Power calculations for the transmission/disequilibrium and affected sib pair tests using elementary probability methods

The transmission/disequilibrium test (TDT) and the affected sib pair test (ASP) both test for the association of a marker allele with some conditions. Here, we present methods for calculating the probability of detecting the association (power) for a study examining a fixed number of families for suitability for the study and for calculating the number of such families to be examined. Both calculations use a genetic model for the association. The model considered posits a bi-allelic marker locus that is linked to a bi-allelic disease locus with a possibly nonzero recombination fraction between the loci. The penetrance of the disease is an increasing function of the number of disease alleles. The TDT tests whether the transmission by a heterozygous parent of a particular allele at a marker locus to an affected offspring occurs with probability greater than 0.5. The ASP tests whether transmission of the same allele to two affected sibs occurs with probability greater than 0.5. In either case, evidence that the probability is greater than 0.5 is evidence for association between the marker and the disease. Study inclusion criteria (IC) can greatly affect the necessary sample size of a TDT or ASP study. IC considered by us include a randomly selected parent at least one parent or both parents required to be heterozygous. It also allows a specified minimum number of affected offspring to be required (TDT only). We use elementary probability calculations rather than complex mathematical manipulations or asymptotic methods (large sample size approximations) to compute power and requisite sample size for a proposed study. The advantages of these methods are simplicity and generality.     

Analytic power and sample size calculation for the genotypic transmission/disequilibrium test in case‐parent trio studies

Case‐parent trio studies considering genotype data from children affected by a disease and their parents are frequently used to detect single nucleotide polymorphisms (SNPs) associated with disease. The most popular statistical tests for this study design are transmission/disequilibrium tests (TDTs). Several types of these tests have been developed, for example, procedures based on alleles or genotypes. Therefore, it is of great interest to examine which of these tests have the highest statistical power to detect SNPs associated with disease. Comparisons of the allelic and the genotypic TDT for individual SNPs have so far been conducted based on simulation studies, since the test statistic of the genotypic TDT was determined numerically. Recently, however, it has been shown that this test statistic can be presented in closed form. In this article, we employ this analytic solution to derive equations for calculating the statistical power and the required sample size for different types of the genotypic TDT. The power of this test is then compared with the one of the corresponding score test assuming the same mode of inheritance as well as the allelic TDT based on a multiplicative mode of inheritance, which is equivalent to the score test assuming an additive mode of inheritance. This is, thus, the first time the power of these tests are compared based on equations, yielding instant results and omitting the need for time‐consuming simulation studies. This comparison reveals that these tests have almost the same power, with the score test being slightly more powerful.     

A Discordant-Sibship Test for Disequilibrium and Linkage: No Need for Parental Data

The sibship disequilibrium test (SDT) is designed to detect both linkage in the presence of association and association in the presence of linkage (linkage disequilibrium). The test does not require parental data but requires discordant sibships with at least one affected and one unaffected sibling. The SDT has many desirable properties: it uses all the siblings in the sibship; it remains valid if there are misclassifications of the affectation status; it does not detect spurious associations due to population stratification; asymptotically it has a chi2 distribution under the null hypothesis; and exact P values can be easily computed for a biallelic marker. We show how to extend the SDT to markers with multiple alleles and how to combine families with parents and data from discordant sibships. We discuss the power of the test by presenting sample-size calculations involving a complex disease model, and we present formulas for the asymptotic relative efficiency (which is approximately the ratio of sample sizes) between SDT and the transmission/disequilibrium test (TDT) for special family structures. For sib pairs, we compare the SDT to a test proposed both by Curtis and, independently, by Spielman and Ewens. We show that, for discordant sib pairs, the SDT has good power for testing linkage disequilibrium relative both to Curtis''s tests and to the TDT using trios comprising an affected sib and its parents. With additional sibs, we show that the SDT can be more powerful than the TDT for testing linkage disequilibrium, especially for disease prevalence >.3.     

Linkage and association studies of QTL for nuclear families by mixed models

The transmission disequilibrium test (TDT) has been utilized to test the linkage and association between a genetic trait locus and a marker. Spielman et al. (1993) introduced TDT to test linkage between a qualitative trait and a marker in the presence of association. In the presence of linkage, TDT can be applied to test for association for fine mapping (Martin et al., 1997; Spielman and Ewens, 1996). In recent years, extensive research has been carried out on the TDT between a quantitative trait and a marker locus (Allison, 1997; Fan et al., 2002; George et al., 1999; Rabinowitz, 1997; Xiong et al., 1998; Zhu and Elston, 2000, 2001). The original TDT for both qualitative and quantitative traits requires unrelated offspring of heterozygous parents for analysis, and much research has been carried out to extend it to fit for different settings. For nuclear families with multiple offspring, one approach is to treat each child independently for analysis. Obviously, this may not be a valid method since offspring of one family are related to each other. Another approach is to select one offspring randomly from each family for analysis. However, with this method much information may be lost. Martin et al. (1997, 2000) constructed useful statistical tests to analyse the data for qualitative traits. In this paper, we propose to use mixed models to analyse sample data of nuclear families with multiple offspring for quantitative traits according to the models in Amos (1994). The method uses data of all offspring by taking into account their trait mean and variance-covariance structures, which contain all the effects of major gene locus, polygenic loci and environment. A test statistic based on mixed models is shown to be more powerful than the test statistic proposed by George et al. (1999) under moderate disequilibrium for nuclear families. Moreover, it has higher power than the TDT statistic which is constructed by randomly choosing a single offspring from each nuclear family.     

The transmission/disequilibrium test: history, subdivision, and admixture.

Disease association with a genetic marker is often taken as a preliminary indication of linkage with disease susceptibility. However, population subdivision and admixture may lead to disease association even in the absence of linkage. In a previous paper, we described a test for linkage (and linkage disequilibrium) between a genetic marker and disease susceptibility; linkage is detected by this test only if association is also present. This transmission/disequilibrium test (TDT) is carried out with data on transmission of marker alleles from parents heterozygous for the marker to affected offspring. The TDT is a valid test for linkage and association, even when the association is caused by population subdivision and admixture. In the previous paper, we did not explicitly consider the effect of recent history on population structure. Here we extend the previous results by examining in detail the effects of subdivision and admixture, viewed as processes in population history. We describe two models for these processes. For both models, we analyze the properties of (a) the TDT as a test for linkage (and association) between marker and disease and (b) the conventional contingency statistic used with family data to test for population association. We show that the contingency test statistic does not have a chi 2 distribution if subdivision or admixture is present. In contrast, the TDT remains a valid chi 2 statistic for the linkage hypothesis, regardless of population history.     

A note on power approximations for the transmission/disequilibrium test

The transmission/disequilibrium test (TDT) is a popular method for detection of the genetic basis of a disease. Investigators planning such studies require computation of sample size and power, allowing for a general genetic model. Here, a rigorous method is presented for obtaining the power approximations of the TDT for samples consisting of families with either a single affected child or affected sib pairs. Power calculations based on simulation show that these approximations are quite precise. By this method, it is also shown that a previously published power approximation of the TDT is erroneous.     

Marker selection for the transmission/disequilibrium test, in recently admixed populations.

Recent admixture between genetically differentiated populations can result in high levels of association between alleles at loci that are <=10 cM apart. The transmission/disequilibrium test (TDT) proposed by Spielman et al. (1993) can be a powerful test of linkage between disease and marker loci in the presence of association and therefore could be a useful test of linkage in admixed populations. The degree of association between alleles at two loci depends on the differences in allele frequencies, at the two loci, in the founding populations; therefore, the choice of marker is important. For a multiallelic marker, one strategy that may improve the power of the TDT is to group marker alleles within a locus, on the basis of information about the founding populations and the admixed population, thereby collapsing the marker into one with fewer alleles. We have examined the consequences of collapsing a microsatellite into a two-allele marker, when two founding populations are assumed for the admixed population, and have found that if there is random mating in the admixed population, then typically there is a collapsing for which the power of the TDT is greater than that for the original microsatellite marker. A method is presented for finding the optimal collapsing that has minimal dependence on the disease and that uses estimates either of marker allele frequencies in the two founding populations or of marker allele frequencies in the current, admixed population and in one of the founding populations. Furthermore, this optimal collapsing is not always the collapsing with the largest difference in allele frequencies in the founding populations. To demonstrate this strategy, we considered a recent data set, published previously, that provides frequency estimates for 30 microsatellites in 13 populations.     

Tests for linkage and association in nuclear families.

The transmission/disequilibrium test (TDT) originally was introduced to test for linkage between a genetic marker and a disease-susceptibility locus, in the presence of association. Recently, the TDT has been used to test for association in the presence of linkage. The motivation for this is that linkage analysis typically identifies large candidate regions, and further refinement is necessary before a search for the disease gene is begun, on the molecular level. Evidence of association and linkage may indicate which markers in the region are closest to a disease locus. As a test of linkage, transmissions from heterozygous parents to all of their affected children can be included in the TDT; however, the TDT is a valid chi2 test of association only if transmissions to unrelated affected children are used in the analysis. If the sample contains independent nuclear families with multiple affected children, then one procedure that has been used to test for association is to select randomly a single affected child from each sibship and to apply the TDT to those data. As an alternative, we propose two statistics that use data from all of the affected children. The statistics give valid chi2 tests of the null hypothesis of no association or no linkage and generally are more powerful than the TDT with a single, randomly chosen, affected child from each family.     

Transmission disequilibrium test with discordant sib pairs when parents are available

The transmission disequilibrium test (TDT) has been employed to map disease susceptibility loci (DSL), while being immune to the problem of population admixture. The customary TDT test (TDT(D)) was developed for affected child(ren) and their parents and was most often applied to case-parent trios. Recently, the TDT has been extended to the situations when (1) parents are not available but affected and nonaffected sibs from each family are available, (2) unrelated control-parent trios are available for combined analyses with case-parent trios (TDT(DC)), and (3) large pedigrees. For many diseases, affected children in the case-parent trios enlisted into the TDT(D) have unaffected sibs who can be recruited. We present an extension of the TDT by effectively incorporating one unaffected sib of each of the affected children in the case-parent trios into a single analysis (TDT(DS), where DS denotes discordant sib pairs). We have developed a general analytical method for computing the statistical power of the TDT(DS) under any genetic model, the accuracy of which is validated by computer simulations. We compare the power of the TDT(D), TDT(DC), and TDT(DS) under a range of parameter space and genetic models. We find that the TDT(DS) is generally more powerful than the TDT(DC) and TDT(D), particularly when the disease is prevalent (>30%) in the population. The relative power of the TDT(D) and the TDT(DS) largely depends upon the allele frequencies and genetic effects at the DSL, whereas the recombination rate, the degree of linkage disequilibrium, and the marker allele frequencies have little effect. Importantly, the TDT(DS) not only may be more powerful, it also has the advantage of being able to test for segregation distortion that may yield false linkage/association in the TDT(D).     

Screening and replication using the same data set: testing strategies for family-based studies in which all probands are affected

For genome-wide association studies in family-based designs, we propose a powerful two-stage testing strategy that can be applied in situations in which parent-offspring trio data are available and all offspring are affected with the trait or disease under study. In the first step of the testing strategy, we construct estimators of genetic effect size in the completely ascertained sample of affected offspring and their parents that are statistically independent of the family-based association/transmission disequilibrium tests (FBATs/TDTs) that are calculated in the second step of the testing strategy. For each marker, the genetic effect is estimated (without requiring an estimate of the SNP allele frequency) and the conditional power of the corresponding FBAT/TDT is computed. Based on the power estimates, a weighted Bonferroni procedure assigns an individually adjusted significance level to each SNP. In the second stage, the SNPs are tested with the FBAT/TDT statistic at the individually adjusted significance levels. Using simulation studies for scenarios with up to 1,000,000 SNPs, varying allele frequencies and genetic effect sizes, the power of the strategy is compared with standard methodology (e.g., FBATs/TDTs with Bonferroni correction). In all considered situations, the proposed testing strategy demonstrates substantial power increases over the standard approach, even when the true genetic model is unknown and must be selected based on the conditional power estimates. The practical relevance of our methodology is illustrated by an application to a genome-wide association study for childhood asthma, in which we detect two markers meeting genome-wide significance that would not have been detected using standard methodology.     

The power of genomic control

Although association analysis is a useful tool for uncovering the genetic underpinnings of complex traits, its utility is diminished by population substructure, which can produce spurious association between phenotype and genotype within population-based samples. Because family-based designs are robust against substructure, they have risen to the fore of association analysis. Yet, if population substructure could be ignored, this robustness can come at the price of power. Unfortunately it is rarely evident when population substructure can be ignored. Devlin and Roeder recently have proposed a method, termed "genomic control" (GC), which has the robustness of family-based designs even though it uses population-based data. GC uses the genome itself to determine appropriate corrections for population-based association tests. Using the GC method, we contrast the power of two study designs, family trios (i.e., father, mother, and affected progeny) versus case-control. For analysis of trios, we use the TDT test. When population substructure is absent, we find GC is always more powerful than TDT; furthermore, contrary to previous results, we show that as a disease becomes more prevalent the discrepancy in power becomes more extreme. When population substructure is present, however, the results are more complex: TDT is more powerful when population substructure is substantial, and GC is more powerful otherwise. We also explore general issues of power and implementation of GC within the case-control setting and find that, economically, GC is at least comparable to and often less expensive than family-based methods. Therefore, GC methods should prove a useful complement to family-based methods for the genetic analysis of complex traits.     

A test of transmission/disequilibrium for quantitative traits in pedigree data, by multiple regression.

The transmission/disequilibrium (TD) test (TDT), proposed, by Spielman et al., for binary traits is a powerful method for detection of linkage between a marker locus and a disease locus, in the presence of allelic association. As a test for linkage disequilibrium, the TDT makes the assumption that any allelic association present is due to linkage. Allison proposed a series of TD-type tests for quantitative traits and calculated their power, assuming that the marker locus is the disease locus. All these tests assume that the observations are independent, and therefore they are applicable, as a test for linkage, only for nuclear-family data. In this report, we propose a regression-based TD-type test for linkage between a marker locus and a quantitative trait locus, using information on the parent-to-offspring transmission status of the associated allele at the marker locus. This method does not require independence of observations, thus allowing for analysis of pedigree data as well, and allows adjustment for covariates. We investigate the statistical power and validity of the test by simulating markers at various recombination fractions from the disease locus.     

An entropy-based genome-wide transmission/disequilibrium test

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

一种有效的复杂疾病基因定位的检测法

连锁不平衡（LD）应用于某些复杂疾病基因的定位,近年来发展了许多LD定位方法,除TDT外,大多数LD定位方法须先假定无人群混和,人群混合可增大在疾病基因定位时犯Ⅰ类错误的机率,产生无效结果。此方法利用LD来检测标记位点和疾病敏感位点（DSL）的连锁（有连锁不平衡）相关（有连锁）。分析时采用不相关样本,已知其父母基因型和至少父母之一为杂合子,再将随机样本依基因型不同分类,然后对来自不同类的数据应用有力的统计方法进行单独和联合分析。此LD定位法不仅适用于患病和正常个体,而且有效消除据父母基因分类的样本定位时人群混合的影响,分析结果和模拟结果也表明此方法解决了在检测标记位点和疾病敏感位点之间的连锁和相关时人群混和的问题,但与TDT比,此法在检测的位点为DSL时丙能有效和充分地利用矫正数据,检测位点不是DSL时,此法和TDT法可相互补充更有效地检测连锁的DSL。     

How do homozygous parents affect TDT as a test for association?

The traditional transmission disequilibrium test (TDT) (Spielman et al., 1993) is a powerful test for association only in the presence of linkage. Since allele transmissions from homozygous parents do not carry any information on linkage, the TDT statistic uses data only on heterozygous parents. However, homozygous parents carry information on association between alleles at a marker locus and a disease locus. In this article, we explore whether inclusion of homozygous parents increases the power to detect association. The resultant test statistic follows a chi(2) distribution with 2 degrees of freedom. Monte-Carlo simulations are included to compare the performance of this test with the traditional TDT under different disease models.     

Mapping genes for polygenic disorders: considerations for study design in the complex trait of inflammatory bowel disease

While the methodology for the mapping of Mendelian disorders is well established, the practical and theoretical steps required for successful gene identification in a complex trait are still difficult to predict. A number of analytical models and simulations based on repetitive drawings from predefined statistical distributions are available. To supplement these analytical models, we developed an integrated simulation approach by directly simulating entire populations under a disease model based on epidemiological data. Random mating, nonoverlapping populations and the absence of differential fitness were assumed. Samples were drawn from these homogeneous and heterogeneous populations and analyzed with established analysis tools. We investigated the properties of linkage and association studies in inflammatory bowel disease - modeled as a six-locus polygenic disorder - as an example of this approach. In nonparametric linkage studies, lod scores varied widely, with the median required sample size depending on the locus-specific relative sibling risk. A fine mapping resolution <4 cM was found to require nonparametric lod scores >10. Family-based association studies (TDT test) and case-control studies showed a similar sensitivity and can identify risk loci in populations with moderate levels of linkage disequilibrium in sample sizes of 500-800 triplets. Case-control association studies were prone to false-positive results if applied in heterogeneous populations, with the false-positive rate increasing with sample size because population heterogeneity is detected with increasing power.     

Sample reproducibility of genetic association using different multimarker TDTs in genome-wide association studies: characterization and a new approach

Multimarker Transmission/Disequilibrium Tests (TDTs) are very robust association tests to population admixture and structure which may be used to identify susceptibility loci in genome-wide association studies. Multimarker TDTs using several markers may increase power by capturing high-degree associations. However, there is also a risk of spurious associations and power reduction due to the increase in degrees of freedom. In this study we show that associations found by tests built on simple null hypotheses are highly reproducible in a second independent data set regardless the number of markers. As a test exhibiting this feature to its maximum, we introduce the multimarker 2-Groups TDT (mTDT(2G)), a test which under the hypothesis of no linkage, asymptotically follows a χ2 distribution with 1 degree of freedom regardless the number of markers. The statistic requires the division of parental haplotypes into two groups: disease susceptibility and disease protective haplotype groups. We assessed the test behavior by performing an extensive simulation study as well as a real-data study using several data sets of two complex diseases. We show that mTDT(2G) test is highly efficient and it achieves the highest power among all the tests used, even when the null hypothesis is tested in a second independent data set. Therefore, mTDT(2G) turns out to be a very promising multimarker TDT to perform genome-wide searches for disease susceptibility loci that may be used as a preprocessing step in the construction of more accurate genetic models to predict individual susceptibility to complex diseases.     

Testing for association in the presence of population stratification: a simulation study comparing the S-TDT, STRAT and the GC

A novel approach for association testing in the presence of population stratification has been introduced by Pritchard et al. (2000a) and Pritchard et al. (2000b). The structured association approach is a two-tiered procedure that first estimates the population structure and then tests the null hypothesis H0: 'no association within subpopulations' in the second step. A power comparison of the stratified test for association (STRAT) (Pritchard et al., 2000b) and the Transmission-Disequilibrium-Test (TDT) (Spielman and Ewens, 1993a) in a simulation framework showed superiority of STRAT if allele frequencies or associations between allele and disease differ strongly in subpopulations. In more homogeneous situations, the TDT had greater power than STRAT. However, the TDT, based on family trios,that uses population controls, needs 50% more genotyping compared to STRAT. The Sib-Transmission-Disequilibrium-Test (S-TDT) needs the same amount of genotyping since it relays in its minimal configuration on pairs of siblings. This raises the question how the S-TDT (Spielman and Ewens, 1998a) performs compared to the population based methods STRAT and Genomic Controls (GC). In this paper, we present a simulation study accounting for two different models of population stratification in different settings of allele frequencies and under different risk models. The results showed that under a discrete as well as under an admixed population model, STRAT strongly outperformed the S-TDT and the GC when different alleles were associated in different subpopulations. In contrast, the S-TDT had greater power than STRAT when the same allele was associated in both subpopulations. Here, the GC was sometimes even more powerful than the S-TDT, depending on the population model and the allele frequency differences. A general recommendation for the use of one of the tests can therefore not be given.     

Detailed analysis of the relative power of direct and indirect association studies and the implications for their interpretation

OBJECTIVES: Genetic association studies are usually based upon restricted sets of 'tag' markers selected to represent the total sequence variation. Tag selection is often determined by some threshold for the r(2) coefficients of linkage disequilibrium (LD) between tag and untyped markers, it being widely assumed that power to detect an effect at the untyped sites is retained by typing the tag marker in a sample scaled by the inverse of the selected threshold (1/r(2)). However, unless only a single causal variant occurs at a locus, it has been shown [Eur J Hum Genet 2006;14:426-437] that significant power loss can occur if this principle is applied. We sought to investigate whether unexpected loss of power might be an exceptional case or more general concern. In the absence of detailed knowledge about the genetic architecture at complex disease loci, we developed a mathematical approach to test all possible situations. METHODS: We derived mathematical formulae allowing the calculation of all possible odds ratios (OR) at a tag marker locus given the effect size that would be observed by typing a second locus and the r(2) between the two loci. For a range of allele frequencies, r(2) between loci, and strengths of association at the causal locus (OR from 0.5 to 2) that we consider realistic for complex disease loci, we next determined the sample sizes that would be necessary to give equivalent power to detect association by genotyping tag and causal loci and compared these with the sample sizes predicted by applying 1/r(2). RESULTS: Under most of the hypothetical scenarios we examined, the calculated sample sizes required to maintain power by typing markers that tag the causal locus at even moderately high r(2) (0.8) were greater than that calculated by applying 1/r(2). Even in populations with apparently similar measurements of allele frequency, LD structure, and effect size at the susceptibility allele, the required sample size to detect association with a tag marker can vary substantially. We also show that in apparently similar populations, associations to either allele at the tag site are possible. CONCLUSIONS: Indirect tests of association are less powered than sizes predicted by applying 1/r(2) in the majority of hypothetical scenarios we examined. Our findings pertain even for what we consider likely to be larger than average effect sizes in complex diseases (OR = 1.5-2) and even for moderately high r(2) values between the markers. Until a substantial number of disease genes have been identified through methods that are not based on tagging, and therefore biased towards those situations most favourable to tagging, it is impossible to know how the true scenarios are distributed across the range of possible scenarios. Nevertheless, while association designs based upon tag marker selection by necessity are the tool of choice for de novo gene discovery, our data suggest power to initially detect association may often be less than assumed. Moreover, our data suggest that to avoid genuine findings being subsequently discarded by unpredictable losses of power, follow up studies in other samples should be based upon more detailed analyses of the gene rather than simply on the tag SNPs showing association in the discovery study.