HANDOUT: Basic Statistics for Quantitative Genetics

I.  Approach to modeling evolution of quantitative trait

    A.  Cannot model change in allele frequences at all loci involved

    B.  Instead, model how evolutionary change in mean value of trait is expected to occur based on knowledge
            of small number of measurable parameters:

            1.  Variance formula:   Var(x) = S f_xi(xi - xmean)²

            2.  Additive genetic variance, V_a

            3.  Phenotypic variance, V_p

            4.  heritability:  h² = V_a / V_p

    C.  Experimental methods for estimating these parameters

            1.  Half-sib breeding experiment

            2.  Parent-offspring regression and mid-parent-offsprfing regression

                          a. regression is the best-fit line

                          b.  Relationship between student and parent height

    D.  The selection differential, s

          1.  When all selection occurs as differential survival, s is the difference between mean of character
                    after selection and the mean of the character before selection.

            2.  More generally, s is the covariance of the character and fitness

                    a.  Definition: Cov(x,y) = S f_xi,yi(xi - xmean)(yi - ymean)

                    b.  For a trait x, s = Cov(W,x)

    E.   The response to selection, R

          1.  R is the change in mean of character from one generation to next

            2.  R = h²s  (Breeder's equation)

II.  Testing accuracy of Breeder's equation

    A.  Experience of breeders indicates qualitative accuracy

    B.  Test of quantitative accuracy (Clayton, Morris and Robertson 19XX)

            1. Trait: bristle number in Drosophila melanogaster

          2. Heritability estimates:  h² = 0.52

            3. Selection experiment

                    a.  Response to selection in 20/100 replicate lines

                  b.  Mean response to selection in different treatments

III. Selection on quantitative traits in natural populations

    A.  Beak size in Galapagos finches (Geospiza fortis).  Grant and Grant, XXX

                1.  Natural history

                2.  Heritability of beak depth

                3. Effects of 1976-1977 drought

                4.  Evolutionary response to selection
 

       B .  Alpine skypilots (see Textbook)
 
 
 

IV.  Multivariate evolution

     A.  Matrix algebra, etc.

          1.  vector representation of multivariate phenotype

          2.  matrix representation of additive genetic variances and covariances

          3.  vector representation of selection gradients

          4.   Multiplying a matrix by a vector.

     B.  Generalization of breeder's equation:

          1.  Multivariate equation for response to selection

          2.  The effect of genetic correlations on response to selection