1. Lecture 22: Quantitative Traits II
Date: 11/07/02
Review: quantitative genetics terms
Simple tests for QTL localization
Review project 1

2. Quantitative Genetics Terminology
Definition
Under HWE
additive effect
least-squares regression coefficients
a1 = p1a
a2 = -p2a
average excess
difference in mean genotypic values of carriers of at least one copy of allele and random individual
breeding value
sum of additive effects of an individual’s genes
2a1
a1+a2
2a2

3. Average Genotypic Value of Offspring
Consider a random mating population and consider the average genotypic value of offspring of each genotype.
Subtract mG
B1B1
B1B2
B2B2
0p1 + a(1+k)p2
0.5a(1+k)p a(1+k)p1 + ap2
a(1+k)p1 + 2ap2 a1
0.5(a1+ a2) a2

4. Breeding Value and Random Mating
Consider the expected genotypic values of progeny produced by parental genotypes.
Genotype
Breeding Value (A)
Progeny Expected Genotypic Value
Deviation
(A/2)
B1B1
2a1
ap2(1+k) a1
B1B2
a1+a2
a[p2 + (1+k)/2]
0.5(a1+a2)
B2B2
2a2
a[2p2 + p1(1+k)] a2

5. Breeding Value Estimation
In random mating populations, therefore, one can estimate the breeding value of an individual as twice the difference between their offspring average genotypic value and the population genotypic value.
The factor of two results because a parent can only pass on ½ of their genetic information (half of their total breeding value) to their offspring.

6. Multiple Allele Average Excess
The average excess of allele i is given by
When random mating can be assumed, then

7. Multiple Allele Additive Effect
Define the multiple regression for multiple alleles:
Then, the additive effects are the regression coefficients that minimize the mean squared error.

8. Finding Regression Coefficients
The general solution is involved.
However, when random mating is assumed, recall that the additive effects (or these regression coefficients that we seek) are equal to average excess. Therefore,

9. Variance Components First, we consider the total variance:
The model component of variance is given by:
Additive genetic variance is the model component:

10. Covariance of Genotypic Value and Gene Content
We seek an expression for the s(G, Ni). To find it, we need E(Ni), E(G), and E(NiG).
E(G) = mG.
E(Ni) = 2pi.

11. Covariance of Genotypic Value and Gene Content (cont)
Putting the results of the previous slide together, we have
But that implies
And under random mating, gives
mean square additive effect
factor accounts for diploidy

12. Multiple Allele Breeding Value
The breeding value remains unchanged. It is just the sum of the additive effects of the two alleles of the parent.
In general, then, we can write
and the variance can be written as
Under random mating, the additive genetic effect is the variance of breeding values in the population.

13. Summary a homozygous effect intrinsic properties of alleles k
dominance coefficient ai
additive effect
properties of allele in population
ai*
average excess A
breeding value
property of individual in population
additive genetic variance
property of a population

14. Basic Overview of QTL Hunt
Genotype individuals at markers and classify them into groups based on marker genotype.
The average phenotypic values within each group can be taken as estimates of genotypic values. They are averaged over environmental effects.
Estimate the total genotypic variance that is associated with the marker locus. From this we get an idea of how close the marker is to the QTL.

15. Basic Overview of QTL Hunt (cont)D
QTL
Single marker: No locus order required, but no precise QTL locations obtained. Used when QTL detection, not location, is key.
Interval mapping: Require locus order and do provide estimates of QTL location.

16. Interval Mapping Variants
Interval mapping or flanking-marker analysis considers each pair of markers in turn. Slightly higher power and more precise estimates of location and QTL effect.
Composite interval mapping uses two markers plus a few other selected markers.
Multipoint mapping considers all linked markers at once.

17. Inbred vs. Outbred
QTL mapping also varies depending on the population you study.
Inbred line crosses are controlled crosses (F2, backcross and the like).
Outbred populations are natural populations (such as humans) where the experimenter cannot control the cross.

18. Single Locus – General Model
Let yi be the trait value for the ith individual in your sample.
Let mi be the marker genotype for the ith individual in your sample.
Then, it is generally assumed that

19. Single Locus – Analysis
The analysis can proceed in multiple ways
t-test (when there are two marker classes)
analysis of variance
linear regression
likelihood estimation and testing

20. Backcross Model – Setup
A and Q are in coupling phase
Marker Genotype
Count
Marg. Freq.
QTL Genotype
Trait Value QQ Qq AA n1
0.5
0.5(1-q)
0.5q Aa n2
1-q q qg
(1-q)g

21. Backcross – t-test H0: mean genotypic values are the
same for both marker classes.

22. Backcross – Interpreting Significant Result
Find the expected value under HA:
In this case, g = (1+k)a or g = (1-k)a
A significant result therefore implies |k| ¹ 1 OR a ¹ 0 OR q ¹ 0.5
|k| ¹ 1
a ¹ 0
q ¹ 0.5

23. Backcross – Linear Model
A linear model for this scenario is
The linear model yields a total sum of squares between the observations and mean:
for m marker
classes

24. Backcross – ANOVA
ANOVA partitions the total sum of squares SST into between marker class and within marker class components:

25. Backcross – ANOVA (cont)
We now seek expressions for the expected sums of squares:
where N = n1 + n nm

26. Backcross – ANOVA (cont)
And for the expectation of the marker sum of squares:
where we use
with
to obtain

27. Backcross – ANOVA Table
Factor df SS MS
E(MS)
B/W marker class
2-1
SSM
MSM
W/I marker class
N-2
SSe
MSe
Total
N-1
SST
MST

28. Backcross – Estimation of Variance Components
Using method of moments, we can obtain estimates of the variance components.

29. Backcross – ANOVA Statistical Test
The F statistic for the ANOVA table can be calculated and used to test the null hypothesis of no linkage.

30. Backcross – Regression Model
In the regression framework, a significant squared correlation r2 indicates that the marker and QTL are linked.
Recall that r2 is the trait variance that is explained/predicted by marker genotype.

31. Backcross – Summary
The two explanations for a significant results confound QTL location and QTL effect.
Genetic effect is estimated with bias.
The QTL location cannot be identified with accuracy.
The method has low power when linkage is loose.
Some power gained in other methods, but more-or-less minimal. This impacts power calculations.

32. Project #1 Genotype Observed Count AABB 5 AaBB 1 aaBB AABb 11 AaBb 7
AABb 11
AaBb 7
aaBb 3
AAbb
Aabb
aabb 4

33. Coupling or Repulsion??
Codominant loci means the log likelihoods are symmetric about 0.5.
Assume coupling.
Use EM algorithm to estimate q mle. Estimate:

34. Significant Linkage?
Calculate the G statistic to determine if there is significant linkage. Statistic:
The test for linkage is a one-sided test. Thus, the critical value is 2.71 and we reject H0: q=0.5 with a significance value of 0.05.

35. Do we have enough power?
Calculate the expected G statistic: EG = 13.5.
The power to detect linkage assuming all assumptions are met is given by the probability that a chi-square with noncentrality parameter EG exceeds the central chi-squared critical value.

36. Segregation Analysis Genotype Observed Count Marginal Counts AABB 51
aaBB
AA = 16 (0.53)
AABb 11
Aa = 13 (0.43)
AaBb 7
aa = 7 (0.23)
aaBb 3
AAbb
Chi-squared: 23.6
Aabb
aabb 4

37. The Actual Simulation Conditions
The project 1 dataset was generated assuming the following parameter values:
0.28, 0.39, 0.19, 0.22, 0.19, 0.28, 0.39, 0.40
0.21, 0.32, 0.18, 0.21, 0.18, 0.25, 0.39, 0.26

38. Accounting for Segregation Distortion
Gen.
Expected Freq.
P(R|G)
P( |G)
AABB
AaBB
aaBB
AABb
AaBb
aaBb
AAbb
Aabb
aabb