{1} i_{2} }} (bP_{{j_{1} \hbox{D}j_{2} }} + aP_{{j_{1} j_{2} }}) \\ \end{aligned} $$

(16)

Thus, the probability that the haplotype $H_{i_{1}i_{2}}$ is transmitted to an affected child is given by

$$ \begin{aligned} P(TH & = H_{{i_{1} i_{2} }} |\hbox{Affected}) = {\sum\limits_{j_{1} } {{\sum\limits_{j_{2} } {P(\hbox{TH} = H_{{i_{1} i_{2} }} , \hbox{NH} = H_{{j_{1} j_{2} }} |\hbox{Affected})} }} } \\ & = (1 - \theta _{1})(1 - \theta _{2})(bP_{{i_{1} \hbox{D}i_{2} }} + aP_{{i_{1} i_{2} }}) + \theta _{1} (1 - \theta _{2})P_{{i_{1} }} (bP_{{\hbox{D}i_{2} }} + aP_{{i_{2} }}) \\ & \quad + (1 - \theta _{1})\theta _{2} P_{{i_{2} }} (bP_{{i_{1} \hbox{D}_{{}} }} + aP_{{i_{1} }}) + \theta _{1} \theta _{2} P_{{i_{1} i_{2} }} \\ \end{aligned} $$

(17)

(Note bP _D +  a = 1)

Similarly, we have

$$\begin{aligned}\,& P(\hbox{NH} = H_{{j_{1} j_{2}}} |\hbox{Affected}) = (1 - \theta_{1})(1 - \theta_{2})P_{{j_{1} j_{2}}} + \theta_{1} (1 - \theta_{2})P_{{j_{2}}} (bP_{{j\hbox{D}}} + aP_{{j_{1}}}) \\& \begin{array}{*{20}c} & \\ \end{array} \begin{array}{*{20}c} & \\ \end{array} + (1 - \theta_{1})\theta_{2} P_{{j_{1}}} (bP_{{\hbox{D}j_{2}}} + aP_{{j_{2}}}) + \theta_{1} \theta_{2} (bP_{{j_{1} \hbox{D}j_{2}}} + aP_{{j_{1} j_{2}}}) \\ \end{aligned}$$

(18)

Using the following relationship between the haplotype and the measure of LD:

$$\begin{aligned} &P_{{i_{1} \hbox{D}i_{2}}} = \delta_{{i_{1} \hbox{D}i_{2}}} + P_{{i_{1}}} \delta_{{\hbox{D}i_{2}}} + P_{{i_{2}}} \delta_{{i_{1} \hbox{D}}} + P_{{i_{1}}} P_{{\rm D}} P_{{i_{2}}},\\ &\,\hbox{and } \delta_{{i_{1} \hbox{D}}} = P_{{i_{1} \hbox{D}}} - P_{{i_{1}}} P_{{\rm D}}, \delta_{{\hbox{D}i_{2}}} = P_{{\hbox{D}i_{2}}} - P_{{\rm D}} P_{{i_{2}}}, \delta_{{i_{1} i_{2}}} = P_{{i_{1} i_{2}}} - P_{{i_{1}}} P_{{i_{2}}}, \end{aligned} $$

We obtain

$$ \begin{aligned} P(\hbox{TH} & = H_{{i_{1} i_{2} }} |\hbox{Affected}) = [a + (1 - a)\theta _{1} \theta _{2} ]P_{{i_{1} }} P_{{i_{2} }} + b(1 - \theta _{1})(1 - \theta _{2})\delta _{{i_{1} \hbox{D}i_{2} }} + bP_{{i_{2} }} (1 - \theta _{1})\delta _{{i_{1} \hbox{D}}} \\ & \quad + bP_{{i_{1} }} (1 - \theta _{2})\delta _{{\hbox{D}i_{2} }} + [\theta _{1} \theta _{2} + (1 - \theta _{1})(1 - \theta _{2})a]\delta _{{i_{1} i_{2} }} \\ \end{aligned} $$

$$ \begin{aligned} P(\hbox{NH} & = H_{{j_{1} j_{2} }} |\hbox{Affected}) = [a\theta _{2} + (1 - \theta _{2})(1 - \theta _{1} + a\theta _{1})]P_{{j_{1} }} P_{{j_{2} }} + b\theta _{1} \theta _{2} \delta _{{j_{1} \hbox{D}j_{2} }} + bP_{{j_{2} }} \theta _{1} \delta _{{j_{1} \hbox{D}}} \\ & \quad + bP_{{j_{1} }} \theta _{2} \delta _{{\hbox{D}j_{2} }} + [a\theta _{1} \theta _{2} + (1 - \theta _{1})(1 - \theta _{2})]\delta _{{j_{1} j_{2} }} \\ \end{aligned} $$

(19)

Measures of LD are random variables. The expectation of $\delta_{i_{1}si_{2}}$ is equal to

$$E[\delta_{{i_{1} si_{2}}}] = E[P_{{i_{1} si_{2}}}] - P_{{i_{1}}} E[\delta_{{si_{2}}}] - P_{{i_{2}}} E[\delta_{{i_{1} s}}] - P_{{i_{1}}} P_{{{\rm D}_{s}}} P_{{i_{2}}}.$$

But, it was shown that ${E[P_{{i_{1} si_{2}}}] = \delta_{{i_{1} si_{2}}} (0)(1 - \theta_{1})^{t} (1 - \theta_{2})^{t} + P_{{i_{1}}} E[\delta_{{si_{2}}}] + P_{{i_{2}}} E[\delta_{{i_{1} s}}] + P_{{i_{1}}} P_{{{\rm D}_{s}}} P_{{i_{2}}}},$ where $\delta_{i_{1}si_{2}} (0)$ is the measure of the initial LD at three loci $M_{i_{1}}D_{s}M_{i_{2}}.$ Thus, we have

$$E[\delta_{{i_{1} si_{2}}}] = \delta_{{i_{1} si_{2}}} (0)(1 - \theta_{1})^{t} (1 - \theta_{2})^{t}$$

(20)

Substituting ${E\left[\delta_{i_{1}s}\right]},\;{E\left[\delta_{si_{2}}\right]}$ in Eq. 14 and ${E\left[\delta_{i_{1}si_{2}}\right]}$ in Eq. 20 into Eq. 19, we obtain

$$ \begin{aligned} P(\hbox{TH} & = H_{{i_{1} i_{2} }} |\hbox{Affected}) = [a + (1 - a)\theta _{1} \theta _{2} ]P_{{i_{1} }} P_{{i_{2} }} + b\delta _{{i_{1} \hbox{D}i_{2} }} (0)(1 - \theta _{1})^{{t + 1}} (1 - \theta _{2})^{{t + 1}} + bP_{{i_{2} }} \delta _{{i_{1} \hbox{D}}} (0)(1 - \theta _{1})^{{t + 1}} \\ & \quad + bP_{{i_{1} }} \delta _{{\hbox{D}i_{2} }} (0)(1 - \theta _{2})^{{t + 1}} + [\theta _{1} \theta _{2} + (1 - \theta _{1})(1 - \theta _{2} )a]\delta _{{i_{1} i_{2} }} (0)(1 - \theta _{1} - \theta _{2})^{t} \\ \end{aligned} $$

$$ \begin{aligned} P(\hbox{NH} & = H_{{j_{1} j_{2} }} |\hbox{Affected}) = [a\theta _{2} + (1 - \theta _{2})(1 - \theta _{1} + a\theta _{1})]P_{{j_{1} }} P_{{j_{2} }} + b\delta _{{j_{1} \hbox{D}j_{2} }} (0)(1 - \theta _{1})^{t} (1 - \theta _{2})^{t} + bP_{{j_{2} }} \delta _{{j_{1} \hbox{D}}} (0)(1 - \theta _{1})^{t} \\ & \quad + bP_{{j_{1} }} \delta _{{\hbox{D}j_{2} }} (0)\theta _{2} (1 - \theta _{2})^{t} + [\theta _{1} \theta _{2} a + (1 - \theta _{1} )(1 - \theta _{2})]\delta _{{j_{1} j_{2} }} (0)(1 - \theta _{1} - \theta _{2})^{t} \\ \end{aligned} $$

Let ${P_{{i_{1} i_{2} \cdot}} = P(\hbox{TH} = H_{{i_{1} i_{2}}} |\hbox{Affected})}$ and ${P_{{\cdot i_{1} i_{2}}} = P(\hbox{NH} = H_{{i_{1} i_{2}}} |\hbox{Affected}),}$ we obtain equations:

$$ \begin{aligned} P_{{i_{1} i_{2} \cdot }} & = P_{{i_{1} }} P_{{i_{2} }} + b\delta _{{i_{1} \hbox{D}i_{2} }} (0)(1 - \theta _{1})^{{t + 1}} (1 - \theta _{2})^{{t + 1}} + bP_{{i_{2} }} \delta _{{i_{1} \hbox{D}}} (0)(1 - \theta _{1})^{{t + 1}} + bP_{{i_{1} }} \delta _{{\hbox{D}i_{2} }} (1 - \theta _{2})^{{t + 1}} \\ & \quad + [\theta _{1} \theta _{2} + (1 - \theta _{1})(1 - \theta _{2})a]\delta _{{i_{1} i_{2} }} (0)(1 - \theta _{1} - \theta _{2})^{t} \\ \end{aligned} $$

$$ \begin{aligned} P_{{ \cdot i_{1} i_{2} }} & = P_{{i_{1} }} P_{{i_{2} }} + b\delta _{{i_{1} \hbox{D}i_{2} }} (0)\theta _{1} \theta _{2} (1 - \theta _{1})^{t} (1 - \theta _{2})^{t} + bP_{{i_{2} }} \delta _{{i_{1} \hbox{D}}} (0)\theta _{1} (1 - \theta _{1})^{t} + bP_{{i_{1} }} \delta _{{\hbox{D}i_{2} }} (0)\theta _{2} (1 - \theta _{2})^{t} \\ & +\quad [(1 - \theta _{1} )(1 - \theta _{2}) + \theta _{1} \theta _{2} a]\delta _{{i_{1} i_{2} }} (0)(1 - \theta _{1} - \theta _{2})^{t} \\ \end{aligned} $$

Based on the above equations, we can calculate

$$\mu_{T} = [- P_{{11 \cdot}} \log P_{{11 \cdot}}, - P_{{12 \cdot}} \log P_{{12 \cdot}}, - P_{{21 \cdot}} \log P_{{21 \cdot}}, - P_{{22 \cdot}} \log P_{{22 \cdot}}]^{T}$$

$$\mu_{{NT}} = [- P_{{\cdot 11}} \log P_{{\cdot 11}}, - P_{{\cdot 12}} \log P_{{\cdot 12}}, - P_{{\cdot 21}} \log P_{{\cdot 21}}, - P_{{\cdot 22}} \log P_{{\cdot 22}}]^{\hbox{T}}.$$

The matrices Σ _p , Σ _q , B, C, and Λ can be similarly defined. Then, for haplotypes produced by two SNPs marker loci flanking a disease locus, substituting μ _T , μ _NT and other parameters, Σ _p , Σ _q , B, C, and Λ into Eq. 9, we obtain the noncentrality parameter λ _HE . Using these analytic formulas for computing the noncentrality parameters of the distribution of the test statistics, we can calculate the power of the test statistics under specified alternative hypothesis.