Abstract
For population casecontrol association studies, the falsepositive rates can be high due to inappropriate controls, which can occur if there is population admixture or stratification. Moreover, it is not always clear how to choose appropriate controls. Alternatively, the parents or normal sibs can be used as controls of affected sibs. For lateonset complex diseases, parental data are not usually available. One way to study lateonset disorders is to perform sibpair or sibship analyses. This paper proposes sibshipbased Hotelling's T^{2 }test statistics for highresolution linkage disequilibrium mapping of complex diseases. For a sample of sibships, suppose that each sibship consists of at least one affected sib and at least one normal sib. Assume that genotype data of multiple tightly linked markers/haplotypes are available for each individual in the sample. Paired Hotelling's T^{2 }test statistics are proposed for highresolution association studies using normal sibs as controls for affected sibs, based on two coding methods: 'haplotype/allele coding' and 'genotype coding'. The paired Hotelling's T^{2 }tests take into account not only the correlation among the markers, but also take the correlation within each sibpair. The validity of the proposed method is justified by rigorous mathematical and statistical proofs under the large sample theory. The noncentrality parameter approximations of the test statistics are calculated for power and sample size calculations. By carrying out power and simulation studies, it was found that the noncentrality parameter approximations of the test statistics were accurate. By power and type I error analysis, the test statistics based on the 'haplotype/allele coding' method were found to be advantageous in comparison to the test statistics based on the 'genotype coding' method. The test statistics based on multiple markers can have higher power than those based on a single marker. The test statistics can be applied not only for biallelic markers, but also for multiallelic markers. In addition, the test statistics can be applied to analyse the genetic data of multiple markers which contain double heterozygotes  that is, unknown linkage phase data. An SAS macro, Hotel_sibs.sas, is written to implement the method for data analysis.
Keywords:
linkage disequilibrium mapping; complex diseasesIntroduction
In recent years, there has been great interest in the research of association studies of complex diseases [16]. By association studies, we mean linkage disequilibrium (LD) mapping of genetic traits. For population casecontrol studies, the marker allele frequency in cases can be compared with that of controls using χ^{2 }test statistics [711]. If there is association between one marker and the trait locus, it is expected that the χ^{2 }tests would lead to significant results. Essentially, this method can be applied to analyse the data for one marker at a time. For multiple markers, the linkage phase may be unknown, [12] and the method cannot be applied simultaneously to analyse the data of multiple markers which contain double heterozygotes. With the development of dense maps such as single nucleotide polymorphisms (SNPs), haplotype maps and highresolution microsatellites in the human genome, enormous amounts of genetic data on human chromosomes are becoming available [1315]. It is interesting when building appropriate models and useful algorithms in association mapping of complex diseases to have the ability to use multiple markers/haplotypes simultaneously.
For tightly linked genetic markers, one may perform association studies of complex diseases based on the Hotelling's T^{2 }test statistics [16]. For population casecontrol data, Xiong et al. proposed two sample Hotelling's T^{2 }test statistics to analyse genotype data of multiple biallelic markers such as SNPs; [17] in addition, logistic regression models were proposed [2,18]. To analyse the multiallelic microsatellite or haplotype data, Fan and Knapp extended Xiong et al. method using two coding methods  'haplotype/allele coding' and 'genotype coding' [19]. For the genetic data of nuclear families or parentoffspring pairs, paired Hotelling's T^{2 }test statistics were proposed, in order to perform association studies based on multiple markers/haplotypes [20].
For lateonset complex diseases, parental data are usually not available. One way to study lateonset disorders is to perform sibpair or sibship analyses [21,22]. This paper proposes sibshipbased paired Hotelling's T^{2 }test statistics for highresolution LD mapping of complex diseases. For a sample of sibships, suppose that each sibship consists of at least one affected sib and at least one normal sib. Assume that genotype data for multiple markers are available for each individual in the sample. Paired Hotelling's T^{2 }test statistics are proposed for highresolution association studies, using normal sibs as controls for affected sibs. The paired Hotelling's T^{2 }tests not only take the correlation among the markers into account, but also the correlation within each sibpair. The validity of the proposed method is justified by rigorous mathematical and statistical proofs under the large sample theory. The noncentrality parameter approximations of the test statistics are calculated for power calculations and comparisons; these are included in the section: Supplementary information: Noncentrality parameters. Type I error rates are calculated by simulations to evaluate the performance of the proposed test statistics. In the section: Supplementary information: Simulation study, the results from the simulation study are presented, to show that the noncentrality parameter approximations of the test statistics are accurate. An SAS macro, Hotel_sibs.sas, was written to implement the method and can be downloaded from the authors' website http://www.stat.tamu.edu/~rfan/software.html/ webcite.
Methods
We assume that a disease locus D is located in a chromosome region. Suppose that the disease locus has two alleles D and d. Allele D is disease susceptible and d is normal. Assume that the diseasesusceptible allele D has population frequency P_{D}, and the normal allele d has population frequency P_{d}.
Paired Hotelling's T^{2 }test statistics
In the region of the disease locus D, assume that J tightly linked markers H_{1}, ..., H_{J }are typed. By tightly linked, we mean that the markers are so close to each other that the recombination fractions among markers are 0. Let us denote the alleles of marker H_{j }by , where n_{j }denotes the number of its alleles. Here, markers can be microsatellites or diallelic markers such as SNPs or haplotypes. If H_{1}, ..., H_{J }are phaseknown haplotypes, the methods developed in this paper are still valid, since the haplotypes can be treated as markers; but the related terminology needs to be changed accordingly. Usually, haplotypes consist of phaseunknown markers; in these cases, we prefer to analyse the genotype marker data directly, instead of estimating the haplotypes first and then analysing the haplotype data. The method developed in this paper can be used to analyse phaseunknown genotype data directly. Consider N sibpairs, each consisting of an affected sibling and a normal sibling. We define coding vectors and for the affected sibling and normal sibling of the ith sibpair, respectively, by one of the following two ways [19,20].
(i) Haplotype/allele coding: For the affected sibling of the ith sibpair, let be his/her genotype at marker H_{j}. Define , where is the number of alleles H_{jk }for the affected sibling of the ith sibpair  that is,
Here and hereafter, the superscript τ denotes the transposition of a matrix or a vector. The dimension of is , which is usually smaller than dimension of the following genotype coding method.
(ii) Genotype coding: Note that can be one of n_{j}(n_{j }+ 1)/2 possible choices: n_{j }homozygous genotypes H_{jk}H_{jk}, and n_{j}(n_{j } 1)/2 heterozygous genotypes H_{jk}H_{jl}, k < l. Depending on the genotype, let us define an indicator vector . Here, is the indicator variable of genotype H_{jk}H_{jk }defined by ; and , k <l is the indicator variable of genotype H_{jk}H_{jl }defined by . The dimension of is n_{j}(n_{j }+ 1)/2  1  that is, the total number n_{j}(n_{j }+ 1)/2 of genotypes of marker H_{j }minus 1 to remove the redundancy. Let be the combined genotype coding of the J markers H_{1}, ... H_{J }. The dimension of is .
For the unaffected sibling of the ith sibpair, let be his/her genotype at marker H_{j}. One may define a vector in the same way, based on either the 'genotype coding' or 'haplotype/allele coding' method. Table 1 in reference 19 gives an example of 'genotype coding' and 'haplotype/allele coding' for a marker with three alleles, to illustrate the above two coding methods.
Table 1. Type I error rates of N = 200 or 300 sibpairs at a significance level α = 0.01 using one marker, H_{1}, or two markers, H_{1 }and H_{2}.
Let and be average coding vectors of affected and unaffected siblings, respectively. Intuitively, and should be similar vectors if the disease locus D is not associated with markers H_{j}, j = 1, ..., J. In the Appendix we prove that the expected value of is 0 if there is no association. Hence, one may build a test statistic based on the difference to test the association between disease locus D and markers H_{j}. To do this, one needs to consider the variancecovariance matrix of . Since siblings' marker genotypes are related to each other, and are not independent. Moreover, and are paired with each other in a sibpair. Therefore, paired T^{2 }test statistics can be used to test the association between disease locus D and markers H_{j }as follows. Define a pairedsample variancecovariance matrix by
A paired Hotelling's T^{2 }statistic can be defined as [16,23]. Let us denote the above Hotelling's T^{2 }statistic for 'haplotype/allele coding' as T_{H}, and the Hotelling's T^{2 }statistic for 'genotype coding' as T_{G}. Assume that the sample size N is sufficiently large that the largesample theory applies. Under the null hypothesis of no association, the statistic T_{H }(or T_{G}) is asymptotically distributed as central χ^{2 }with degrees of freedom. Under the alternative hypothesis of association, T_{H }(or T_{G}) is asymptotically distributed as noncentral χ^{2}. For power calculation and comparison, the noncentrality parameter of statistic T_{H }or T_{G }can be derived under the alternative hypothesis of association.
For general sibships each containing at least one affected sibling and at least one normal sibling, the Hotelling's T^{2 }test statistics T_{H }and T_{G }above can be generalised as follows. Assume that N sibships are available. In the ith sibship, assume that n_{i }siblings are affected and m_{i }siblings are normal. Let and be average coding vectors of affected and normal siblings, respectively. To be precise, let , j = 1, ⋯, n_{i }be the coding vectors of the affected siblings of the ith sibship. Then, ; is defined, accordingly. Utilising to replace and to replace in the above paragraph and defining and , we may define the related Hotelling's T^{2 }test statistics T_{H }and T_{G}.
Noncentrality parameters
The derivation of noncentrality parameters of sibpairs is provided in the section Supplementary information: Noncentrality parameters.
Results
Type I errors
Tables 1, 2 and 3 show type I error rates of test statistics T_{H }and T_{G }at a significance level α = 0.01, using one marker H_{1 }or two markers H_{1 }and H_{2}. Three models are considered. In model I, one marker H_{1 }is used in analysis: H_{1 }is a biallelic marker with equal allele frequency P(H_{11}) = P(H_{12}) = 0.50. In model II, two biallelic markers H_{1 }and H_{2 }are used in analysis, where P(H_{ij}) = 0.5, i, j = 1, 2, . In model III, one marker H_{1 }is used in analysis, where H_{1 }is a quadriallelic marker with allele frequencies P(H_{21}) = P(H_{22}) = 0.35, P(H_{23}) = P(H_{24}) = 0.15.
Table 2. Type I error rates of N = 200 or 300 sibships at a significance level α = 0.01 using one marker, H_{1}, or two markers, H_{1 }and H_{2}.
Table 3. Type I error rates of N = 200 or 300 sibships at a significance level α = 0.01 using one marker, H_{1}, or two markers, H_{1 }and H_{2}.
Each time, 5,000 simulated datasets are generated and each dataset contains N = 200 or 300 sibships under the assumption that there is no association between the marker(s) and the disease locus; a type I error rate is then calculated as the proportion of the 5,000 datasets for which the empirical test statistics are greater than, or equal to, the cutoff point at the significance level α = 0.01. The process is repeated 100 times. Thus, 100 type I error rates are calculated. The mean, standard deviation, minimum and maximum of the 100 type I error rates are presented in the entries of Tables 1, 2 and 3. Since the disease locus is not associated with the marker(s), the empirical test statistics which are greater than or equal to the cutoff point at the significance level α = 0.01 are treated as false positives. Thus, the type I error rates of Tables 1, 2 and 3 are empirical results.
In Table 1, only sibpairs are used in the calculations. In each sibpair, one sibling is affected and the other one is normal. In Table 2, combinations of both sibpairs and sibships of size 3 are used: the number of sibpairs is equal to N/2; the number of sibships of size 3 is N/2; in each of N/4 sibships of size 3, one is affected and the other two are normal; in the remaining N/4 sibships of size 3, two are affected and the other one is normal. In Table 3, combinations of sibpairs and sibships of sizes 3 and 4 are used: the number of sibpairs is equal to N/2; the number of sibships of size 3 is N/5; and the number of sibships of size 4 is 3N/10; in each of N/10 sibships of size 3, one is affected and the other two are normal; in the remaining N/10 sibships of size 3, two are affected and the other one is normal; in each of N/10 sibships of size 4, one is affected and the other three are normal; in each of N/10 sibships of size 4, two are affected and the other two are normal; in the remaining N/10 sibships of size 4, three are affected and the other one is normal.
From the results presented in Tables 1, 2 and 3, it is clear that T_{H }has a lower type I error than T_{G}. That is, the test statistic of the 'haplotype/allele coding' method has a lower type I error than the test statistic of the 'genotype coding' method. The 'haplotype/allele coding' method leads to more robust and reliable test statistics. The type I error rates of the test statistic of the 'haplotype/allele coding' method are reasonable for models I, II and III when N = 200. In addition, the type I error rates of the test statistic of the 'genotype coding' method are reasonable for models I and II when N = 200. The type I error rates of the test statistic for the 'genotype coding' method are slightly higher than the nominal level 0.01 for model III when N = 200 and become lower when N = 300. Note that the number of degrees of freedom for tests T_{G }and T_{H }is 3 and 9, respectively, for model III. Hence, the number of degrees of freedom for test T_{G }is large for model III. When the number of degrees of freedom for tests is large, the asymptotic criteria can be problematic. In this case, a large sample is necessary to keep the type I error rates in a reasonable range.
The results are similar in Tables 1, 2 and 3. Thus, the type I error rates are little affected by the varying structure of the sibships. The reason for this is that we basically take averages of the coding vectors for sibships whose size is larger than 2.
Power calculation and comparison
To make power comparisons, we consider four genetic models: heterogeneous recessive, heterogeneous dominant, additive and multiplicative. For optimistic models, Table 4 gives penetrance probabilities taken from Nielsen et al. or Fan and Knapp [11,19]. For less optimistic models, Table 5 lists penetrance probabilities taken from Fan and Knapp [19]. For j = 1, ..., J, let us denote the measures of LD between allele H_{jk }of the marker H_{j }and the disease locus D by Δ_{jk }= P(H_{jk}D)  P(H_{jk})P_{D}, k = 1, ..., n_{i}. Here, P(H_{jk}D) is the frequency of haplotype H_{jk}D, and P(H_{jk}) is the frequency of allele H_{jk}. For two biallelic markers H_{1 }and H_{2}, let be the measure of LD between the two markers, where P(H_{11}H_{21}) is the frequency of haplotype H_{11}H_{21}. Assume that the two markers H_{1 }and H_{2 }flank the disease locus D in the order H_{1}DH_{2}. Let be the measure of the third order LD [24]. Here, P(H_{11}DH_{21}) is the frequency of haplotype H_{11}DH_{21}.
Table 4. First set of parameters of simulated genetic models.
Table 5. Second set of parameters of simulated genetic models.
Figure 1 shows power curves of T_{H }and T_{G }against the measure of LD Δ_{11 }at a significance level α = 0.05 using two biallelic marker H_{1 }and H_{2}, when P(H_{i1}) = P(H_{i2}) = 0.50, i = 1, 2, P_{D }= 0.15 and N = 200 sibpairs for the first set of parameters of the four genetic models of Table 4. The power curves of T_{H1 }and T_{G1 }are calculated based on one marker H_{1}. In the graphs, Delta_11 = Δ_{11}; the other parameters are given in the legend of the Figure. Figure 2 shows power curves of T_{H }and T_{G }against the measure of LD Δ_{11 }at a significance level α = 0.05 using two biallelic marker H_{1 }and H_{2}, when P(H_{i1}) = P(H_{i2}) = 0.50, i = 1, 2, P_{D }= 0.15 and N = 600 sibpairs for the second set of parameters of the four genetic models listed in Table 5. Similarly to Figure 1, the power curves of T_{H1 }and T_{G1 }are calculated based on one marker H_{1}. The other parameters are the same as those of Figure 1.
Figure 1. Power curves of T_{H }and T_{G }at a significance level α = 0.05, using two biallelic markers H_{1 }and H_{2}, when P(H_{i1}) = P(H_{i2}) = 0.50, i = 1,2, P_{D }= 0.15, and N = 200 sibpairs for the first set of parameters of the four genetic models of Table 4. The power curves of T_{H1 }and T_{G1 }are calculated based on one marker H_{1}. In the graphs, Delta_11 = Δ_{11 }= P(H_{11}D)  P(H_{11})P_{D }is a measure of linkage disequilibrium (LD) between marker H_{1 }and disease locus D; in addition, the other parameters are given by Δ_{21 }= P(H_{21}D) 2 P(H_{21})P_{D }= Δ_{11}, Δ_{H1H2 }= P(H_{11}H_{21})  P(H_{11})P(H_{21}) = 0.05; and .
Figure 2. Power curves of T_{H }and T_{G }at a significance level α = 0.05, using two biallelic markers H_{1 }and H_{2}, when P(H_{i1}) = P(H_{i2}) = 0.50, i = 1,2, P_{D }= 0.15 and N = 600 sibpairs for the second set of parameters of the four genetic models of Table 5. The power curves of T_{H1 }and T_{G1 }are calculated based on one marker H_{1}. In the graphs, Delta_11 = Δ_{11 }= P(H_{11}D)  P(H_{11})P_{D }is a measure of linkage disequilibrium (LD) between marker H_{1 }and disease locus D; in addition, the other parameters are given by Δ_{21 }= P(H_{21}D)  P(H_{21})P_{D }= Δ_{11}, , and .
From Figures 1 and 2, it is clear that T_{H }generally has a higher power than that of T_{G}. This is consistent with the results of Fan and Knapp for population casecontrol studies and Fan et al. for nuclear family data [19,20]. This is most likely due to the large number of degrees of freedom of the test statistic T_{G}. The power of T_{H }(or T_{G}) based on two markers H_{1 }and H_{2 }is generally higher than that of T_{H1 }(or T_{G1}), which is only based on one marker H_{1}. Hence, it is advantageous to use two markers rather than one marker in the analysis. This observation can be generalised  that is, it is advantageous to use multiple tightly linked markers in analysis. Note that the number of degrees of freedom of test statistic T_{G }can increase rapidly as the number of markers increases. This is particularly true when multiallelic markers are used in analysis; but the number of degrees of freedom of T_{H }only increases by one if one more biallelic marker is added to the analysis. Thus, T_{H }has the advantage of high power when multiple markers are used; in addition, the number of degrees of freedom of T_{H }would be not very large. For optimistic models in Table 4, the sample sizes required to achieve certain power levels are lower than those of the less optimistic models in Table 5.
Not only can the test statistics T_{H }and T_{G }be applied to analyse the genetic data of the biallelic markers, but they can also be applied to analyse the genetic data of the multiallelic markers. Figure 3 shows the power curves of T_{H }and T_{G }against the measure of LD Δ_{11 }at a significance level α = 0.05 using a quadriallelic marker H_{1}, when P(H_{11}) = P(H_{12}) = 0.35, P(H_{13}) = P(H_{14}) = 0.15, P_{D }= 0.15 and N = 200 sibpairs for the first set of parameters of the four genetic models of Table 4. The other parameters are given in the legend of the Figure. Figure 4 shows power curves of T_{H }and T_{G }at a significance level α = 0.05 using a quadriallelic marker H_{1}, when P(H_{11}) = P(H_{12}) = 0.35, P(H_{13}) = P(H_{14}) = 0.15, P_{D }= 0.15 and N = 600 sibpairs for the second set of parameters of the four genetic models of Table 5. Similarly to Figures 1 and 2, T_{H }generally has a higher power than that of T_{G}.
Figure 3. Power curves of T_{H }and T_{G }at a significance level α = 0.05 using a quadricallelic marker H_{1}, when P(H_{11}) = P(H_{12}) = 0.35, P(H_{13}) = P(H_{14}) = 0.15 P_{D }= 0.15 and N = 200 sibpairs for the first set of parameters of the four genetic models of Table 4. Delta_11 = Δ_{11 }= P(H_{11}D)  P(H_{11})P_{D }is a measure of linkage disequilibrium (LD) between marker H_{1 }and disease locus D. In addition, Δ_{12 }= Δ_{11}, Δ_{13 }= Δ_{14 }= Δ_{11}/2. The simulated power curves of ST_{H }and ST_{G }are calculated using combinations of both sibpairs and sibships of size 3: the number of sibpairs is equal to N/2 = 100; the number of sibships of size 3 is N/2 = 100; in each of N/4 = 50 sibships of size 3, one is affected and the other two are normal; in the remaining N/4 = 50 sibships of size 3, two are affected and the other one is normal.
Figure 4. Power curves of T_{H }and T_{G }at a significance level α = 0.05 using a quadricallelic marker H_{1}, when P(H_{11}) = P(H_{12}) = 0.35, P(H_{13}) = P(H_{14}) = 0.15, P_{D }= 0.15 and N = 600 sibpairs for the second set of parameters of the four genetic models of Table 5. Delta_11 = Δ_{11 }= P(H_{11}D)  P(H_{11})P_{D }is a measure of linkage disequilibrium (LD) between marker H_{1 }and disease locus D. In addition, Δ_{12 }= Δ_{11}, Δ_{13 }=  Δ_{14 }= Δ_{11}/2. The simulated power curves of ST_{H }and ST_{G }are calculated using combinations of both sibpairs and sibships of size 3 and sibships of size 4; the number of sibpairs is equal to N/2 = 300; the number of sibships of size 3 is N/2 = 120; and the number of sibships of size 4 is 3N/10 = 180; in each of N/10 = 60 sibships of size 3, one is affected and the other two are normal; in the remaining N/10 = 60 sibships of size 3, two are affected and the other one is normal; in each of N/10 = 60 sibships of size 4, one is affected and the other three are normal; in each of N/10 = 60 sibships of size 4, two are affected and the other two are normal; in the remaining N/10 = 60 sibships of size 4, three are affected and the other one is normal.
In addition to the power curves of T_{H }and T_{G}, which are based on sibpair data, Figures 3 and 4 show the simulated power curves of ST_{H }and ST_{G}, which are based on sibships of varying structures. In Figure 3, combinations of both sibpairs and sibships of size 3 are used to calculate the simulated power curves of ST_{H }and ST_{G}: the number of sibpairs is equal to N/2 = 100; the number of sibships of size 3 is N/2 = 100; in each of N/4 = 50 sibships of size 3, one is affected and the other two are normal; in the remaining N/4 = 50 sibships of size 3, two are affected and the other one is normal. In Figure 4, combinations of sibpairs and sibships of sizes 3 and 4 are used to calculate the simulated power curves of ST_{H }and ST_{G}: the number of sibpairs is equal to N/2 = 300; the number of sibships of size 3 is N/5 = 120; and the number of sibships of size 4 is 3N/10 = 180; in each of N/10 = 60 sibships of size 3, one is affected and the other two are normal; in the remaining N/10 = 60 sibships of size 3, two are affected and the other one is normal; in each of N/10 = 60 sibships of size 4, one is affected and the other three are normal; in each of N/10 = 60 sibships of size 4, two are affected and the other two are normal; in the remaining N/10 = 60 sibships of size 4, three are affected and the other one is normal.
To calculate the simulated power curves ST_{H }and ST_{G}, the interval (0, 0.045) of the LD measure Δ_{11 }of LD is uniformly divided into 20 subintervals in Figures 3 and 4. Correspondingly, the 20 subintervals lead to 21 endpoints. For each endpoint, there is a set of parameters for each power curve. Using the set of parameters, 2,500 datasets are simulated for each endpoint. For each dataset, the empirical statistics T_{H }and T_{G }were calculated. The simulated power is the proportion of the 2,500 simulated datasets for which the empirical statistic is larger than the cutoff point of the corresponding χ^{2}distribution at a 0.05 significance level.
From Figures 3 and 4, it can be seen that the simulated power ST_{H }is generally higher than the power of T_{H}, and the simulated power ST_{G }is generally higher than the power of T_{G}. Intuitively, sibships of large size contain more information than that of a sibpair. The test statistics T_{H }and T_{G }can accurately capture the information contained in sibships of large size. Moreover, it can also be seen in Tables 1, 2 and 3 that the type I error is not inflated by including sibships of varying structure.
Simulation study
To evaluate the accuracy of the noncentrality parameter approximations, we performed simulations for the power curves in Figures 1, 2, 3 and 4. The results are presented in the section: Supplementary information: Simulation study. It can be seen that the approximations are excellent.
Discussion
The goal of this study was to develop sibshipbased Hotelling's T^{2 }test statistics for highresolution association mapping of complex diseases. This extends our previous research of paired Hotelling's T^{2 }test statistics of nuclear family data or parentoffspring pairs [20]. For lateonset complex diseases, parental data are usually not available. This motivated us to perform sibpair or sibship analyses to study lateonset disorders. Based an two coding methods'haplotype/allele coding' and 'genotype coding'paired Hotelling's T^{2 }test statistics T_{H }and T_{G }are proposed for highresolution association studies, using normal sibs as controls for affected sibs. The test statistics can be applied to any number of markers, which can be either biallelic or multiallelic. After power calculation and comparison, it was found that it is advantageous to use two markers rather than one marker in the analysis. This observation can be generalised  that is, it is advantageous to use multiple tightly linked markers in analysis. The test statistic T_{H }based on the 'haplotype/allele coding' method is generally more powerful than the test statistic T_{G }based on the 'genotype coding' method. This is most likely due to the large number of degrees of freedom of T_{G}. Moreover, the type I error rates of the test statistic T_{H }are lower than those of test statistic T_{G}.
For population casecontrol association studies, falsepositive rates can be high due to inappropriate controls, which can occur if there is population admixture or stratification [25]. Moreover, it is not always clear how to choose the appropriate controls. Alternatively, the parents or normal sibs can be used as controls of affected sibs [22,2629]. For parental/sibling controls, the methods proposed by Fan and Knapp [19] and Xiong et al. [17] are not valid, since cases and controls are correlated with each other. The two sample Hotelling's T^{2 }test statistics only take into account the correlation among markers [17,19]. For sibship data, not only the correlation among the markers but also the correlation within each sibpair needs to be taken into account. The paired Hotelling's T^{2 }test statistics T_{H }and T_{G }developed in this paper correctly take both the correlation among the markers and the correlation within each sibpair into account. The proposed method is potentially useful in association mapping of lateonset complex diseases.
Cordell and Clayton [2] and Chapman et al. [18] proposed logistic regression models for populationbased case control studies or family studies. Both our proposed method and the logistic regression models can be used in association studies of multilocus marker data. One advantage of the logistic regression models is that it is easy to add covariates to model the environmental effects, in addition to the genetic effects; however, it is not clear how to incorporate the environmental effects into our Hotelling's T^{2 }test statistics. While we are able to calculate the noncentrality parameters for our T^{2 }test statistics for power and sample size calculations, it is not clear if one might get similar results for the logistic regression models. In the study by Cordell and Clayton [2], the authors mainly discuss the analysis of SNP data and only briefly describe a way to analyse the multiallelic markers data. We feel that more investigations are necessary in order for multiallelic markers data to be used in the logistic regression models. By contrast, our proposed T^{2 }can be used to analyse either biallelic or multiallelic marker data, or both simultaneously. Moreover, more investigations are needed to make power comparisons of the two methods.
In Figures 3 and 4, we show that the power of test statistics T_{H }and T_{G }based on combinations of sibships of varying structures are generally higher than the power of the test statistics based on sibpairs. This is because the test statistics T_{H }and T_{G }use the average coding vectors for sibships whose sizes are larger than 2. This averaging strategy does not affect the mean of the coding vectors and , but it will lead to a variancecovariance matrix S, which increases the test statistics. Moreover, it can be seen from Tables 1, 2 and 3 that the type I error is not inflated by including sibships of varying structure. Although the proposed test statistics benefit from this, it is unlikely that they are optimal. One way would be to use weighted sibships in constructing test statistics. In this paper, we assume that there are no missing data. For practical genotype data, genotypic information may be missing at some markers for a portion of the sample [26]. As a result, the methods used here need to be updated to address the problem of missing data. Another issue is that it is not clear how to combine population data, the nuclear family data and sibship data in one single analysis. In practice, the three types of genetic data can be available. They can be analysed separately, but it would be preferable to combine them in a unified analysis, which may lead to higher power. These issues needs more indepth investigation.
Appendix
Consider a sibpair in which one sibling is affected and the other is unaffected/normal. For convenience, assume that the first sibling is affected and the second sibling is normal. Let us denote A_{1 }= (the first sibling is affected), U_{2 }= (the second sibling is unaffected). Let f_{DD}, f_{Dd }= f_{dD }and f_{dd }be the probabilities that an individual with genotypes DD, Dd and dd is affected with the disease, respectively. Since allele D is disease susceptible, one may assume that f_{DD }≥ f_{Dd }≥ f_{dd}. Let , and . Denote the disease prevalence in population by , and . Assume that the affected status of an individual depends only on his/her own genotype at the disease locus. Let us denote the event (i IBD) = the sibpair share i gene identical by descent (IBD) at the disease locus D. Then the joint probability
where s, t, q, r take values of disease allele D and d. To calculate the above equations, we consider the three partitions (2 IBD), (1 IBD) and (0 IBD). These three partitions have probabilities 1/4, 1/2 and 1/4, respectively. Conditional on each partition, the corresponding conditional probabilities are then calculated. The frequency of homozygous genotype H_{jk}H_{jk }in an affected sibling is given by:
Similarly, the frequency of homozygous genotype H_{jk}H_{jk }in an unaffected sibling is given by:
Note that can be calculated by the formula for a_{jkk }by substituting f_{st }with and vice versa. Note that the haplotype frequencies P(H_{jk}D) = Δ_{jk }+ P(H_{jk})P_{D}, P(H_{jk}d) = Δ_{jk }+ P(H_{jk})P_{d}. Under the null hypothesis of no association between the markers H_{i}, i = 1, 2, ..., J, and the disease locus D  that is, Δ_{ij }= 0 for all j, the haplotype frequencies are equal to the product of allele frequencies; for example, P(H_{jk}D) = P(H_{jk})P_{D }and P(H_{jk}d) = P(H_{jk})P_{d}. From equations (4) and (5), .
Similarly, the frequency of the heterozygous genotype H_{jk}H_{jl}, k ≠ l, in an affected sibling can be calculated as follows:
The frequency of the heterozygous genotype H_{jk}H_{jl}, k ≠ l, in an unaffected sibling can be calculated as follows:
Note that can be calculated by the formula for a_{jkl }by substituting f_{st }using and vice versa. Under the null hypothesis of no association between the markers H_{i}, i = 1, 2, ..., J, and the disease locus D  that is, Δ_{ij }= 0 for all j, the haplotype frequencies are equal to the product of the allele frequencies; for example, P(H_{jk}D) = P(H_{jk})P_{D}, P(H_{jk}d) = P(H_{jk})P_{d}, P(H_{jl}D) = P(H_{jl})P_{D }and P(H_{jl}d) = P(H_{jl})P_{d}. From equations (4) and (5), . Therefore, the expectation for the 'genotype coding' method.
For the 'haplotype/allele coding' method, equations (2), (3), (4) and (5) imply
From equation (6), expectation by 'haplotype/allele coding' method, under the null hypothesis of no association between the markers H_{i}, j = 1, ..., J and disease locus D.
Supplementary information: Noncentrality parameters
Consider N sibpairs, each consisting of an affected sibling and a normal sibling. For convenience, assume that the first sibling is affected and the second sibling is normal in each sibpair. Let us denote A_{1 }= (the first sibling is affected), U_{2 }= (the second sibling is unaffected). For 'haplotype/allele coding', the coding vector of the affected sibling in the ith sibpair is . Similarly, is the coding vector of the normal sibling. Denote the variancecovariance matrix of by . The elements of the above variancecovariance matrices are given in Appendices A, B, and C: and in Appendix A, and in Appendices B and C. Using quantities of and in the Appendix to the manuscript, can be calculated. The noncentrality parameter λ_{H }of Hotelling's statistics T_{H }is given by .
For the 'genotype coding' method, the coding vector of the affected sibling in the ith sibpair is j = 1, ..., J. Similarly, is the coding vector of the normal sibling. Let a_{jkl }and be the frequencies of genotype H_{jk}H_{jl }in affected and unaffected siblings given in the Appendix to the manuscript. Then,
Using and , one may calculate the expectation . Let be the variancecovariance matrix of . Then the noncentrality parameter λ_{G }of Hotelling's statistics T_{G }is given by . The elements of the above variancecovariance matrices are given in Appendices D and E: and in Appendix D, and in Appendix E.
Appendix A
Consider the 'haplotype/allele coding' method. The variancecovariance matrices are
The variance of the number of the alleles H_{jk }in the affected sibling and unaffected sibling can be calculated as
Similarly, the covariance between the number of alleles H_{jk }and the number of alleles H_{jl}, l ≠ k, in the affected sibling and unaffected sibling can be calculated as
For j ≠ g, assume that markers H_{j }and H_{g }flank disease locus D in the order of H_{j}DH_{g}. Let P(H_{jk}DH_{gh}) be frequencies of haplotype H_{jk}DH_{gh}. The frequencies of other haplotypes are denoted accordingly. For the ith sibpair, let be the disease genotype of the unaffected sibling and be the disease genotype of the affected sibling. To calculate the covariance between , , denote for j ≠ g, k ≠ k', h ≠ h',
For k = 1,..., n_{j } 1 and h = 1,..., n_{g } 1, j ≠ g, the covariance
Similarly, for k = 1,..., n_{j } 1 and h = 1,..., n_{g } 1, j ≠ g, the covariance
where , , and are the expected genotype frequencies in the normal sibling as follows:
To calculate , , and , one may use the formulae of , , and by substituting f_{st }using .
Appendix B
The conditional covariance
For the 'haplotype/allele coding' method, the expectations and are given by two quantities and (see Appendix to the paper). To get , we will calculate and , l ≠ k in this Appendix. In Appendix C, we will calculate the expectation for j ≠ g. Note that:
Since the siblings can share 2, 1 and 0 genes identical by descent (IBD) at the disease locus D with probabilities 1/4, 1/2 and 1/4, respectively, the expectation
For l ≠ k, one may calculate the expectation
Similarly, one has the following expectation
For l ≠ k, one may calculate the expectation
For l_{1 }≠ l_{2}, l_{1 }≠ k and l_{2 }≠ k, one may calculate the expectation
By using equations (4), (5), (6), (7) and (8), we may calculate in (3). If k ≠ l, then
First, one may calculate the expectation
For n ≠ k, l, one may have the following expectation
For m ≠ k, l, one may have the following expectation
For m ≠ k, l, n ≠ m, k, l, one way have the following expectation:
Using equations (5) (6), (7), (8), (9), (10), (11) and (13), we may calculate terms of equation (7).
Appendix C
Appendix D
For the 'genotype coding' method, the coding vector of the affected sibling in the ith sibpair is , j = 1,..., J. Similarly, j = 1,..., J is the coding vector of the normal sibling in the ith sibpair. Using the expectations and given in equations (1) and (2), one may calculate the following variancecovariance matrices:
The covariances between x_{ijk}, x_{ijkk' }and x_{igh}, x_{ighh}' are given by
Similarly,
Using results of equations (19), (20) and (21), one may calculate and for the 'genotype coding' method.
Appendix E
In this Appendix, we calculate the following covariance matrix for the 'genotype coding' method
The probability P(A_{1}, U_{2}) is given in the Appendix to the manuscript, and the components of expectations and are given in equations (1) and (2). For , we note the following results:
the expectation is given by (4); For l ≠ k, the expectation is given by (5); For l ≠ k, is given by (6); For l ≠ k, is given by (7); For l_{1 }≠ l_{2}, l_{1 }≠ k, l_{2 }≠ k, is given by (8); For l ≠ k, is given by (10); For l ≠ k, n ≠ k, l, is given by (11); For l ≠ k, m ≠ k, l, is given by (12); For l ≠ k, m ≠ k, l, n ≠ m, k, l, is given by (13). In addition, is given by (15); is given by (16); is given by (17); Finally, is given by (18).
Supplementary information: Simulation study
In order to evaluate the accuracy of the noncentrality parameter approximations, we performed simulations for power curves in Figures 1, 2, 3 and 4 of the paper. To do this, we divided the interval (0, 0.065) (or (0, 0.045)) of the LD measure Δ_{11 }of LD uniformly into 20 subintervals for Figures 1 and 2 (or Figures 3 and 4). Correspondingly, the 20 subintervals lead to 21 endpoints. For each endpoint, there is a set of parameters for each power curve. Using the set of parameters, 2,500 datasets are simulated for each endpoint. For each dataset, the empirical statistics T_{H}, T_{G}, T_{H1 }and T_{G1 }were calculated. The simulated power is the proportion of the 2,500 simulated datasets for which the empirical statistic is larger than the cutoff point of the corresponding χ^{2}distribution at a 0.05 significance level.
From Figures 1, 2, 3 and 4, it can be seen that the theoretical power curves of T_{H}, T_{G}, T_{H1 }and T_{G1 }are perfectly close to the simulated power curves. Thus, the noncentrality parameter approximations are very accurate.
Figure 1. The simulated power curves T_{H}, T_{G}, T_{H1 }and T_{G1 }are plotted. The corresponding parameters are the same as those in Figure 1 of the paper. Abbreviation: LD = linkage disequilibrium.
Figure 2. The simulated power curves T_{H}, T_{G}, T_{H1 }and T_{G1 }are plotted. The corresponding parameters are the same as those in Figure 2 of the paper. Abbreviation: LD = linkage disequilibrium.
Figure 3. The simulated power curves T_{H }and T_{G }are plotted. The corresponding parameters are the same as those of Figure 3 in the paper. Abbreviation: LD = linkage disequilibrium.
Figure 4. The simulated power curves T_{H }and T_{G }are plotted. The corresponding parameters are the same as those of Figure 4 of the paper. Abbreviation: LD = linkage disequilibrium.
Acknowledgements
M. Knapp was supported by grant KN 370/11 (Project D1 of FOR 423) from the Deutsche Forschungsgemeinschaft. R. Fan was supported by the National Science Foundation Grant DMS0505025.
References

Botstein D, Risch N: Discovering genotypes underlying human phenotypes: Past successes for Mendelian disease, future approaches for complex disease.
Nat Genet 2003, 33(Suppl):228237. PubMed Abstract  Publisher Full Text

Cordell HJ, Clayton DG: A unified stepwise regression procedure for evaluating the relative effects of polymorphisms within a gene using case/control or family data: Application to HLA in type 1 diabetes.
Am J Hum Genet 2002, 70:124141. PubMed Abstract  Publisher Full Text  PubMed Central Full Text

Rannala B, Reeve JP: Highresolution multipoint linkagedisequilibrium mapping in the context of a human genome sequence.
Am J Hum Genet 2001, 69:159178.
p. 672
PubMed Abstract  Publisher Full Text  PubMed Central Full Text 
Risch N: Implications of multilocus inheritance for genedisease association studies.
Theor Popul Biol 2001, 60:215220. PubMed Abstract  Publisher Full Text

Risch N, Merikangas K: The future of genetic studies of complex human diseases.
Science 1996, 273:15161517. PubMed Abstract  Publisher Full Text

Spielman RS, McGinnis RE, Ewens WJ: Transmission test for linkage disequilibrium: The insulin gene region and insulindependent diabetes mellitus (IDDM).
Am J Hum Genet 1993, 52:506516. PubMed Abstract  PubMed Central Full Text

Chapman NH, Wijsman EM: Genome screens using linkage disequilibrium tests: Optimal marker characteristics and feasibility.
Am J Hum Genet 1998, 63:18721885. PubMed Abstract  Publisher Full Text  PubMed Central Full Text

Olson JM, Wijsman EM: Design and sample size considerations in the detection of linkage disequilibrium with a disease locus.
Am J Hum Genet 1994, 55:574580. PubMed Abstract  PubMed Central Full Text

Kaplan N, Martin ER: Power calculations for a general class of tests of linkage and association that use nuclear families with affected and unaffected sibs.
Theor Popul Biol 2001, 60:193201. PubMed Abstract  Publisher Full Text

Kaplan N, Morris R: Issues concerning association studies for fine mapping a susceptibility gene for a complex disease.
Genet Epidemiol 2001, 20:432457. PubMed Abstract  Publisher Full Text

Nielsen DM, Ehm MG, Weir BS: Detecting markerdisease association by testing for HardyWeinberg disequilibrium at a marker locus.
Am J Hum Genet 1998, 63:15311540. PubMed Abstract  Publisher Full Text  PubMed Central Full Text

Ott J: Analysis of human genetic linkage. 3rd edition. Johns Hopkins University Press, Baltimore and London; 1999.

The International HapMap Consortium: The International HapMap Project.
Nature 2003, 426:789796. PubMed Abstract  Publisher Full Text

The International SNP Map Working Group: A map of human genome sequence variation containing 1.42 million single nucleotide polymorphisms.
Nature 2001, 409:928933. PubMed Abstract  Publisher Full Text

Kong A, Gudbjartsson DF, Sainz J, et al.: A high resolution recombination map of the human genome.
Nat Genet 2002, 31:241247. PubMed Abstract  Publisher Full Text

Hotelling H: The generalization of Student's ratio.
Ann Math Stat 1931, 2:360378. Publisher Full Text

Xiong MM, Zhao J, Boerwinkle E: Generalized T^{2 }test for genome association studies.
Am J Hum Genet 2002, 70:12571268. PubMed Abstract  Publisher Full Text  PubMed Central Full Text

Chapman JM, Cooper JD, Todd J, Clayton D: Detecting disease associations due to linkage disequilibrium using haplotype tags: A class of tests and the determinants of the statistical power.
Hum Hered 2003, 56:1831. PubMed Abstract  Publisher Full Text

Fan RZ, Knapp M: Genome association studies of complex diseases by casecontrol designs.
Am J Hum Genet 2003, 72:850868. PubMed Abstract  Publisher Full Text  PubMed Central Full Text

Fan RZ, Knapp M, Wjst M, et al.: High resolution T^{2 }association tests of complex diseases based on family data.
Ann Hum Genet 2005, 69:187208. PubMed Abstract  Publisher Full Text

Loukola A, Chadha M, Penn SG, et al.: Comprehensive evaluation of the association between prostate cancer and genotype/haplotypes in CYP17A1, CYP3A4, and SRD5A2.
Eur J Hum Genet 2004, 12:321332. PubMed Abstract  Publisher Full Text

Spielman RS, Ewens WJ: A sibship test for linkage in the presence of association: The sib transmission/disequilibrium test.
Am J Hum Genet 1998, 62:450458. PubMed Abstract  Publisher Full Text  PubMed Central Full Text

Anderson TW: An introduction to multivariate statistical analysis. 2nd edition. Wiley, New York; 1984.

Thomson G, Baur MP: Third order linkage disequilibrium.
Tissue Antigens 1984, 24:250255. PubMed Abstract  Publisher Full Text

Ewens WJ, Spielman RS: The transmission/disequilibrium test: History, subdivision, and admixture.
Am J Hum Genet 1995, 57:455464. PubMed Abstract  PubMed Central Full Text

Allen AS, Rathouz PJ, Satten GA: Informative missingness in genetic association studies: Caseparent designs.
Am J Hum Genet 2003, 72:671680. PubMed Abstract  Publisher Full Text  PubMed Central Full Text

Curtis D: Use of siblings as controls in casecontrol association studies.
Ann Hum Genet 1997, 61:319333. PubMed Abstract  Publisher Full Text

Falk CT, Rubinstein P: Haplotype relative risk: An easy reliable way to construct a proper control sample for risk calculations.
Ann Hum Genet 1987, 51:227233. PubMed Abstract  Publisher Full Text

Zhao HY, Zhang SL, Merikangas KR, et al.: Transmission/disequilibrium tests using multiple tightly linked markers.
Am J Hum Genet 2000, 67:936946. PubMed Abstract  Publisher Full Text  PubMed Central Full Text