Complex genetic diseases exhibit varying degrees of familial aggregation but do not fit simple Mendelian patterns of inheritance. Presumably, genes and environment both contribute to the occurrence of these diseases. Since genes act in only a few, well-understood ways, it seems logical to identify genetic factors first, then use them to sort out environmental contributions. Genetic linkage analysis and segregation analysis represent two powerful tools for identifying specific genes, detecting genetic heterogeneity and predicting recurrence risk. Greater understanding of both environmental and genetic contributions to disease will lead in turn to improved counseling, prevention, and eventually cure. The work in this Continuation will attack critical methodological and mathematical problems relating to the genetic epidemiology of common complex diseases, such as diabetes and other autoimmune disorders. These problems and complications, arising in linkage and segregation analysis, are: (I) Two-locus models are of increasing importance for a number of human diseases, including IDDM (insulin-dependent diabetes). Part I examines linkage analysis of two-locus models and determines the extent to which a disease-marker association can mimic linkage. It also investigates two- locus models of genetic heterogeneity. (II) How families are selected ("ascertained") for segregation analysis can profoundly affect the results of that analysis, yet how to correct for ascertainment remains a difficult, only partly solved problem. Part II develops a new nonparametric ascertainment correction and considers two other critical ascertainment problems as well. (III) Numerous complications arise when performing linkage analysis on complex diseases. Part III focuses on linkage strategies, including informativeness of different types of families and the sensitivity of lod scores to changes in the data, and the examination of generally accepted myths in the field so as to evaluate their accuracy. The final section of the proposal focuses on linkage strategies and will tie together all other parts. Throughout, the work will both (a) determine the extent of problems and (b)develop solutions to these problems. Solutions will require both theoretical tools and computer simulation. Moreover, the project will not necessarily be restricted to the specific areas detailed above but will also devote effort to new problems of pressing importance if they arise during the grant period.