1. Family-Based Association Tests
“If you cannot get rid of the family skeleton, you may as well make it dance” (G.B. Shaw)

2. Outline Overview Trios: Transmission Disequilibrium Test (TDT)
Discordant sibships: Conditional logistic regression
General Pedigree: FBAT test
Comparisons and extensions

3. Family-based designs Discordant sibpairs, sibships
Affected offspring and their parents
Trios (2 parents, child) common design
Complex nuclear families
Extended pedigrees
Leftovers from linkage (next lecture)

4. Family-based vs. Case-control

5. Family-based vs. Case-control
Completely robust to population substructure
Robust to HWE failure
More powerful for very rare highly penetrant diseases (e.g., arguments coming back for sequencing)
Pseudo-controls (e.g., longevity study…), but much harder to recruit (esp. late onset diseases, children generally not difficult)
Adjusting for PC’s/AIMs does well in practice, now
Test for HWE in controls
More powerful in most other situations
More careful selection of good controls (sort of)

6. Family-based vs. Case-control
Detect genotyping error (Mendel error)
More complex analysis (but doable)
Cryptics, maybe
Standard regression methods

7. Mendel’s laws Recall the playing cards example...
One allele from each parent for each gene
Many family based tests based on this, rather than estimating allele frequencies (case-control)

8. Mendelian transmission: Ex
E.g., parents are Aa, Aa:
P(offspring=AA | Mother=Aa,Father=Aa)=?
P(offspring=Aa | Mother=Aa,Father=Aa)=?
P(offpsring=aa | Mother=Aa, Father=Aa)=?

9. Mendelian transmission: Ex
E.g., parents are Aa, Aa:
P(offspring=AA | Mother=Aa,Father=Aa)=1/4
P(offspring=Aa | Mother=Aa,Father=Aa)=1/2
P(offpsring=aa | Mother=Aa, Father=Aa)=1/4
Conditioning on parents...

10. Mendelian transmission: Ex
E.g., parents are AA, Aa:
P(offspring=AA | Mother=AA,Father=Aa)=?
P(offspring=Aa | Mother=AA,Father=Aa)=?
P(offpsring=aa | Mother=AA, Father=Aa)=?

11. Mendelian transmission: Ex
E.g., parents are AA, Aa:
P(offspring=AA | Mother=AA,Father=Aa)=1/2
P(offspring=Aa | Mother=AA,Father=Aa)=1/2
P(offpsring=aa | Mother=AA, Father=Aa)=0

12. Mendelian transmission: Ex
E.g., parents are AA, AA:
P(offspring=AA | Mother=AA,Father=AA)=?
P(offspring=Aa | Mother=AA,Father=AA)=?
P(offpsring=aa | Mother=AA, Father=AA)=?

13. Mendelian transmission: Ex
E.g., parents are AA, AA:
P(offspring=AA | Mother=AA,Father=AA)=1
P(offspring=Aa | Mother=AA,Father=AA)=0
P(offpsring=aa | Mother=AA, Father=AA)=0
Homozygote parents are “non-informative” (no variation in offspring’s conditional genotype distribution)

14. Outline Overview Trios: Transmission Disequilibrium Test (TDT)
Discordant sibships: Conditional logistic regression
General Pedigree: FBAT test
Comparisons and extensions

15. Trios: Transmission Disequilibrium Test (TDT)
Test based on transmissions from parents to offspring
Assumptions
Parents’ and offspring genotypes known
dichotomous phenotype (though Q-TDT), only affected offspring
Count transmissions from heterozygote parents, and compare to expected transmissions
Mendel’s laws of segregation (previous slides), not control group
test for over/under-transmission of alleles in cases (intuition…)
Conditional test
offspring affection status
Parental genotypes (conditions out allele frequencies, which is what case-control is based on testing)
Spielmen et al., AJHG 1993

16. Trios: Transmission Disequilibrium Test (TDT)
Non-transmitted parental allele
Transmitted parental allele
w AA parents (transmit one A, do not transmit other A)
z aa parents (transmit one a, do not transmit other a)
x Aa parents that transmit A, do not transmit a
y Aa parents that transmit a, do not transmit A

17. Possible Parental Configurations
AA-AA, AA-Aa, AA-aa, Aa-AA, Aa-Aa, Aa-aa, aa-AA, aa-Aa, aa-aa
(Ones not bolded are symmetric for what we will do next, e.g., AA-Aa == Aa-AA
Six possible configurations

18. Both parents homozygous
Non-transmitted parental allele
AA-AA | AA
Transmitted parental allele
Offspring genotype is deterministic, no variation, not informative!

19. Both parents homozygous
Non-transmitted parental allele
aa-aa | aa
Transmitted parental allele
Offspring genotype is deterministic, no variation, not informative!

20. Both parents homozygous
Non-transmitted parental allele
AA-aa | Aa
Transmitted parental allele
Offspring genotype is deterministic, no variation, not informative!

21. One parent heterozygous
Non-transmitted parental allele
AA-Aa |
AA,Aa
← Pr
Transmitted parental allele
Non-transmitted parental allele
Transmitted parental allele
Variation from one parent

22. One parent heterozygous
Non-transmitted parental allele
Aa-aa |
Aa,aa
← Pr
Transmitted parental allele
Non-transmitted parental allele
Transmitted parental allele
Variation from one parent

23. Both parents heterozygous
Non-transmitted parental allele
Aa-Aa |
AA,Aa,aa
← Pr
Transmitted parental allele
Non-transmitted parental allele
Non-transmitted parental allele
Transmitted parental allele
Transmitted parental allele
Variation from both parents

24. Trios: Transmission Disequilibrium Test (TDT)
Non-transmitted parental allele
Transmitted parental allele
w AA parents (transmit one A, do not transmit other A)
z aa parents (transmit one a, do not transmit other a)
x Aa parents that transmit A, do not transmit a
y Aa parents that transmit a, do not transmit A

25. Transmission Disequilibrium Test (TDT)
Non-transmitted parental allele
Transmitted parental allele
No variation in w or z (recall homozygous parents non informative)
(x-y)2/(x+y) ~ 12; it’s just special case of McNemar’s test
Think of it as testing are there an excess of the A allele in the affected offspring than would happen by Mendel's laws?

26. Transmission Disequilibrium Test (TDT)
Non-transmitted parental allele
Insulin Dependent Diabetes Mellitus (IDDM)
Transmitted parental allele
Example from the text: 94 families, 78 parents transmit allele A, 46 transmit allele a
(78-46)2/(78+46)=8.26, p- value=0.004
Spielman et al., 1993

27. Limitations of TDT Only affected offspring Only dichotomous phenotypes
Bi-allelic markers
Additive genetic model
No missing parents
Incorporating siblings assumes no linkage (more next time)
Can’t do multiple markers, multiple phenotypes

28. Key features of the TDT
Random variable in analysis is offspring genotype
Parental genotypes fixed
Trait fixed (condition on affected offspring)

29. Outline Overview Trios: Transmission Disequilibrium Test (TDT)
Discordant sibships: Conditional logistic regression
General Pedigree: FBAT test
Extra

30. Discordant sibships Conditional logistic regression
P(Y1=1|Y1+Y2=1,g1,g2,…)
Matching each sib together, conditions on the fact that they have discordant phenotypes
Standard model for disease as in logistic regression, just matching based on family strata
Can also use FBAT framework
Similar power for main effects
Greater power for GxE (Witte, AJE 1999; Chatterjee et al., Gen Epi 2005; Hoffmann et al., Biometrics 2011)
You will go through an example in the homework

31. Outline Overview Trios: Transmission Disequilibrium Test (TDT)
Discordant sibships: Conditional logistic regression
General Pedigree: FBAT test
Comparisons and extensions

32. FBAT: More general methodology
Maintains general principals of TDT
Other genetic models (dominant, recessive, …)
Additional siblings, extended pedigrees, missing parents
Multiple markers, (haplotypes)
Test statistic intuition: covariance between offspring trait and genotype

33. FBAT: Extending TDT to more general families
For the moment, assume parents are genotyped
Let i index across families, j offspring
Score test of f({offspring genotype}ij|traitij,parentsi), use Mendel’s laws, Bayes rule
U=i,j (traitij-offset) x ({offspring genotype}ij - E[{offspring genotype}ij|parentsi])
Assume trait is continuous or binary
Assume offset is mean (continuous) or population prevalence (dichotomous)
Condition on Parents (avoid specification of allele distribution)
Condition on offspring phenotypes (avoid specification of trait distribution)

34. FBAT: Extending the TDT to more general families (cont.)
U=i,j (traitij-offset) x ({offspring genotype}ij - E[{offspring genotype}ij|parentsi])
Intuition: Like a sample covariance between trait and genotype
ZFBAT=U/sqrt(var(U)) ~ N(0,1)

35. FBAT: Extending the TDT to more general families (cont.)
U=i,j (traitij-offset) x ({offspring genotype}ij - E[{offspring genotype}ij|parentsi])
Let oij={offspring genotype}ij
Let Pi=parentsi
E[oij|Pi]
= X(AA)P(oij=AA|Pi) + X(AA)P(oij=AA|Pi)
+ X(AA)P(oij=AA|Pi)
Essentially using Mendel’s laws, as we calculated earlier

36. FBAT computations X = Additive coding of A alleles
Parents AA, aa: E(X|P) = 0*P(AA|P)+1*P(Aa|P)+2*P(aa|P) = 0*0+1*1+2*0=1
Child:
X Pr(X) (X-E(X|P))
1 1 0
Parents Aa, Aa (E(X|P)=0*(1/4)+1*(1/2)+2*(1/4)=1
Child
0 1/4 1/4
1 1/2 0
2 1/4 1/4
(Over/under-transmissions)
AA-aa | Aa
Uninformative families still contribute nothing!
Aa-Aa |
AA,Aa,aa

37. Seem familiar? FBAT=TDT
If Y=affection status (1=affected, 0=unaffected), offset=0, then FBAT==TDT
Similarly conditional logistic regression roughly equivalent to TDT in terms of power for main effects

38. FBAT offset for dichotomous traits
If all offspring are affected, then it does not matter
For rare diseases, affected most informative
For more common, can get some information from unaffecteds
Population prevalence, allows one to gain a little information from unaffecteds

39. Offset choice
Disease prevalence K = 0.05, allele frequency of the disease gene p=0.05, attributable fraction of the disease due to carrying at least one disease gene AF=0.3, significance level α=10−4 and sample size 100
Lange and Laird (2002)
Disease prevalence K=0.3, allele frequency of the disease gene p=0.143, attributable fraction of the disease due to carrying at least one disease gene AF=0.25, significance
level α=0.01 and sample size 100.

40. Offset choice

41. FBAT offset for continuous traits
The trait mean
(Optimal choice is E(Y), depends on ascertainment)
Residual from the trait adjusted for covariates
e.g., regress gender on bmi, use residual
Suppose Y is your phenotype of interest, Z covariate
Linear regression Y = 0 + 1Z
Compute residual R=Y- (0 + 1Z)
Use R as trait in FBAT

42. Continuous vs. Dichotomous trait
Modeling as continuous trait -- more powerful
With highly selected traits, dichotomizing may be preferable
Using mean for offset is a poor choice here
Results very sensitive to offset choice
Dichotomizing will lose power compared to best offset choice

43. Offset general comments
Very poor choice -- poor power
More complicated slightly more efficient offsets are also available

44. Childhood asthma management program (CAMP) example
696 trios
bi-allelic locus in IL13 gene
five groups of 22 quantitative phenotypes

45. DeMeo Gen Epi, 2006

46. Can also do a multi-marker (gene-based) test...
DeMeo Gen Epi, 2006

47. Obesity GWAS example BMI follow-up for 24 years 86,604 SNPs
694 participants
One of the first GWAS successes

48. GWAS example uses clever screening approach, longitudinal phenotype data...

49. Obesity example: Longitudinal phenotype

50. Obesity example: Screening based on “conditional mean model”
Prioritizes SNPs based on modeling X imputed from parental genotypes (PBAT software)
f(X,P)=f(X|P)f(P)
Screening not robust to population substructure, but later testing is (so doesn’t matter)

51. Obesity example: Results

52. Screening based on “conditional power”...
Started with only analyze “top k” (Lange et al.)
Criticized, not looking at all SNPs, and in practice...
Prior distribution for type I error (Iulianna et al, AJHG 2007)
Bayesian (Naylor et al, Gen Epi 2010)

53. Critiques
Only modeling the offspring conditional on parents, not using parents?
Other models do, not robust to population stratification (but could adjust for covariates…)
Are used in conditional mean model screening approach

54. Outline Overview Trios: Transmission Disequilibrium Test (TDT)
Discordant sibships: Conditional logistic regression
General Pedigree: FBAT test
Comparisons and extensions

55. Power of FBAT, CACO, Rare disease
In your book, but not necessarily a fair comparison...
200 trios (600 genotypes)
200 DSP (400 genotypes)
200 sibtrios (3 offspring, no parents, 600 genotypes)
200 cases/200 controls (400 genotypes)
OR=1.5

56. Power of common disease
200 trios (600 genotypes)
200 DSP (400 genotypes)
200 sibtrios (3 offspring, no parents, 600 genotypes)
200 cases/200 controls (400 genotypes)
OR=1.5

57. Final thoughts FBAT also extended to X chromosome Survival Analysis
Multi-marker
Multi-phenotype
Haplotypes
Missing parents
Gene-environment/gene-gene interactions
Meta-analysis

58. Final thoughts
Other likelihood approaches, Shaid 1989, Cordell 2000, Dudbridge 2010 (software unphased)
Other approach by Allison 1997, Abecasis for quantitative traits
Also simultaneous modeling of family and case-control data (Sage/Mendel software)
If large enough sample, maybe cryptics?

59. Software FBAT http://www.biostat.harvard.edu/fbat/fbat.htm
PBAT
P2BAT
Dudbridge's UNPHASED
ased/
Clayton's software