1. Lecture 21: Quantitative Traits I Date: 11/05/02  Review: covariance, regression, etc  Introduction to quantitative genetics

2. Joint Density  Suppose you have two random variables X and Y. For each entity in the sample you take from the population you obtain a random pair (X i, Y i ).  The joint density function for two random variables is given by p(x,y) such that

3. Conditional Density  The conditional density function for the random variable Y conditional on a particular realization x of the random variable X is p(y|x).  The joint and conditional densities are related: p(x,y) = p(y|x)p(x), where p(x) is the marginal density

4. Independent Random Variables  If the random variables X and Y are independent, then p(x,y) = p(x)p(y).  As a consequence, we also have p(y|x) = p(y) and p(x|y) = p(x).  If X and Y are dependent, then the relationship between then is either linear or nonlinear. In either case, a linear relationship is a first approximation to the true relationship.

5. Conditional Expectation  The expectation of the product is given by  The conditional expectation is given by  If X and Y are independent, then

6. Covariance  Definition: The covariance of two random variables is defined as  and an covariance estimator is given by

7. Meaning of Covariance  The sign of the covariance implies something about how Y responds to changes in X or vice versa.

8. Regression and History  The use of a linear function as a first approximation of the relationship between two random variables is termed regression.  In fact, regression was first introduced in a genetics context.  Galton (1889) studied the average height of parents X and the height of offspring Y.

9. Least Squares Method  The least squares method finds estimates a and b for the coefficients of the assumed linear relationship by minimizing the mean squared error.  Because means, variances and covariances are available from phenotypic data, these estimates are particularly useful.

10. Homoscedasticity and Heteroscedasticity  Definition: If the variance in the residual error is constant regardless of the dependent variable x, then E(Y|X) is homoscedastic.  Definition: There is heteroscedasticity in the data if the residual variance depends on the value of the dependent variable x.  Transformations exist to achieve homoscedasticity.

11. Example: Regression  Suppose Cov(X,Y) = 10, Var(X) = 10, Var(Y) = 15 and the means E(X) = E(Y) = 0.  Regress X on Y and Y on X.  Then the intercept estimates is  and the slope estimates are

12. Correlation  Definition: The correlation of two random variables X and Y is defined as  with estimator  Correlations are scale independent:

13.  Correlations imply linear association.  r is the standardized regression coefficient that is obtained if x and y are scaled to have unit variance.  r 2 measures the proportion of the Var(Y) that is explained if E(Y|X) is linear. Correlation (cont)

14. Example: rats

15. Quantitative Traits  Definition: A quantitative trait is one with a continuous distribution. In other words, it is a trait that is measured not counted.  It is assumed that quantitative traits are controlled by many genes, each with small effect. Environmental effects are also important.  Definition: A quantitative trait locus (QTL) is locus controlling a quantitative trait.

16. Quantitative Trait – A Model  We start with a very simple scenario. Suppose there is one locus determining a quantitative trait. Suppose that there are only two alleles at this locus. We seek a model for this scenario. This model will have two parameters to account for the two degrees of freedom (when location is removed) among the 3 possible outcomes (genotypes).

17.  Let the phenotypic value of a particular genotype be z. When environment has an effect, z is a consequence of both the underlying genotype and the environment. We can write z = G + E. Here G is the genotypic value and it is the expected phenotypic value averaged over all environments.  Each of the three genotypes has an associated genotypic value. Phenotypic and Genotypic Value

18. Quantitative Trait – Model A 0(1+k)a2a2a

19. Quantitative Trait – Model B -a-ada

20. Model A – Parameter Meanings 0(1+k)a2a2a Value of kGenetic Interpretation 0 1 >1overdominance <-1underdominance

21. Example – Scaling Quantitative Trait  The Booroola (B) gene influences fecundity in Merino sheep. GenotypeMean Litter Size BB2.66 Bb2.17 bb1.48

22. Gene Content  Definition: The B 1 gene content of a genotype is the number of copies of the B 1 allele. The gene content for allele B 1 in genotype B 1 B 2 is 1.  At a single locus, the genotypic value is not a linear relationship on gene content, unless k=0. 0 a 2a2a 1 2 0 (1+k)a 2a2a

23. Partitioning Genotypic Value  Let N 1 be the number of B 1 alleles in the genotype.  Let N 2 be the number of B 2 alleles in the genotype.  Then, multiply regress the genotypic value on independent variables N 1 and N 2.  Assume again only two alleles, then N 1 = 2 – N 2. Call N 2 = N.

24. Predicted Genotypic Values  The predicted genotypic values are given as

25. Weighted Mean of  ’s is 0

26. Slope of Regression Line  Recalling the formula for the slope of a regression line, we have  We will now find expressions for the covariance and variance.

27. Derivation of Slope

28. Average Effect of Allelic Substitution  The previous derivations were completed under the assumption of random mating and HWE.  The slope  is the change in genotypic value associated with the addition of one more allele. To add one more B 2 allele, one must replace another B 1 allele with B 2, so it is also called the average effect of allelic substitution.  Except under additivity (k=0), this substitution effect can only be defined in terms of the population.

29. Partitioning Genetic Variance  Because we now have a linear function for genotypic value G  we can write the total genetic variance as  but there is no covariance term.

30. Additive and Dominance Components  The first term is additive genetic variance: the amount of variance of G that is explained by regression on N.  The second term is dominance genetic variance: the residual variance for the regression.  We seek an expression for both terms.

31. Derivation of Slope GenotypeB1B1B1B1 B1B2B1B2 B2B2B2B2 N012 G0(1+k)a2a2a Freq.* GN0(1+k)a4a4a N2N2 014 

32. Derivation of Genetic Variance Components

33. Genetic Variance Components  Both components depend on gene frequencies (conditional on population from which they are derived).  When k=0 (purely additive effects), then additive genetic variance is maximized when heterozygosity is maximized.  With dominance k>0, additive genetic variance is maximized at higher frequencies of the recessive allele. Rare recessive alleles cause little genetic variance because they are not often expressed.

34. Why?  Why have we partitioned the genotypic value into additive and dominance components?  When a parent transmits alleles to the offspring, the dominance deviation in the parent is irrelevant because only one gamete is transmitted.  May think of additive genetic component as the heritable component of an individual’s genotypic value.

35. Average Excess  There are multiple ways to measure the effect of an allele. The effect of allelic substitution  is one. The additive effect  i is another.  Definition: The average excess of an allele is the difference between the mean genotypic value of individuals carrying at least one copy of the allele and the mean genotypic value of a random individual form the entire population.

36. Average Excess with Random Mating

37. Breeding Value  Definition: The breeding value of an individual is the sum of the additive effects. GenotypeBreeding Value B1B1B1B1 2121 B1B2B1B2 1+21+2 B2B2B2B2 22

38. Breeding Value and Random Mating  Consider the expected genotypic values of progeny produced by parental genotypes. GenotypeBreeding Value Progeny Expected Genotypic Value Deviation B1B1B1B1 2121 ap 2 (1+k) 11 B1B2B1B2 1+21+2 a[p 2 + (1+k)/2]  1 +  2 )/2 B2B2B2B2 22 a[2p 2 + p 1 (1+k)] 22

39. How to Use this Analysis  A common approach today.  Identify candidate loci that are potential contributors to the variation of the trait of interest.  Genotype a random selection of individuals are identified by molecular markers.  Determine average phenotypic values within each genotypic class.  Estimate fraction of total phenotypic variation associated with candidate locus.

40. An Example  Consider the Booroola gene example in two random mating populations where the gene B is present at gene frequencies 0.5. BBBbbb G ij 2.662.171.48 P ij 0.250.500.25

41. Booroola (cont) ValueEstimate Mean genotypic value 2.120 Additive Effects  B = 0.295  b = -0.295 Breeding ValuesA BB = 0.59; A Bb = 0; A bb = 0.59 Dominance Deviations D BB = -0.05; D Bb = 0.05; D bb = -0.05

42. Booroola (cont) ValueEstimate Additive Genetic Variance 0.1740 Dominance Genetic Variance  Total Genetic Variance0.1765