Haseman-Elston weighted by marker informativity

In the Haseman-Elston approach the squared phenotypic difference is regressed on the proportion of alleles shared identical by descent (IBD) to map a quantitative trait to a genetic marker. In applications the IBD distribution is estimated and usually cannot be determined uniquely owing to incomplete marker information. At Genetic Analysis Workshop (GAW) 13, Jacobs et al. [BMC Genet 2003, 4(Suppl 1):S82] proposed to improve the power of the Haseman-Elston algorithm by weighting for information available from marker genotypes. The authors did not show, however, the validity of the employed asymptotic distribution. In this paper, we use the simulated data provided for GAW 14 and show that weighting Haseman-Elston by marker information results in increased type I error rates. Specifically, we demonstrate that the number of significant findings throughout the chromosome is significantly increased with weighting schemes. Furthermore, we show that the classical Haseman-Elston method keeps its nominal significance level when applied to the same data. We therefore recommend to use Haseman-Elston with marker informativity weights only in conjunction with empirical p-values. Whether this approach in fact yields an increase in power needs to be investigated further.     

Construction of a confidence set of markers for the location of a disease gene using affected sib pair data

We have previously proposed a confidence set approach for finding tightly linked genomic regions under the setting of parametric linkage analysis. In this article, we extend the confidence set approach to nonparametric linkage analysis of affected sib pair (ASP) data based on their identity-by-descent (IBD) information. Two well-known statistics in nonparametric linkage analysis, the Two-IBD test (proportion of ASPs sharing two alleles IBD), and the Mean test (average number of alleles shared IBD in the ASPs), are used for constructing confidence sets. Some numerical analyses as well as a simulation study were carried out to demonstrate the utility of the methods. Our results show that the fundamental advantages of the confidence set approach in parametric linkage analysis are retained when the method is generalized to nonparametric analysis. Our study on the accuracy of confidence sets, in terms of choice of tests, underlying disease incidence data, and amount of data available, leads us to conclude, among other things, that the Mean test outperforms the Two-IBD test in most situations, with the reverse being true only for traits with small additive variance. Although we describe how to construct confidence sets based on only two familiar tests, one can construct confidence sets similarly using other allele sharing statistics.     

The affected-/discordant-sib-pair design can guarantee validity of multipoint model-free linkage analysis of incomplete pedigrees when there is marker-marker disequilibrium

Genomewide linkage studies are tending toward the use of single-nucleotide polymorphisms (SNPs) as the markers of choice. However, linkage disequilibrium (LD) between tightly linked SNPs violates the fundamental assumption of linkage equilibrium (LE) between markers that underlies most multipoint calculation algorithms currently available, and this leads to inflated affected-relative-pair allele-sharing statistics when founders' multilocus genotypes are unknown. In this study, we investigate the impact that the degree of LD, marker allele frequency, and association type have on estimating the probabilities of sharing alleles identical by descent in multipoint calculations and hence on type I error rates of different sib-pair linkage approaches that assume LE. We show that marker-marker LD does not inflate type I error rates of affected sib pair (ASP) statistics in the whole parameter space, and that, in any case, discordant sib pairs (DSPs) can be used to control for marker-marker LD in ASPs. We advocate the ASP/DSP design with appropriate sib-pair statistics that test the difference in allele sharing between ASPs and DSPs.     

Linkage analysis of alcohol dependence using both affected and discordant sib pairs

The basic idea of affected-sib-pair (ASP) linkage analysis is to test whether the inheritance pattern of a marker deviates from Mendelian expectation in a sample of ASPs. The test depends on an assumed Mendelian control distribution of the number of marker alleles shared identical by descent (IBD), i.e., 1/4, 1/2, and 1/4 for 2, 1, and 0 allele(s) IBD, respectively. However, Mendelian transmission may not always hold, for example because of inbreeding or meiotic drive at the marker or a nearby locus. A more robust and valid approach is to incorporate discordant-sib-pairs (DSPs) as controls to avoid possible false-positive results. To be robust to deviation from Mendelian transmission, here we analyzed Collaborative Study on the Genetics of Alcoholism data by modifying the ASP LOD score method to contrast the estimated distribution of the number of allele(s) shared IBD by ASPs with that by DSPs, instead of with the expected distribution under the Mendelian assumption. This strategy assesses the difference in IBD sharing between ASPs and the IBD sharing between DSPs. Further, it works better than the conventional LOD score ASP linkage method in these data in the sense of avoiding false-positive linkage evidence.     

Exploiting excess sharing: a more powerful test of linkage for affected sib pairs than the transmission/disequilibrium test

The transmission/disequilibrium test (TDT) is a popular, simple, and powerful test of linkage, which can be used to analyze data consisting of transmissions to the affected members of families with any kind pedigree structure, including affected sib pairs (ASPs). Although it is based on the preferential transmission of a particular marker allele across families, it is not a valid test of association for ASPs. Martin et al. devised a similar statistic for ASPs, Tsp, which is also based on preferential transmission of a marker allele but which is a valid test of both linkage and association for ASPs. It is, however, less powerful than the TDT as a test of linkage for ASPs. What I show is that the differences between the TDT and Tsp are due to the fact that, although both statistics are based on preferential transmission of a marker allele, the TDT also exploits excess sharing in identity-by-descent transmissions to ASPs. Furthermore, I show that both of these statistics are members of a family of "TDT-like" statistics for ASPs. The statistics in this family are based on preferential transmission but also, to varying extents, exploit excess sharing. From this family of statistics, we see that, although the TDT exploits excess sharing to some extent, it is possible to do so to a greater extent-and thus produce a more powerful test of linkage, for ASPs, than is provided by the TDT. Power simulations conducted under a number of disease models are used to verify that the most powerful member of this family of TDT-like statistics is more powerful than the TDT for ASPs.     

A powerful and robust test statistic for randomization inference in group-randomized trials with matched pairs of groups

For group-randomized trials, randomization inference based on rank statistics provides robust, exact inference against nonnormal distributions. However, in a matched-pair design, the currently available rank-based statistics lose significant power compared to normal linear mixed model (LMM) test statistics when the LMM is true. In this article, we investigate and develop an optimal test statistic over all statistics in the form of the weighted sum of signed Mann-Whitney-Wilcoxon statistics under certain assumptions. This test is almost as powerful as the LMM even when the LMM is true, but it is much more powerful for heavy tailed distributions. A simulation study is conducted to examine the power.     

Using discordant sib pairs to map loci for qualitative traits with high sibling recurrence risk.

A common approach for detecting genetic linkage using siblings is to collect affected sib pairs (ASPs) and to identify markers where allele sharing exceeds expectation. Alternatively, markers can be analyzed in discordant sib pairs (DSPs) for allele sharing below expectation. Relative to the ASP approach, according to Risch, the power of the DSP approach increases with sibling recurrence risk, the two approaches being equally effective at 50% recurrence risk. However, with many diseases associated with more moderate sibling recurrence risk, less emphasis has been placed on the use of DSPs and the development of the underlying theory. In this paper, we expand the work of Risch to provide a more general foundation for DSP studies, since power can be quite high under the appropriate conditions. For example, in some highly affected populations, such as the diabetes-prone Pima Indians, sibling recurrence risk can be very large and, thus, DSPs ideal. Similarly, as we show through simulation, DSPs are preferable for diabetic nephropathy due to a 70% recurrence rate among siblings with insulin-dependent diabetes mellitus. Following the diabetic nephropathy example, we consider more systematically the situations in which DSPs can provide an efficient alternative to ASPs.     

Linkage detection adaptive to linkage disequilibrium: the disequilibrium maximum-likelihood-binomial test for affected-sibship data

It has been demonstrated in the literature that the transmission/disequilibrium test (TDT) has higher power than the affected-sib-pair (ASP) mean test when linkage disequilibrium (LD) is strong but that the mean test has higher power when LD is weak. Thus, for ASP data, it seems clear that the TDT should be used when LD is strong but that the mean test or other linkage tests should be used when LD is weak or absent. However, in practice, it may be difficult to follow such a guideline, because the extent of LD is often unknown. Even with a highly dense genetic-marker map, in which some markers should be located near the disease-predisposing mutation, strong LD is not inevitable. Besides the genetic distance, LD is also affected by many factors, such as the allelic heterogeneity at the disease locus, the initial LD, the allelic frequencies at both disease locus and marker locus, and the age of the mutation. Therefore, it is of interest to develop methods that are adaptive to the extent of LD. In this report, we propose a disequilibrium maximum-binomial-likelihood (DMLB) test that incorporates LD in the maximum-binomial-likelihood (MLB) test. Examination of the corresponding score statistics shows that this method adaptively combines two sources of information: (a) the identity-by-descent (IBD) sharing score, which is informative for linkage regardless of the existence of LD, and (b) the contrast between allele-specific IBD sharing score, which is informative for linkage only in the presence of LD. For ASP data, the proposed test has higher power than either the TDT or the mean test when the extent of LD ranges from moderate to strong. Only when LD is very weak or absent is the DMLB slightly less powerful than the mean test; in such cases, the TDT has essentially no power to detect linkage. Therefore, the DMLB test is an interesting approach to linkage detection when the extent of LD is unknown.     

The future is now - will the real disease gene please stand up?

The transmission/disequilibrium test (TDT) [Spielman et al.: Am J Hum Genet 1993;52:506-516] has been postulated as the future of gene mapping for complex diseases, provided one is able to genotype a dense enough map of markers across the genome. Risch and Merikangas [Science 1996;273:1516-1517] suggested a million-marker screen in affected sibpair (ASP) families, demonstrating that the TDT is a more powerful test of linkage than traditional linkage tests based on allele-sharing when there is also association between marker and disease alleles. While the future of genotyping has arrived, successes in family-based association studies have been modest. This is often attributed to excessive false positives in candidate gene studies. This problem is only exacerbated by the increasing numbers of whole genome association (WGA) screens. When applied in ASPs, the TDT statistic, which assumes transmissions to siblings are independent, is not expected to have a constant variance in the presence of variable linkage. This results in generally more extreme statistics, hence will further aggravate the problem of having a large number of positive results to sort through. So an important question is how many positive TDT results will show up on a chromosome containing a disease gene due only to linkage, and will they obfuscate the true disease gene location. To answer this question we combined theory and computer simulations. These studies show that in ASPs the normal version of the TDT statistic has a mean of 0 and a variance of 1 in unlinked regions, but has a variance larger than 1 in linked regions. In contrast, the pedigree disequilibrium test (PDT) statistic adjusts for correlation between siblings due to linkage and maintains a constant variance of 1 at unassociated markers irrespective of linkage. The TDT statistic is generally larger than the PDT statistic across linked regions. This is true for unassociated as well as associated markers. To compare the two tests we ranked both statistics at the disease locus, or an associated marker, among statistics at all other markers. The TDT did better job than PDT placing the score of the associated marker near the top. Though, strictly speaking, the TDT in ASPs should be interpreted as a test of linkage and not a test of association, there is a good chance that if a marker stands out, the marker is associated as well as linked. In conclusion, our results suggest that TDT is an effective screening tool for WGA studies, especially in multiplex families.     

A method for incorporating ages at onset in affected sibpair linkage studies

Data with varying age at disease onset arise frequently in studies of mapping disease associated genes. Naively combining affected subjects with different ages at onset may result in a much reduced power in detecting the disease genes. In this paper we present a weighted score test statistic to detect the linkage between marker and latent disease loci using affected sibpairs, where the weight is used for assigning differential contribution due to the varying age at onset of each affected sibpair to the test statistic. We show that the weighted test has a correct type I error rate asymptotically. For an illustrative purpose, we analyze a data set from the 12th Genetic Analysis Workshop. The result shows that the weighted tests appear to be able to pinpoint the location of latent disease genes better than the mean IBD test with equal weight with respect to the age at onset. To avoid the potential power loss due to the improper weight, we propose to use a combined test statistic, taking the maximum of two tests, one that is weighted by the age-dependent penetrance function and the other that may be invariant to the age. We conduct an analytical study, comparing the combined test with weighted and equal weight with respect to age test. It shows that the combined test retains the most power of the better one of the two tests being combined.     

Genome-wide linkage scan for genes affecting longitudinal trends in systolic blood pressure

Only one genome scan to date has attempted to make use of the longitudinal data available in the Framingham Heart Study, and this attempt yielded evidence of linkage to a gene for mean systolic blood pressure. We show how the additional information available in these longitudinal data can be utilized to examine linkages for not only mean systolic blood pressure (SBP), but also for its trend with age and its variability. Prior to linkage analysis, individuals treated for hypertension were adjusted to account for right-censoring of SBP. Regressions on age were fitted to obtain orthogonal measures of slope, curvature, and residual variance of SBP that were then used as dependent variables in the model-free linkage program SIBPAL. We included mean age, gender, and cohort as covariates in the analysis. To improve power, sibling pairs were weighted for informativity using weights derived from both the marker and trait. The most significant results from our analyses were found on chromosomes 12, 15, and 17 for mean SBP, and chromosome 20 for both SBP slope and curvature.     

Linkage analysis for diseases with variable age of onset

We present a method for the multivariate linkage analysis of the age of onset of a disease. The approach allows the incorporation of covariates for the study of gene by environment interactions. It is applicable to general pedigrees. The likelihood of the data is expressed as a function of the number of alleles identical by descent at a marker, the censored ages of onset and disease status, and environmental exposures. In a simulation study, we compare the power to detect linkage under different sampling schemes for either a dominant or recessive trait when approximately 10% of individuals are gene carriers. The majority of the linkage information from a sample of randomly selected sib pairs was retained when the analyses were limited to sibships with one sibling having early-onset disease (<59 years old). Incorporating parental phenotypes could improve the power to detect the gene. When the sample consists of affected sib pairs (ASPs) having variable age of onset, the likelihood ratio (LR) test had higher power than the means (t(2)) test for detecting a locus with a large genetic relative risk (R(g) = 20). However, the power of the two tests was similar when ASPs are selected so that the proband has an early onset of disease. Lastly, the LR test had more power than the t(2) test to detect linkage in the presence of gene by environment interactions.     

Weighted least-squares approach for comparing correlated kappa

In the medical sciences, studies are often designed to assess the agreement between different raters or different instruments. The kappa coefficient is a popular index of agreement for binary and categorical ratings. Here we focus on testing for the equality of two dependent kappa coefficients. We use the weighted least-squares (WLS) approach of Koch et al. (1977, Biometrics 33, 133-158) to take into account the correlation between the estimated kappa statistics. We demonstrate how the SAS PROC CATMOD can be used to test for the equality of dependent Cohen's kappa coefficients and dependent intraclass kappa coefficients with nominal categorical ratings. We also test for the equality of dependent Cohen's kappa and dependent weighted kappa with ordinal ratings. The major advantage of the WLS approach is that it allows the data analyst a way of testing dependent kappa with popular SAS software. The WLS approach can handle any number of categories. Analyses of three biomedical studies are used for illustration.     

A general approach for sample size and power calculations based on the Haseman-Elston method

Risch and Zhang (1995; Science 268: 1584-9) reported a simple sample size and power calculation approach for the Haseman-Elston method and based their computations on the null hypothesis of no genetic effect. We argue that the more reasonable null hypothesis is that of no recombination. For this null hypothesis, we provide a general approach for sample size and power calculations within the Haseman-Elston framework. We demonstrate the validity of our approach in a Monte-Carlo simulation study and illustrate the differences using data from published segregation analyses on body weight and heritability estimates on carotid artery artherosclerotic lesions.     

Adaptive design and estimation in randomized clinical trials with correlated observations

Clinical trial designs involving correlated data often arise in biomedical research. The intracluster correlation needs to be taken into account to ensure the validity of sample size and power calculations. In contrast to the fixed-sample designs, we propose a flexible trial design with adaptive monitoring and inference procedures. The total sample size is not predetermined, but adaptively re-estimated using observed data via a systematic mechanism. The final inference is based on a weighted average of the block-wise test statistics using generalized estimating equations, where the weight for each block depends on cumulated data from the ongoing trial. When there are no significant treatment effects, the devised stopping rule allows for early termination of the trial and acceptance of the null hypothesis. The proposed design updates information regarding both the effect size and within-cluster correlation based on the cumulated data in order to achieve a desired power. Estimation of the parameter of interest and its confidence interval are proposed. We conduct simulation studies to examine the operating characteristics and illustrate the proposed method with an example.     

Theoretical and empirical power of regression and maximum-likelihood methods to map quantitative trait loci in general pedigrees

Both theoretical calculations and simulation studies have been used to compare and contrast the statistical power of methods for mapping quantitative trait loci (QTLs) in simple and complex pedigrees. A widely used approach in such studies is to derive or simulate the expected mean test statistic under the alternative hypothesis of a segregating QTL and to equate a larger mean test statistic with larger power. In the present study, we show that, even when the test statistic under the null hypothesis of no linkage follows a known asymptotic distribution (the standard being chi(2)), it cannot be assumed that the distribution under the alternative hypothesis is noncentral chi(2). Hence, mean test statistics cannot be used to indicate power differences, and a comparison between methods that are based on simulated average test statistics may lead to the wrong conclusion. We illustrate this important finding, through simulations and analytical derivations, for a recently proposed new regression method for the analysis of general pedigrees to map quantitative trait loci. We show that this regression method is not necessarily more powerful nor computationally more efficient than a maximum-likelihood variance-component approach. We advocate the use of empirical power to compare trait-mapping methods.     

Trimmed Weighted Simes' Test for Two One‐Sided Hypotheses With Arbitrarily Correlated Test Statistics

The two‐sided Simes test is known to control the type I error rate with bivariate normal test statistics. For one‐sided hypotheses, control of the type I error rate requires that the correlation between the bivariate normal test statistics is non‐negative. In this article, we introduce a trimmed version of the one‐sided weighted Simes test for two hypotheses which rejects if (i) the one‐sided weighted Simes test rejects and (ii) both p‐values are below one minus the respective weighted Bonferroni adjusted level. We show that the trimmed version controls the type I error rate at nominal significance level α if (i) the common distribution of test statistics is point symmetric and (ii) the two‐sided weighted Simes test at level 2α controls the level. These assumptions apply, for instance, to bivariate normal test statistics with arbitrary correlation. In a simulation study, we compare the power of the trimmed weighted Simes test with the power of the weighted Bonferroni test and the untrimmed weighted Simes test. An additional result of this article ensures type I error rate control of the usual weighted Simes test under a weak version of the positive regression dependence condition for the case of two hypotheses. This condition is shown to apply to the two‐sided p‐values of one‐ or two‐sample t‐tests for bivariate normal endpoints with arbitrary correlation and to the corresponding one‐sided p‐values if the correlation is non‐negative. The Simes test for such types of bivariate t‐tests has not been considered before. According to our main result, the trimmed version of the weighted Simes test then also applies to the one‐sided bivariate t‐test with arbitrary correlation.     

Models and tests of linkage and association studies of quantitative trait locus for multi-allele marker Loci

In this paper, we explore models and tests for association and linkage studies of a quantitative trait locus (QTL) linked to a multi-allele marker locus. Based on the difference between an offspring's conditional trait means of receiving and not receiving an allele from a parent at marker locus, we propose three statistics T(m), T(m,row) and T(m,col) to test association or linkage disequilibrium between the marker locus and the QTL. These tests are composite tests, and use the offspring marginal sample means including offspring data of both homozygous and heterozygous parents. For the linkage study, we calculate the offspring's conditional trait mean given the allele transmission status of a heterozygous parent at the marker locus. Based on the difference between the conditional means of a transmitted and a nontransmitted allele from a heterozygous parent, we propose statistics T(parsi), T(satur), T(gen) and T(m,het) to perform composite tests of linkage between the marker locus and the quantitative trait locus in the presence of association. These tests only use the offspring data that are related to the heterozygous parents at the marker locus. T(parsi) is a parsimonious or allele-wise statistic, T(satur) and T(gen )are satured or genotype-wise statistics, and T(m,het) compares the row and column sample means for offspring data of heterozygous parents. After comparing the powers and the sample sizes, we conclude that T(parsi) has higher power than those of the bi-allele tests, T(satur), T(gen), and T(m,het). If there is tight linkage between the marker and the trait locus, T(parsi) is powerful in detecting linkage between the marker and the trait locus in the presence of association. By investigating the goodness-of-fit of T(parsi), we find that T(satur) does not gain much power compared to that of T(parsi). Moreover, T(parsi) takes into account the pattern of the data that is consistent with linkage and linkage disequilibrium. As the number of alleles at the marker locus increases, T(parsi) is very conservative, and can be useful even for sparse data. To illustrate the usefulness and the power of the methods proposed in this paper, we analyze the chromosome 6 data of the Oxford asthma data, Genetic Analysis Workshop 12.     

The score statistic of the LD-lod analysis: detecting linkage adaptive to linkage disequilibrium

We study the properties of a modified lod score method for testing linkage that incorporates linkage disequilibrium (LD-lod). By examination of its score statistic, we show that the LD-lod score method adaptively combines two sources of information: (a) the IBD sharing score which is informative for linkage regardless of the existence of LD and (b) the contrast between allele-specific IBD sharing scores which is informative for linkage only in the presence of LD. We also consider the connection between the LD-lod score method and the transmission-disequilibrium test (TDT) for triad data and the mean test for affected sib pair (ASP) data. We show that, for triad data, the recessive LD-lod test is asymptotically equivalent to the TDT; and for ASP data, it is an adaptive combination of the TDT and the ASP mean test. We demonstrate that the LD-lod score method has relatively good statistical efficiency in comparison with the ASP mean test and the TDT for a broad range of LD and the genetic models considered in this report. Therefore, the LD-lod score method is an interesting approach for detecting linkage when the extent of LD is unknown, such as in a genome-wide screen with a dense set of genetic markers.     

Atomic solvation parameters in the analysis of protein-protein docking results.

Several sets of amino acid surface areas and transfer free energies were used to derive a total of nine sets of atomic solvation parameters (ASPs). We tested the accuracy of each of these sets of parameters in predicting the experimentally determined transfer free energies of the amino acid derivatives from which the parameters were derived. In all cases, the calculated and experimental values correlated well. We then chose three parameter sets and examined the effect of adding an energetic correction for desolvation based on these three parameter sets to the simple potential function used in our multiple start Monte Carlo docking method. A variety of protein-protein interactions and docking results were examined. In the docking simulations studied, the desolvation correction was only applied during the final energy calculation of each simulation. For most of the docking results we analyzed, the use of an octanol-water-based ASP set marginally improved the energetic ranking of the low-energy dockings, whereas the other ASP sets we tested disturbed the ranking of the low-energy dockings in many of the same systems. We also examined the correlation between the experimental free energies of association and our calculated interaction energies for a series of proteinase-inhibitor complexes. Again, the octanol-water-based ASP set was compatible with our standard potential function, whereas ASP sets derived from other solvent systems were not.