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5. 0192. ✷ More generally, if n discrete random variables X1 , . . , Xn are independent, it follows that P(X1 = x1 , X2 = x2 , . . , Xn = xn ) = P(X1 = x1 )P(X2 = x2 ) . . 24) for any sequence x1 , . . , xn of observed values. For independent continuous random variables X1 , X2 , . . , Xn , we must use probabilities instead of intervals. If h1 , . . , hn are small positive numbers, then12 P(X1 ∈ [x1 , x1 + h1 ], X2 = [x2 , x2 + h2 ], . . , Xn = [xn , xn + hn ]) ≈ fX1 (x1 )fX2 (x2 ) . . fXn (xn )h1 h2 .

No genetic model needs to be specified, and this is an advantage for complex diseases such as inheritable diabetes and psychiatric disorders. 3 Linear regression Depending on the causal connections between two variables, X and Y , their true relationship may be linear or nonlinear. In any case, a linear model can always be used as a first approximation to the true pattern of association. e. 7) ✕ where ✡ is the y-intercept, is the slope of the line, the regression coefficient, and e is the residual error.

2 Two-Point Linkage Analysis We use the term two-point linkage analysis for analysis of linkage between two genes, usually, but not necessarily, a disease gene and a marker gene. The parameter of interest is the recombination fraction θ. g. 5. The co-segregation of disease- and marker alleles in a pedigree can be summarized in the likelihood function which measures the support, given by the data, for different θ-values. 1 Analytical likelihood and lod score calculations In this section we define the basis for parametric linkage analysis, and to keep things mathematically tractable we deliberately make assumptions that may not always be 59 60 CHAPTER 4.

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