Transcript Confounding from Cryptic Relatedness in Association Studies

Confounding from
Cryptic Relatedness in
Association Studies
Benjamin F. Voight
(work jointly with JK Pritchard)
Importance

Case/control association tests are becoming increasingly popular
to identify genes contributing to human disease.

These tests can be susceptible to false positives if the underlying
statistical assumptions are violated, i.e. independence among all
sampled alleles used in the test for association.

It is well appreciated that population structure results in false
positives (Knowler et al., 1988; Lander and Schork, 1994).

Methods exist which correct for this effect (Devlin and Roeder,
1999; Pritchard and Rosenberg, 1999; Pritchard et al. 2000).
Your (favorite)
Population
Obtain a sample of
affected cases from the
population.
Cases are not independent
draws from the population
allele frequencies.
Problem: the relatedness is
cryptic, so the investigator
does not know about the
relationships in advance.
Importance

Devlin and Roeder (1999) have argued that if one is doing a
genetic association study, then surely one must believe that the
trait of interest has a genetic basis that is at least (partially)
shared among affected individuals.

Given that cases share a set of risk factors by descent, then presumably
they are more related to one another than to random controls.

These authors presented numerical examples which suggested
that this effect may be an important factor, in practice.

However, these examples were artificially constructed, and not
modeled on any population-based process.

Few empirical data to suggest if cryptic relatedness negatively
impacts association studies. In a founder population, nonindependence resulting from relatedness does matter. (Newman
et al., 2001).
Goals

Determine whether, or when,
cryptic relatedness is likely to be a
problem for general applications.

Develop a formal model for
cryptic relatedness in a population
genetics framework.

In a founder population, estimate
the inflation factor due to (cryptic)
relatedness, and compare to
analytical results.

Avoid staring at “x” in front of a
chalkboard.
Modeling Definitions

m affected individuals and m random controls, sampled in the
current generation.

Pairs of chromosomes coalesce in a previous generation t = 1, 2,
~
… t with the usual probabilities.
~
t 1

t 1


1
1 
 2N
t

 1

 2N ~

t
All samples are typed at a single bi-allelic locus, unlinked to
disease, with alleles B and b, at frequencies p and (1-p) in the
population.
Definitions

Define:



Kp – population prevalence of disease.
Kt – probability that an relative of type t (or~t ) of an affected proband is
also affected.
lt – recurrence risk ratio, Kt/Kp (Risch, 1990).

Gi(a) – indicator (0 or 1) for the B allele on homologous chromosome a
for the i-th case. (with a  for diploid individuals)

Hj(a) – as above, but for a j-th random control.

Define a test statistic which measure the difference in allele counts between
cases and controls (slightly modified from Devlin and Roeder, 1999):

(1)
( 2)
T
Gi  Gi  

i 1
i 1

m


m

m
m
H H
(1)
j
j 1
j 1
( 2)
j

.


Under the null hypothesis of no association between the marker and
phenotype, an allele has a genotype B with probability p, independently for
all alleles in the sample. If so,
Var[T ]  4mp (1  p).

If cryptic relatedness exists in the sample, then the variance of the test – call
this Var*[T ] – may exceed the variance under the null. We measure the
deviation from the null variance using the “inflation factor” d:
Var * [T ]relatedness
d
4mp (1  p)
0.25
1% error rate
d = 1.0 (No Inflation Factor)
Probability Density
0.2
0.15
d = 1.5
0.1
d = 2.0
0.05
0
0
2
4
6
8
10
12
14
16
Chi Squared Value
d
1.0
1.5
2.0
Type-I nominal (a)
Fold-Error Rate
.05
~2.19
~3.32
1.00
.01
~3.55
~6.88
1.00
18
20

Recall that we want the variance to our test, T, under a model of cryptic
relatedness:
 m
Var * [T ]  Var 
Gi(1) 
 i 1


m
G
( 2)
i
i 1




m
H
j 1
m
(1)
j

H
j 1
( 2)
j

.


Use the following non-dodgy assumptions:
1. Draws of alleles from the population are simple Bernoulli trials. (Variance terms)
2. Controls are a random sample from the population. (Covariance terms with Hj’s
are 0)
3. Allow the possibility that cases and controls depart from Hardy-Weinberg
proportions by some factor, call this F. (Covariance terms for alleles in the same
individual)
4. For the mutational model,
a. Suppose the mutation process is the same for cases and random controls.
b. Conditional on a case and random chromosome having a very recent coalescent
time (on the order of 1-10 generations), assume that the chance that the alleles are
in different states is 0.
Then after …
Smoke from
my brain
JKP attempts
desperately
to keep me honest.
Me, after many hours
of intensive thought
processing

Var*[T ] can be simplified to:
Var*[T ]  4mp(1  p)(1  F )  4m(m  1)  Cov[Gi( a ) , Gi(a) ]
where i≠i´.

And now, we evaluate the covariance term under a model of cryptic
relatedness. This covariance term is fairly complicated, but it is related to the
following probability:
~ ~
 aff
P
t
|i
aff ]]
P[[tt(ii,ia),(
i  i
i , at
) | i
which denotes the probability that allele copy a and a´ from individuals i and
~
i´ coalesce in time t , conditional on the proposition that individuals i and i´
are both affected (with i≠i´). So what’s this probability?

Apply some Bayesian Trickery:
~
~

P
[
t

t
]

P
[
i

i

aff
|
t

t]
~
ii 
ii 
P[tii  t | i  i   aff ] 
P[i  i   aff ]
~

P
[
i

aff
|
i

aff
,
t

t ]  P[i   aff ]
~
ii 
 P[tii  t ] 
Depends on the
P[i  aff ]  P[i   aff ]
population model
(not on phenotype)

~ K ~t  K p
 P[tii  t ] 
K p2
~
 P[tii  t ]  l~t
Depends on the
genetic model
… and after some plug and play we finally get:
d  1  F  (m  1) 


~
t 1
~
P[tii  t ]  (l~t  1)
Under an additive model

Handy relationship between any lr’s and the sibling recurrence
risk ratio, a single parameter under an additive model (Risch,
1990):
(lr  1)  4  r  (ls  1)
where r is the kinship coefficient for type-r relatives, which is ¼
for r = 1, and decays by ½ for each increment to r. Using this
relationship we can simplify
(m  1)  (ls  1)  
1 
d  1 F 
 1 

~ 
2N
2N 
t 1

~
t 1
1
 
2
~
t 1
Simulations

Use Wright-Fisher forward simulation to assess analytical results:

Simulate 1,000 bi-allelic unlinked loci forward in time 4N generations,
with mutation parameter q = 4Nm = 1. (†)

Choose a single locus with the desired disease allele frequency, and assign
phenotypes to all members of the population under an additive genetic
model.

Select m cases and m random controls, use all non-disease loci to infer the
inflation factor based on the mean of all tests.
(†) because
WF simulations are notoriously slow to simulate, we use a speed-up by simulating a smaller
population with a proportionally higher mutation rate, and then rescale the population size and
mutation rate to the desired levels.
Simulation Results
95% central interval about the mean was at least .001 in each case.
“Tautological” Hutterite Analysis

Quick-note on the Hutterites

13,000 member pedigree where the genealogy is
known, with ~800 members phenotyped/genotyped
at many markers across the genome.

Target (for each phenotype):
a. Estimate coalescent probabilities for cases and random
controls based on the genealogy – “allele-walking”
simulations
b. Calculate the inflation factor (d) for each phenotype, and
compare to the analytic prediction.
Note increased probabilities in
cases over random controls
for recent coalescent times
Hutterite Analysis

Quick-note on the Hutterites

13,000 member pedigree where the genealogy is
known, with ~800 members phenotyped/genotyped
at many markers across the genome.

Target (for each phenotype):
a. Estimate coalescent probabilities for cases and random
controls based on the genealogy – “allele-walking”
simulations
b. Calculate the inflation factor (d) for each phenotype, and
compare to the analytic prediction.
Empirical d’s in a Founder Population
The inbreeding coefficient (F) was estimated at .048 and was included in the calculation.
Summary

We modeled cryptic relatedness using population-based
processes. Surprisingly, these expressions are functions of
directly observable parameters (population size, sample size, and
the genetic model parameterized by lr).

Our analytical results indicate that increased false positives due
to cryptic relatedness will usually be negligible for outbred
populations.

We applied out technique to a founder population as an example.
For six different phenotypes we found evidence for inflation,
which matched analytic predictions.
Acknowledgements

JK Pritchard and NJ Cox
(thesis advisors)

Carole Ober (access to
the empirical data)

$/£ :
NIH, NIH/NIGMS
Genetics Training Grant
Fine, name that tune: from
memory, recite of the first
1677 words of Kingman’s
1982 paper and I’ll get the
next round.
In the bar at the conference
during the week