TDT statistics for mapping quantitative trait loci

The original transmission disequilibrium test (TDT), was introduced to test for linkage between a marker and a disease-susceptibility locus (Spielman et al . 1993). Allison (1997) extended the TDT procedure to quantitative traits. Allison's test, however, is restrictive in that it requires family trios consisting of one heterozygous parent, one homozygous parent and one child, and considers only the situation where the marker locus is analogous to the quantitative trait locus itself. In this paper, we propose, investigate and apply a general TDT for quantitative traits that permits more than one child per family, does not require only one parent to be heterozygous, and allows for the fact that the various alleles at the marker and trait loci may be at varying degree of linkage disequilibrium. We also show that this TDT for quantitative traits is still a valid test of linkage in the presence of population substructure. To provide guidelines for study design, we develop analytic formulae for calculation of the power of the TDT for mapping quantitative trait loci and investigate the impact of various factors on the power. Power calculations show that the proposed TDT for quantitative traits is more powerful than Allison's basic test statistic and the extreme discordant sib pair linkage method. The proposed TDT statistic for quantitative traits is applied to systolic blood pressure variation in the Rochester Family Heart Study using an extremely discordant sibling pair design.     

Simple approximations for the maximal transmission/disequilibrium test with a multi-allelic marker

Spielman et al . (1993) popularized the transmission/disequilibrium test (TDT) to test for linkage between disease and marker loci that show a population association. Several authors have proposed extensions to the TDT for multi-allelic markers. Many of these approaches exhibit a 'swamping' effect in which a marker with a strong effect is not detected by a global test that includes many markers with no effect. To avoid this effect, Schaid (1996) proposed using the maximum of the bi-allelic TDT statistics computed for each allele versus all others combined. The maximal TDT statistic, however, no longer follows a chi-square distribution. Here, a refinement to Bonferroni's correction for multiple testing provided by Worsley (1982) based on maximal spanning trees is applied to calculate accurate upper bounds for the type I error and p -values for the maximal TDT. In addition, an accurate lower Bonferroni bound is applied to calculate power. This approach does not require any simulation-based analysis and is less conservative than the standard Bonferroni correction. The bounds are given for both the exact probability calculations and for those based on the normal approximation. The results are assessed through simulations.     

A CORRECTION TO TDT STATISTICS FOR MAPPING QUANTITATIVE TRAIT LOCI

Xiong et al. (1998) proposed a general transmission disequilibrium test (TDT) for quantitative traits. Their TDT for quantitative traits permits more than one child and more than one heterozygous parent per family. This TDT is also appropriate for the situation where the trait and marker loci are at varying degrees of linkage and linkage disequilibrium. Their generalized TDT statistic for a multi-allele marker locus may be more useful in practice. However, there are several mistakes in their mathematical presentation for this generalized TDT statistic for a multi-allele marker locus, and it confuses readers. Here, we present a corrected TDT statistic for a multi-allele marker locus with a clearer presentation. We hope that this new presentation will remove some confusion for readers.     

An extended transmission/disequilibrium test (TDT) for multi-allele marker loci

The transmission/disequilibrium test (TDT) was recently introduced by Spielman et al. (1993) as a test for linkage and linkage disequilibrium. The test is based on the unequal probability of transmission of two different marker alleles from parents to affected offspring, when the marker locus and the hypothetical disease locus are linked and are in linkage disequilibrium. The probabilities of marker allele transmission to affected offspring conditional on parental genotype have been derived by Ott (1989) for a biallelic marker and a recessive disorder with no phenocopies. Here, we derive the transmission probabilities for a multi-allele marker locus and a generalized single locus disease model in a random sample of affected individuals from a randomly mating population. The form of these transmission probabilities suggests an extension of the TDT to multi-allele marker loci, in which the alternative hypothesis is restricted to take account of the likely pattern of unequal transmission when the recombination fraction is near 0. We show how our extended TDT can be implemented by standard software for logistic regression, although we have also written our own program which is available on request. We have evaluated the approximate power of the test under a range of realistic assumptions, and it appears that the test will often have good power when linkage disequilibrium is strong and if the disease is recessive.     

Hidden linkage: a comparison of the affected sib pair (ASP) test and transmission/disequilibrium test (TDT)

I compare the transmission/disequilibrium test (TDT) and affected sib pair (ASP) test under a general algebraic model describing a bi-allelic disease locus. Assuming linkage to a bi-allelic marker, I derive two binomial probabilities, one for parental allele 'transmission' (Pt ) which determines the magnitude of the TDT χ² statistic (χ²_tdt), and a second for identity-by-descent (ibd) marker allele 'sharing' ( P _s) which determines the magnitude of the ASP test statistic (χ²_asp). I also consider the ASP test applied to a completely polymorphic marker and demonstrate that the probability of ASP marker allele sharing ( P _s) is identical to P _s observed for a bi-allelic marker in equilibrium with the disease locus. I present a general framework for determining the power of the TDT and ASP test based on expressions for P _t, P _s and the proportion ( H / F ) of ascertained parents who are informative at the marker. Two previous analytic investigations of TDT power based on the work of Ott (1989), and Risch & Merikangas (1996) are shown to be special cases of this general framework. In addition, I show the relationship between the framework I present and a third analytic investigation of TDT power for multi-allelic markers based on the work of Sham & Curtis (1995).     

Use of siblings as controls in case-control association studies

The transmission disequilibrium test (TDT) is a simple means to detect association which should only be positive if the marker allele is linked to the disease locus, but it cannot be used if parents of affected subjects are unavailable, as can occur when the disease has a late age of onset. Although one can sometimes deduce parental genotypes using the siblings of affected cases, reliance on this procedure can introduce bias and may also result in discarding many families which could provide useful information. Instead, it is shown that the use of unaffected siblings as controls is, like the TDT, robust against bias due to population stratification and other sources, and is expected only to produce positive results when a marker is both associated and linked with the disease locus. The method can have much less power than a case-control study using unrelated controls, but this can be guarded against by seeking unaffected siblings which are genotypically distinct from cases and by focusing only on alleles which are different within pairs of cases and controls. This yields a pair-wise test for association which can be used for multiallelic markers in a manner exactly analogous to the extended TDT (ETDT). The use of siblings as controls is simple, robust, practical and worthy of further consideration.     

On extending the transmission/disequilibrium test (TDT)

The transmission/disequilibrium test (TDT), for evaluation of the null hypothesis of neither linkage nor association between a marker locus and disease, is extended to the more general situation of transmission of two multi-allele marker loci from parents to affected offspring. Transmission probabilities are derived for a generalized single locus disease model, where the disease locus is taken to lie between the two marker loci. There could be unlinked modifier loci for the disease. Examples of the extended TDT are given and it is shown how the contribution from each locus can be evaluated, both separately and jointly.     

Linkage and family-based association study of schizophrenia and the synapsin III locus that maps to chromosome 22q13

The human synapsin III gene (synapsin III) is a member of a neuron-specific phosphoprotein gene family involved in short-term neurotransmitter release. We mapped synapsin III to chromosomal region 22q13 (13.1-13.31) by fluorescence in situ hybridization, a region that has been identified as a potential schizophrenia susceptibility locus. The dinucleotide repeat marker D22S280 located in intron 5 of synapsin III was genotyped in a linkage and family-based association study to assess the role of the synapsin III locus in the etiology of schizophrenia. In 12 pedigrees with periodic catatonia comprising 135 individuals, we found exclusion of linkage of marker D22S280 using lod score analysis with autosomal dominant/recessive models as well as affected only LOD score methods with dominant/recessive models. In a family-based association study of 61 unrelated parent-offspring trios with schizophrenia (according to the the Diagnostic and Statistical Manual of Mental Disorders, fourth edition [DSM-IV, American Psychiatric Association, 1994]), we found no association of individual D22S280 alleles to disease. Results of a multiallelic transmission/disequilibrium test (TDT(max) = 3.00; P = 0.55) challenged the possibility that D22S280 alleles appear with DSM-IV schizophrenia more frequently than expected. In addition, no evidence for gender differences or parent-of-origin effects were found. Thus, the synapsin III locus at chromosome 22q13 is not likely to contain a schizophrenia susceptibility gene.     

Combining the case-control methodology with the small size transmission/disequilibrium test for multiallelic markers

Case-control studies compare marker-allele distributions in affected and unaffected individuals, and significant results may be due to linkage but can also simply reflect population structure. To test for linkage after obtaining a significant case-control finding, within-family analysis can be performed. In a transmission/disequilibrium test (TDT), genotypes of cases are compared to those of their parents to explore whether a specific allele, or marker, at a locus of interest is transmitted to a greater degree than Mendelian inheritance would warrant. For multiallelic markers, several authors have proposed extensions to the TDT. In this article, we propose a TDT test, utilizing the available information of a case-control study in the grouping of alleles for multiallelic markers, and thereby increase the statistical power of a TDT test with a small sample size.     

No evidence for transmission disequilibrium between a new marker at the myelin basic protein locus and multiple sclerosis in French patients

The myelin basic protein (MBP) gene is a candidate locus for susceptibility to multiple sclerosis. Several groups have tested a complex (TGGA)n repeat in the 5' region of this gene for association/linkage with multiple sclerosis, with divergent results. This region of tandem repetitive sequence has been subjected to complex rearrangements, and there is a possibility that alleles of the same size have different internal structures, which reduces the interest of this marker for linkage disequilibrium studies and may at least partly explain the conflicting results obtained so far. To overcome this problem, we isolated a new polymorphic (CA)n repeat within the Golli-MBP locus. The limited number of alleles identified makes this other marker suitable for transmission disequilibrium studies. We tested this marker for linkage with multiple sclerosis, using the transmission-disequilibrium test (TDT) on a sample of 196 nuclear families in which the genotypes of both parents could be unambiguously defined. We found no evidence of transmission disequilibrium between multiple sclerosis and any of the three alleles of this marker, even when the patients were subdivided according to their HLA-DRB1*1501 status. The present data thus provide no evidence for a contribution of the MBP gene to multiple sclerosis susceptibility in French patients.     

Evaluating linkage and linkage disequilibrium: use of excess sharing and transmission disequilibrium methods in affected sib pairs

Two popular and robust approaches to analysing affected sib pair (ASP) data for linkage are the traditional excess sharing methods and the transmission/disequilibrium test (TDT). Here we derive an overall test of linkage for multi-allelic ASP marker data which comprises two component tests: one for excess sharing and one for transmission disequilibrium. This method has several advantages. Firstly the overall test of linkage is often more powerful than either of the two component tests. Secondly the method makes it possible to determine the contribution of linkage disequilibrium (LD), in addition to linkage, to an overall positive linkage result. This is useful because the presence of LD in addition to linkage may suggest that the marker locus is in very close proximity to a disease susceptibility gene. Thirdly the method provides estimates of the risk associated with transmission of the different marker alleles.     

Two-sample Comparison Based on Prediction Error, with Applications to Candidate Gene Association Studies

To take advantage of the increasingly available high-density SNP maps across the genome, various tests that compare multilocus genotypes or estimated haplotypes between cases and controls have been developed for candidate gene association studies. Here we view this two-sample testing problem from the perspective of supervised machine learning and propose a new association test. The approach adopts the flexible and easy-to-understand classification tree model as the learning machine, and uses the estimated prediction error of the resulting prediction rule as the test statistic. This procedure not only provides an association test but also generates a prediction rule that can be useful in understanding the mechanisms underlying complex disease. Under the set-up of a haplotype-based transmission/disequilibrium test (TDT) type of analysis, we find through simulation studies that the proposed procedure has the correct type I error rates and is robust to population stratification. The power of the proposed procedure is sensitive to the chosen prediction error estimator. Among commonly used prediction error estimators, the .632+ estimator results in a test that has the best overall performance. We also find that the test using the .632+ estimator is more powerful than the standard single-point TDT analysis, the Pearson's goodness-of-fit test based on estimated haplotype frequencies, and two haplotype-based global tests implemented in the genetic analysis package FBAT. To illustrate the application of the proposed method in population-based association studies, we use the procedure to study the association between non-Hodgkin lymphoma and the IL10 gene.     

Evidence by allelic association-dependent methods for a type 1 diabetes polygene (IDDM6) on chromosome 18q21

Type 1 diabetes is a common polygenic disease. Fine mapping of polygenes by affected sibpair linkage analysis is not practical and allelic association or linkage disequilibrium mapping will have to be employed to attempt to detect founder chromosomes. Given prior evidence of linkage of the Jk-D18S64 region of chromosome 18q12-q21 to type 1 diabetes, we evaluated the 12 informative microsatellite markers in the region for linkage with disease by the transmission disequilibrium test (TDT) in a UK data set of type 1 diabetic families (n = 195). Increased transmission of allele 4 of marker D18S487 to affected children was detected (P = 0.02). Support for this was extended in a total of 1067 families from four different countries by isolating, and evaluating by the TDT, two novel microsatellites within 70 kb of D18S487. Evidence for linkage and association was P = 5 x 10(-5) and 3 x 10(-4), respectively. There was no evidence for increased transmission of associated alleles to nonaffected siblings. Analysis of an additional 390 families by the TDT did not extend the evidence further, and reduced support in the total 1457 families to P = 0.001 for linkage and P = 0.003 for association. However, evidence for linkage by affected sibpair allele sharing was strong (P = 3.2 x 10(-5)) in the second data set. Heterogeneity in TDT results between data sets was, in part, accounted for by the presence of more than one common disease- associated haplotype (allelic heterogeneity) which confounds the analysis of individual alleles by the TDT. Guidelines for strategies for the mapping of polygenes are suggested with the emphasis on collections of large numbers of families from multiple populations that should be as genetically homogeneous as possible.      

Transmission/disequilibrium tests for quantitative traits

The transmission/disequilibrium test (TDT) is a powerful method of locating disease genes. The TDT was originally proposed for use in studies of qualitative traits in families with both parents available. Recently, the TDT has been extended to studies of qualitative traits in sibships without parents available and in families with one parent available. It has also been extended for use in studies of quantitative traits in families with both parents available and in sibships with multiple offspring. In this paper, we first propose a new class of TDT-type tests for linkage in the presence of linkage disequilibrium for use in studies of families with both parents available. The TDT of Spielman et al . (1993) for qualitative traits and the TDT of Rabinowitz (1997) for quantitative traits are special cases of the new tests. Second, we propose a new class of TDT-type tests for linkage for use in studies of families with one parent available. Third, we study the validity and the power of the tests using simulations. Finally, we propose a method of combining data from different types of families. The combined test is valuable and allows researchers full use of the available data in detecting linkage between a marker locus and an unobservable quantitative trait locus. An important feature of the tests proposed in this paper is that no assumptions on the distribution of the quantitative traits are needed.     

SNPs and snails and puppy dogs' tails: analysis of SNP haplotype data using the gamete competition model

The gamete competition model is a likelihood version of the transmission disequilibrium test (TDT) that is inspired by conditional logistic regression and the Bradley–Terry ranking procedure. In family-based association studies, both the TDT and the gamete competition model apply directly to data on a single nucleotide polymorphism (SNP). Because any given SNP has limited polymorphism, it is tempting to collect several SNPs within a gene into a single super marker whose alleles are haplotypes. Unfortunately, this tactic wreaks havoc with the traditional TDT, which requires codominant markers (Spielman et al . 1993; Terwilliger & Ott, 1992). Eliminating phase ambiguities by assigning haplotypes to individuals before conducting the TDT may give misleading results because only the most probable haplotypes are then considered. Because pedigree implementations of the gamete competition model can accommodate dominant as well as codominant markers, they circumvent the phase problem by including all possible phases weighted by their estimated frequencies.     

A logistic regression based extension of the TDT for continuous and categorical traits

The transmission disequilibrium test (TDT), designed as a test of linkage in the presence of association (i.e. linkage disequilibrium), has received considerable attention in the recent statistical genetics literature due to its advantages over other within-family analytic methods. One limitation of the conventional TDT is its application solely to linkage disequilibrium between a genetic marker and a single categorical trait (e.g. presence or absence of a disease). In this paper, we present an extension of the TDT using logistic regression to examine the relation between a candidate gene or genetic marker and one or more continuous or categorical explanatory variables. This logistic regression extension of the TDT possesses all of the desirable features of the conventional TDT, as well as many advantages associated with traditional regression analysis. We describe the model and its properties, as well as a number of its possible applications, and apply it to examine linkage disequilibrium between the dopamine receptor D2 gene (DRD2) and symptoms of childhood attention deficit hyperactivity disorder (ADHD). We also briefly compare the logistic regression TDT to other quantitative TDTs that have been proposed in the literature, and highlight the advantages of a regression-based approach for examining the relation between a candidate gene and one or more continuous or categorical traits. Given its features, we regard the logistic regression extension of the TDT as a flexible new data analytic method with extensive potential applications to problems in medical, psychiatric, and behavioral genetics.     

A transmission disequilibrium test for general pedigrees that is robust to the presence of random genotyping errors and any number of untyped parents

Two issues regarding the robustness of the original transmission disequilibrium test (TDT) developed by Spielman et al are: (i) missing parental genotype data and (ii) the presence of undetected genotype errors. While extensions of the TDT that are robust to items (i) and (ii) have been developed, there is to date no single TDT statistic that is robust to both for general pedigrees. We present here a likelihood method, the TDT(ae), which is robust to these issues in general pedigrees. The TDT(ae) assumes a more general disease model than the traditional TDT, which assumes a multiplicative inheritance model for genotypic relative risk. Our model is based on Weinberg's work. To assess robustness, we perform simulations. Also, we apply our method to two data sets from actual diseases: psoriasis and sitosterolemia. Maximization under alternative and null hypotheses is performed using Powell's method. Results of our simulations indicate that our method maintains correct type I error rates at the 1, 5, and 10% levels of significance. Furthermore, a Kolmorogov-Smirnoff Goodness of Fit test suggests that the data are drawn from a central chi2 with 2 df, the correct asymptotic null distribution. The psoriasis results suggest two loci as being significantly linked to the disease, even in the presence of genotyping errors and missing data, and the sitosterolemia results show a P-value of 1.5 x 10(-9) for the marker locus nearest to the sitosterolemia disease genes. We have developed software to perform TDT(ae) calculations, which may be accessed from our ftp site.     

Comparative association analysis reveals that corneodesmosin is more closely associated with psoriasis than HLA-Cw*0602-B*5701 in German families

HLA antigens are associated with psoriasis vulgaris across populations with different ethnic background. We have previously shown that in Caucasians this association is primarily based on the class I alleles of the extended HLA haplotype 57.1 (EH57.1/I), HLA-Cw6-HLA-B57. However, it remained unclear whether HLA-Cw6 itself or a closely linked locus predisposes to the disease. An interesting candidate for this presumed locus is corneodesmosin, which is exclusively synthesized in keratinocytes. The corneodesmosin gene locus (CDSN) is only 160 kb telomeric to HLA-C and tightly associated with psoriasis. In order to find out whether EH57.1/I or a corneodesmosin variant are the susceptibility determinants on 6p, HLA class I alleles and single-nucleotide polymorphisms (SNPs) of corneodesmosin were investigated at the sequence level and analyzed by comparative association tests. Transmission disequilibrium tests (TDT) were performed in 52 nuclear families, of which 36 were fully informative for a joint comparison of HLA and CDSN with regard to association to psoriasis. The extended TDT according to Wilson was employed to test for locus interaction. Using the HLA haplotype EH57.1/I and the CDSN haplotype formed by three intragenic variant sites at nt=619 (T), 1236 (T), and 1243 (C), we obtained the best resolution of parental transmission to index cases in the trio families. On direct comparison of the contributions of the HLA and the CDSN haplotypes, there was a markedly stronger association of the corneodesmosin TTC haplotype, which is not apparent in single locus analysis. We show furthermore that there is no higher order interaction between psoriasis, HLA, and CDSN. This lack of three-locus interaction is suggestive of two independent genetic contributions to psoriasis within the major histocompatibility complex (MHC).     

Cost-effective analysis of candidate genes using htSNPs: a staged approach

We have previously shown that the selection of haplotype tag single nucleotide polymorphisms (htSNPs) and their statistical analysis in a multi-locus transmission/disequilibrium test (TDT) results in a more cost-effective genotyping strategy in disease association studies of genes by minimising redundancy due to linkage disequilibrium between SNPs. Further savings can be achieved by the use of a two-stage genotyping strategy. This approach is illustrated here in conjunction with the multi-locus TDT in determining whether common alleles of the immune regulatory genes RANK and its ligand TRANCE (RANKL) are associated with type 1 diabetes (T1D). A saving of approximately 75% of potential genotyping reactions could be made with minimal loss of power. There was little evidence from our analysis for association between the TRANCE and RANK genes and T1D in the populations tested.     

Effects of age at onset on the power of the affected sib pair and transmission/disequilibrium tests

Information on the age of a patient at disease onset, an important feature of complex diseases, is often collected in studies designed to map the disease genes. Penetrance-model-free methods, requiring no specification of penetrance functions, have been used extensively for detecting linkage and association between marker and disease loci. In this paper, we conduct an analytical study to examine the effects of incorporation of age at onset information on the power of two commonly used penetrance-model-free methods, the affected sib-pair (ASP) and transmission/disequilibrium tests (TDT). Assuming a Cox model with a major gene effect for the age at onset, we quantify analytically how age at onset affects the identity by descent (IBD) probabilities, the mean IBD values, and the expected numbers of alleles transmitted from heterozygous parents to affected children under various genetic models. We show that the power of the mean IBD test and the TDT can be greatly affected by the ages at onset of affected siblings or children used in the study. Generally, the most powerful test for ASPs is that based on affected sib pairs both having early disease onset and for TDT analyses is that based on trios with early-onset children. Naively combining affected sib pairs with different ages at onset or parent-children trios with different ages at onset of affected children can result in reduced power for detecting linkage or association. These results may be used to guide collection and analysis of sib pairs or families for diseases with variable age at onset.