Lecture 27: Quantitative Genetics I. The Basic Statistical Foundations

(version 24 April 2003)

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Quantitative genetics: the nature of continuous characters

Fisher (1918) building on the work of plant breeders (1900's - 1910) suggested the polygenetic model.

• Many characters have a large number of loci underlying them.

• The underlying genetic variation between individuals, coupled with environmental variation generates a continuous distribution of character values.

As an example, suppose each capital letter allele adds one to the character value. For one and two loci, the distribution of genotypic values are

For 10 loci of equal effect, each with the frequency of the capital letter allele being 1/2,

If we cannot determine the genotype of an individual, we would at least like to predict what the resemblance between relatives should be.

Why? We wish to predict the response to selection

The Basic Model

z = G + E

• Phenotypic value (z) = genetic value (G) + environmental value (E)

• Nature versus nurture: how much of the resemblance is due to shared genetic values (shared G values) and how much due to shared environmental values (shared E values)?

• As we show shortly, the genetic value G can be broken down into several components, i.e., G = u + A + D

• Key is that we can use the resemblance between relatives to estimate the population variances of the genetic and environmental components. From these we can then use these variances to predict the resemblance between different sets of relatives.

Before moving on, we need ...

A Brief Statistical Aside: Variances, Covariances, and Regressions

The Variance, Var(x)

The Variance of a random variable x provides a measure of the spread of the data around its mean value (E[x]), with

• A small variance implies that data the falls closely around the mean

• A large variance implies that the data is widely distributed about the mean

If the variance is large, there is poor predictability.

Example of computing Var

Suppose x takes on the values 1, 2, 3 with probability 0.2, 0.3, and 0.5 (respectively).

• E [ x ] = 1*0.2 + 2*0.3+ 3*0.5 = 2.3
• E [ x² ] = 1²*0.2 + 2²*0.3+ 3²*0.5 = 5.9
• Var (x) = E [ x² ] - ( E [ x ] )² = 5.9 - 2.3² = 0.61

The Covariance, Cov(x,y)

Define the Covariance between two random variables x and y as

• Cov(x,y) > 0. Individuals with a large x value also tend to have a large y value

• Cov(x,y) < 0. Individuals with a large x value also tend to have a small y value

• Cov(x,y) = 0. No linear association between x and y

• E [ x ] = 0*(0.1+0.3) + 1*(0.2 + 0.4) = 0.6
• E [ y ] = 0*(0.1+0.2) + 1*(0.3 + 0.4) = 0.7
• E [ x*y ] = 0.1*0*0 + 0.3*0*1 + 0.2*1*0 + 0.4*1*1 = 0.4
• Cov(x,y) = E [ x*y ] - E [ x ]*E [ y ] = 0.4 - 0.6*0.7 = -0.02

An example of a situation where Cov(x,y) = 0, but x is entirely predicted by knowing y is a parabola. Note here, however, that there is no linear association between x and y.

If the random variables x and y are uncorrelated, then Cov(x,y) = 0.

Two variables that are independent are uncorrelated .

Properties of Covariances

1. Cov(x,y) = Cov(y,x)

2. Cov(a*x,y) = a Cov(x,y) (for any constant a)

3. Cov(x,y+z) = Cov(x,y) + Cov(x,z)

4. Cov(a+x,y) = Cov(x,y) (for any constant a)

5. Cov(x,x) = Var(x)
   

   Correlation, p(x,y)

   Two very closely related statistics are the regression of one variable on another and the correlation between two variables. The correlation p(x,y) provides a scaled measure of association

   p(x,y) = Cov(x,y) / [ Var(x) * Var(y) ] ^1/2

   Note that this implies

   Cov(x,y) ² = p(x,y) ² [ Var(x) * Var(y) ]

   The correlation ranges from +1 (perfectly positively correlated) to -1 (perfectly negatively correlated).

   The advantage of using the correlation is that we can compare the strength of linear relationships between different pairs of variables.

   Compare the scatterplot below with the one shown under the covariance discussion. Clearly, the linear association between the two variables is much stronger in the plot below. However, the plot above could easily have a larger covariance if the associated variances are larger. However, the correlation for the below graph is much greater than the correlation for the above graph.

   
   

   The Regression of y on x

   Finally, we can make explicit use of the covariance between two variables in obtaining the best linear predictor of the value of y given we know x. This is also called the regression of y on x. For example, suppose we know an individual's height (x). What can we say about their weight (y)?

   As a second example, consider the following data where each point represents the average value of a character in the parents and the average value of the character in their offspring, giving a parent-offspring regression

   

   Mathematically, we express the best linear relationship of y given we know x as

   y = a + b _{y | x} x

   For example, with human height, if your parents have height x, the predicted (average) height in your sibs is y = 23.4 + 0.65 x.

   • Hence, if your parents are 60 and 66 inches, their average height is 63 inches
   • The predicted height of their offspring is 23.4 + 0.65*63 = 64.35 inches
   • Obviously, there is a distribution of values around this expected value.

   The slope of the regression of y on x follows from the covariance, with

   b _{y | x} = Cov(x,y) / Var(x)

   Likewise the intercept a is given by

   a = E[y] - b _{y | x} E[x]

   

   How well does the regression predict the value of y?

   Recall that Var(y) provides a measure of its predictability.

   It can be shown that the variation in y given we observe x is

   Var(y | know x) = (1-p(x,y)²)Var(y)

   Hence, the fraction of the total variance in y accounted for by knowing x is p(x,y)².

   For example, for the human-height data, it can be shown that the correlation is 0.65.

   • Hence, 0.65 ² = 42.25%.
   • Thus knowing the height of your parents reduces the uncertainly in your height by 42.25%
   

   Fisher's Decomposition of the Genetic Value

   R. A. Fisher (1918) showed that the genetic value G can be further decomposed as

   G = u +A + D

   where

   • u = Population mean

   • A = Additive genetic value, also called the Breeding value

   • D = Dominance value

   • There are additional terms if epistasis is present (for example, if the genotypic value of a two locus genotype AaBb does not equal the average value of Aa genotypes + the average value of Bb genotypes).

   Hence,

   z = G + E = u + (A+D) + E

   Fisher's great insight :

   Information for relatives allows us to estimate Var(A), Var(D). Using these estimates, we can predict the resemblance between different sets of relatives.

   

   Genetic Covariance between relatives

   Consider the covariance between the genotypic values of two relatives (R1 and R2). By construction A and D are uncorrelated.. Hence,

   Cov(G_R1, G_R2) = Cov(A_R1+D_R1, A_R2+ D_R2)

   = Cov(A_R1, A_R2) + Cov(D_R1, D_R2)

   If two relatives share only one allele ibd, then

   • Cov(A_R1, A_R2) = Var(A)/2
   • Cov(D_R1, D_R2) = 0

   If two relatives share both alleles ibd, then

   • Cov(A_R1, A_R2) = Var(A)
   • Cov(D_R1, D_R2) =Var(D)

   Just what do A and D represent?

   

   Additive effects

   Consider the genotypic value for a given locus with (potentially) many alleles, B₁, ..., B_k.

   (In a random-mating population), The additive effect of an allele B_i is simply the mean genotypic value for an individual carrying a copy of allele B_i.

   Hence, the predicted genotypic value of B_i B_j is

   Predicted[G_ij]= u + a_i + a_j

   The variance Var(A) = 2 Var(a), is called the Additive genetic Variance.

   Of course, the above are predicted values. The difference between the predicted and actual values for the genotypic value at each locus is defined as the dominance deviation

   D _ij = G_ij - Predicted[G_ij] = G _ij - ( u + a_i + a_j)

   The variance in the dominance deviations, Var(D), is called the Dominance Variance.

   Thus to obtain A and D, we simply sum over all loci.

   

   Breeding Values

   A is called the breeding value (BV).

   • The average breeding value of a random individual is zero

   • The expected offspring mean for two parents is the population mean plus the average of the parental breeding values.
     • If Fred's BV is 20 and Mary's is -15, then their offspring are expected to have an average value that is (20 - 15)/2 = 2.5 units above the population mean

   • The expected offspring mean for one parent is the population mean plus the half the parental breeding value
     • Offspring with Fred as their father are expected to be 20/2 = 10 units above the population mean
     • Offspring with Mary as their mother are expected to be (-15)/2 = 7.5 units below the population mean