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Hum Genomics. 2005; 2(2): 90–112.
Published online 2005 June 1. doi:  10.1186/1479-7364-2-2-90
PMCID: PMC3530186

Sibship T2 association tests of complex diseases for tightly linked markers

Abstract

For population case-control association studies, the false-positive 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 late-onset complex diseases, parental data are not usually available. One way to study late-onset disorders is to perform sib-pair or sibship analyses. This paper proposes sibship-based Hotelling's T2 test statistics for high-resolution 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 T2 test statistics are proposed for high-resolution 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 T2 tests take into account not only the correlation among the markers, but also take the correlation within each sib-pair. The validity of the proposed method is justified by rigorous mathematical and statistical proofs under the large sample theory. The non-centrality 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 non-centrality 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 bi-allelic markers, but also for multi-allelic 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 diseases

Introduction

In recent years, there has been great interest in the research of association studies of complex diseases [1-6]. By association studies, we mean linkage disequilibrium (LD) mapping of genetic traits. For population case-control studies, the marker allele frequency in cases can be compared with that of controls using χ2 test statistics [7-11]. If there is association between one marker and the trait locus, it is expected that the χ2 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 high-resolution micro-satellites in the human genome, enormous amounts of genetic data on human chromosomes are becoming available [13-15]. 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 T2 test statistics [16]. For population case-control data, Xiong et al. proposed two sample Hotelling's T2 test statistics to analyse genotype data of multiple bi-allelic markers such as SNPs; [17] in addition, logistic regression models were proposed [2,18]. To analyse the multi-allelic micro-satellite or hap-lotype 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 parent-offspring pairs, paired Hotelling's T2 test statistics were proposed, in order to perform association studies based on multiple markers/haplotypes [20].

For late-onset complex diseases, parental data are usually not available. One way to study late-onset disorders is to perform sib-pair or sibship analyses [21,22]. This paper proposes sibship-based paired Hotelling's T2 test statistics for high-resolution 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 T2 test statistics are proposed for high-resolution association studies, using normal sibs as controls for affected sibs. The paired Hotelling's T2 tests not only take the correlation among the markers into account, but also the correlation within each sib-pair. The validity of the proposed method is justified by rigorous mathematical and statistical proofs under the large sample theory. The non-centrality parameter approximations of the test statistics are calculated for power calculations and comparisons; these are included in the section: Supplementary information: Non-centrality 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 non-centrality 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/.

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 disease-susceptible allele D has population frequency PD, and the normal allele d has population frequency Pd.

Paired Hotelling's T2 test statistics

In the region of the disease locus D, assume that J tightly linked markers H1, ..., HJ 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 Hj by equation M1, where nj denotes the number of its alleles. Here, markers can be micro-satellites or di-allelic markers such as SNPs or haplotypes. If H1, ..., HJ are phase-known 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 phase-unknown 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 phase-unknown genotype data directly. Consider N sib-pairs, each consisting of an affected sibling and a normal sibling. We define coding vectors equation M2 and equation M3 for the affected sibling and normal sibling of the i-th sib-pair, respectively, by one of the following two ways [19,20].

(i) Haplotype/allele coding: For the affected sibling of the i-th sib-pair, let equation M4 be his/her genotype at marker Hj. Define equation M5, where equation M6 is the number of alleles Hjk for the affected sibling of the i-th sib-pair -- that is,

equation M7

Here and hereafter, the superscript τ denotes the transposition of a matrix or a vector. The dimension of equation M8 is equation M9, which is usually smaller than dimension equation M10 of the following genotype coding method.

(ii) Genotype coding: Note that equation M11 can be one of nj(nj + 1)/2 possible choices: nj homozygous genotypes HjkHjk, and nj(nj - 1)/2 heterozygous genotypes HjkHjl, k < l. Depending on the genotype, let us define an indicator vector equation M12. Here, equation M13 is the indicator variable of genotype HjkHjk defined by equation M14; and equation M15, k <l is the indicator variable of genotype HjkHjl defined by equation M16. The dimension of equation M17 is nj(nj + 1)/2 - 1 -- that is, the total number nj(nj + 1)/2 of genotypes of marker Hj minus 1 to remove the redundancy. Let equation M18 be the combined genotype coding of the J markers H1, ... HJ . The dimension of equation M19is equation M20.

For the unaffected sibling of the i-th sib-pair, let equation M21 be his/her genotype at marker Hj. One may define a vector equation M22in the same way, based on either the 'genotype coding' or 'haplotype/allele coding' method. Table Table11 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 sib-pairs at a significance level α = 0.01 using one marker, H1, or two markers, H1 and H2.

Let equation M24 and equation M25 be average coding vectors of affected and unaffected siblings, respectively. Intuitively, equation M26and equation M27 should be similar vectors if the disease locus D is not associated with markers Hj, j = 1, ..., J. In the Appendix we prove that the expected value of equation M28 is 0 if there is no association. Hence, one may build a test statistic based on the difference equation M29 to test the association between disease locus D and markers Hj. To do this, one needs to consider the variance-covariance matrix of equation M30. Since siblings' marker genotypes are related to each other, equation M31 and equation M32 are not independent. Moreover, equation M33 and equation M34 are paired with each other in a sib-pair. Therefore, paired T2 test statistics can be used to test the association between disease locus D and markers Hj as follows. Define a paired-sample variance-covariance matrix by

equation M35

A paired Hotelling's T2 statistic can be defined as equation M36[16,23]. Let us denote the above Hotelling's T2 statistic for 'haplotype/allele coding' as TH, and the Hotelling's T2 statistic for 'genotype coding' as TG. Assume that the sample size N is sufficiently large that the large-sample theory applies. Under the null hypothesis of no association, the statistic TH (or TG) is asymptotically distributed as central χ2 with equation M37equation M38degrees of freedom. Under the alternative hypothesis of association, TH (or TG) is asymptotically distributed as non-central χ2. For power calculation and comparison, the non-centrality parameter of statistic TH or TG 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 T2 test statistics TH and TG above can be generalised as follows. Assume that N sibships are available. In the i-th sibship, assume that ni siblings are affected and mi siblings are normal. Let equation M39and equation M40 be average coding vectors of affected and normal siblings, respectively. To be precise, let equation M41, j = 1, (...), ni be the coding vectors of the affected siblings of the i-th sibship. Then, equation M42; equation M43 is defined, accordingly. Utilising equation M44 to replace equation M45 and equation M46 to replace equation M47 in the above paragraph and defining equation M48 and equation M49, we may define the related Hotelling's T2 test statistics TH and TG.

Non-centrality parameters

The derivation of non-centrality parameters of sib-pairs is provided in the section Supplementary information: Non-centrality parameters.

Results

Type I errors

Tables Tables1,1, ,22 and and33 show type I error rates of test statistics TH and TG at a significance level α = 0.01, using one marker H1 or two markers H1 and H2. Three models are considered. In model I, one marker H1 is used in analysis: H1 is a bi-allelic marker with equal allele frequency P(H11) = P(H12) = 0.50. In model II, two bi-allelic markers H1 and H2 are used in analysis, where P(Hij) = 0.5, i, j = 1, 2, equation M50. In model III, one marker H1 is used in analysis, where H1 is a quadri-allelic marker with allele frequencies P(H21) = P(H22) = 0.35, P(H23) = P(H24) = 0.15.

Table 2
Type I error rates of N = 200 or 300 sibships at a significance level α = 0.01 using one marker, H1, or two markers, H1 and H2.
Table 3
Type I error rates of N = 200 or 300 sibships at a significance level α = 0.01 using one marker, H1, or two markers, H1 and H2.

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 cut-off 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 Tables1,1, ,22 and and3.3. Since the disease locus is not associated with the marker(s), the empirical test statistics which are greater than or equal to the cut-off point at the significance level α = 0.01 are treated as false positives. Thus, the type I error rates of Tables Tables1,1, ,22 and and33 are empirical results.

In Table Table1,1, only sib-pairs are used in the calculations. In each sib-pair, one sibling is affected and the other one is normal. In Table Table2,2, combinations of both sib-pairs and sibships of size 3 are used: the number of sib-pairs 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 Table3,3, combinations of sib-pairs and sibships of sizes 3 and 4 are used: the number of sib-pairs 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 Tables1,1, ,22 and and3,3, it is clear that TH has a lower type I error than TG. 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 TG and TH is 3 and 9, respectively, for model III. Hence, the number of degrees of freedom for test TG 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 Tables1,1, ,22 and and3.3. 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 Table44 gives penetrance probabilities taken from Nielsen et al. or Fan and Knapp [11,19]. For less optimistic models, Table Table55 lists penetrance probabilities taken from Fan and Knapp [19]. For j = 1, ..., J, let us denote the measures of LD between allele Hjk of the marker Hj and the disease locus D by Δjk = P(HjkD) - P(Hjk)PD, k = 1, ..., ni. Here, P(HjkD) is the frequency of haplotype HjkD, and P(Hjk) is the frequency of allele Hjk. For two bi-allelic markers H1 and H2, let equation M51 be the measure of LD between the two markers, where P(H11H21) is the frequency of haplotype H11H21. Assume that the two markers H1 and H2 flank the disease locus D in the order H1DH2. Let equation M52 be the measure of the third order LD [24]. Here, P(H11DH21) is the frequency of haplotype H11DH21.

Table 4
First set of parameters of simulated genetic models.
Table 5
Second set of parameters of simulated genetic models.

Figure Figure11 shows power curves of TH and TG against the measure of LD Δ11 at a significance level α = 0.05 using two bi-allelic marker H1 and H2, when P(Hi1) = P(Hi2) = 0.50, i = 1, 2, PD = 0.15 and N = 200 sib-pairs for the first set of parameters of the four genetic models of Table Table4.4. The power curves of TH1 and TG1 are calculated based on one marker H1. In the graphs, Delta_11 = Δ11; the other parameters are given in the legend of the Figure. Figure Figure22 shows power curves of TH and TG against the measure of LD Δ11 at a significance level α = 0.05 using two bi-allelic marker H1 and H2, when P(Hi1) = P(Hi2) = 0.50, i = 1, 2, PD = 0.15 and N = 600 sib-pairs for the second set of parameters of the four genetic models listed in Table Table5.5. Similarly to Figure Figure1,1, the power curves of TH1 and TG1 are calculated based on one marker H1. The other parameters are the same as those of Figure Figure11.

Figure 1
Power curves of TH and TG at a significance level α = 0.05, using two bi-allelic markers H1 and H2, when P(Hi1) = P(Hi2) = 0.50, i = 1,2, PD = 0.15, and N = 200 sib-pairs for the first set of parameters of the four genetic models of Table 4. The ...
Figure 2
Power curves of TH and TG at a significance level α = 0.05, using two bi-allelic markers H1 and H2, when P(Hi1) = P(Hi2) = 0.50, i = 1,2, PD = 0.15 and N = 600 sib-pairs for the second set of parameters of the four genetic models of Table 5. The ...

From Figures Figures11 and and2,2, it is clear that TH generally has a higher power than that of TG. This is consistent with the results of Fan and Knapp for population case-control 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 TG. The power of TH (or TG) based on two markers H1 and H2 is generally higher than that of TH1 (or TG1), which is only based on one marker H1. 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 TG can increase rapidly as the number of markers increases. This is particularly true when multi-allelic markers are used in analysis; but the number of degrees of freedom of TH only increases by one if one more bi-allelic marker is added to the analysis. Thus, TH has the advantage of high power when multiple markers are used; in addition, the number of degrees of freedom of TH would be not very large. For optimistic models in Table Table4,4, the sample sizes required to achieve certain power levels are lower than those of the less optimistic models in Table Table55.

Not only can the test statistics TH and TG be applied to analyse the genetic data of the bi-allelic markers, but they can also be applied to analyse the genetic data of the multi-allelic markers. Figure Figure33 shows the power curves of TH and TG against the measure of LD Δ11 at a significance level α = 0.05 using a quadri-allelic marker H1, when P(H11) = P(H12) = 0.35, P(H13) = P(H14) = 0.15, PD = 0.15 and N = 200 sib-pairs for the first set of parameters of the four genetic models of Table Table4.4. The other parameters are given in the legend of the Figure. Figure Figure44 shows power curves of TH and TG at a significance level α = 0.05 using a quadri-allelic marker H1, when P(H11) = P(H12) = 0.35, P(H13) = P(H14) = 0.15, PD = 0.15 and N = 600 sib-pairs for the second set of parameters of the four genetic models of Table Table5.5. Similarly to Figures Figures11 and and2,2, TH generally has a higher power than that of TG.

Figure 3
Power curves of TH and TG at a significance level α = 0.05 using a quadric-allelic marker H1, when P(H11) = P(H12) = 0.35, P(H13) = P(H14) = 0.15 PD = 0.15 and N = 200 sib-pairs for the first set of parameters of the four genetic models of Table ...
Figure 4
Power curves of TH and TG at a significance level α = 0.05 using a quadric-allelic marker H1, when P(H11) = P(H12) = 0.35, P(H13) = P(H14) = 0.15, PD = 0.15 and N = 600 sib-pairs for the second set of parameters of the four genetic models of Table ...

In addition to the power curves of TH and TG, which are based on sib-pair data, Figures Figures33 and and44 show the simulated power curves of STH and STG, which are based on sibships of varying structures. In Figure Figure3,3, combinations of both sib-pairs and sibships of size 3 are used to calculate the simulated power curves of STH and STG: the number of sib-pairs 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 Figure4,4, combinations of sib-pairs and sibships of sizes 3 and 4 are used to calculate the simulated power curves of STH and STG: the number of sib-pairs 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 STH and STG, the interval (0, 0.045) of the LD measure Δ11 of LD is uniformly divided into 20 subintervals in Figures Figures33 and and4.4. 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 TH and TG were calculated. The simulated power is the proportion of the 2,500 simulated datasets for which the empirical statistic is larger than the cut-off point of the corresponding χ2-distribution at a 0.05 significance level.

From Figures Figures33 and and4,4, it can be seen that the simulated power STH is generally higher than the power of TH, and the simulated power STG is generally higher than the power of TG. Intuitively, sibships of large size contain more information than that of a sib-pair. The test statistics TH and TG can accurately capture the information contained in sibships of large size. Moreover, it can also be seen in Tables Tables1,1, ,22 and and33 that the type I error is not inflated by including sibships of varying structure.

Simulation study

To evaluate the accuracy of the non-centrality parameter approximations, we performed simulations for the power curves in Figures Figures1,1, ,2,2, ,33 and and4.4. 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 sibship-based Hotelling's T2 test statistics for high-resolution association mapping of complex diseases. This extends our previous research of paired Hotelling's T2 test statistics of nuclear family data or parent-offspring pairs [20]. For late-onset complex diseases, parental data are usually not available. This motivated us to perform sib-pair or sibship analyses to study late-onset disorders. Based an two coding methods--'haplotype/allele coding' and 'genotype coding'--paired Hotelling's T2 test statistics TH and TG are proposed for high-resolution 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 bi-allelic or multi-allelic. 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 TH based on the 'haplotype/allele coding' method is generally more powerful than the test statistic TG based on the 'genotype coding' method. This is most likely due to the large number of degrees of freedom of TG. Moreover, the type I error rates of the test statistic TH are lower than those of test statistic TG.

For population case-control association studies, false-positive 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 [22,26-29]. 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 T2 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 sib-pair needs to be taken into account. The paired Hotelling's T2 test statistics TH and TG developed in this paper correctly take both the correlation among the markers and the correlation within each sib-pair into account. The proposed method is potentially useful in association mapping of late-onset complex diseases.

Cordell and Clayton [2] and Chapman et al. [18] proposed logistic regression models for population-based case control studies or family studies. Both our proposed method and the logistic regression models can be used in association studies of multi-locus 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 T2 test statistics. While we are able to calculate the non-centrality parameters for our T2 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 multi-allelic markers data. We feel that more investigations are necessary in order for multi-allelic markers data to be used in the logistic regression models. By contrast, our proposed T2 can be used to analyse either bi-allelic or multi-allelic marker data, or both simultaneously. Moreover, more investigations are needed to make power comparisons of the two methods.

In Figures Figures33 and and4,4, we show that the power of test statistics TH and TG based on combinations of sibships of varying structures are generally higher than the power of the test statistics based on sib-pairs. This is because the test statistics TH and TG 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 equation M56 and equation M57, but it will lead to a variance-covariance matrix S, which increases the test statistics. Moreover, it can be seen from Tables Tables1,1, ,22 and and33 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 in-depth investigation.

Appendix

Consider a sib-pair 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 A1 = (the first sibling is affected), U2 = (the second sibling is unaffected). Let fDD, fDd = fdD and fdd 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 fDD fDd fdd. Let equation M58, equation M59 and equation M60. Denote the disease prevalence in population by equation M61, and equation M62. 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 sib-pair share i gene identical by descent (IBD) at the disease locus D. Then the joint probability

equation M63
(1)

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 HjkHjk in an affected sibling is given by:

equation M64
(2)

Similarly, the frequency of homozygous genotype HjkHjk in an unaffected sibling is given by:

equation M65
(3)

Note that equation M66 can be calculated by the formula for ajkk by substituting fst with equation M67 and vice versa. Note that the haplotype frequencies P(HjkD) = Δjk + P(Hjk)PD, P(Hjkd) = -Δjk + P(Hjk)Pd. Under the null hypothesis of no association between the markers Hi, i = 1, 2, ..., J, and the disease locus D -- that is, Δij = 0 for all j, the haplotype frequencies are equal to the product of allele frequencies; for example, P(HjkD) = P(Hjk)PD and P(Hjkd) = P(Hjk)Pd. From equations (4) and (5), equation M68.

Similarly, the frequency of the heterozygous genotype HjkHjl, k l, in an affected sibling can be calculated as follows:

equation M69
(4)

The frequency of the heterozygous genotype HjkHjl, k l, in an unaffected sibling can be calculated as follows:

equation M70
(5)

Note that equation M71 can be calculated by the formula for ajkl by substituting fst using equation M72 and vice versa. Under the null hypothesis of no association between the markers Hi, i = 1, 2, ..., J, and the disease locus D -- that is, Δij = 0 for all j, the haplotype frequencies are equal to the product of the allele frequencies; for example, P(HjkD) = P(Hjk)PD, P(Hjkd) = P(Hjk)Pd, P(HjlD) = P(Hjl)PD and P(Hjld) = P(Hjl)Pd. From equations (4) and (5), equation M73. Therefore, the expectation equation M74 for the 'genotype coding' method.

For the 'haplotype/allele coding' method, equations (2), (3), (4) and (5) imply

equation M75
(6)

From equation (6), expectation equation M76 by 'haplotype/allele coding' method, under the null hypothesis of no association between the markers Hi, j = 1, ..., J and disease locus D.

Supplementary information: Non-centrality parameters

Consider N sib-pairs, 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 sib-pair. Let us denote A1 = (the first sibling is affected), U2 = (the second sibling is unaffected). For 'haplotype/allele coding', the coding vector of the affected sibling in the i-th sib-pair is equation M77. Similarly, equation M78 is the coding vector of the normal sibling. Denote the variance-covariance matrix of equation M79 by equation M80. The elements of the above variance-covariance matrices are given in Appendices A, B, and C: equation M81 and equation M82 in Appendix A, and equation M83 in Appendices B and C. Using quantities of equation M84 and equation M85 in the Appendix to the manuscript, equation M86 can be calculated. The non-centrality parameter λH of Hotelling's statistics TH is given by equation M87.

For the 'genotype coding' method, the coding vector of the affected sibling in the i-th sib-pair is equation M88j = 1, ..., J. Similarly, equation M89 is the coding vector of the normal sibling. Let ajkl and equation M90 be the frequencies of genotype HjkHjl in affected and unaffected siblings given in the Appendix to the manuscript. Then,

equation M91
(-1)
equation M92
(-2)

Using equation M93 and equation M94, one may calculate the expectation equation M95. Let equation M96 be the variance-covariance matrix of equation M97. Then the non-centrality parameter λG of Hotelling's statistics TG is given by equation M98. The elements of the above variance-covariance matrices are given in Appendices D and E: equation M99 and equation M100 in Appendix D, and equation M101 in Appendix E.

Appendix A

Consider the 'haplotype/allele coding' method. The variance-covariance matrices are

equation M102
equation M103

The variance of the number of the alleles Hjk in the affected sibling and unaffected sibling can be calculated as

equation M104
equation M105

Similarly, the covariance between the number of alleles Hjk and the number of alleles Hjl, l k, in the affected sibling and unaffected sibling can be calculated as

equation M106
equation M107

For j g, assume that markers Hj and Hg flank disease locus D in the order of HjDHg. Let P(HjkDHgh) be frequencies of haplotype HjkDHgh. The frequencies of other haplotypes are denoted accordingly. For the i-th sib-pair, let equation M108 be the disease genotype of the unaffected sibling and equation M109 be the disease genotype of the affected sibling. To calculate the covariance between equation M110, equation M111, denote for j g, k k', h h',

equation M112
equation M113
equation M114
equation M115

For k = 1,..., nj - 1 and h = 1,..., ng - 1, j g, the covariance

equation M116

Similarly, for k = 1,..., nj - 1 and h = 1,..., ng - 1, j g, the covariance

equation M117

where equation M118, equation M119, equation M120 and equation M121 are the expected genotype frequencies in the normal sibling as follows:

equation M122

To calculate equation M123, equation M124, equation M125 and equation M126, one may use the formulae of equation M127, equation M128, equation M129 and equation M130 by substituting fst using equation M131.

Appendix B

The conditional covariance

equation M132

For the 'haplotype/allele coding' method, the expectations equation M133 and equation M134 are given by two quantities equation M135 and equation M136 (see Appendix to the paper). To get equation M137, we will calculate equation M138 and equation M139, l k in this Appendix. In Appendix C, we will calculate the expectation equation M140 for j g. Note that:

equation M141
(-3)

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

equation M142
(-4)

For l k, one may calculate the expectation

equation M143
(-5)

Similarly, one has the following expectation

equation M144
(-6)

For l k, one may calculate the expectation

equation M145
(7)

For l1 l2, l1 k and l2 k, one may calculate the expectation

equation M146
(8)

By using equations (4), (5), (6), (7) and (8), we may calculate equation M147in (3). If k ≠ l, then

equation M148
(9)

First, one may calculate the expectation

equation M149
(10)

For n k, l, one may have the following expectation

equation M150
(11)

For m k, l, one may have the following expectation

equation M151
(12)

For m k, l, n m, k, l, one way have the following expectation:

equation M152
(13)

Using equations (5) (6), (7), (8), (9), (10), (11) and (13), we may calculate terms of equation (7).

Appendix C

For j g, the expectation

equation M153
(14)

Suppose that blocks/markers Hj and Hg flank disease locus D in the order HjDHg. The expectation

equation M154
(15)

If h' h, the expectation

equation M155
(16)

If k k', the expectation

equation M156
(17)

If k k', h ≠ h' the expectation

equation M157
(18)

Appendix D

For the 'genotype coding' method, the coding vector of the affected sibling in the i-th sib-pair is equation M158, j = 1,..., J. Similarly, equation M159j = 1,..., J is the coding vector of the normal sibling in the i-th sib-pair. Using the expectations equation M160 and equation M161 given in equations (1) and (2), one may calculate the following variance-covariance matrices:

equation M162
(19)

The covariances between xijk, xijkk' and xigh, xighh' are given by

equation M163
(20)

Similarly,

equation M164
(21)

Using results of equations (19), (20) and (21), one may calculate equation M165 and equation M166 for the 'genotype coding' method.

Appendix E

In this Appendix, we calculate the following covariance matrix for the 'genotype coding' method

equation M167

The probability P(A1, U2) is given in the Appendix to the manuscript, and the components of expectations equation M168 and equation M169 are given in equations (1) and (2). For equation M170, we note the following results:

the expectation equation M171 is given by (4); For l k, the expectation equation M172 is given by (5); For l k, equation M173 is given by (6); For l k, equation M174 is given by (7); For l1 l2, l1 k, l2 k, equation M175 is given by (8); For l k, equation M176 is given by (10); For l k, n k, l, equation M177 is given by (11); For l k, m k, l, equation M178 is given by (12); For l k, m k, l, n m, k, l, equation M179 is given by (13). In addition, equation M180 is given by (15); equation M181 is given by (16); equation M182 is given by (17); Finally, equation M183 is given by (18).

Supplementary information: Simulation study

In order to evaluate the accuracy of the non-centrality parameter approximations, we performed simulations for power curves in Figures Figures1,1, ,2,2, ,33 and and44 of the paper. To do this, we divided the interval (0, 0.065) (or (0, 0.045)) of the LD measure Δ11 of LD uniformly into 20 subintervals for Figures Figures11 and and22 (or Figures Figures33 and and4).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 TH, TG, TH1 and TG1 were calculated. The simulated power is the proportion of the 2,500 simulated datasets for which the empirical statistic is larger than the cut-off point of the corresponding χ2-distribution at a 0.05 significance level.

From Figures Figures1,1, ,2,2, ,33 and and4,4, it can be seen that the theoretical power curves of TH, TG, TH1 and TG1 are perfectly close to the simulated power curves. Thus, the non-centrality parameter approximations are very accurate.

Figure 1. The simulated power curves TH, TG, TH1 and TG1 are plotted. The corresponding parameters are the same as those in Figure Figure11 of the paper. Abbreviation: LD = linkage disequilibrium.

Figure 2. The simulated power curves TH, TG, TH1 and TG1 are plotted. The corresponding parameters are the same as those in Figure Figure22 of the paper. Abbreviation: LD = linkage disequilibrium.

Figure 3. The simulated power curves TH and TG are plotted. The corresponding parameters are the same as those of Figure Figure33 in the paper. Abbreviation: LD = linkage disequilibrium.

Figure 4. The simulated power curves TH and TG are plotted. The corresponding parameters are the same as those of Figure Figure44 of the paper. Abbreviation: LD = linkage disequilibrium.

Acknowledgements

M. Knapp was supported by grant KN 370/1-1 (Project D1 of FOR 423) from the Deutsche Forschungsgemeinschaft. R. Fan was supported by the National Science Foundation Grant DMS-0505025.

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