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Statistics is the science that formalizes the process of making inferences from observations. Basic to this process of formalization is the concept of a statistical model. In general, a statistical model is an attempt to provide a mathematical simplification of the mechanism that produced the observations. Statistical models are useful since they allow investigation and optimization of the process of analyzing the observations in a context that is wider than the context of the outcome of the specific trial that is being analyzed. For example, it opens the door to considerations such as: “Had we had the opportunity to try a specific inferential procedure on other datasets, all generated by the same statistical mechanism as the one we observe, how would our procedure perform on the average? What would be the probability of drawing an incorrect conclusion?” Such questions are impossible to address unless we adopt a wider point of view.

Variability in observed phenotypes may result from a variety of factors — inherited as well as environmental. The blend of all these influences gives rise to the unique being every living creature is. Still, the main role of science is to identify the rules which unite seemingly unrelated phenomena. The role of genetics is no different. Its first, and most important, task is to identify the major factors that give rise to different phenotypical characteristics. Once these major factors have been identified, the investigation can be carried on in order to identify genes having secondary effects.

This chapter introduces some concepts of population genetics that are used in various places later in the book. In addition it allows us to expand our knowledge of R functions and to practice conditional probability arguments, which are used here to analyze the genetic makeup of the present generation by conditioning on that of the parental generation. The reader who has a working knowledge of R may wish to skip Sect. 3.1 and the latter part of Sect. 3.2 that refers to Sect. 3.1. The first part of Sect. 3.2 (including the programs “meiosis.rec” and “cross.rec”) introduces the concept of recombination, which plays a very important role throughout the book. It can be read independently of Sect. 3.1. Sections 3.3 and 3.4 are important for the third part of the book that deals with human genetics. This chapter contains some relatively difficult mathematical material, which is marked by an asterisk (*) and can be omitted.

In this chapter we introduce the basic principles behind testing a genetic marker for linkage to a quantitative phenotype. We split the presentation into two parts. In the first part we deal with the case of a marker located at a quantitative trait locus (QTL). The parameters that determine the statistical properties of the test are the parameters of the regression model that relates the observed phenotype to the QTL as presented in Chap. 2. Later in this chapter we extend the investigation to the case of a marker in the vicinity of, but not necessarily on top of, the QTL, so the recombination fraction also plays a role. Fortunately the effect of recombination can be separated from the effect of the QTL. From Chaps. 5 onward we discuss the case, which is the common practice today, of testing a set of markers for linkage.

The task of mapping loci related to a trait is carried out in a number of steps. The aim of the first step is the identification of the chromosome carrying the trait-related polymorphism. Within that chromosome, one finds a wide region which is likely to contain the genetic factor. The second step involves the narrowing down of the region. Finally, once the functional polymorphism has been barricaded within a small enough chromosomal region, the direct examination of polymorphic loci is carried out. An experimental design, which is efficient for one step, may be less efficient for a subsequent step. Therefore, it is useful to apply different experimental resources for each step. Likewise, the statistical tools and the analytic theory may vary from one step to the next. In this and the next chapters we will concentrate on the statistical considerations that are important for the first step: a whole genome scan aimed at the identification of a (relatively wide) chromosomal region containing the functional polymorphism.

The significance level is the determining factor in the specification of the rejection region of a statistical test. Only the distribution under the null assumption of no signal plays a role in setting the level of the threshold, once the test statistic and the general form of test are decided upon. However, after setting that threshold, one can examine other statistical properties of the resulting test. A central property is statistical power of the test — the probability to reject the null hypothesis when a signal is present. Since this probability depends on the values of the parameters, one often speaks of the power function to emphasize this dependence. For a test at a single marker, this probability is obtained approximately from the normal distribution; it is a function of the noncentrality parameter given by (4.6). In this chapter we will examine the concept of power for a whole-genome scan.

In the preceding chapters we have assumed that all marker genotypes are known exactly. In practice there are often missing genotypes; and even when the genotypes of actual markers are not missing, one can regard the positions between actual markers as potential markers, the genotypes of which are missing. One can then make an attempt to reconstruct the missing information by statistical means. For important reasons that will become apparent in Chap. 9, problems of missing information are even more acute in outbred populations, and in particular humans. In this chapter we introduce, in the relatively simple situation of crosses between inbred strains, some basic techniques that are particularly useful for dealing with different kinds of missing information that arise in problems of gene mapping. In later chapters we will discuss more general approaches that can handle more complex situations.

In this chapter we introduce briefly a number of more advanced topics, some of which are of current research interest. Some of the topics involve multidimensional parameters and are closely related to Probs. 5.4 and 6.4, which are concerned with detection of linkage in an intercross when the model permits both an additive and a dominance effect. (See also Prob. 4.3.) In these cases the new problems can be solved by conceptually straightforward extensions of results derived previously. In other cases the analysis becomes substantially more difficult and the results less conclusive, so the discussion here is at best an introduction.

In humans we cannot create inbred lines, backcrosses, etc. Consequently, it is more difficult to study directly the correlation of phenotypes and genetic markers. We can proceed indirectly by noting that relatives frequently have more similar phenotypes than non-relatives, presumably because they have more similar genotypes. For studying human diseases, particularly convenient units are affected sib pairs (ASP), which are the subject of this chapter. We delay until Chap. 11 a discussion of the substantially more complex problem of pedigrees involving variable numbers and relationships of affecteds.

Admixture occurs when two or more populations merge to form a new population. A classical example in humans is the African-American population. The African population and the European Caucasian population have diverged during thousands of years of largely separate evolution. During the late eighteenth and early nineteenth centuries, substantial numbers of Africans were brought to the region that became the United States of America, where most lived as slaves of the Caucasian population. Although interracial marriages were rare, substantial genetic blending between those two populations did occur (in addition to some blending with the Native American population). It is estimated that about 20% of the genetic material in today’s African-American population originated from a non-African, predominantly a Caucasian, source.