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Confounding from Cryptic Relatedness

in Case-Control Association Studies

Benjamin F. Voight*, Jonathan K. Pritchard

Department of Human Genetics, University of Chicago, Chicago, Illinois, United States of America

Case-control association studies are widely used in the search for genetic variants that contribute to human diseases. It

has long been known that such studies may suffer from high rates of false positives if there is unrecognized population

structure. It is perhaps less widely appreciated that so-called ‘‘cryptic relatedness’’ (i.e., kinship among the cases or

controls that is not known to the investigator) might also potentially inflate the false positive rate. Until now there has

been little work to assess how serious this problem is likely to be in practice. In this paper, we develop a formal model

of cryptic relatedness, and study its impact on association studies. We provide simple expressions that predict the

extent of confounding due to cryptic relatedness. Surprisingly, these expressions are functions of directly observable

parameters. Our analytical results show that, for well-designed studies in outbred populations, the degree of

confounding due to cryptic relatedness will usually be negligible. However, in contrast, studies where there is a

sampling bias toward collecting relatives may indeed suffer from excessive rates of false positives. Furthermore,

cryptic relatedness may be a serious concern in founder populations that have grown rapidly and recently from a small

size. As an example, we analyze the impact of excess relatedness among cases for six phenotypes measured in the

Hutterite population.

Citation: Voight BF, Pritchard JK (2005) Confounding from cryptic relatedness in case-control association studies. PLoS Genet 1(3): e32.

Introduction

Case-control association studies are a popular, convenient,

and potentially powerful strategy for identifying genes of

small effect that contribute to complex traits [1]. However,

case-control studies may be susceptible to high rates of false

positives if the underlying statistical assumptions are not

satisfied. In particular, it has long been a source of concern

that population structure might cause confounding in such

studies [2,3], and a number of statistical methods have been

developed to detect and correct for unrecognized population

structure [4–9].

However, in their 1999 paper, Devlin and Roeder argued

that another source of confounding, ‘‘cryptic relatedness,’’

might actually be a more serious source of error for case-

control studies. Cryptic relatedness refers to the idea that

some members of a case-control sample might actually be

close relatives, in which case their genotypes are not

independent draws from the population frequencies. When

that happens, the allele frequency estimates in the case and

control samples are unbiased but may have greater variance

than expected, and tests of association that ignore the excess

relatedness have inflated type-1 error rates. Devlin and

Roeder [4] pointed out that if one is doing a genetic

association study, then one surely believes that the disease

has an underlying genetic basis that is at least partially shared

among affected individuals. If the cases share a set of genetic

risk factors then, presumably, this means that the cases will be

somewhat more closely related to each other, on average,

than they are to control individuals. Devlin and Roeder then

presented some numerical examples that suggested that

cryptic relatedness may be an important effect in practice.

However, it is difficult to know how realistic those examples

are because they were constructed artificially, and were not

based on a population genetic model.

At this time, there are few empirical data that bear on

whether cryptic relatedness is a serious problem in practice.

One study of association mapping in a founder population

concluded that in that population, cryptic relatedness did

have a significant impact on tests of association [10]. Methods

exist that can incorporate kinship relationships into the test

for association if such information is known [11–14]. If

relationships are not known in advance, then genomic

control methods can correct for cryptic relatedness [4,6,8],

while structured association methods (developed for the

population structure problem) cannot [7,9].

In this article, we aim to address the question of whether,

and when, cryptic relatedness is likely to be a serious issue for

case-control association studies. Our approach is to develop a

formal model of cryptic relatedness within a population

framework. We show that a natural measure of the impact of

cryptic relatedness, that we will denote d, depends on the

population size, the genetic model parameterized by the

recurrence risk ratio [15], and the number of sampled cases

and controls. Our initial model assumes that studies are ‘‘well

designed’’ in the sense that they do not have serious sampling

biases, such as a bias toward enrolling related cases into a

study. For that model, our results indicate that for association

studies in large outbred populations, the confounding effect

due to cryptic relatedness is expected to be negligible, but

that it may well be a more serious issue in small, growing

Received May 18, 2005; Accepted August 2, 2005; Published September 2, 2005

DOI: 10.1371/journal.pgen.0010032

Copyright: ? 2005 Voight and Pritchard. This is an open-access article distributed

under the terms of the Creative Commons Attribution License, which permits

unrestricted use, distribution, and reproduction in any medium, provided the

original author and source are credited.

Editor: Goncalo Abecasis, University of Michigan, United States of America

*To whom correspondence should be addressed. E-mail: bvoight@uchicago.edu

A previous version of this article appeared as an Early Online Release on August 2,

2005 (DOI: 10.1371/journal.pgen.0010032.eor).

PLoS Genetics | www.plosgenetics.orgSeptember 2005 | Volume 1 | Issue 3 | e320302

Page 2

populations. We also consider two simple scenarios in which

the sampling is biased toward collecting relatives among the

cases. Such sampling can lead to non-trivial inflation.

Results

A Model of Cryptic Relatedness

Consider a study in which m cases affected with a disease

and m random controls are genotyped at a single bi-allelic

locus with alleles B and b that are at frequencies p and 1 ? p,

respectively. We aim to model the impact of cryptic related-

ness on a test of association at this locus, assuming that the

locus is not in fact linked to any disease-associated genes. The

starting point for our notation and modeling is taken, with

some modification, from [4].

We suppose that cases and controls are sampled from a

single population (i.e., without population structure) of finite

size, with discrete generations, and that mating is independ-

ent of the phenotype of interest. All individuals are sampled

from the current generation. Since the impact of cryptic

relatedness is due to alleles that are identical by descent, it

will be necessary to model the coalescence times of

chromosomes. We will use T 2 f1;2;3;...g to denote the

random time at which a particular pair of chromosomes in

the current generation coalesces. (That is, T is the number of

generations before the present at which the copies of the

marker locus on each of the two chromosomes in question

trace their ancestry back to a single ancestral chromosome.)

According to standard models, for randomly chosen chro-

mosomes(i.e., unconditionalon phenotype)

P½T ¼ t? ¼ 1=ð2NtÞ ? ½Pt?1

number of diploid individuals in generation x [16].

We will also assume that affected individuals have the same

distribution of family sizes as do unaffected individuals, and

that selection against the disease phenotype is negligible.

Hence, chromosomes from affected individuals coalesce with

chromosomes from random individuals at the same rate as do

chromosomesfrompairsofrandomindividuals.Tobeprecise,

let T(i,a)(i9,a9)denote the coalescence time between chromo-

somes a and a9 from individuals i and i9. (Here, a and a9 denote

one of the two copies of each chromosome, chosen at random

in individuals i and i9, respectively.) Then by assumption,

x¼1ð1 ? 1=ð2NxÞÞ?, where Nx is the

P½Tði;aÞði9;a9Þ¼ tj/i¼ aff;/i¼ rand?

¼ P½Tði;aÞði9;a9Þ¼ tj/i¼ rand;/i¼ rand?;

where /i¼ aff and /i¼ rand indicate that individuals i and i9

carry affected and random (unknown) phenotypes, respec-

tively. In contrast, we will show that chromosomes from pairs

of affected individuals have an excess probability of very

recent coalescence. The extra relatedness of cases occurs

because they share a heritable trait, and not from average

differences in the family sizes of affected and unaffected

individuals. Under the assumption in Equation 1, it follows

that P½/i¼ affjTði;aÞði9;a9Þ¼ t;/i¼ rand? ¼ Kp, where Kp de-

notes the overall population prevalence of the disease of

interest. This is reasonable, because simply knowing that

individual i has a relative i9 whose affection status is unknown,

should not alter the probability that i is affected.

We also define a quantity Kt that is analogous to the

standard relative recurrence risk Kr[15]. Specifically, for a

pair of individuals i and i9, where i is affected, Ktis defined as

the probability that i9 is also affected, given that a specific

pair of alleles from the two individuals coalesces to a common

ancestral chromosome t generations before the present

(where the alleles are at a locus unlinked to any disease loci):

Kt¼ P½/i9¼ affj/i¼ aff;Tði;aÞði9;a9Þ¼ t?.

Notice, however, that the definition of Kt implies some

ambiguity in the actual relationship between the two

individuals in question: e.g., T can be 1 either for siblings or

for half-siblings, and 2 for cousins or half-cousins. Therefore,

to evaluate Kt, it will be necessary to be specific about mating

patterns in the population. Later in the paper, we describe

results for two particular models of random mating.

The ratio Kt/Kpwill be denoted kt. This is closely related to

the standard recurrence risk ratio kr[15], and measures the

proportional increase in risk for an individual given that one

of his/her chromosomes coalesces with the chromosomes of

an affected individual t generations before the present. Due

to shared genetic or environmental factors, kr(and hence kt)

is often ? 1 for close relatives; this means that even random

sampling of affected individuals can lead to a sample that

contains an excess of related cases.

Let GðaÞ

i

or absence (GðaÞ

i

¼ 0) of the B allele on the ath copy of this

locus in affected individual i. (Here, a 2 f1;2g labels the two

homologous copies of a marker in a diploid individual.)

Similarly, HðaÞ

j

denotes the analogous indicator variable for

the ath copy in control individual j.

Then we define a test statistic, D, which measures the

difference in the overall allele counts between case and

control samples at a given marker:

ð1Þ

be an indicator variable for the presence (GðaÞ

i

¼ 1)

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Confounding from Cryptic Relatedness

Synopsis

There has long been concern in the human genetics community

that case-control association studies may be subject to high rates of

false positives if there is unrecognized population structure. After

being considered rather suspect in the 1990s for this reason, case-

control studies are regaining popularity, and will no doubt be used

widely in future genome-wide association studies.

Therefore, it is important to fully understand the types of factors

that can lead to excess rates of false positives in case-control studies.

Virtually all of the previous discussion in the literature of excess false

positives (confounding) in case-control studies has focused on the

role of population structure. Yet a widely cited 1999 paper by Devlin

and Roeder (that introduced the genomic control concept) argued

that, in fact, ‘‘cryptic relatedness’’ (referring to the idea that some

members of a case-control sample might actually be close relatives,

unbeknownst to the investigator) is likely to be a far more important

confounder than population structure. Moreover, one of the two

main types of statistical approaches for dealing with confounding in

case-control studies (i.e., structured association methods) does not

correct for cryptic relatedness.

This work provides the first careful model of cryptic relatedness, and

outlines exactly when cryptic relatedness is and is not likely to be a

problem. The authors provide simple expressions that predict the

extent of confounding due to cryptic relatedness. Surprisingly, these

expressions are functions of directly observable parameters. The

analytical results show that, for well-designed studies in outbred

populations, the degree of confounding due to cryptic relatedness

will usually be negligible. However, in contrast, studies where there

is a sampling bias toward collecting relatives may indeed suffer from

excessive rates of false positives.

Page 3

D ¼

X

m

i¼1

Gð1Þ

i

þ

X

m

i¼1

Gð2Þ

i

? ð

X

m

j¼1

Hð1Þ

j

þ

X

m

j¼1

Hð2Þ

j Þ:

ð2Þ

When appropriately normalized, D forms the basis of

familiar tests of association. Under the null hypothesis,

D2/Var[D] is v2distributed with one degree of freedom [4].

D is proportional to both the trend test [17] and to the allele

test [18].

Under the standard null hypothesis, an allele copy at a

given marker is type B with probability p, independently for all

allele copies in the sample. The independence assumption

implies that there is no population structure, no inbreeding,

and that all cases and controls are mutually unrelated. If all

alleles are mutually independent, then the variance of D is

4mp(1 ? p). If, however, cryptic relatedness exists in the

sample, then the actual variance of the test—call this

Var*[D]—will exceed the variance predicted under the null

hypothesis. We will measure the deviation from the null

variance using the ‘‘inflation factor’’ d, defined as follows:

Var?½D?

4mpð1 ? pÞ:

In the absence of true association between the marker and

the genotype, the commonly used test of association,

D2/[4mp(1 ? p)], has a distribution that is the product of d

and a v2random variable [4].

Values of the inflation factor, d, near 1.0 imply that the

standard test of association is correctly calibrated, or nearly

so. Values of d substantially larger than 1.0 indicate that there

will be an excess of false positive signals. Our target here is to

derive an expected value for d under a model of cryptic

relatedness. These general results do not rely on a particular

genetic model, but we do present examples using an additive

model. We consider models of constant population size and

of recent population expansion.

d ¼

ð3Þ

Theory

We now characterize the extra variance that is caused by

relatedness within a given case-control study, and use this to

compute the expected inflation factor d. Starting from the

definition of D, in Equation 2, we can write Var*[D] as

Var?½D? ¼ m ? f2 ? Var½GðaÞ

þ 2 ? Cov½Hð1Þ

X

where i 6¼ i9, j 6¼ j9. We now need to determine how the value

of this expression depends on cryptic relatedness.

Since Giand Hjare Bernoulli trials, we have:

Var½Gi? ¼ Var½Hj? ¼ pð1 ? pÞ:

The following two terms in Equation 4 account for the

possibility of departures from Hardy-Weinberg equilibrium

in the sample. Assuming that these factors are independent of

case-control status, we can write these as

i? þ 2 ? Var½HðaÞ

j ;Hð2Þ

j ? þ 2 ? Cov½Gð1Þ

X

X

i;Gð2Þ

i?

j ? þ ðm ? 1Þ ?

a;a9

Cov½GðaÞ

i;Gða9Þ

i9?

þ ðm ? 1Þ ?

a;a9

Cov½HðaÞ

j ;Hða9Þ

j9? ? 2m ?

a;a9

Cov½GðaÞ

i;Hða9Þ

j

?gð4Þ

ð5Þ

Cov½Gð1Þ

i;Gð2Þ

i? ¼ Cov½Hð1Þ

j ;Hð2Þ

j

? ¼ pð1 ? pÞ ? F:

ð6Þ

where F measures the extent of the departure from Hardy-

Weinberg equilibrium [4,19]. If, in fact, there is a different

average level of inbreeding in cases than in controls [20], then

we would replace F in Equation 7 and thereafter, with an

average F across the cases and controls. (Notice that, unlike

here, the inflation factor used by Devlin and Roeder was

defined relative to the trend test, so that Hardy-Weinberg

departures cancel out in their formulation.)

In our model, the controls are sampled randomly from the

population. This means that the terms Cov½HðaÞ

Cov½GðaÞ

p, the fact that a random allele in the population is B, or b,

provides no additional information about the genotype of

another case or control in the sample. The assumption that

controls are sampled randomly will usually be a good

approximation, even if controls are specifically ascertained

as not having the disease. As we will show below, the size of

these covariance terms depends on the recurrence risk ratio

for the phenotype, and the recurrence risk ratio for being

unaffected is typically near one.

Next, since case alleles Giare each similarly distributed, we

can reduce Equation 4 by characterizing a single covariance

between case alleles and then collecting the sum of all

covariance terms that contain only case alleles. Given this, the

Hardy-Weinberg equilibrium terms, and Equation 5, Equa-

tion 4 simplifies to:

j;Hða9Þ

j9? and

i;Hða9Þ

j

? are zero. This follows because, conditional on

Var?½D? ¼ 4mpð1 ? pÞð1 þ FÞ þ 4mðm ? 1Þ ? Cov½GðaÞ

where i 6¼ i9. And now, finally, we need to evaluate

Cov½GðaÞ

to do this, we first need to evaluate the probability that alleles

in affected individuals share a common ancestor in gen-

eration t before the present. This will allow us to calculate the

extra relatedness in cases due to the phenotype.

Recall that Kp is the population prevalence of the

disease; Ktis the probability that a relative of an affected

individual is also affected, given that the two individuals

share a common ancestor t generations before the present;

and that kt¼ Kt/Kpis the corresponding ratio of risks [15].

Next, let T(i,a)(i9,a9) denote the coalescent time of allele

copies a and a9 from individuals i and i9. In a slight abuse

of notation, we will abbreviate T(i,a)(i9,a9) as Tii9. In what

follows, individuals i and i9 are random (unphenotyped)

draws from the population, except when specifically noted

(e.g., /i¼ aff indicates that i is affected). Then, using Bayes’

rule, we can compute the coalescence rates for two

chromosomes sampled from affected cases in the popula-

tion as follows:

i;Gða9Þ

i9?; ð7Þ

i;Gða9Þ

i9? under a model of cryptic relatedness. In order

P½Tii9¼ tj/i¼ aff;/i9¼ aff?

¼P½/i¼ aff;/i9¼ affjTii9¼ t?

P½/i¼ aff;/i9¼ aff?

? P½Tii9¼ t?ð8Þ

¼P½/i¼ affjTii9¼ t? ? P½/i9¼ affj/i¼ aff;Tii9¼ t?

P½/i¼ aff? ? P½/i9¼ aff?

where P[Tii9¼ t] denotes the prior probability of coalescence

in generation t, for random (unphenotyped) individuals.

Next, using the assumption that affected and unaffected

individuals coalesce with random chromosomes at the same

rate (Equation 1), it follows that P½/i¼ affjTii9¼ t? ¼ Kp, and

hence

? P½Tii9¼ t?

PLoS Genetics | www.plosgenetics.orgSeptember 2005 | Volume 1 | Issue 3 | e32 0304

Confounding from Cryptic Relatedness

Page 4

P½Tii9¼ tj/i¼ aff;/i9¼ aff? ¼KpKt

K2

p

? P½Tii9¼ t?

¼ kt? P½Tii9¼ t?:

ð9Þ

Equation 9 produces a pleasingly simple result: the

coalescence rate for chromosomes from affected individuals

is increased by a factor that is closely related to the standard

recurrence risk ratio.

Recurrence Risk for Relatives

The recurrence risk ratio is an important quantity in

genetic epidemiology, and is widely measured [1]. For siblings,

typical recurrence risk ratios for complex diseases range from

around 2 to 50. For more distant relationships, the risk ratio

declines approximately geometrically toward 1 as the number

of meioses separating two relatives increases.

In our theoretical development, we will assume that disease

inheritance is governed by a single additive gene [15],

unlinked to the marker locus of interest. Other genetic

models, including more complex models, behave similarly to

this, except that the rate of decay of ktwith increasing t may

differ somewhat [15], leading to different coefficients in the

cryptic relatedness term in Equation 16 below.

For the additive model, [15] obtained an expression for the

recurrence risk ratio, kr, for any possible relationship, r, in

terms of the recurrence risk ratio for full siblings, ks:

kr? 1 ¼ 4 ? /r? ðks? 1Þð10Þ

where /r is the kinship coefficient between rth-degree

relatives. For example, /r¼ 1=4 between sibs, and decays by

1/2 for each increment to r. To connect krto our model—

which is written in terms of coalescent time t instead of r—we

need to be more explicit about the mating patterns in the

population model.

For example, under the standard Wright-Fisher model

where individuals select their parents independently at

random, most relatives are ‘‘half-relations’’: half-siblings,

half-first cousins, half-second cousins, etc. In that case, for t

¼ 1, 2, 3, ... , the corresponding kinship coefficients are

/r¼ 1=8; 1=32; 1=128, and so on. Then for example, for t¼

2, kt ? 1 ¼ 4(ks ? 1)/32. If instead, mating is purely

monogamous, but partners are still chosen at random, then

all relationships are ‘‘full’’: full siblings, full cousins, etc. That

is, for t ¼ 1, 2, 3, ... , the corresponding kinship coefficients

are /r¼ 1=4; 1=16; 1=64; ... .

In summary, ktmay be much larger than 1 for the closest

relatives, but it becomes approximately 1 if the common

ancestor is more than just a few generations ago (. 10 or 15,

say). This qualitative conclusion does not depend strongly on

the assumed genetic model. Referring to Equation 9, this

means that chromosomes from affected individuals have an

excess probability of coalescing extremely rapidly (within the

past few generations). If they do not, then they behave

essentially like random chromosomes, for which coalescence

takes place on timescales of thousands of generations in

typical populations (Figure 1).

The dynamics of this process are reminiscent of structured

coalescent models with many demes [21–23]. In those models,

two chromosomes from the same deme either coalesce with

each other very quickly or escape into the population at large,

and coalesce on a much longer time scale. These two phases

have been described by John Wakeley as the ‘‘scattering

phase’’ and the ‘‘collecting phase,’’ respectively [24]. An

extreme example of this type of process (with selfing) was

illustrated by Rousset [25].

Calculating the Inflation Factor

As described above, ancestral chromosomes of affected

individuals coalesce at an increased rate during the most

recent few generations (Figure 1), and otherwise behave

essentially like random chromosomes. We now provide a

heuristic derivation of the inflation factor d; later we show

that our expression closely approximates the results obtained

in simulations. For simplicity, we consider the following

approximation.

Let R be the excess probability of very recent coalescence

for affected chromosomes relative to random chromosomes.

That is,

X

where n might be taken as 10 or 15, say. Then write:

Cov½GðaÞ

R’

n

t¼1

P½Tii9¼ t? ? ðkt? 1Þ;

ð11Þ

i;Gða9Þ

i9? ¼ E½GðaÞ

ijGða9Þ

i9

¼ 1?P½Gða9Þ

i9

¼ 1? ? E½GðaÞ

i?E½Gða9Þ

i9?

¼ E½GðaÞ

ijGða9Þ

ijGða9Þ

¼ 1?, notice that there are two

i9

¼ 1?p ? p2:

ð12Þ

To evaluate E½GðaÞ

cases. With probability R, the two chromosomes coalesce very

rapidly due to their shared phenotype. In that case, they share

i9

Figure 1. Coalescence Rates for Pairs of Random Chromosomes (Red)

and for Pairs of Chromosomes from Affected Individuals (Green)

Notice that chromosomes from affected individuals have a small excess

probability of coalescing very rapidly (i.e., in the most recent ten

generations or so). Otherwise, their coalescence rates are essentially like

those of random chromosomes. The region at the left-hand side of the

graph between the red and green lines represents the excess probability

of very recent coalescence among case chromosomes (denoted R in the

text). This is what gives rise to the effect of cryptic relatedness. For larger

t, the line for cases drops slightly below the line for random individuals,

since both distributions integrate to 1. These plots assume an additive

genetic model, with ks¼60, the ‘‘half’’-relationships mating model, and

a population size of 2,000. The line for cases was generated under the

approximation that the excess relatedness is completely limited to the

first n¼10 generations. In this case, the maximum coalescent probability

for case chromosomes is 0.00275, when t¼1; R ’ 0.00334. As expected,

the mean coalescence time is ’ 4,000 generations for both distributions.

Alterations in n yield similar results (unpublished data).

DOI: 10.1371/journal.pgen.0010032.g001

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Confounding from Cryptic Relatedness

Page 5

such a recent common ancestor that they are almost certainly

identical by descent. In the second case, with probability 1 ?

R, the two chromosomes behave as random chromosomes,

and their genotypes are independent Bernoulli draws from

the population frequencies:

Cov½GðaÞ

And finally, substituting Equations 11, 13, and 7 into

Equation 3, we obtain

d’1 þ F þ ðm ? 1ÞR

’1 þ F þ ðm ? 1Þ ?

t¼1

Equation 14 is worthy of discussion. When the simplest

model of independence among sampled alleles holds, then d¼

1. The term containing F corresponds to Hardy-Weinberg

departures, due to inbreeding for instance. The summation

term corresponds to the effect of cryptic relatedness; the sum

itself can be thought of as calculating the excess probability of

identity by descent between chromosomes from affected

individuals. Overall, the effect of cryptic relatedness increases

linearly with the sample size m (for a given population size

and kt).

i;Gða9Þ

i9?’½1 ? R þ pð1 ? RÞ?p ? p2¼ pð1 ? pÞR:

ð13Þ

X

n

P½Tii9¼ t? ? ðkt? 1Þ:

ð14Þ

Applications to Specific Models

In this section, we evaluate Equation 14 under a range of

specific models, in order to determine when cryptic related-

ness is likely to have a substantial impact on case-control

studies. The models presented assume an additive genetic

model, as described above. At first, we will assume that the

population is of constant size N, so that the probability of

coalescence in generation t, P[Tii9¼ t], is (1 ? 1/2N)t?1/(2N).

After that, we turn to models with population growth. For

simplicity, we set F ¼ 0.

The Inflation Factor in Populations of Constant Size

Recall from Equation 10 that kt? 1 ¼ 4/rðks? 1Þ. Recall

also, that when individuals select their parents independently

at random, as in the standard Wright-Fisher model, that most

relatives are ‘‘half-relations’’ (e.g., half-siblings, half-cousins,

etc.), and then the kinship coefficients /rare 1/8, 1/32, 1/128,

... for t ¼ 1, 2, 3, etc. Using dhalfto indicate this situation

where individuals are related via ‘‘half-relationships,’’ it

follows that

X

Noting that ð1 ?

small), and thatP2?2tþ1converges quickly to 2/3; Equation

dhalf’1 þðm ? 1Þ ? ðks? 1Þ

dhalf’1 þðm ? 1Þ ? ðks? 1Þ

2N

n

t¼1

1 ?

1

2N

??t?1

? 2?2tþ1: ð15Þ

1

2NÞt?1’1 for small t (provided that N is not

15 can be further approximated as

3N

:

ð16Þ

If instead, mating is purely monogamous, but partners are

still chosen at random, then all relationships are ‘‘full’’—e.g.,

full siblings, full cousins, etc., and the kinship coefficients are

two-fold higher. The corresponding inflation factor, dfull, is

dfull’1 þ2ðm ? 1Þ ? ðks? 1Þ

3N

;

ð17Þ

indicating that the impact of cryptic relatedness is approx-

imately doubled when there is fully monogamous pairing of

parents, compared to when there is independent pairing of

parents for each offspring.

Simulations

To check the accuracy of our analytical results, we

generated population histories via Wright-Fisher simulation

and estimated the inflation factor, d, for a given disease and

population genetic model, as described in the Materials and

Methods section. Results are presented in Table 1, and

compared to predicted results from Equation 16. The results

show close agreement between the analytical prediction and

the simulation results. In some cases, the analytical results

slightly overestimate the inflation factor, probably due to the

approximations used in relating Equation 9 to d.

While the choice of an additive model for the phenotype

(i.e., a heterozygote has exactly one-half the penetrance for

the phenotype as a homozygote for the risk allele does) is

mathematically convenient, alternative modes of inheritance

(including multilocus models, or models with dominance

components) are certainly likely in practice. Such models will

have the impact of changing the rate of decay of kt, and hence

the coefficient of the cryptic relatedness term in Equations 16

or 17. While we do not present a complete exploration of

such models, we have performed a modest number of

additional simulations under non-additive models. We have

found that those results are qualitatively similar to the results

presented above (unpublished data).

Intrinsic Constraints on d

Table 1 shows the predicted impact of cryptic relatedness

for a range of possible disease parameters. The magnitude of

the inflation factor is fairly small for all parameter

combinations shown, with a maximum value of 1.07. To

make this more concrete, an inflation factor of 1.07 implies a

quite modest excess of false positives: for instance, a fraction

1.5310?3of tests would be significant at the p¼10?3level. As

another example, consider a genetic model based loosely on a

study of autism [26], where ks ¼ 75, and Kp of 0.0004.

Table 1. Values of the Inflation Factor as a Function of Model

Parameters, and a Comparison of the Simulated (^dmean) and

Analytical (dA) Predictions, for Populations of Constant Size

ks

Sample Size (m) Population Size

(N)

Simulation

^dmean

Analytical

dA

1.5

3.0

5.0

9.1

3.0

3.0

3.0

3.0

2.0

2.0

2.0

2.0

350

350

350

350

300

600

900

1,200

900

900

900

900

12,500

12,500

12,500

12,500

12,500

12,500

12,500

12,500

6,250

12,500

25,000

50,000

1.000

1.008

1.028

1.061

1.005

1.024

1.041

1.057

1.044

1.022

1.010

1.004

1.005

1.019

1.037

1.073

1.016

1.032

1.048

1.064

1.048

1.024

1.012

1.006

Notice that the magnitude of the inflation is quite small for all parameter values shown. Estimates of d based on the

median were essentially identical (unpublished data). The standard errors on the simulation estimates of d are ,

0.001; the slight differences between those and the analytical results are probably due to some approximations in

the theory.

DOI: 10.1371/journal.pgen.0010032.t001

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Confounding from Cryptic Relatedness

Page 6

Assuming the full-sibling model of relatedness, a sample size

of 1,000, and a population size of 2.5 million (i.e., the number

required to find that many cases), d is just 1.02.

These examples notwithstanding, however, Equations 16

and 17 seem to suggest that d can be made arbitrarily large

simply by increasing the sample size m. But in fact, the space

of sensible models is actually rather constrained. Since m

cannot exceed Kp times the population size, there is a

practical limit on m for a given ks and population size.

Because of this constraint, it is difficult to construct

biologically plausible parameter combinations that result in

substantial inflation factors for randomly mating populations

of constant size.

To be more specific, let Ksbe the rate of disease in full

siblings of an affected proband, i.e., Ks¼ ksKp. Furthermore,

let f be the fraction of all affected individuals in the

population that are included in the sample. Then, noting

that f ¼ m/NKp, Equation 17 can be rewritten as

dfull’1 þ2ðm ? 1Þ

3N

’1 þ2

?

Ks? Kp

Kp

??

3fðKs? KpÞ:

ð18Þ

Therefore, since f ? 1, for diseases where Ksis smaller than,

say, about 1%, the inflation factor is negligible. The only way

to get large values of d is to have high values of Ks? Kpand

nearly complete ascertainment of cases (high f ). For instance,

if Kswere 0.2 and kswere 4, then the inflation factor could be

as large as 1.1, producing a small excess of false positives. But

the latter calculation assumes complete sampling of affected

individuals ( f ¼ 1), which would usually be difficult for a

common disease.

In summary, in populations of constant size, the impact of

cryptic relatedness is generally very small, unless (1) Ks is

quite large—more than 0.2, say, and (2) f is near 1, meaning

that there is nearly complete ascertainment of cases from the

population. Hence, cryptic relatedness should not be a

serious concern for most complex trait studies in stable

populations, assuming random sampling of cases. As we will

show in the next section, the situation is more serious for

models with population growth.

The Inflation Factor with Changes in Population Size

We now consider a model that allows for changes in

population size. Let Ntrepresent the population size at time t.

Then, provided that the coalescent probability 1/2Ntis not

especially large in any of the recent generations, and since kt

?1 decays as t increases, we can rewrite and simplify Equation

14 to

X

where again kt refers to the recurrence risk ratio for

coalescence time t. Because (kt? 1) decays quickly toward

zero, it is apparent that only changes in population size

during the last few generations will impact d. Moreover, for

given values of m and kt, smaller population sizes in the past

will produce higher inflation factors.

To check the accuracy of our results regarding demo-

graphic expansion, we modified the forward simulation

procedure used above such that instead of a single N, we

simulated exponential growth that began at time tonsetin the

d’1 þ ðm ? 1Þ ?

n

t¼1

1

2Nt

??

? ðkt? 1Þ;

ð19Þ

recent past starting at an initial population size NA. For each

subsequent generation t, the population size was determined

by the equation Ntþ1¼Nt? eafor a growth rate a such that the

population size in the final generation is Nf. We performed at

least 10,000 repetitions for each parameter combination, and

the 95% standard error about the mean for each estimated d

was no greater than 0.01. In our analytic calculation, we

Table 2. Values of the Inflation Factor in Very Recently Expanded

Populations

Generations from

Past (tonset)

Pre-Growth

Size (NA)

Post-Growth

Size (Nf)

^dmean^dmedianAnalytical

Result

2 125

250

500

125

250

500

125

250

500

125

250

500

125

250

500

125

250

500

25,0002.52

1.83

1.46

1.17

1.12

1.09

1.05

1.05

1.04

2.44

1.77

1.42

1.12

1.08

1.06

1.03

1.03

1.02

2.56

1.84

1.46

1.16

1.12

1.09

1.05

1.05

1.04

2.48

1.78

1.42

1.12

1.08

1.06

1.03

1.03

1.02

2.62

1.87

1.47

1.17

1.12

1.09

1.06

1.05

1.05

2.53

1.81

1.43

1.12

1.09

1.06

1.03

1.03

1.02

5 25,000

1025,000

2 50,000

5 50,000

10 50,000

In this model there has been exponential growth from size NAto Nfstarting tonsetgenerations ago. Notice that

extreme models of expansion can produce non-negligible inflation factors. The genetic model was constructed such

that ks¼2, and the sample size was 2,000 cases and 2,000 random controls. The standard errors for the simulation

estimates (^dmeanand^dmedian) are ? 0.01.

DOI: 10.1371/journal.pgen.0010032.t002

Figure 2. Cumulative Probability of Coalescence within the Last n

Meioses in the Hutterite Founder Population

Each line plots the estimated probability that two chromosomes drawn

at random, from different individuals affected with a given phenotype, or

from two random control individuals, descend from a single ancestral

chromosome within the last n meioses. These estimates are based on the

recorded Hutterite genealogy. The x-axis plots the average number of

meioses along the two lineages back to the common ancestor. Notice

that in the most recent generations, the case samples coalesce at higher

rates than do random controls.

DOI: 10.1371/journal.pgen.0010032.g002

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Confounding from Cryptic Relatedness

Page 7

assumed the ‘‘half’’ relationships model, as in Equation 15

and 16.

Results of the simulations, for a range of parameter values,

are summarized in Table 2. Again, the theoretical prediction

in Equation 19 is close to the simulated values. Under very

recent growth models,^d can be substantial (as much as 2.5 for

the extreme growth scenario shown). Under more realistic

models of population growth, the effect of cryptic relatedness

is smaller, but still non-trivial. Based on these results, it seems

clear that the magnitude of growth is an important factor for

determining d. In populations that have grown rapidly from

small size in the past few generations, cryptic relatedness may

indeed lead to high inflation factors. It should be noted that

many of the models presented have extreme growth; hence,

the higher levels of cryptic relatedness shown here are likely

to exceed anything seen in practice in human populations.

The qualitative difference between the equilibrium model

and the population-growth model can be understood as

follows. Consider two studies in which m affected individuals

are sampled from each of two populations that have the same

current size. If one population is of fixed size, while the other

has grown rapidly from a smaller size, then the probability

that two individuals are closely related is much higher in the

growing population than in the equilibrium population. It

follows from Equation 19 that this produces a higher inflation

factor in the growing population than in the stable one.

Cryptic Relatedness with Biased Sampling

Thus far, we have considered models that assume ‘‘good’’

sampling design, in the sense that the sample of cases

represents a random sample of the affected individuals in a

population. We now consider the impact of sampling schemes

that bias toward enrolling close relatives as cases in a study.

For the previous models, we showed that with random

ascertainment of cases, the inflation factor d is maximized

with complete ascertainment of cases from a population. The

following models are instead motivated by the scenario in

which a study enrolls only a small fraction of the affected

individuals in a large population but, due to sampling biases,

tends to recruit close relatives. Such situations might arise in

practice if, for example, a patient at a clinic or in a study

encouraged affected family members to visit the same clinic,

or also to enroll in the study.

As an extreme, but simple example, consider first the

situation in which the case sample consists of m(1 ? r)

unrelated affected individuals, plus mr/2 pairs of affected

siblings (r 2 ½0;1?). The controls are all unrelated to anyone

else. Assume furthermore that there is not inbreeding, so that

F ¼ 0 and the probability of recent identity-by-descent for

chromosomes in siblings is 0.5. (For simplicity, we assume

both in this and the next model that the sampling is from a

sufficiently large population relative to m that we can

approximately ignore the impact of cryptic relatedness apart

from that induced by the biased sampling of siblings.) Then

recall from Equation 14 that d ’ 1 þ F þ (m ? 1)R where R is

the (average) excess probability of recent coalescence,

computed across all pairs of case chromosomes. In this

model, a fraction r/(m ? 1) of the pairs of individuals are

siblings. The probability that a randomly selected chromo-

some a in one sibling and a9 in the other sibling descend from

the same parental chromosome is R ¼ 1/4. Hence, for this

model we obtain d ’ 1 þ r/4. At most, if the entire case

sample is made up of sibling pairs, d ¼ 1.25. Any relatedness

among the controls would further increase d.

As a second simple example, suppose that a study recruits

only a small fraction of affected individuals from a large

population, but that recruits sometimes then encourage their

siblings to enroll. Let the number of siblings of a recruited

individual be Poisson with mean g, and let h be the probability

that an affected sibling goes on to enroll in the study,

independently for each affected sibling. Then the number of

siblings of the initial recruit who enroll as patients in the

study is Pois(ghKs). After some algebra, it follows that the

expected fraction of pairs of case individuals in the sample

who are siblings is c(c þ 2)/[(m ? 1)(c þ 1)], where c ¼ ghKs.

Hence (again taking F ¼ 0), we obtain

d’1 þcðc þ 2Þ

4ðc þ 1Þ:

ð20Þ

From these examples, it seems that biased sampling of cases

can have a substantial effect on inflating the test statistics—

though this is less dramatic perhaps than might have been

expected. For example, suppose that index cases have an

average of g¼2 siblings, that they refer affected siblings with

probabilityh¼0.5,andthatKs¼0.4.Thentheinflationfactord

’ 1.17.

Cryptic Relatedness in the Hutterites

We have used data collected from a founder population,

the Schmiedeleut (S-leut) Hutterites of South Dakota, to

illustrate the impact of cryptic relatedness on association

studies for phenotypes measured in that population [27]. The

S-leut Hutterite population consists of 13,000 members

connected by a single, known, multigenerational pedigree

that goes back to 64 founder individuals about 12–13

generations ago. Approximately 800 members of this pop-

ulation have been phenotyped for many traits and genotyped

at a large number of microsatellite markers [27,28]. We

considered six phenotypes: asthma, atopy, diabetes, hyper-

tension, obesity (. 33% body fat for males, . 38% body fat

for females), and stuttering (ever stuttered), all of which we

treated as binary traits. We are grateful to C. Ober, who

kindly allowed us access to these data.

It has previously been reported that naı ¨ve tests of

association produce an excess of false positive signals in this

population [10,14]. Our aim in this section is to further

explore the impact of relatedness among cases in the context

of the theory developed here. In particular, we set out to

determine (1) whether we could detect excess relatedness

among affected individuals, (2) the empirical level of

confounding at random markers, and (3) whether we could

predict the observed level of confounding based on the

pedigree.

The fact that we have complete genealogical information

for the Hutterites allows us to estimate the coalescence

probabilities for pairs of alleles in any two individuals at any

time since the founding of this population. These proba-

bilities were estimated as described in the Materials and

Methods section. The data do not provide information about

coalescent events more than about 12 generations before the

present, but the theory presented above suggests that the

impact of cryptic relatedness is due to very recent coalescent

events (and this is supported by our results, as follows).

The results of this analysis are presented in Figure 2. For all

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Confounding from Cryptic Relatedness

Page 8

six phenotypes, there is an excess rate of coalescence within

the pedigree, relative to random controls. Moreover, most of

the increased probability of coalescence is due to rather close

relatedness among cases (i.e., mainly for ? 4 meioses). This is

consistent with the theoretical prediction that kt?1 declines

rapidly to zero.

We next used the genotype data to obtain an empirical

estimate of d for each phenotype, under the assumption that

most random markers are not genuinely associated with

disease loci. We considered 437 microsatellite markers typed

in approximately 800 members of this population and

estimated d as described in the simulation methods above.

The procedure for estimating d in this data is described in the

Materials and Methods section.

Table 3 summarizes the results from this analysis. For all six

phenotypes, there is a non-trivial inflation to the test for

association under the null hypothesis, in the range of about

1.2–1.3. This is consistent with the previous report by

Newman et al. of an excess of positive signals at a set of

microsatellite markers in this population [10]. An inflation

factor of 1.2 implies a rejection rate that is ’ 1.5-fold too

high at the 5% level, and ’ 2.7-fold too high at the 0.001 level.

A d of 1.3 implies a rejection rate that is ’ 1.7-fold too high at

the 0.05 level, and ’ 3.8-fold too high at the 0.001 level. In a

majority of cases, the predicted level of inflation matches

empirical estimates, and the analytical result in all cases

predicts a non-trivial inflation factor for each phenotype. For

related subsets of phenotypes (asthma/atopy and obesity/

hypertension/diabetes), the observed inflation factor appears

similar. However, this is partly coincidental: d depends on

both the coalescent time and the sample size, which are

different for each phenotype.

Discussion

Should one be concerned about confounding from cryptic

relatedness in association studies? To address this question,

we have developed theory to predict the amount of cryptic

relatedness expected in a random-mating population. Our

results demonstrate that confounding effects of this kind are

expected to be substantial only under rather special

conditions. The bulk of the effect is due to the occurrence

of quite close relationships among sampled individuals.

Except in small populations, random pairs of affected

individuals are unlikely to be closely related. Our results in

Equation 14 show that for a given genetic model and

population size, the impact of cryptic relatedness grows

linearly with sample size. However, this obscures the fact that

in practice, the maximum number of cases m that can be

sampled from a given population size, N, is constrained by the

population prevalence (Kp), and hence is inversely related to

kr. That is to say, assuming constant population size, it is

difficult to construct examples in which cryptic relatedness

has an appreciable effect.

In contrast, studies of populations in which there has been

rapid and recent population growth, and where the total

study population is small, should indeed be concerned about

cryptic relatedness. This scenario produces higher levels of

relatedness than are possible for the same values of m and kr

in stable populations. Studies in populations that meet these

conditions—especially founder populations—should use ped-

igree-based methods or genomic control to minimize false

positives due to cryptic relatedness [4,10,12].

Another situation in which cryptic relatedness may be

important is when there is extensive inbreeding. A model in

which individuals are likely to mate with relatives will

increase d relative to the models analyzed in this paper.

When there is inbreeding, if two individuals share one recent

common ancestor, they are likely to share other recent

ancestors. That is, conditional on having a recent common

ancestor, the expected kinship coefficient between two

individuals would be higher than modeled in Equations 16

and 17. With modest inbreeding, this is likely to be a small

effect, but the effect may be important in some populations

with extensive inbreeding. Indeed, population structure may

be viewed as a strong form of inbreeding, and that is often

suspected to be a non-trivial source of confounding [29]. In

contrast, sampling schemes that draw both cases and controls

equally from just a segment of a population (e.g., from part of

a city) should not induce particular problems. Even if there is

extra covariance among sampled individuals, this should

occur both within and between cases and controls equally,

and thus cancel (Equation 4).

It should be noted that our results assume that the disease

phenotype is selectively neutral (see discussion surrounding

Equation 1). If, in fact, affected individuals or mutation

carriers have fewer offspring than normal, then this will mean

that affected individuals tend to have fewer close relatives

than do random individuals. This effect would in many cases

lower the probability of recent coalescence of case chromo-

somes, thus reducing the size of d. This situation would

reduce the level of cryptic relatedness relative to the models

Table 3. Observed ð^dobs) and Predicted (dA) Inflation Factors for Six Phenotypes Measured in the Hutterite Founder Population

PhenotypeSample Size (m)P[Coal] (Cases) P[Coal] (Randoms)

dA

^dobs

95% CI about^dobs

Asthma

Atopy

Diabetes

Hypertension

Obesity

Stuttering

67

174

36

53

152

30

0.0467

0.0464

0.0425

0.0481

0.0453

0.0471

0.0454

0.0451

0.0384

0.0444

0.0444

0.0449

1.13

1.27

1.19

1.23

1.18

1.10

1.30

1.32

1.19

1.22

1.21

1.19

1.29–1.32

1.30–1.34

1.18–1.20

1.21–1.23

1.20–1.23

1.18–1.20

The predicted inflation factors were estimated by computing the probability that pairs of case chromosomes, or pairs of random control chromosomes coalesce (P[Coal]) within the Hutterite pedigree (see Figure 2). The mean inbreeding

coefficient (F) for the set of Hutterites with phenotype data was estimated to be 0.038 in the sample [27] and was included when calculating the analytical result (dA). The confidence intervals on dobs(last column) show the central 95% interval

about^dobs.

DOI: 10.1371/journal.pgen.0010032.t003

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Confounding from Cryptic Relatedness

Page 9

presented here. Conversely, a phenotype that increased

fitness (perhaps in carriers of genes responding to selection

only) might lead to increased d.

Lastly, it should be noted that our primary model assumed

a ‘‘good’’ epidemiological design in which individuals are

ascertained randomly from the population. However, cryptic

relatedness can also result from the non-random ascertain-

ment of family members in a case-control study. For instance,

affected family members might be more likely to seek

treatment in the same clinic, or affected individuals might

encourage their affected relatives to enroll in a study. These

types of situations may be difficult to detect at the time of

enrollment, but can have non-trivial consequences even in

large outbred populations. We have shown that these

situations indeed result in excess false positive rates. After

data collection, we recommend the use of techniques for

identifying cryptic relative pairs based on genetic data [30–

33]. Genomic control [4] can then be helpful for identifying

any residual inflation.

Materials and Methods

Simulations. To check the accuracy of our initial analytical results,

we generated population histories via Wright-Fisher simulation and

estimated the inflation factor, d. A population of size N was advanced

forward in time 4N generations, with non-overlapping generations

and random pairing of parents, independently for each offspring. For

each simulation, 1,000 bi-allelic sites separated by a recombination

fraction of 0.5 (i.e., freely recombining) were simulated with a

mutation rate of h ¼ 4Nl ¼ 1. After 4N generations, a random site

with the desired allele frequency was selected as the true disease

locus, and affection status was assigned to all members of the

population based on an additive genetic model. To shorten the

computational time, we initiated the simulations such that a smaller

population with proportionally higher mutation rate was advanced

forward in time until a given point in the distant past, and then the

population size and mutation rate were rescaled to the desired levels.

Samples of m random controls and m affected cases were then drawn

from the simulated population. Then, for each marker, apart from

the disease locus, we constructed the 2 3 2 contingency table

containing the allele counts for cases and controls, respectively;

provided that the expected count for each cell in the table was at least

five, we computed the standard Pearson’s v2test statistic. We then

estimated the inflation factor d using estimators based on both the

mean and median values of the v2statistics [4,6]. For each estimated d,

95% standard errors about the mean were based on 10,000 replicate

simulations.

Estimating coalescent probabilities in the Hutterites. We estimated

the coalescent probabilities for pairs of alleles in two individual

Hutterites by the following. Starting from the affected individuals in

the population, or from a matched random sample of individuals

from the current population, we simulated the inheritance of a pair

of randomly chosen chromosomes from different individuals, back-

ward through time, from the present to the founders of the

population. If the two chromosomes coalesced to a common ancestral

chromosome within the pedigree, we counted the number of meioses

back to that common ancestor, reporting the average number if the

number of meioses was different on the two lineages. We repeated

this procedure until we observed at least 500,000 coalescence events

within the simulation. To estimate the mean inbreeding coefficient

(F) in this sample, we used the same procedure as above except that

we picked the two chromosomes from the same random individual,

traced them backward in time, and determined how frequently those

two chromosomes coalesced within the pedigree.

Calculating the inflation factor in the Hutterites. For each marker,

we constructed a 23k contingency table, where k was the number of

alleles for this marker. Then, we pooled the smallest allele counts in

the table with the second smallest allele counts until a 2 3 2

contingency table was formed. These artificial 2 3 2 tables should

mimic the results that would be obtained using bi-allelic markers. The

depth of the pedigree is short enough that mutation within the

pedigree should have minimal impact on d. For each phenotype, we

selected a random sample of controls with data collected for the

analyzed phenotype and then treated the remaining affected

individuals in the sample as cases. The list of random controls was

then truncated (randomly) so that the sample sizes were equal in the

two groups. For this set of cases and controls, we estimated d based on

the mean of tests from these 437 markers. This procedure was

performed 1,000 times.

To be more careful about the possibility that some loci might be

genuinely associated with a phenotype or in various degrees of

linkage, we repeated the analysis using approximately 40 micro-

satellite markers, unlinked either to one another or to candidate gene

regions showing evidence of linkage. The resulting^ds based on the

mean were almost identical for all phenotypes to the larger marker

sample (unpublished data). Finally, we generated a semi-analytical

result for the phenotype by plugging the coalescent probabilities

estimated from the pedigree, along with estimated inbreeding

coefficients, and the average number of cases selected across all

replicates, into Equation 14.

Acknowledgments

We thank Carole Ober for providing the marker, phenotypic, and

genealogical data used for the Hutterite data analysis and for

comments on the manuscript; Rebecca Anderson and Natasha

Phillips for additional assistance in organizing and interpreting the

phenotype data; and Catherine Bourgain, Graham Coop, William

Wen, Sebastian Zo ¨llner, and the anonymous reviewers for helpful

comments or discussion. This work was supported in part by the

National Institutes of Health (HG002772) and a Hitchings-Elion

award from Burroughs Wellcome Fund to JKP; BFV received support

from the above grant to JKP as well as NIH DK55889 to Nancy J. Cox

and from a Genetics Regulation Training Grant NIH/NIGMS NRSA 5

T32 GM07197.

Competing interests. The authors have declared that no competing

interests exist.

Author contributions. BFV and JKP both conceived of and

designed the model, and wrote the paper. In addition, BFV also

performed the simulations and analyzed the data.

&

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PLoS Genetics | www.plosgenetics.orgSeptember 2005 | Volume 1 | Issue 3 | e320311

Confounding from Cryptic Relatedness