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Biostatistics (2001), 2, 2, pp. 173–181
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Referent sampling, family history and relative
risk: the role of length-biased sampling
ORI DAVIDOV
Department of Statistics, University of Haifa, Haifa 31905, Israel
E-mail: [email protected]
MARVIN ZELEN
Department of Biostatistics, Harvard School of Public Health, and the Dana Farber Cancer Institute,
655 Huntington Avenue, Boston, MA 02115, USA
E-mail: [email protected]
S UMMARY
Familial risk of disease is often assessed using case control studies based on referent databases. A
referent database is a collection of family histories of cases typically assembled as a result of one
family member being diagnosed with disease. This sampling scheme is equivalent to sampling families
proportional to their size. The larger the family, the greater the probability of finding the family in the
referent registry. This phenomena is known as length-biased sampling. The consequence of this kind of
sampling is to bias the regression estimate associated with family history. The estimate is typically inflated
in comparison to what is true for the actual population.
Keywords: Case control study; Family history; Length biased sampling; Referent registries; Relative risk.
1. I NTRODUCTION
There is accumulating evidence that some diseases cluster in families. Familial aggregation of disease
may be due to genetic (Claus, 1995) and/or environmental risk factors. It is well known that environmental
risk factors, such as dietary intake, correlate within families, and with some diseases (Doll and Peto,
1981). More recently, the discovery of breast cancer susceptibility genes (e.g. BRCA1 and BRCA2) firmly
established the role of genetics in a common type of cancer (Claus et al., 1998). A method of assessing
the risk of carrying these genes was recently proposed by Parmigiani et al. (1998). The investigations of
familial aggregation were initially motivated by the analysis of registries of families. There are two general
ways to assemble such registries. One method is to randomly sample families from the general population.
This method is inefficient when the disease under study is rare. The other method is to assemble the
registry by collecting family histories from individuals diagnosed with the disease. We refer to the latter
as a referent registry and the sampling process for creating such a registry as referent sampling. Referent
sampling can be embedded in both retrospective and prospective study designs. However, case control
studies for assessing the role of family history are more common (Adami, 1981; Bain, 1980; Slattery and
Kerber, 1993). Typically, individuals sampled from a referent registry are compared with a control group.
Often, positive family history is defined as the existence of disease in a prespecified group of relatives.
The random variable indicating positive family history is then treated like any other ‘exposure’ (Breslow
and Day, 1980). Potential problems associated with this type of design have been pointed out by Weiss et
al. (1982). The paper by Khoury and Flanders (1995) compared the odds ratio derived using case control
c Oxford University Press (2001)
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O. DAVIDOV AND M. Z ELEN
designs to those derived from comparing lifetime risks among first-degree relatives. They concluded that
using positive family history as a risk factor in case control studies may overestimate the true risk of
positive family history. This paper investigates the bias inherent in using an aggregate measure of positive
family history in the context of logistic regression models for case control data when the cases are drawn
from a referent registry. In particular, we quantify the bias in estimating the parameters associated with
the relative risk of disease as well as the odds ratio. A consequence of this bias is that the current methods
for analysing referent registries result in finding excess risk within families even if no excess risk actually
exists. We also show how this bias can be avoided.
The paper is organized in the following way. In the next section we introduce our notation and
formulation. In Section 3 we derive exact and approximate formulae for the magnitude of the bias. The
numerical properties of the bias and their practical implications are examined in Section 4. In Section 5
we examine extensions to correlated outcomes and covariate dependent probabilities. Finally, Section 6
serves as a brief summary and discussion.
2. N OTATION AND FORMULATION
Consider a family of size N . A minimum family size of two is assumed. The term ‘family’ can be
defined in any
Nway which is appropriate. Let Di be an indicator of disease status for the ith family member.
Then D = i=1
Di is the number of diseased individuals in the family. The individual D1 is designated
as the proband, i.e. the individual through which the family was ascertained. The proband may be diseased
or healthy. Let R = 1 if the family is in a referent registry and R = 0 otherwise. If R = 0, the family is
obtained from a random sample of control families from the general population. We discuss three ways
in which a random sample of control families may be collected. One way is to take a random sample of
healthy individuals (probands) and collect their family histories. Alternatively, data can be collected by
drawing a random sample of families from a registry of families. There is no proband when the family
unit is sampled in this way. A closely related method is to randomly select individuals from the population
and generate their family histories. These probands may, or may not, have the disease. This method is
equivalent to randomly sampling from a family registry. The case-probands are randomly selected cases
from the referent registry. The referent registry is a random sample of families with at least one case. Let
D O and N O refer to the number of disease individuals and family size after discounting the proband. If
a family of size n has d diseased individuals, then D O = d − 1 and N O = n − 1 for a referent registry;
a control group generated by sampling individuals without disease will have D O = d, N O = n − 1; and
a sample from a family registry will have D O = d, N O = n. Our developments focus on case control
studies in which the family history for the case is obtained from a referent registry and information on
control families is obtained from a random sample of non-diseased individuals. Results for other sampling
methods will also be discussed. Typically, the case and control are matched for covariates believed to be
related to the occurrence of disease. In practice, case control studies for familial association are analysed
using the logistic regression model
Pr(D1 = 1|x, y) =
exp(α + βx + γ t y)
.
1 + exp(α + βx + γ t y)
(2.1)
The parameter α is associated with the base line risk and sampling fraction. The parameter β is associated
with x which we take to be an indicator of positive family history, i.e. X = I{D O 1} . The parameter γ is
associated with other covariates such as age, gender and ethnicity which may be associated with disease.
It is well known that exp(β) is the odds ratio for both prospective and retrospective studies. Since the
case-proband is drawn from a referent registry and the control-proband is from a random sample, we can
Referent sampling, family history and relative risk: the role of length-biased sampling
175
write
Pr(X = 1|D1 = 1) = Pr(X = 1|D1 = 1, R = 1) = Pr(D O 1|R = 1),
Pr(X = 1|D1 = 0) = Pr(X = 1|D1 = 0, R = 0) = Pr(D O 1|R = 0).
Hence the retrospective relative risk of disease is
ψ=
Pr(D O 1 | R = 1)
.
Pr(D O 1 | R = 0)
(2.2)
For diseases with very low incidence ψ is approximately equal to exp(β). The relative risk is the ratio of
the probabilities of having additional disease in a family given the method used to collect the data. The
denominator of (2.2) can be estimated by the proportion of families from a random registry with disease.
The numerator can be estimated by the proportion of families in the referent registry having two or more
diseased individuals. These probabilities are of interest because they provide the basis for assessing the
risk associated with family history (Bain, 1980). The relationship between the relative risk ψ and the
log odds ratio β implies that if β (or ψ) is estimated with bias so will ψ (or β). Working directly with
probabilities, rather than odds ratios is simpler, and therefore we demonstrate the bias on the relative risk
scale. Finally, define θ = Pr(D = 1) to be the probability of disease for an individual. For the moment
we are ignoring age and gender or any other covariates that may be associated with disease. We assume
that all family members have the same chance of being incident with disease and that the probability of
disease is independent among family members and of family size. Later we relax these assumptions. It is
worth mentioning that a similar model is discussed by Khoury et al. (1993) in the context of segregation
analysis and ascertainment bias.
3. T HE BIAS
In this section we derive exact and approximate formulae for the probabilities of having additional
cases in a family drawn from either a random or referent registry, assuming a case control type design.
Throughout this section disease incidence is assumed independent within families. The expression for the
relative risk (2.2) is an easy consequence. The relative risk is a function of the marginal probability of
disease and the moments of the family size distribution. Our results quantify the magnitude of the bias
arising when case-referent sampling is used without consideration of the family size.
The probability of at least one diseased individual in a family of size N = n is given by
Pr(D 1 | N = n) = 1 − (1 − θ)n
n(n − 1) 2
= nθ −
θ + O(θ 3 ).
2
(3.1)
Note that (3.1) remains true when (D, N ) is replaced with (D O , N O ). Moreover, this equality holds in
both the referent and random registries. More formally,
Pr(D O 1 | N O = n, R = k) = Pr(D 1 | N = n)
for k = 0, 1.
Consider a family drawn from the random registry (R = 0). The family size is a random variable with
probability mass function p(n) = Pr(N = n|R = 0). We assume that the family is ascertained through a
healthy proband, hence N = N O + 1 which implies that
Pr(N O = n | R = 0) = p(n + 1).
(3.2)
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O. DAVIDOV AND M. Z ELEN
Therefore,
Pr(D 0 1 | R = 0) =
Pr(D 1 | N = n) Pr(N O = n | R = 0)
n
=
[1 − (1 − θ)n ] p(n + 1)
n
=1−
1
G N (1 − θ)
1−θ
(3.3)
where G N (·) is the generating function for the random variable N . It is easily shown that the coefficient
of θ k in the expansion of G N (1 − θ ) is µ[k] (−1)k /k!, where µ[k] denotes the kth factorial moment of N .
Expanding (3.3) in a Taylor series we have
Pr(D 0 1 | R = 0) = θ (µ − 1) −
θ2 2
[σ + (µ − 1)(µ − 2)] + O(θ 3 )
2
where µ and σ 2 are the mean and variance of the family size. In order to calculate the equivalent quantity
for the referent registry, it is necessary to calculate the family size distribution within the referent registry.
Recall that a family is in the referent registry if at least one family member has the disease. Therefore, the
events {R = 1} and {D 1} are equivalent. Hence
Pr(N = n | R = 1) =
Pr(D 1|N = n) Pr(N = n)
[1 − (1 − θ)n ]
= p(n)
.
Pr(D 1)
1 − G N (1 − θ)
(3.4)
We point out that (3.4) can be written as
n
k
k
k=0 (−1) n [k+1] θ /k!
p(n) ∞
k
k
k=0 (−1) µ[k+1] θ /k!
2
2
n
1 2n [3] µ + 3nµ[2] − 2nµµ[3] − 3n [2] µµ[2] 2
1 n [2] µ − nµ[2]
= p(n)
+ O(θ 3 )
θ
+
θ
−
µ 2
12
µ2
µ3
where n [k] = n(n − 1) · · · (n − k + 1) has expectation µ[k] . Note that for small values of θ the above
expression can be approximated by np(n)/µ which is the classical form of the length-biased sampling
distribution (cf. Cox and Isham, 1980). For a general treatment of weighted distributions, of which (3.4)
and its approximation are special cases, see Gill et al. (1988) and Kirmani and Ahsanullaha (1987).
Consequently, the conditional distribution of N O given R = 1 is obtained by replacing n with n + 1
in (3.4). Hence the quantity Pr(D O 1 | R = 1) is evaluated by multiplying (3.1) by equation (3.4),
corrected for the proband, and summing over n, resulting in
Pr(D 0 1 | R = 1) =
(1 − θ )(1 − G N (1 − θ)) − G N (1 − θ) + G N (θ 2 − 2θ + 1)
(1 − θ)(1 − G N (1 − θ))
(3.5)
which, expanded in a Taylor series, equals
Pr(D 0 1 | R = 1) =
µ[2]
θ+
µ
2
µ[3] − µ[2] /2 µ[2] /4 2
θ + O(θ 3 ).
−
µ
µ2
Finally, the relative risk is given by the ratio of (3.5) to (3.3). The first-order approximation for the relative
risk is
ψ=
Pr(D 0 1 | R = 1)
σ2
+ O(θ ).
=
1
+
µ(µ − 1)
Pr(D 0 1 | R = 0)
(3.6)
Referent sampling, family history and relative risk: the role of length-biased sampling
177
The coefficients of terms involving θ are complicated functions of the moments and are therefore omitted.
Note that (3.6) provides an accurate approximation for small values of θ . It can be shown that ψ is
decreasing in θ and therefore it attains its supremum at θ = 0. The relative risk and the odds ratio, exp(β),
associated with positive family history are approximately equal for small values of θ . However, the odds
ratio is slightly larger than the relative risk for all θ > 0. Additionally, note that for small values of θ and
a fixed coefficient of variation the bias increases as the mean family size decreases.
Our calculations show that contrary to (perhaps naive) intuition, the relative risk is biased away
from unity. The bias arises because the marginal probabilities of disease are calculated by averaging
over the family size. The family size, however, depends on the sampling plan. The magnitude of the
bias is determined by the ratio of the effective family sizes. The effective family size is equal to the
mean number of family members susceptible to disease and corresponds to the first term in the Taylor
expansion of the probability of additional cases in a family. In the referent registry the effective family
size is (σ 2 + µ2 )/µ − 1 and in the random registry it is µ − 1. The ratio of these expected family sizes
is equal to (3.6). Although it is well known that the analysis of family history data should account for the
family size (Elandt-Johnson, 1971; Lynch, 1977) the subtle effect of length-biased sampling has not been
recognized.
Finally, if the control family is obtained from a family registry there is no proband. In this case D O =
D and N O = N . Consequently, we have
Pr(D 0 1|R = 0) = µθ −
µ[2] θ 2
+ O(θ 3 )
2
which implies that the relative risk is
ψ =1+
σ2 − µ
+ O(θ )
µ2
which may be smaller or larger than unity according to the sign of σ 2 − µ.
4. Q UANTIFYING THE BIAS
The mean and variance of the number of siblings, denoted µs and σs2 , were estimated from data on
married couple families (US Bureau of Census, 1998). If the mean and variance of sibsizes is known it is
then possible to calculate µ and σ 2 for different definitions of the term ‘family’. For example, in breast
cancer research, a ‘family’ is sometimes defined as the first-degree female relatives (mother and sisters).
In this case the mean family size equals the mean number of sisters plus their mother. The addition of
the mother does not alter the variability. Thus µ = µs /2 + 1 and σ 2 = σs2 /2. The latter calculations
are also appropriate for prostate cancer. Estimating µ and σ 2 for families defined differently is done
in a similar manner. Adjusting our calculations to deal with families including second-degree relatives
is straightforward. Three types of families are considered: sibling families, single-gender families, and
maternal line families. Single-gender families are defined as having all first-degree relatives of the same
gender. A maternal line family consists of female siblings, their maternal aunts and maternal grandmother.
Table 1 summarizes estimates of family size as well as the associated bias 1 + σ 2 /µ(µ − 1).
A quick glance reveals that the bias is considerable, especially when the mean family size is small.
Recall that relative risks larger than one are interpreted as evidence for a familial or heritable component
for the disease. Investigators, using referent registries, have reported relative risks for breast cancer ranging
from 1.5 to 6.9 depending on the nature of the disease and familial relationship. Sattin (1985) reports that
the relative risk is 2.3 for first-degree relatives and 1.5 for second-degree relatives. Adami (1981) reports
relative risks for breast cancer of 1.7 and 2.2 for the same categories. Byrne (1991) reports a relative risk
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O. DAVIDOV AND M. Z ELEN
Table 1. Mean, variance and bias for various generic families
µs
2.01
Siblings
σs2
bias
2.99 2.49
Single gender families
µ
σ2
bias
2.01 1.49
1.75
Maternal line
µ
σ2
bias
3.01 2.99
1.5
of 2.3 for sisters which was increased to 4.7 for bilateral disease and 6.9 for premenopausal women. In
the light of our results, odds ratios obtained through case-referent sampling should be interpreted with
caution because the correct null value is no longer unity. It is given by (3.6), which may be estimated from
the data. Although our results do not automatically invalidate positive associations established using casereferent sampling it is plausible that a portion of the relative risk reported is an artifact of the sampling
method. In fact, in Section 5 we show that the relative risk of disease is estimated with error even when
relatives are positively correlated. In addition, it can be shown that the regression parameters associated
with other covariates (i.e. γ in (2.1)) cannot be consistently estimated whenever the indicator of positive
family history is included in the model.
5. E XTENSIONS
In this section we briefly describe two extensions of our previous results. We show that the relative risk
is similarly inflated when family members are positively correlated and when the probability of disease
depends on a covariate such as age. We omit the mathematical derivations as those can be patterned on
Section 3.
5.1
Correlated outcomes
In this section we consider models in which the disease events are positively correlated. In particular we
are interested in correlation induced by an (unobserved) genetic mechanism. Consider N siblings. Let
G i equal 1 if individual i is the carrier of certain genes and zero otherwise. We assume that these genes
are associated with the disease. The probability of disease conditional on the genes, i.e. the penetrance,
is defined by θk = Pr(Di = 1|G i = k) for k = 0, 1, with θ1 θ0 . We classify families by their risk
status. We consider three risk categories which correspond to the number of parents (founders) carrying
the disease gene. Thus F = 0, 1 or 2. Define qk = Pr(G i = 1|F = k) for k = 0, 1, 2, to be the
probability of transmitting the disease genes from parents to their children. Obviously q2 q1 q0 = 0.
The probability of risk status k is given by δk = Pr(F = k) and is related to the prevalence of the
gene in the general population which mates randomly. Most standard genetic models, such as single gene
models, multiple gene models, as well as dominant and recessive gene models can be expressed within
this formulation. For more details see Lange (1997). The vector D is the only observable in this latent
variable model. Its distribution is an easy consequence. Risk assessment based on a similar model is given
in Davidov (1996).
Let πk = Pr(Di = 1|F = k) be the marginal probability of disease given the risk status of the
parents. It is easily seen that πk = θ0 + qk (θ
1 − θ0 ). Consequently, the marginal probability of disease
, equals δk πk . Similarly, the marginal probability of disease in the
in the random registry, denoted ζ (0)
referent registry is given by ζ (1) = δk+ πk where δk+ = Pr(F = k|D 1) is the conditional probability
of the risk status given a positive family history. Naturally ζ (1) > ζ (0) because high-risk families are more
heavily represented in the referent registry. The family size distribution in the random registry is, again,
p(n). It immediately follows that Pr(D 0 1 | R = 0) is given by (3.3) with ζ (0) replacing θ. The family
Referent sampling, family history and relative risk: the role of length-biased sampling
179
size distribution in the referent registry is given by
p(n)
3
[1 − (1 − πk )n ] +
δ .
1 − G N (1 − πk ) k
k=1
(5.1)
Note that the first-order approximation for (5.1) is np(n)/µ as in Section 3. Therefore, Pr(D 0 1 | R =
1) may be approximated by the first term of the expansion of (3.5) with ζ (1) replacing θ. Therefore, the
relative risk can be approximated by
(1)
σ2
ζ
1+
.
µ(µ − 1) ζ (0)
This expression for the relative risk is a product of two terms. The first term is exactly equation (3.6)
which is associated with the different family size distribution between the two registries. The second term
is associated with the differential probabilities of disease.
5.2
Covariate dependent probabilities
Let z denote a covariate which is related to the probability of disease. For example z may denote the age
of an individual. It is assumed that θ (z) is increasing in z and that 0 θmin θ j θmax 1 where
θ j = θ (z j ) is the probability that the jth individual has disease prior to age z j . Let Z be a vector denoting
the covariate values for all family member. We have that
n
Pr(D 1 | N = n) = · · ·
1−
(1 − θ(z j )) f Z|N (z|n) dz
(5.2)
j=1
where f Z|N (z|n) is the joint conditional distribution of the covariates given a family of size n. Note
that (5.2) is bounded below by 1 − (1 − θmin )n and above by 1 − (1 − θmax )n . Typically, covariates are
dependent within families (e.g. diet) but the may be independent (e.g. gender). If N and the covariate
Z 1 , . . . , Z N are independent within a family then
considerable simplifications arise. It is easily seen
that (5.2) reduces to 1 − (1 − θ )n where θ = θ(z) f (z) dz denotes the mean probability of disease.
Repeating the calculations of Section 3 by replacing θ by θ we find the bias in the relative risk to
be unchanged. The implication is that an unmeasured exposure which is related to disease but affects
individuals independently does not impact the bias of the relative risk with case-referent sampling.
More generally, it can be shown that the first-order approximation for (5.2) is
Pr(D 1 | N = n, R = r ) = nζ (r ) + O(ζ (r )2 )
(5.3)
where ζ (r ) is the marginal mean probability of disease in the registry R = r . Clearly ζ (0) is the mean
probability of disease in a random sample of families. Note that Pr(D 1 | N = n, Z = z) is increasing
in z. A family is in the referent registry if at least one member has disease. Therefore large families with
large z values will be more heavily represented in the registry. Consequently, the marginal probability
of disease in the referent registry is greater than the marginal mean probability of disease in the random
registry, i.e. ζ (1) ζ (0) . The quantities Pr(D 0 1 | R = r ), up to a first-order approximation, are an
easy consequence. Repeating the appropriate calculations we get
(1)
σ2
ζ
ψ∼
= 1+
µ(µ − 1) ζ (0)
as in the previous section. The exact expressions for the relative risk are quite complicated. The key idea in
our simplifications is expressing Pr(D 0 1 | R = r ) as a function of ζ (r ) (r = 0, 1), the registry-specific
marginal probability of disease, and repeating the calculations of Section 3.
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O. DAVIDOV AND M. Z ELEN
6. S UMMARY AND DISCUSSION
The study of family history and disease has been extensively investigated using referent family
registries. This is especially true for many cancer sites. Our calculations indicate that the relative risk,
associated with an aggregate measure of positive family history, and estimated from a case control study
in which the case-probands are sampled from a referent registry, may be significantly inflated. The
relationship between the odds ratio and the relative risk imply that the odds ratio estimated using the
appropriate logistic regression model is also biased. In addition, the parameters associated with other
covariates included in the logistic model cannot be consistently estimated. The bias arises because the
marginal probability of additional cases in a family is calculated by averaging over the family size. The
family size, however, depends on the type of registry from which the family is drawn. The larger the
family, the greater the probability of finding the family in the referent registry. This phenomena is known
as length-biased sampling. Consequently, the mean family size in the referent registry is larger than the
mean family size in the population. The magnitude of the bias is determined by the ratio of the effective
family sizes (which is the family size minus one for the proband). The effective family size corresponds
to the first term in the Taylor expansion of the probability of additional cases in the family. The bias
is considerable, especially for small families, and exists irrespective of the correlation among family
members. Hence, the relative risk is biased away from unity even if disease incidence is independent.
If, on the other hand, disease incidence is covariate dependent and/or correlated within families (due to
latent genes, say), then the relative risk is equal, approximately, to a product of two terms: one term
reflects the differential probabilities of disease for random and referent registries, the other is associated
with different family size distribution within the registries. The implications for data analysis are clear. A
finding of excessive risk of positive family history in the referent registry compared with a random registry
is often interpreted as if the disease is positively correlated within families. We have shown that this is
not the case. The bias can be eliminated with a proper analysis. The correct analysis always conditions
on family size and structure. Hence, in a case control study the familial history of cases and controls
can be compared by matching on family size. A referee has pointed out that family size may also act
as a confounder when its distribution is correlated with covariates which relate to disease. Consequently,
matching will also reduce confounding.
7. ACKNOWLEDGEMENT
This investigation was partially supported by Public Health Service Grant CA-78607 from the
National Cancer Institute, National Institute of Health, Department of Health and Human Service.
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[Received March 13, 2000; revised August 9, 2000; accepted for publication August 10, 2000]