1. Human Genetics
Genetic Epidemiology

2. Family trees can have a lot of nuts

3. Genetic Epidemiology - Aims
Gene detection
Gene characterization
mode of inheritance
allele frequencies
→ prevalence, attributable risk

4. Genetic Epidemiology - Methods
Aggregation
Segregation
Co-segregation
Association

5. Segregation affected and unaffected or two distributions:
determined by a dominant or recessive allele
Also possible: three distributions:
Can the dichotomy or trichotomy be explained by Mendelian segregation?

6. Likelihood (parameter(s); data)  Probability (data | parameter(s))
The joint probability of the genotypes and phenotypes of all the members of a pedigree can be written as

7. Transmission Probabilities Value if there is Mendelian segregation1
P(AA transmits A) = τ AA A
P(Aa transmits A) = τ Aa A
P(aa transmits A) = τ aa A

8. Ascertainment We examine segregating sibships
The proportion of sibs affected is larger than expected on the basis of Mendelian inheritance
The likelihood must be conditional on the mode of ascertainment
We need to know the proband sampling frame

9. Cosegregation Chromosome segments are transmitted
Cosegregation is caused by linked loci
ultimate statistical proof of genetic etiology

10. Methods of Linkage Analysis
Trait model-based – assume a genetic model underlying the trait
Trait model-free - no assumptions about the genetic model underlying the trait
(parametric)
(non-parametric)
Ascertainment is often not an issue for locus detection by linkage analysis

11. Model-based Linkage Analysis
If founder marker genotypes are known or can be inferred exactly,
→ no increase in Type 1 error
→ smallest Type 2 error when the model is correct
If founder marker genotypes are unknown, we can
1) estimate them
2) use a database
All parameters other than the recombination fraction are assumed known

13. Model-free Linkage Analysis Identity-in-state versus Identity-by-descent
Two alleles are identical by descent if they are copies of the same parental allele
A1A1
A1A2
IBD

14. Sib pairs share
0, 1 or 2 alleles identical by descent at a marker locus
0, 1 or 2 alleles identical by descent at a trait locus
Linkage
The average proportion shared at any particular locus is 1/2

15. Relative Pair Model-Free Linkage Analysis
We correlate relative-pair similarity (dissimilarity) for the trait of interest with relative-pair similarity (dissimilarity) for a marker
Linkage between a trait locus and a marker locus
→ positive correlation
Affected relative pair analysis: Do affected relative pairs share more marker alleles than expected if there is no linkage?
No controls!

16. Association Causes of association between a marker and a disease
chance
stratification, population heterogeneity
very close linkage
pleiotropy

17. Causes of Allelic Association
Heterogeneity/stratification
This allelic association is nuisance association
Simpson's paradox: If we mix two populations that have both different disease prevalence and different marker allele prevalence, and there is no association between the disease and marker allele in each population, there will be an association between the disease and the marker allele in the mixed population.
The best solution to avoid this confounding is to study only ethnically homogeneous populations

18. (Tight) Linkage
Imagine a number of generations ago, a normal allele d mutated to a disease allele D on a particular chromosome on which the allele at a marker locus was A1
mutation
A d
A D
This chromosome is passed down through the generations, and now there are many copies. If the distance between D and A1 is small, recombinations are unlikely, so most D chromosomes carry A1
This is the type of allelic association we are interested in

19. Guarding Against Stratification
Three solutions:
use a homogenous population
use family-based controls
use genomic control

20. Matching on Ethnicity
Close relatives are the best controls, but can lead to overmatching
Cases and control family members must have the same family history of disease
Siblings
Cousins

21. Transmission Disequilibrium Test (TDT)
A design that uses pseudosibs as controls
Cases and their parents are typed for markers
A1A2
A2A2
Transmitted genotype is A1A2
Untransmitted genotype is A2A2
Father transmits A1, does not transmit A2
Mother transmits A2, does not transmit A2
(uninformative in terms of alleles)

22. Build up a 2 x 2 table: Transmitted A1 A2 A1 Untransmitted A2 •c a b d
The counts a and d come from homozygous parents
The counts b and c come from heterozygous parents
McNemar's test : χ12
(b - c)2
b + c

23. Genomic Control
Calculate an association statistic for a candidate locus
Calculate the same association statistic, from the same sample, for a set of unlinked loci
Determine significance by reference to the results for the unlinked loci

24. Linkage Between a Marker and a Disease
Intrafamilial association
Typically no population association
Not affected by population stratification
Population association if very close

25. Association versus Linkage
Allelic Association
Linkage
Association at the population level
Intrafamilial association
Pinpoints alleles
Pinpoints loci
More powerful
Less powerful
More tests required
Fewer tests required
More sensitive to mistyping
Less sensitive to mistyping
Sensitive to population stratification
Not sensitive to population stratification
Which is better?

26. What is the Best Design and Analysis?
If heterogeneity / stratification is a non-issue,
unrelated cases and controls for association analysis
(genome scan?)
If heterogeneity / stratification could be an issue, genome scan desired,
large extended pedigrees, type all (founders and non- founders) for equi-spaced markers, for linkage analysis
Note: cost, burden of multiple testing
A wise investigator, like a wise investor, would hedge bets with a judicious mix

27. Case-Control Data
Consider a particular marker allele, A1, sample of cases and controls: N n2 n1 n0
Total S s2 s1 s0
Controls R r2 r1 r0
Cases 2 1
Number of A1 alleles

28. Consider the probability structure:q2 q1 q0
Controls p2 p1 p0
Cases 2 1
Number of A1 Alleles
Cochran-Armitage trend: test the null hypothesis
p2 + ½p1 = q2 + ½q1
without assuming the two alleles a person has are independent
Sasieni (1997) Biometrics 53:

29. asymptotically has a χ2 distribution with 1 d.f

30. Cochran-Armitage Trend Test
Does not assume independence of alleles within a person
Does assume independence of genotypes from person to person
Is not valid if there is population stratification
The increased variance due to stratification can be estimated from a random set of markers that are independent of the disease
genomic control.
Devlin and Roeder (1999) Biometrics 55:

31. Case-only Studies Look at departure from (1-p)2 2p(1-p) p2 A*A* A1A*
where p = P(A1) = p2 + ½p1
Suggested as
more powerful (only cases needed)
more precise (signal decreases faster with distance from the causative locus)
Hardy-Weinberg Disequilibrium (HWD) test statistic:

32. Case - only Studies No power in the case of a multiplicative model
No controls
there must be a difference in HWD between cases and controls
therefore we consider this HWD trend test:

34. Weighted average of the Cochran-Armitage trend test and the HWD trend test statistics
We want to give more weight to b or d, whichever yields the larger signal
Therefore take

35. To investigate the null distribution of this average we simulate many different situations – sample sizes up to 10,000 cases and 10,000 controls - and generate
For all situations considered, the distribution is well approximated by a Gamma distribution

36. As the sample size and marker allele frequency increase, the largest mean and the smallest variance occur for 10,000 cases and 10,000 controls, and for a marker allele frequency 0.5
For 10,000 cases and 10,000 controls, and marker allele frequency 0.5, the upper tail of the distribution is well approximated by a Gamma distribution with mean μ = 1.78 and variance σ2 = 3.45

37. We develop a prediction equation to determine percentiles of the null distribution for smaller sample sizes and marker allele frequencies
We base goodness of fit on the root mean squared error (RMSE) of logeα, calculated for various sample size combinations, from the variance among 50 replicate samples:

38. With ~90% confidence, the true loge α lies in the. interval logeα + 1
With ~90% confidence, the true loge α lies in the interval logeα (RSME), i.e., α is within e+1.645(RSME) - fold of the true α
For total sample size (R + S) 200 or larger and α = or larger, in the very worst case (R = S = 100, α = ) with 90% confidence α could differ from the true α by a factor of at most ~ 4.8
The average RMSE is 0.35, corresponding to being between 78% and 122% of the true α with 90% confidence

39. Genetic Models Simulated Probability of being affected given
POWER
Genetic Models Simulated
Probability of being affected given
A1A1
A1A*
A*A*
1 Recessive 1
1.00
0.10
2 Recessive 2
0.05
3 Additive
0.50
0.00
4Multiplicative
0.81
0.045
0.0025
Each simulated population contains 500,000 individuals allowed to randomly mate for 50 generations after the appearance of a disease mutation
Marker loci placed at distances 0 – 6 cM from the disease susceptibility locus
For type I error, no association between the disease and marker loci

40. Tests Performed Homogeneous populations HWD, cases only Allele test
Allele test x HWD in cases
HWD trend test
Cochran-Armitage trend test
Cochran-Armitage trend test x HWD trend test
Weighted average
Population stratification
Cochran-Armitage trend test with genomic control
Product of this and the HWD trend test
Weighted average with genomic control

41. Type I error, homogeneous population
∆ HWD test, cases only
▲ product of the allele test and HWD test

42. Type I error, population stratification
○ allele test
◊ Cochran-Armitage trend test
▲ product of the allele test and HWD test
■ weighted average test
● product of the Cochrn-Armitage trend test and the HWD test

43. Power, homogeneous population
■ weighted average test

44. Power, population stratification
□ HWD trend test
♦ CA test with genomic control
■ weighted average with genomic control

45. Conclusions
Under recessive inheritance, the weighted average has better performance than either the Cochran-Armitage trend test or the HWD trend test
Has good performance for other models as well
The product of the Cochran-Armitage trend test statistic and the HWD test statistic (cases only) has better power, but has inflated Type I error if there is population stratification
The weighted average has good overall properties, automatically controls for marker mistyping

46. With acknowledgment to
Kijoung Song

47. Can we use evolutionary models, when we have large amounts of genetic data on a sample of cases and controls, to obtain a more powerful way of detecting loci involved in the etiology of disease?
Will these models bear fruit or nuts?