Introduction to Pedigree Analysis II
Biostatistics 666
Elements of Pedigree Likelihoods
n
Prior Probabilities
n
For founder genotypes For offspring genotypes For individual phenotypes
n
Segregation probabilities
n
n
Penetrances
n
Parameters in Pedigree Likelihood
n
Prior Probabilities
n
Allele Frequencies Genetic Map and Recombination Fractions Relationships Among Individuals Model for Relating Genes and Disease Error Model for Genotype Data
n
Segregation Probabilities
n n
n
Penetrance
n n
Prior Probabilities for Founders
n n
P(Gfounder) Assume hardy-weinberg equilibrium
n
Based on allele frequencies Assume linkage equilibrium Frequencies at multiple loci are products of frequencies at individual loci
n
May be multilocus frequencies
n n
Segregation Probabilities
n n
P(Go | Gf , Gm) Probability of offspring genotype conditional on parental genotypes
n
Follows from Mendel’s laws
n
If two loci are considered, depends on recombination fraction
Penetrances
n n
P(Xi | Gi) Probability of observed phenotype conditional on genotype
n
If penetrances are 1 the genotypes are known
n
Generally, assume that phenotypes are independent within families
Overall Pedigree Likelihood
L = å ...å P( X 1 ,..., X n | G1 ,..., Gn )P(G1 ,..., Gn )
G1 Gn
= å ...åÕ P( X i | Gi )P(G1 ,..., Gn )
G1 Gn i
= å ...åÕ P( X i | Gi )
G1 Gn i
founder
Õ
P(G founder )
{o , f ,m}
Õ P (G
o
| G f , Gm )
Typical calculation
n n
List all possible genotypes Create reduced lists
n n
Eliminate those where P(X|G) = 0 Eliminate those where P(Go|Gf,Gm) = 0
n
Iterate over all possibilities
Example Pedigree
? O
?
A
A
A
A
AB
A
Condition on Phenotype
Person I-1 I-2 II-1 II-2 II-3 II-4 III-1 III-2 III-3 Genotypes {AA, AO, BB, BO, AB, OO} {OO} {AA, AO, BB, BO, AB, OO} {AA, AO} {AA, AO} {AA, AO} {AA, AO} {AB} {AA, AO} #Genotypes 6 1 6 2 2 2 2 1 2
1152 possibilities to consider
Condition on Family Members
Person I-1 I-2 II-1 II-2 II-3 II-4 III-1 III-2 III-3 Genotypes {AA, AO, AB} {OO} {BO, AB} {AO} {AO} {AA, AO} {AA, AO} {AB} {AA, AO} #Genotypes 3 1 2 1 1 2 2 1 2
48 possibilities
Segregation Probabilities
n
P(Go | Gf , Gm)
n n n
Go = (Ho1, Ho2) Gf = (Hf1, Hf2) Gf = (Hm1, Hm2)
n
P(Go | Gf , Gm) =
P(Ho1| Hf1, Hf2)P(Ho2| Hm1, Hm2) + P(Ho2| Hf1, Hf2)P(Ho1| Hm1, Hm2)
Genetic Map
n n
Set of n ordered loci Set of n-1 distances between consecutive loci
B xAB xBC C xCD D xDE E xEF F
A
The Morgan
n
Distance along which one crossing over is expected per generation The basic unit of genetic distance
n n
n
Usually, distances are reported in cM 1 cM = 0.01 M
n
Total human genetic map about 3.3 M
Recombination
Non-Recombinant Gametes
Recombinant Gametes
1-q
q
Recombination
Non-Recombinant Gametes / /
Recombinant Gametes / /
1-q
Even Number of Events
Odd Number of Events
q
Recombination and Map Distance
1.00
Observed Recombination
0.80
0.60
0.40
0.20
0.00 0.00
0.20
0.40
0.60
0.80
1.00
Distance
Haldane Map Function
n
Assume crossovers are independent
n
Follow a Poisson distribution in each interval
ì- 1 2 ln(1 - 2q ) if 0