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Introduction to GWAS analysis
Analysis of linkage patterns among chromosome-specific
genetic markers to infer genotype / phenotype correlations
Based on DH Hamer et al. (1993) Science 261: 321
Disclaimer: The study and data are presented as a conceptual framework for understanding linkage and LOD analysis.
No opinion is expressed as to any inferences from the results of that study.
Steven M. Carr
Dept of Biology
Memorial University of Newfoundland
St John’s NL A1B 3X9
Conceptual considerations of Study Design
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QTL (Quantitative Trait Loci) Linkage
• phenotype influenced by multiple gene loci in linked region
Linkage: genetic markers in region ought to be linked with phenotype
Hypothesis of X-linked inheritance
• => simplifies search to X chromosome
Pedigree analysis of pairs of brothers
• Given a heterozygous mother
• Identity by Descent [D] inferred from same X alleles
• Discordance [n] inferred from different X alleles
• Otherwise, Similarity by State [S] may be same X allele
Logarithm of Odds (LOD) calculation
• Z1 = likelihood(D), given observed data (allele frequencies)
• LOD = ln[ Z1 / prob(null)], where p(null) = random expectation
• Likelihood: Conditional probability, given a priori evidence
• Roughly: is I by D significantly higher than expected by chance
• Cf. Poker Hands
Study Design of Hamer et al. (1993)
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QTL (Quantitative Trait Loci) study:
Possible genetic influence on (behavioral) phenotype
Linkage between phenotype & genotype markers
E.g.: Same-Sex orientation of gay male s [one subtype]
• Behavior shared with gay uncle but not father
• => mother-side X chromosome inheritance
Pedigree analysis provides data for and against hypothesis
• Genetic concordance of X inherited from mother
• Identity by Descent [D]: copies of same X
• Discordance [n]: copies of different X
• Similarity by State [S]: a similar X may be same X allele
• Uninformative [-]: not enough information
Logarithm of Odds (LOD) calculation
• Z1 = likelihood(D), given observed data (allele frequencies)
• LOD = log10 [Z1 / prob(null)], where p(null) = random expectation
X chromosome: Xq27~28 region
Pedigree of an X-linked male trait
(e.g., Male Homosexuality: Hamer et al. (1993))
X-linkage indicated if uncles & nephews share trait:
Mother (sister of uncle) implicated as carrier
Identity by Descent vs State
vs discordant (-) & un-informative (n) sites
Identity by Descent vs State
vs discordant (-)
Identical
by Descent
Identical
by State
Discordant
(-)
LOD score analysis of Five-Card Draw Poker
(Logarithm of Odds)
Given 52 cards, there are (52 choose 5) = 2,598,960 = 2.6 x 106 possible five-card hands
The probability of any random hand = 3.85 x 10-7
P(flush) = (13 choose 5) = 1,287 for each of 4 suits = 5,148 / 2,598,960 = 1.98 x 10-3
P(4-of-a-Kind): 13 4s-of-a-kind, and 48 other cards to fill = (13*48) / 2,598,960 = 2.40 x 10-4
log10 (1.98 x 10-3) = -2.70
log10 [(2.40 x 10-4) / (1.98 x 10-3)] = -0.92
log10 (2.40 x 10-4) = -3.62
A flush or 4-of-a-kind, though unlikely, are far more likely (~ 103~4) than a random hand
4-of-a-kind is less likely (and therefore beats) a flush
Note: evaluate (n choose k) = n! / k! (n-k)!
LOD score analysis (hypothetical, simplified)
Consider 21 possible concordance ratios: What odds?
Consider 21 successive markers with observed ratios: Is there Linkage?
Linkage Analysis (hypothetical result)
over 22 X-chromosome molecular markers (1-21)
LOD > 3 (markers 12~15) exceeds the threshold for inference of association
between genes in this region and the correlated phenotype
Hamer et al. (1993) Trait mapping to RSTUV region of Xq28
Proband
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5
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Marker Locus
A B C D
G
H
I
J
K
O
P
Q
R
S
T
U
V
n D D D D D n
D
n
n
n
n D D D
D D
n
D
D
n D n
D n D n D D n
n
n D
D D
n
D D D S S D S
S S S S S S S
S
S S S
S S S
S S S
S S
S
S S
S S
S S S
S S
S S
S
S
S S S S
S
S S S
S S
S S S
S
E
F
D
n
n
D
n
n
D
n
D
D
D
n
D
D
n
D D
n
D S
S S
S S
S S
S
S S
S S
S S
S S
S S
S
n
n
D
S
S
S
S
S
S
S
S
S
D
n
D
n
D
D
D
n
n
S S
S S S
S S
S
S S
S
S
S S S
S S
S
S S S S
n
n n n n
n
D D n D
S D D n n D
S S S S S S
S S S S
S S S S S
S
S S
S
S
S S
S S
S S
n n
D
D n n D n
n D n n n
D n D D D
D D n n n
n D D D D
D D D n
D n D
D D n
D
n
S
S S S S S
S S S S S
S S S S
S S S
S S S S S
S S S S S
S S S S S
S S S S S
S S S S S
S S S S
S S S S
S S S S
S S S S
S S S
S
S
S
S S
n
n
n
n
D
n
n
n
n
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
n
D
D
D
n
D
D
D
D
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
n
n
D
n
D
D
n
n
D
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
n
n
D
n
n
n
D
D
n
D
n
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
n
D
D
n
n
D
D
D
D
n
D
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
n
D
D
D
D
D
n
D
n
n
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
n
n
S
S
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
S
S
S
S
S
D
D
D
S
S
S
n
S
S
S
n
S
S
S
S
L
M
N
S
S
S S S S
D D D
n
n
D D
n n
S
S S S S
S S S S S
S S S S S S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
n
n
S
S
S
S
n
n
n n
S
S
S
n
n S n
S S S
S
LOD scores (long calculation)
= The ratio of the unlikely observed event to the unlikely random event,
(where brothers share any allele with p = z1 = ½ )
Simple formula [red box] is modified by marker allele frequencies,
corrections for loci Identical by State, etc.
Hamer et al. (1993): Linkage Analysis
over 22 X-chromosome molecular markers (A-V)
LOD score assignments over X:
high scores (=> linkage) in RSTUV region of Xq28
LOD = log10 of odds
LOD > 3
< 1/1000