Download A New Method for Estimating the Risk Ratio in Studies Using Case

American Journal of Epidemiology
Copyright © 1998 by The Johns Hopkins University School of Hygiene and Public Health
All rights reserved
Vol. 148, No. 9
Printed in U.S.A.
A New Method for Estimating the Risk Ratio in Studies Using Case-Parental
Control Design
Fengzhu Sun,1 W. Dana Flanders,2 Quanhe Yang,3 and Muin J. Khoury4
The authors describe a new simple noniterative, yet efficient method to estimate the risk ratio in studies
using case-parental control design. The new method is compared with two other noniterative methods,
Khoury's method and Flanders and Khoury's method, and with a maximum likelihood-based method of Schaid
and Sommer. The authors found that the variance of the new estimation method is usually smaller than that
of Khoury's method or Flanders and Khoury's method and that it is slightly larger than that of the maximum
likelihood-based method of Schaid and Sommer. Despite the slightly large variance of the new estimator
compared with that of the maximum likelihood-based method, the simplicity of the new estimator and its
variance makes the new method appealing. When genotypic information for only one parent is available, the
authors also describe a method to estimate the risk ratio without assuming Hardy-Weinberg equilibrium or
random mating. A simple formula for the variance of the estimator is given. Am J Epidemiol 1998; 148:902-9.
case-control studies; gene frequency; genes; genetic markers; genetics; odds ratio; risk
Association studies are widely used in epidemiology. One such approach is the case-control design in
which case subjects and appropriate control subjects
are selected from the population, and the fraction of
case subjects with a risk factor is then compared with
the fraction of control subjects with that risk factor. A
potential problem with the case-control design involves the selection of appropriate control subjects.
Rubinstein et al. (1) and Falk and Rubinstein (2)
proposed a case-parental control design to avoid this
problem. With the case-parental control design, case
subjects are randomly sampled and the hypothetical
control subjects are assumed to carry the genotypes
formed by the nontransmitted parental alleles. If the
candidate locus for disease susceptibility has alleles
"AT and "AT with "M" the susceptible allele, the
fraction of case subjects with allele "M" is compared
with the fraction of hypothetical control subjects with
that allele. The haplotype relative risk is the odds ratio
used with the case-parental control design. Ott (3)
studied the statistical properties of the haplotype relative risk under the recessive model, and Knapp et al.
(4) showed that the distribution of the estimator for
haplotype relative risk is stochastically smaller than
that for risk ratio in the presence of positive linkage
disequilibrium between a marker and disease allele.
For epidemiologic purposes, it is important to estimate the risk ratio of individuals carrying genotypes
"MM" and "MN" versus those carrying "AW." Schaid
and Sommer (5) gave maximum likelihood approaches
to estimate the risk ratio with and without the HardyWeinberg equilibrium assumption. Because the main
purpose of the case-parental control design is to avoid
population stratification, the Hardy-Weinberg equilibrium assumption may not hold in some situations.
Schaid and Sommer (5) proposed a maximum
likelihood-based conditional on parental genotype
(CPG) method to estimate the risk ratio without using
the Hardy-Weinberg equilibrium assumption. Knapp
et al. (6) showed how to estimate the relative risks
based on the CPG method without using iterative
algorithms.
Khoury (7) and Flanders and Khoury (8) proposed
noniterative methods to estimate the risk ratio. Both
methods provide approximate unbiased estimates.
Here we propose another simple noniterative method
to estimate the risk ratio. The variance of the new
estimator is generally smaller than those obtained using Khoury's method and Flanders and Khoury's
method, and it is slightly larger than that obtained with
the CPG method. We also give a simple formula to
Received for publication October 29, 1997, and accepted for
publication March 25, 1998.
Abbreviations: CPG, conditional on parental genotype.
1
Department of Genetics, Emory University, Atlanta, GA.
2
Department of Epidemiology, Emory University, Atlanta, GA.
3
Birth Defects and Genetic Diseases Branch, Centers for Disease Control and Prevention, Atlanta, GA.
4
Office of Genetics and Disease Prevention, Centers for Disease
Control and Prevention, Atlanta, GA.
Reprint requests to Fengzhu Sun, Department of Genetics,
Emory University School of Medicine, 1462 Clifton Road, 4th Floor,
Atlanta, GA 30322.
902
Risk Ratio in Case-Parental Control Design Studies
estimate the variance of the new estimator. The simplicity of the new estimator and its variance makes the
new method appealing.
When genotypic information from only one parent
is available, Schaid and Sommer (5) proposed an estimation method for the risk ratio assuming both
Hardy-Weinberg equilibrium and random mating.
Flanders and Khoury (8) proposed another method that
only assumed random mating. The method we propose
here requires neither of these two assumptions and is
thus applicable to more general situations.
The organization of this paper is as follows. First,
we describe the case-parental control design and
present a new 4 X 4 table that summarizes relevant
data by distinguishing the paternal and maternal alleles
for case subjects and the hypothetical control subjects
formed by the two nontransmitted parental alleles.
Second, we calculate the probabilities for the 16 cells.
Third, we present Khoury's method, Flanders and
Khoury's method, and the CPG method of Schaid and
Sommer in terms of the 16 cells. Fourth, we present
the new estimation method and the variance of the new
estimator. Finally, we compare the new method with
the other three methods using Monte-Carlo simulations.
MATERIALS AND METHODS
The case-parental control design
In the case-parental control design, case subjects are
randomly sampled from all new case subjects in the
population, and both the case subjects and their parents are genotyped at the candidate locus. Relevant
data thus consist of the genotypes of case subjects and
their parents. To facilitate analysis, we summarize the
data using a new 4 X 4 table (table 1) formed by the
genotypes of case subjects and the genotypes of hypothetical control subjects carrying the nontransmitted
parental alleles. In this table, we distinguish between
paternal and maternal alleles; that is, the first allele is
the paternal allele, and the second allele is the maternal allele. We also use the following notation: NN =
0, NM = 1, MN = 1', and MM = 2, with the numbers
corresponding to the number of susceptible alleles
"M" a genotype has. Let Ftj be the number of case
subjects in cell (i, j). The genotypes of parents are
1.
903
completely determined in each cell and are given in
parentheses. The first genotype in each set of parentheses is the father's genotype, and the second is the
mother's genotype. The first allele in each parental
genotype is the transmitted allele, and the second
allele is the nontransmitted allele. All the estimating
methods are based on this table.
Calculation of cell probabilities. First let us calculate the cell probabilities for table 1. We do not assume
the Hardy-Weinberg equilibrium nor do we assume
random mating. There are a total of nine mating types
in the parental generation that can be identified by
distinguishing paternal and maternal genotypes: (AW,
AW), (AW, MAO, (NN, MM), (MN, NN), (MN, MN),
(MN, MM), (MM, NN), (MM, MN), and (MM, MM).
As above, we denote NN = 0, MN = 1, and MM =
2 corresponding to the number of mutant alleles in the
genotype. Let q^ be the probability of mating type (i,j)
in the parental generation. Let pt be the risk of the
disease among people with genotype i. Then it is
possible to calculate the cell probabilities in table 1
conditional on the case subjects being affected. Table
2 gives the cell probabilities corresponding to table 1
multiplied by P(D), the overall risk of the disease in
the general population. The risk ratio for individuals
having genotype "MN" versus those having genotype
"AW" is defined as \x = pxlpQ, and the risk ratio for
individuals having genotype "MM' versus those having genotype "AW" is defined as A2 = p2lpo- We also
define X_x = px/p2- The objective of this study is to
obtain accurate noniterative methods for estimating A,
and A2.
The Khoury, Flanders and Khoury, and CPG
methods
To simplify the presentation, we define the following variables.
vo = mxo, Fu. + Fyi = mu,F]2 + FV2 = mn,
F2o = m20, F2l + F2V = WRI-
Case-parental control design by transmitted and nontransmitted genotypes distinguishing paternal and maternal
alleles
Control
Case
NN(O)
NM(1)
MM(2)
Am J Epidemiol
NN(O)
NM(1)
MN(1')
MM(2)
Foo (NN, NN)
F 10 (NN, MN)
Fro(MN, NN)
F 20 (MN, MN)
F 01 (NN, NM)
F n (NN, MM)
F V 1 (MN, NM)
F21 (MN, MM)
Fov (NM, NN)
F 1V (NM, MN)
Fvr(MM,NN)
F21. (MM, MN)
F 02 (NM, NM)
F12 (NM, MM)
FV2(MM, NM)
F22 (MM, MM)
Vol. 148, No. 9, 1998
904
Sun et al.
TABLE 2. Cell probabilities multiplied by P(D) corresponding
to table 1
estimator for A_, and Aj similar to that of the Khoury
method based on m01, m10, m12, and m21.
Control
Case
NN(O)
NN(0)
NM(1)
QooPo
MN(V)
NM(1)
£(0) = ^ l i )
m21
MM(2)
1/2q01p,
1 / ^.Q -] 2P1
1/4(7, ,p1
1/2q12p
MM(2)
Q 2 oPi
q22p2
All the estimation methods depend on these seven
variables.
Khoury's estimator for A, and A2 is given by
m0l
We can also obtain another approximate unbiased
estimator for A_, and Aj based on m02, mn, and m20.
2m;20
X, = m lo /m O i,
X2 =
By comparing table 1 above with table 1 in the
paper by Flanders and Khoury (8), we calculate
Flanders and Khoury's estimator as
A,=
K2 =
mQl/(mw
m02
mn/(m02 + mu + m20)
+ m01) + 2mO2/(mO2 + mn + m 20 )'
Now we have two estimators for both A_t and A,.
The weighted average of X^, with weight m21 and X_,
with weight 2m20 gives our estimator A_j. Similarly,
the weighted average of Aj0-* with weight m01 and A,
with weight 2m02 gives our estimator X P The final
estimator is
.
Note here that the estimator of A2 is the same as that
given by Khoury (7). Flanders and Khoury (8) also
proposed a modified method to estimate A2 in which
A_, = P\lp2 can be estimated in a similar manner as
A,:
*-,=
2m02'
mnl(mu + m2]) + mu/(m02 +. mu + m20)
m2]/(mn + m2]) + 2m20/(m02 + mu + m20)'
and then A2 can be estimated by A,/A_t.
Similarly, by comparing table 1 above with table 1
in Schaid and Sommer (5), the log-likelihood function
to be maximized is given by
L(A,, A2) = (m20 + m21)log(A2) + (m10 + mn
+ m12)log(A,) - (m12 + m21)log(A, + A2)
2m
m21'
mn
m01
02
Finally, we estimate A2 by X]/X_1. From table 2, it
can be shown that the above estimator is an approximate unbiased estimator of the risk ratio.
We note that Flanders and Khoury's estimator
of A_j can be regarded as a weighted average of
A^, with weight m2\l(mX2 + m 21 ) and A(_!i with
weight 2m2O/(mo2 + mu + m 20 ). Similarly, their
estimator of A, is the weighted average of A(i0)
with weight mQl/(mu + m10) and X(,]) with weight
2mO2/(mo2 + mu + m 20 ).
Next we treat (m^, i,j = 0, 1, 2) as a multinomial
random vector. Using the delta method, the variance of
the logarithm of A_, and A, can be approximated by
2m,
- (m02 + mu + m20)\og(\2 + 2A, + 1)
- (mm + m]0)log(A1 + 1).
20
1
Var(ln(X_,)) -
mu
1
2m20 + m21
The maximum point of this log-likelihood function
yields the CPG estimator. We use the approach of
Knapp et al. (6) to calculate the CPG estimator.
The new estimation method
2m:20
(2m 20 + m 2 1 ) 2 '
Var(ln(X,)) «
1
2m02 + W01
When the number of case subjects is large, table 1
approximately equals a constant times table 2. From
tables 1 and 2, we obtain one approximate unbiased
(2m 02 + m 0 ,) 2 '
Am J Epidemiol
Vol. 148, No. 9, 1998
Risk Ratio in Case-Parental Control Design Studies 905
The variance of the logarithm of A2 can be approximated by
Var(ln(A2))
Var(ln(X,))
2{An-Am-An)
A?,-(A 1 0 -A 1 2 ) 2 -
Var(ln(A2)) * Var(ln(A_,)) + Var(ln(A,))
2m,,
(m]0 + mu)(mn + mn)'
As previously shown (2, 8), we can approximate the
confidence limits of the risk ratio by treating the
logarithm of the estimated risk ratio as though it is
normally distributed.
Only one parental genotype is known. Sometimes
genotypic information from only one parent is available. We summarize the relevant data as in table 3
where Atj is the number of genotype i case subjects
whose available parent has genotype j . We define the
following estimator of A_, and A,, respectively.
X_, = '
x,=
+ A19
A 10
2A 2 1
2A01
Then A2 is estimated by A,/A_j. It can be shown that
the estimator is an approximate unbiased estimator of
the corresponding risk ratio when {qtj) are symmetric.
It is also an approximate unbiased estimator of the
corresponding risk ratio if father or mother is missing
with equal probability Vz.
Using the delta method, the variance of the logarithm of the estimated risk ratio is given by
When (q^) are not symmetric, A_, and A, given
above are not the approximate unbiased estimators of
A_j and A,. Next we give another estimation method
that is unbiased even if qtj are not symmetric. Let P
and M be the numbers of available fathers and mothers, respectively. Let PyiMfj) be the number of genotype i case subjects whose father (mother) has genotype;. Define
A[j = MPtj + PMy.
The new estimator is given as above by replacing Atj
with A-j.
From table 1 we can obtain the number of fathers
and mothers with genotypes 0, 1, and 2 given the
genotypes of case subjects as shown in table 4. From
table 2, the corresponding cell probabilities multiplied
by P{D) for table 4 can be obtained as given in table
5. From tables 4 and 5, we can easily calculate the
expected values of A\y Substitution of the expected
values of A,y into the expression for the estimated risk
ratio shows approximate unbiasedness. We assume
that the availability of the maternal genotype is independent of the individual genotype as is the availability of paternal genotype. The variance of this estimator
is much more complicated and is omitted from the
paper. Interested readers can request the exact formula
from the first author.
Comparisons of the estimation methods
2A 10
(A,,
Var(ln(X,)) - ^ -
A, o
A
+An-A]0)2
A 12
TABLE 4. Case-parental control design when only one
parental genotype is available
2A 12
(A,, +A 1 0 -A 1 2 )2>
TABLE 3. Case-parental control design when only one
parental genotype is available
Case
genotype
NN(0)
NM(1)
MM(2)
Am J Epidemiol
Parental genotype
NN(0)
NM(1)
•^01
0
Vol. 148, No. 9, 1998
Next we use Monte-Carlo simulations to compare
the four estimation methods: the CPG method of
Schaid and Sommer (5), Khoury's method (7),
Flanders and Khoury's method (8), and the new
method. Under the Hardy-Weinberg equilibrium and
random mating, the nine mating probabilities are de-
Case
genotype
Parental genotype
NN(0)
MM(2)
NN(0)
NM(1)
MM(2)
Foo 1- F ol
For + Fo2
0
Fio"l- F n
0
Fiv + F12 + F v 0 + ^ n
F20 + F21
Frr + F r 2
F 2 2 - \-F2v
NN(0)
NM(1)
MM(2)
Foo "• F o v
F r o " f Fr1.
0
MM(2)
0
A12
A22
NM(1)
When only father is available
When only mother is available
^01 + F02
F , i + F r 2 + F 10 + F 1 V
F 20 + F2V
0
Frr + F 12
F 22 -.hF 2 1
906
Sun et al.
TABLE 5.
Cell probabilities multiplied by P{D) corresponding to table 6
Parental genotype
Case genotype
MM(2)
NM(1)
NN(O)
When on/y father is available
NN(0)
NM(1)
MM(2)
1/2(2qOo + 9oi)Po
1/2(Qoi "I" 2q O 2 )p,
1/4(2q10 + q n )p 0
1/2(q12 + qu + q,0)p,
0
1/4(q 11 + 2q 1 2 )p 2
0
1/2(2q 20 + q 2 ,)p.
1/2(q 21 + 2q 2 2 )p 2
IVhen only mother is available
NN(0)
NM(1)
MM(2)
1/2(2q O0 + <7io)Po
V 2 ( q 1 0 "I" 2 q 2 0 ) p ,
1/4(2q01 + q^po
1/2(q21 + qu + qo,)p,
0
1/4(9, •, + 2q 2 1 )p 2
termined by a single parameter, /, the prevalence of
allele ' W in the parental population. The following
equations hold under the assumption of the HardyWeinberg equilibrium and random mating.
ffl
= 4/ (l ~f)\
3
qn = fci = 2/(1 - / ) ,
To simplify the presentation, we assumed the
Hardy-Weinberg equilibrium and random mating in
the parental population in our simulations, although
our results should hold even if these conditions are not
present. We also fixed the three riskspo,px, andp2 for
individuals harboring genotypes 0, 1, and 2, respectively. For all the four estimators to be meaningful,
none of the nty should equal zero. We chose the
number of case subjects such that the minimum expected value of my is about 10 for given/, prevalence
of allele "N" in the general population, and for given
risks pQ, px, and p2. For that number of case subjects,
we ran the simulations 10,000 times and recorded the
estimated risk ratios calculated by each of the four
methods. We calculated the average of the estimated
risk ratios and the square" root of the average squared
differences of the estimated risk ratios from the true
risk ratio for each method. Table 6 gives the averages
of the estimated risk ratios from 10,000 simulations
using Khoury's method, Flanders and Khoury's
method, the new method, and the CPG method. Table
7 gives the square roots of the average squared differences of the estimated risk ratios from the true risk
ratio. In both table 6 and table 7, we let Xt = pxlp0 =
4, A2 = p2/p0 = 10, a n d / = 0.2, 0.5, and 0.8.
RESULTS
In these simulations, the average number of cases
was 550, 480, and 6,400 for / = 0.2, 0.5, and 0.8,
respectively. Some general patterns emerge from our
simulation results (tables 6 and 7), although they may
1/2(2q 02 + q12)Pt
1/2(q, 2 + 2q 2 2 )p 2
TABLE 6. The average estimated risk ratios through 10,000
simulations using Khoury's method (K), Flanders and
Khoury's method (FK), the new method (N), and the
conditional on parental genotype method (CPG), with A1 = 4
and As = 10 and f, allele frequency of N
A
2
0
A,
K
A2
FK
N
CPG
FK
K
N
CPG
0.2 4.56 4.23 4.10 4.09 10.29 10.59 10.25 10.25
0.5 4.56 4.27 4.25 4.23 11.35 10.74 10.68 10.64
0.8 4.12 4.10 4.10 4.09 11.22 10.45 10.32 10.30
TABLE 7. The square roots of the average squared
differences of estimated risk ratios with the true risk ratio
through 10,000 simulations, with A1 = 4 and Aj = 10 using
Khoury's method (K), Flanders and Khoury's method (FK), the
new method (N), and the conditional on parental genotype
method (CPG)
A
A2
A,
W
K
FK
N
CPG
K
FK
N
CPG
0.2
0.5
0.8
2.22
2.31
0.78
1.13
1.27
0.71
0.67
1.21
0.71
0.65
1.15
0.68
1.83
5 .38
5 .13
2.85
3.39
2.55
1.66
3.11
2.11
1.62
3.01
2.10
not hold for all parameters. As seen in table 6, all
simulations yield risk ratios that are close to the true
value on average, although all averages are slightly
above the true value with these sample sizes and the
new and CPG methods are the closest. The variances
of the four estimators are in the order of Khoury's
method > Flanders and Khoury's method > the new
method > CPG. The variance of the new estimator is
only slightly larger than that of the CPG method. We
also simulated the process using other parameters
(data not shown). The patterns are similar.
We also calculated the variance and the lower and
upper confidence limits of the logarithm of the estimated risk ratio. We used A,' = 4, A2 = 10, and/ = 0.2
in these simulations. The histograms of Var(log(A,))
and Var(log(A2)) are given in figure 1. The real simulated variances of log(A,) and log(A2) are 0.026 and
Am J Epidemiol
Vol. 148, No. 9, 1998
Risk Ratio in Case-Parental Control Design Studies
1.9
2.0
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
2.9
3.0
3.1
32
3.3
3.4
3.5
3.6
3.7
3.8
3.9
4.0
4.1
42
4.3
4.6
4.7
U
U
U
it
U
U
54
5.5
907
Estimated variancetOO
3.1
12
3.3
3.4
3.5
3.6
3.7
3.8
19
4.0
••'...*
4.1
4.2
43
4.4
4.S
Estimated varlanc»*100
'
FIGURE 1. The histograms for the estimated variances of logfA,) (a) and logfAJ (b). A, = 4, A2 = 10, and f = 0.2. The real variances are 0.026
(a) and 0.039 (b), respectively.
0.039, respectively. From figure 1, we see that the
estimated variances of logCA,) and log(A2) using our
formulas center around the true variances. The histograms for the upper 10 percent confidence limits of
Am J Epidemiol
Vol. 148, No. 9, 1998
log(A,) and log(A2) using the proposed approach are
given in figure 2. The real values of the upper 10
percent confidence limits of log(A,) and log(A2) are
1.67 and 2.65, respectively. Thus, the estimated con-
908
Sun et al.
1.10
1.15
1.20
IIS
U0
1.35
1.40
1.4S
1.50
1.55
ISO
1.65
1.70
175
1.80
1.85
1.90
1.85
2.00
2.05
2.10
2.15
220
2.25
2.30
Estimated upper 10% confidence limit
1.W
—
-
_
1.92 2.00 2.08
2.16 2.24 2.32 2.40 2.48 2.56 £64 2.72 2M
"
2.88 2.96 3.04 3.12 3.20 3J8 3.36 3.44 3.52 3.60 3.68 3.76
Ettlmtttd upptr 10% confidence IMt
_J&
.
FIGURE 2. The histograms for the estimated upper 10 percent confidence limits for logfA,) (a) and logfAj) (b). A, = 4, A2 = 10, and f = 0.2.
The real upper 10 percent confidence limits are 1.67 (a) and 2.65 (b), respectively.
fidence limits also center around the true values. In the
simulations, log(A,) = log(4) «= 1.39, and log(A2) =
log(10) = 2.30. From the simulated data, we see that
in 94.8 percent of the simulations, the estimated upper
10 percent confidence limits of log(A.,) are greater than
its true value. Similarly, in 95.2 percent of the simulations, the estimated upper 10 percent confidence
limits of log(A2) are greater than its true value.
Am J Epidemiol
Vol. 148, No. 9, 1998
Risk Ratio in Case-Parental Control Design Studies
DISCUSSION
In this paper, we propose a new simple noniterative
method for estimating the risk ratio in studies using
case-parental control design. The new estimation
method has several important features. First, it is an
approximate unbiased estimator of the risk ratio. Second, the variance of the new estimator is smaller than
that of Khoury's estimator and Flanders and Khoury's
estimator, and it is roughly the same as that of the
maximum likelihood-based method of Schaid and
Sommer. Third, there exists a simple approximate
formula for the variance of the logarithm of the estimated risk ratio. The simplicity of the new estimator
and its variance makes the new method appealing.
When information from only one parent is available,
we propose a method to estimate the risk ratio without
assuming the Hardy-Weinberg equilibrium and random mating, thus relaxing the conditions in earlier
studies (5, 7). Therefore, the new method is applicable
to more general situations.
ACKNOWLEDGMENTS
This research is partly supported by a grant from the
Research Council of Emory University and NIH FIRST
award R29-DK53392 (to F. Sun).
Am J Epidemiol
Vol. 148, No. 9, 1998
909
The authors thank Dr. Schaid for bringing their attention
to the noniterative solution of the CPG method by Dr.
Knapp et al. (7). They also acknowledge Dr. Knapp for
suggestions that led to the improvement of the presentation.
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