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**|**Hum Genomics**|**v.2(2); 2005**|**PMC3530186

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- Abstract
- Introduction
- Methods
- Results
- Discussion
- Appendix
- Supplementary information: Non-centrality parameters
- Appendix A
- Appendix B
- Appendix C
- Appendix D
- Appendix E
- Supplementary information: Simulation study
- References

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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

Ruzong Fan: rfan/at/stat.tamu.edu

Received 2005 April 28; Accepted 2005 April 28.

Copyright ©2005 Henry Stewart Publications

This article has been cited by other articles in PMC.

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 *T*^{2 }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 *T*^{2 }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 *T*^{2 }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.

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 *χ*^{2 }test statistics [7-11]. 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 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 *T*^{2 }test statistics [16]. For population case-control data, Xiong *et al*. proposed two sample Hotelling's *T*^{2 }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 *T*^{2 }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 *T*^{2 }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 *T*^{2 }test statistics are proposed for high-resolution 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 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/.

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 *P _{D}*, and the normal allele

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

(i) *Haplotype/allele coding*: For the affected sibling of the *i*-th sib-pair, let be his/her genotype at marker *H _{j}*. Define , where is the number of alleles

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}*(

For the unaffected sibling of the *i*-th sib-pair, 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 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.

Type I error rates of *N *= 200 or 300 sib-pairs 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}*,

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

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

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

Tables Tables1,1, ,22 and and33 show type I error rates of test statistics *T _{H }*and

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}.

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 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 3*N*/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 *T _{H }*has a lower type I error than

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.

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 *H _{jk }*of the marker

Figure Figure11 shows power curves of *T _{H }*and

From Figures Figures11 and and2,2, it is clear that *T _{H }*generally has a higher power than that of

Not only can the test statistics *T _{H }*and

In addition to the power curves of *T _{H }*and

To calculate the simulated power curves *ST _{H }*and

From Figures Figures33 and and4,4, it can be seen that the simulated power *ST _{H }*is generally higher than the power of

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.

The goal of this study was to develop sibship-based Hotelling's *T*^{2 }test statistics for high-resolution association mapping of complex diseases. This extends our previous research of paired Hotelling's *T*^{2 }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 *T*^{2 }test statistics *T _{H }*and

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 *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 sib-pair needs to be taken into account. The paired Hotelling's *T*^{2 }test statistics *T _{H }*and

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 *T*^{2 }test statistics. While we are able to calculate the non-centrality 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 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 *T*^{2 }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 *T _{H }*and

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 *A*_{1 }= *(the first sibling is affected)*, *U*_{2 }= *(the second sibling is unaffected)*. Let *f _{DD}*,

(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 *H _{jk}H_{jk }*in an affected sibling is given by:

(2)

Similarly, the frequency of homozygous genotype *H _{jk}H_{jk }*in an unaffected sibling is given by:

(3)

Note that can be calculated by the formula for *a _{jkk }*by substituting

Similarly, the frequency of the heterozygous genotype *H _{jk}H_{jl}*,

(4)

The frequency of the heterozygous genotype *H _{jk}H_{jl}*,

(5)

Note that can be calculated by the formula for *a _{jkl }*by substituting

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

(6)

From equation (6), expectation by 'haplotype/allele coding' method, under the null hypothesis of no association between the markers *H _{i}*,

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 *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 *i*-th sib-pair is . Similarly, is the coding vector of the normal sibling. Denote the variance-covariance matrix of by . The elements of the above variance-covariance 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 non-centrality parameter *λ _{H }*of Hotelling's statistics

For the 'genotype coding' method, the coding vector of the affected sibling in the *i*-th sib-pair is *j *= 1, ..., *J*. Similarly, is the coding vector of the normal sibling. Let *a _{jkl }*and be the frequencies of genotype

(-1)

(-2)

Using and , one may calculate the expectation . Let be the variance-covariance matrix of . Then the non-centrality parameter *λ _{G }*of Hotelling's statistics

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

For *j *≠ *g*, assume that markers *H _{j }*and

For *k *= 1,..., *n _{j }*- 1 and

Similarly, for *k *= 1,..., *n _{j }*- 1 and

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 .

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:

(-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

(-4)

For *l *≠ *k*, one may calculate the expectation

(-5)

Similarly, one has the following expectation

(-6)

For *l *≠ *k*, one may calculate the expectation

(7)

For *l*_{1 }≠ *l*_{2}, *l*_{1 }≠ *k *and *l*_{2 }≠ *k*, one may calculate the expectation

(8)

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

(9)

First, one may calculate the expectation

(10)

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

(11)

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

(12)

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

(13)

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

For *j *≠ *g*, the expectation

(14)

Suppose that blocks/markers *H _{j }*and

(15)

If *h' *≠ *h*, the expectation

(16)

If *k *≠ *k'*, the expectation

(17)

If *k *≠ *k'*, *h ≠ **h' *the expectation

(18)

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

(19)

The covariances between *x _{ijk}*,

(20)

Similarly,

(21)

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

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).

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 **Δ**_{11 }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 *T _{H}*,

From Figures Figures1,1, ,2,2, ,33 and and4,4, it can be seen that the theoretical power curves of *T _{H}*,

**Figure 1**. The simulated power curves *T _{H}*,

**Figure 2**. The simulated power curves *T _{H}*,

**Figure 3**. The simulated power curves *T _{H }*and

**Figure 4**. The simulated power curves *T _{H }*and

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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