Simulating the component counts of combinatorial structures.

This article describes and compares methods for simulating the component counts of random logarithmic combinatorial structures such as permutations and mappings. We exploit the Feller coupling for simulating permutations to provide a very fast method for simulating logarithmic assemblies more generally. For logarithmic multisets and selections, this approach is replaced by an acceptance/rejection method based on a particular conditioning relationship that represents the distribution of the combinatorial structure as that of independent random variables conditioned on a weighted sum. We show how to improve its acceptance rate. We illustrate the method by estimating the probability that a random mapping has no repeated component sizes, and establish the asymptotic distribution of the difference between the number of components and the number of distinct component sizes for a very general class of logarithmic structures.

Significance testing for direct identity-by-descent mapping.

Direct identity-by-descent mapping is a technique for narrowing down the location of the gene or genes responsible for a given genetic disease to small segments of the genome. The technique involves DNA comparisons between pairs of affected individuals. The data generated are in the form of matching segments of the genome, representing regions likely to be identical-by-descent (IBD). Regions in the genome over which there are significantly more segments aligned than is expected by chance are taken as candidate regions for the disease gene or genes. Due to the complex geometric nature of the data, significance testing involves certain mathematical difficulties. We present here a new method for measuring this significance. This method introduces a novel statistic and is appropriate whether or not the relationships between the paired individuals are known. We give examples that we have calculated by implementing this method, including an application to real data.

A sibship test for linkage in the presence of association: the sib transmission/disequilibrium test.

Linkage analysis with genetic markers has been successful in the localization of genes for many monogenic human diseases. In studies of complex diseases, however, tests that rely on linkage disequilibrium (the simultaneous presence of linkage and association) are often more powerful than those that rely on linkage alone. This advantage is illustrated by the transmission/disequilibrium test (TDT). The TDT requires data (marker genotypes) for affected individuals and their parents; for some diseases, however, data from parents may be difficult or impossible to obtain. In this article, we describe a method, called the "sib TDT" (or "S-TDT"), that overcomes this problem by use of marker data from unaffected sibs instead of from parents, thus allowing application of the principle of the TDT to sibships without parental data. In a single collection of families, there might be some that can be analyzed only by the TDT and others that are suitable for analysis by the S-TDT. We show how all the data may be used jointly in one overall TDT-type procedure that tests for linkage in the presence of association. These extensions of the TDT will be valuable for the study of diseases of late onset, such as non-insulin-dependent diabetes, cardiovascular diseases, and other diseases associated with aging.

Linkage-disequilibrium mapping without genotyping.

Genomic mismatch scanning (GMS) is a technique that enriches for regions of identity by descent (IBD) between two individuals without the need for genotyping or sequencing. Regions of IBD selected by GMS are mapped by hybridization to a microarray containing ordered clones of genomic DNA from chromosomes of interest. Here we demonstrate the feasibility and efficacy of this form of linkage-mapping, using congenital hyperinsulinism (HI), an autosomal recessive disease, whose relatively high frequency in Ashkenazi Jews suggests a founder effect. The gene responsible (SUR1) encodes the sulfonylurea receptor, which maps to chromosome 11p15.1. We show that the combination of GMS and hybridization of IBD products to a chromosome-11 microarray correctly maps the HI gene to a 2-Mb region, thereby demonstrating linkage-disequilibrium mapping without genotyping.

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.

Heterozygosity mapping of partially congenic lines: mapping of a semidominant neurological mutation, Wheels (Whl), on mouse chromosome 4.

We identified a semidominant, chemically induced, mouse mutation with a complex array of abnormal behaviors including bidirectional circling and hyperactivity, abnormal circadian rhythmicity and abnormal responses to light. In this report, we genetically and phenotypically characterized the circling/waltzing component of the abnormal behavior. We mapped the locus controlling this trait by heterozygosity mapping of partially congenic lines carrying the mutagenized chromosome outcrossed to different inbred strains for three generations. Analysis of 68 PCR-based markers in 13 affected individuals indicated that the mutant locus, named Wheels (Whl), resides in the subcentromeric portion of mouse chromosome 4. The statistical evaluation of data obtained by heterozygosity mapping validates this efficient mapping approach. Further characterization of the Whl mutation demonstrated that Whl/Whl homozygotes die during embryonic life and that the penetrance of circling behavior depends on genetic background. Morphological analysis of the inner ears of Whl/+ mice revealed a variable number of abnormalities in the sensory and nonsensory portions of their semicircular canals. Abnormalities ranged from slight atrophy of one or more cristae to complete absence of the lateral crista and canal. The molecular characterization of the gene disrupted in the Whl mutation will provide insight into developmental mechanisms involved in inner ear formation.

The TDT reveals linkage and linkage disequilibrium in a rare disease.

We used the TDT as the basis for our analysis of data from Problem 1 of GAW9. Among the 360 marker loci on six chromosomes, we searched for any that might show both linkage and allelic association with the disease. We applied the TDT to each allele at every marker locus and found strong evidence for linkage in two regions: one on chromosome 1, another on chromosome 5.

Transmission test for linkage disequilibrium: the insulin gene region and insulin-dependent diabetes mellitus (IDDM).

A population association has consistently been observed between insulin-dependent diabetes mellitus (IDDM) and the "class 1" alleles of the region of tandem-repeat DNA (5' flanking polymorphism [5'FP]) adjacent to the insulin gene on chromosome 11p. This finding suggests that the insulin gene region contains a gene or genes contributing to IDDM susceptibility. However, several studies that have sought to show linkage with IDDM by testing for cosegregation in affected sib pairs have failed to find evidence for linkage. As means for identifying genes for complex diseases, both the association and the affected-sib-pairs approaches have limitations. It is well known that population association between a disease and a genetic marker can arise as an artifact of population structure, even in the absence of linkage. On the other hand, linkage studies with modest numbers of affected sib pairs may fail to detect linkage, especially if there is linkage heterogeneity. We consider an alternative method to test for linkage with a genetic marker when population association has been found. Using data from families with at least one affected child, we evaluate the transmission of the associated marker allele from a heterozygous parent to an affected offspring. This approach has been used by several investigators, but the statistical properties of the method as a test for linkage have not been investigated. In the present paper we describe the statistical basis for this "transmission test for linkage disequilibrium" (transmission/disequilibrium test [TDT]). We then show the relationship of this test to tests of cosegregation that are based on the proportion of haplotypes or genes identical by descent in affected sibs. The TDT provides strong evidence for linkage between the 5'FP and susceptibility to IDDM. The conclusions from this analysis apply in general to the study of disease associations, where genetic markers are usually closely linked to candidate genes. When a disease is found to be associated with such a marker, the TDT may detect linkage even when haplotype-sharing tests do not.

An optimizing principle of natural selection in evolutionary population genetics.

This paper brings together two themes in evolutionary population genetics theory. The first concerns Fisher's Fundamental Theorem of Natural Selection: a recent interpretation of this theorem claims that it is an exact result, relating to the so-called "partial" increase in mean fitness. The second theme concerns the desire to find an optimality principle in genetic evolution. Such a principle is found here: of all gene frequency changes which lead to the same partial increase in mean fitness as the natural selection gene frequency changes, the natural selection values minimize a generalized distance measure between parent and daughter generation gene frequency values.

Statistical analysis of in situ hybridization data: derivation and use of the zmax test.

There are many situations in which grain distributions resulting from in situ hybridization of radioactively labeled probes to unique genes should be subjected to a statistical analysis. However, the problems posed by analysis of in situ hybridization data are not straightforward, and no completely satisfying method is currently available. We have developed a procedure in which the major and any number of minor site(s) of hybridization may be specifically located and the significance of each tested. This zmax procedure first tests the overall distribution for departure from randomness and then identifies significantly overlabeled whole chromosomes (or chromosome arms or other large segments), a process that may be repeated to pinpoint significantly overlabeled regions within these chromosomes. We describe in detail the derivation of the zmax statistic, present tables of significant zmax levels, and show with examples how zmax is used in tests of significance of in situ hybridization data.

Genome mapping with anchored clones: theoretical aspects.

As part of our effort to construct a physical map of the genome of Arabidopsis thaliana we have made a mathematical analysis of our experimental approach of anchoring yeast artificial chromosome clones with genetically mapped RFLPs and RAPDs. The details of this analysis are presented and their implications for mapping the Arabidopsis genome are discussed.

An interpretation and proof of the Fundamental Theorem of Natural Selection.

Fisher's "Fundamental Theorem of Natural Selection" has long caused controversy in population genetics theory. Viewed as a statement about the increase, or rate of increase, of mean fitness over time, it encounters difficulties with cases arising in a multi-locus system for which mean fitness can decrease. An interpretation of the theorem is put forward here which implies that it is correct as a mathematical statement, but of less biological value than was claimed by Fisher.

Is knowing the age-order of alleles in a sample useful in testing for selective neutrality?

The most powerful, and most frequently used, test of selective neutrality, based on data consisting of observed allelic frequencies in a sample of genes at some locus, is the procedure of G. A. Watterson. This procedure uses the sample homozygosity F* as the test statistic, and in effect leads to rejection of the hypothesis of selective neutrality if the observed value of F* differs significantly from neutral theory expectations. The homozygosity statistic is invariant under relabeling of the alleles and thus cannot use any further information on the alleles which might be available. We present results which suggest that information concerning the age order of the alleles cannot be used to provide a more powerful testing procedure than that of Watterson.

Remarks on ascertainment.

Genetic Analysis Workshop data come from various sources, with possibly different ascertainment procedures in each. It is plausible that classical ascertainment models do not accurately describe the ascertainment process in each data source. For this reason, we review the biases which can arise from an incorrect specification on the ascertainment process and discuss their relevance for the GAW5 data analysis.

A resolution of the ascertainment sampling problem. II. Generalizations and numerical results.

The ascertainment problem arises when families are sampled by a nonrandom process and some assumption about this sampling process must be made in order to estimate genetic parameters. Under classical ascertainment assumptions, estimation of genetic parameters cannot be separated from estimation of the parameters of the ascertainment process, so that any misspecification of the ascertainment process causes biases in estimation of the genetic parameters. Ewens and Shute proposed a resolution to this problem, involving conditioning the likelihood of the sample on the part of the data which is "relevant to ascertainment." The usefulness of this approach can only be assessed by examining the properties (in particular, bias and standard error) of the estimates which arise by using it for a wide range of parameter values and family size distributions and then comparing these biases and standard errors with those arising under classical ascertainment procedures. These comparisons are carried out in the present paper, and we also compare the proposed method with procedures which condition on, or ignore, parts of the data.

A resolution of the ascertainment sampling problem. III. Pedigrees.

When nuclear families are sampled by an ascertainment procedure whose properties are not known, biased estimates of genetic parameters will arise if an incorrect specification of the ascertainment procedure is made. Elsewhere we have put forward a resolution of this problem by introducing an ascertainment-assumption-free (AAF) method, for nuclear family data, which gives asymptotically unbiased estimators no matter what the true nature of the ascertainment process. In the present paper we extend this method to cover pedigree data. Problems that arise with pedigrees but not with families--for example, the question of which families in a pedigree are "ascertainable"--are also considered. Comparisons of numerical results for pedigrees and nuclear families are also made.