* Correspondence to: Jane M. Olson, Department of Epidemiology and Biostatistics, Rammelkamp Center for Educa-tion and Research, 2500 MetroHealth Drive, Case Western Reserve University, Cleveland, Ohio 44109, U.S.A. E-mail:[email protected]/grant sponsor: National Center for Human Genome ResearchContract/grant number: HG01577Contract/grant sponsor: National Center for Research ResourcesContract/grant number: PR03655Contract/grant sponsor: National Cancer InstituteContract/grant number: CA73270Contract/grant sponsor: National Institute of General Medical SciencesContract/grant number: GM28345CCC 0277}6715/99/212961}21$17.50 Received June 1998Copyright  1999 John Wiley & Sons, Ltd. Accepted January 1999STATISTICS IN MEDICINEStatist. Med. 18, 2961 } 2981 (1999)TUTORIAL IN BIOSTATISTICSGENETIC MAPPING OF COMPLEX TRAITSJANE M. OLSON*, JOHN S. WITTE AND ROBERT C. ELSTONDepartment of Epidemiology and Biostatistics, Rammelkamp Center for Education and Research, MetroHealth Campus,Case Western Reserve University, Cleveland, Ohio, U.S.A.SUMMARYStatistical genetic mapping methods are powerful tools for "nding genes that contribute to complex humantraits. Mapping methods combine knowledge of the biological mechanisms of inheritance and the random-ness inherent in those mechanisms to locate, with increasing precision, trait genes on the human genome.We provide an overview of the two major classes of mapping methods, genetic linkage analysis andlinkage disequilibrium analysis, and related concepts of genetic inheritance. Copyright  1999 John Wiley& Sons, Ltd.1. INTRODUCTIONIn recent years, genetic study of complex human traits has increased dramatically. Most humandiseases are now viewed as having some genetic component, and considerable e!ort is being madeto "nd and study the genes involved. As a result, statistical methods used to "nd disease genes arereceiving a great deal of attention, and improvements in methodology are continually beingproposed. In this article we provide an overview of genetic mapping methods. We "rst explainconcepts in genetic inheritance, focusing primarily on the genetic mechanisms that investigatorsexploit in genetic mapping, and introduce relevant terminology. We then introduce the reader tothe two main areas of mapping methodology: genetic linkage analysis and linkage disequilibriummapping.2. GENETIC TERMINOLOGY AND LINKAGE CONCEPTS2.1. Genetic ModelsSimple genetic models are derived from Mendelian laws of inheritance. Each individual has twosets of 23 chromosomes, one maternal and one paternal in origin. One of the 23 pairs ofchromosomes are the sex chromosomes, and we shall concern ourselves with the remaining 22pairs of autosomal chromosomes in this tutorial. Each chromosome consists of a long strand ofDNA, a linear molecule with units known as base pairs. A chromosomal location (which may bea single base pair or a collection of consecutive base pairs) is termed a genetic locus. At each locus,there may be distinct variants, called alleles. In common parlance, the term gene is often used todenote both locus and allele, but the two should be regarded as distinct concepts by thestatistician. For an individual, the pair of alleles (maternal and paternal) at a locus is called thegenotype. A genotype is called homozygous if the two alleles are the same allelic variant andheterozygous if they are di!erent allelic variants. If more than one locus is involved, the pattern ofalleles for a single chromosome is called a haplotype; together, the two haplotypes for anindividual is still called a (multilocus) genotype. Each o!spring receives at each locus only one ofthe two alleles from a given parent; alleles are transmitted randomly (that is, each with probability1/2), and o!spring genotypes are independent conditional on the parental genotypes. Theprobability that a parental genotype transmits a particular allele or haplotype to an o!spring iscalled the transmission probability, and is the "rst component of a genetic model.The second component of a genetic model concerns the relationship between the (unobserved)genotypes and the observed characteristics, or phenotype, of an individual. The phenotype may bediscrete or continuous. We de"ne penetrance to be the probability (mass or density) of a pheno-type given a genotype; a complete genetic model requires speci"cation of the penetrances of allpossible genotypes. The third component of a genetic model is the (distribution of) relativefrequencies of the alleles in the population. These allele frequencies are used primarily todetermine prior probabilities of genotypes when inferring genotype from phenotype.These three components, taken together, fully describe the genetic model of a trait. Given a setof phenotypic data on pedigrees, one can estimate the genetic model using statistical techniquescollectively known as segregation analysis.  While segregation analysis is beyond the scope ofthis paper, it is helpful to realize that in a segregation analysis, genotypes are latent variablesinferred from trait phenotypes. For simple Mendelian traits, in which only one genetic locus issegregating, estimation of the genetic model is usually straightforward, as only one set of latentvariables (genotypes) is involved. For complex traits, which are the emphasis of most geneticstudies today and which are probably due to the e!ects of more than one locus, estimation of thegenetic model is di$cult to impossible, because each locus represents a di!erent set of (possiblyinteracting) latent variables. As a result, two approaches to genetic linkage analysis have evolved:those that require prior speci"cation of a genetic model for the trait under study (model-basedmethods), and those that do not (model-free methods). For a more detailed review of geneticmodels and genetic likelihoods, see Thompson. We now discuss concepts speci"c to linkageanalysis.2.2. Recombination and LinkageTwo loci that are on the same chromosome are said to be syntenic. If they are close enoughtogether, the alleles at the two loci that are paternal (maternal) in origin tend to pass to the same2962 J. M. OLSON, J. S. WITTE AND R. C. ELSTONCopyright  1999 John Wiley & Sons, Ltd. Statist. Med. 18, 2961}2981 (1999)Figure 1. Diagram showing part of two homologous chromosomes at the time of gamete formation. In (a), the twochromosomes have paired up and each chromosome (parental solid, maternal open) has divided into two identicalchromatids.