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
Citation: Voight BF, Pritchard JK (2005) Confounding from cryptic relatedness in case-control association studies. PLoS Genet 1(3): e32.
Case-control association studies are a popular, convenient,
and potentially powerful strategy for identifying genes of
small effect that contribute to complex traits . 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
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  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 . 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 , 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
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: email@example.com
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
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.
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 .
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 .
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
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. 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, 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.
or absence (GðaÞ
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.)
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:
be an indicator variable for the presence (GðaÞ
PLoS Genetics | www.plosgenetics.orgSeptember 2005 | Volume 1 | Issue 3 | e32 0303
Confounding from Cryptic Relatedness
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.
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 .
D is proportional to both the trend test  and to the allele
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:
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 .
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.
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
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 ? þ 2 ? Cov½Gð1Þ
j ? þ ðm ? 1Þ ?
þ ðm ? 1Þ ?
j9? ? 2m ?
i? ¼ Cov½Hð1Þ
? ¼ pð1 ? pÞ ? F:
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 , 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
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:
? 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
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 .
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:
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
? P½Tii9¼ t?
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Confounding from Cryptic Relatedness
P½Tii9¼ tj/i¼ aff;/i9¼ aff? ¼KpKt
? P½Tii9¼ t?
¼ kt? P½Tii9¼ t?:
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 . 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 ,
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 , leading to different coefficients in the
cryptic relatedness term in Equation 16 below.
For the additive model,  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
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 . An
extreme example of this type of process (with selfing) was
illustrated by Rousset .
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
Let R be the excess probability of very recent coalescence
for affected chromosomes relative to random chromosomes.
where n might be taken as 10 or 15, say. Then write:
P½Tii9¼ t? ? ðkt? 1Þ;
i9? ¼ E½GðaÞ
¼ 1? ? E½GðaÞ
¼ 1?, notice that there are two
¼ 1?p ? p2:
To evaluate E½GðaÞ
cases. With probability R, the two chromosomes coalesce very
rapidly due to their shared phenotype. In that case, they share
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).
PLoS Genetics | www.plosgenetics.org September 2005 | Volume 1 | Issue 3 | e320305
Confounding from Cryptic Relatedness
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:
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 ?
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
i9?’½1 ? R þ pð1 ? RÞ?p ? p2¼ pð1 ? pÞR:
P½Tii9¼ t? ? ðkt? 1Þ:
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
Noting that ð1 ?
small), and thatP2?2tþ1converges quickly to 2/3; Equation
dhalf’1 þðm ? 1Þ ? ðks? 1Þ
dhalf’1 þðm ? 1Þ ? ðks? 1Þ
? 2?2tþ1: ð15Þ
2NÞt?1’1 for small t (provided that N is not
15 can be further approximated as
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
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.
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 , 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
Sample Size (m) Population Size
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
PLoS Genetics | www.plosgenetics.org September 2005 | Volume 1 | Issue 3 | e320306
Confounding from Cryptic Relatedness
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
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
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
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 ?
? ðkt? 1Þ;
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
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.
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.
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Confounding from Cryptic Relatedness
assumed the ‘‘half’’ relationships model, as in Equation 15
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:
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
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 . 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
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
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 . 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.
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 . 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)
95% CI about^dobs
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  and was included when calculating the analytical result (dA). The confidence intervals on dobs(last column) show the central 95% interval
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Confounding from Cryptic Relatedness
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  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
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.
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
Competing interests. The authors have declared that no competing
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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