Multiple Comparisons
Measures of LD
Jess Paulus, ScD
January 29, 2013

1.
2.
Multiple comparisons
Measures of Linkage disequilibrium
• D’ and r 2
• r 2 and power

Multiple testing & significance thresholds
 Concern about multiple testing
 Standard thresholds (p<0.05) will lead to a large number of “significant” results
 Vast majority of which are false positives
 Various approaches to handling this statistically

Reject
H
0
: SNP prevents
DM
Observed in the
Sample Fail to reject H
0
:
No assoc.
Unobserved Truth in the Population
H
0
: No
H a
: SNP prevents DM association
True positive
(1 – β)
False positive
Type I error (α)
False negative
Type II error ( β) :
True negative
(1α)

= Also known as: “Level of significance”
Probability of Type I error – rejecting null hypothesis when it is in fact true
(false positive), typically 5%
p value = The probability of obtaining a result as extreme or more extreme than you found in your study by chance alone

A genome-wide association scan of
500,000 SNPs will yield:
25,000 false positives by chance alone using
α = 0.05
5,000 false positives by chance alone using
α = 0.01
500 false positives by chance alone using
α = 0.001

 Multiple comparisons (or "multiple testing") problem occurs when one considers a set, or family, of statistical inferences simultaneously
 Type I errors are more likely to occur
 Several statistical techniques have been developed to attempt to adjust for multiple comparisons
 Bonferroni adjustment

 Standard Bonferroni correction


Test each SNP at the α* =α /m
1 level
Where m
1
= number of markers tested
 Assuming m
1
= 500,000, a Bonferroni-corrected threshold of α*= 0.05/500,000 = 1x10–7
 Conservative when the tests are correlated
 Permutation or simulation procedures may increase power by accounting for test correlation

Measures of LD
Jess Paulus, ScD
January 29, 2013

 Haplotype: an ordered sequence of alleles at a subset of loci along a chromosome
 Moving from examining single genetic markers to sets of markers

a g a g A G A G
A a
G g
A
A
G g
A
A g
G a
A
A G A G a g a
 Basic data: table of haplotype frequencies
G g
A
8
2
62.5% a
0
6
37.5%
50%
50% g g
G

2


Both measure correlation between two loci
D prime …

 Ranges from 0 [no LD] to 1 [complete LD]
R squared…
 also ranges from 0 to 1
 is correlation between alleles on the same chromosome

 Deviation of the observed frequency of a haplotype from the expected is a quantity called the linkage disequilibrium (D)

If two alleles are in LD, it means D ≠ 0
 If D=1, there is complete dependency between loci
 Linkage equilibrium means D=0



Q
*
G g
Measure
D’

2 = r 2
A n
11 n
01 n

1 a n
10 n
00 n

0
Formula n
11 n
00
 n
10 n
01 min( n

1 n
0

, n

0 n
1

)
 n
11 n
00
 n
10 n
01

2 n

1 n

0 n
1
 n o
 n
11 n
00
 n
10 n
11 n

0 n
01 n
11 n
00 n n
11
11 n n n
10
00
00 n
01
 n
10 n
01
 n
10 n
01 n
1
 n
0

Ref.
Lewontin (1964)
Hill and Weir
(1994)
Levin (1953)
Edwards (1963)
Yule (1900)

a
A a
A g
G g
G a
A
A
A g
G g
G
A
A
A a
G g
G g
A a
A a
G g
G g
D
’
= n
11 n
00
 n
10 n
01 min( n

1 n
0

, n

0 n
1

)
A
G 8 g a
0
2 6
62.5% 37.5%
50%
50%
R 2 =
 n
11 n
00
 n
10 n
01

2 n

1 n

0 n
1
 n o

D’ = (8

6 – 0x2) / (8 
6) =1 r 2 = (8

6 – 0x2) 2 / (10

6

8

8)
= .6

2
 r 2 is directly related to study power
 A low r 2 corresponds to a large sample size that is required to detect the LD between the markers
 r 2 *N is the “effective sample size”
 If a marker M and causal gene G are in LD, then a study with N cases and controls which measures M
(but not G) will have the same power to detect an association as a study with r 2 *N cases and controls that directly measured G

2
 Example:



N = 1000 (500 cases and 500 controls) r 2 = 0.4
If you had genotyped the causal gene directly, would only need a total N=400 (200 cases and
200 controls)

1.
2.
Multiple comparisons
Measures of Linkage disequilibrium
• D’ and r 2
• r 2 and power