Twin Concordance Rates
Two different concordance rates are often used in twin studies in which twins are
not sampled from the general population, the pairwise rate and the probandwise rate.
Actually, these two concordance rates are special cases of a general, maximum likelihood
method for calculating concordance that can easily be done on any twin data. The only key
to using the general formula is to keep track of the doubly ascertained twin pairs during the
data collection phase of a study. First, I present the algebraic formulae, then a numerical
example, and finally provide the proof.
The algebraic formulae
Let C1 denote the number of concordant pairs in the sample where both twins were
independently ascertained. Note that C1 is the number of pairs and not the number of
probands. For example, if both members of the Smith twins and both members of the
Jones twins are doubly ascertained, then C1 = 2 and not 4. Let C2 denote the number of
concordant pairs in which one and only one member was ascertained and let C = C1 + C2 or
the total number of concordant pairs. Finally, let D denote the number of discordant pairs.
The pairwise concordance rate, denoted here as A, is
A
C
C D
=
+
.(1)
The proband concordance rate, denoted as B is
B
C C
C C D
=
+
+ +
2
2
1 2
1 2
.(2)
The first step in the approach outlined here is to calculate the quantity π, the
probability of ascertaining an individual with the trait under study. This quantity can be
estimated directly from the observed twin data by the equation
ˆπ =
+
2
2
1
1 2
C
C C
.
Although one could calculate two estimates of π, one for MZ and the other for DZ twins, it
is preferable to pool the estimate. Hence, the recommended equation becomes
√ ( )
( )
π =
+
+ + +
2
2
1 1
1 1 2 2
C C
C C C C
mz dz
mz dz mz dz
(3)
where subscripts mz and dz denote zygosity.
Let CI denote the total number of concordant pairs for the ith zygosity. That is CI =
C1I + C2I. With π estimated from equation (1), an unbiased estimate of the population
concordance rate (θi) is given by
√
( )
θ
π i
i
i i
C
C D
=
+ −
2
2 2
.(4)
Naturally there will be a separate estimate of θ for MZ and DZ twins. The quantity θ is a
conditional probability that estimates the population probability that a cotwin will be affected
given that his/her partner is affected. It is also equal to the segregation ratio. Hence, it has a
much more important meaning in quantitative genetics than either the pairwise or
probandwise concordance.
When the ascertainment probability is very low, π approaches 0 and equation (4)
reduces to the pairwise rate given in equation (1). In complete ascertainment where all
affected individuals are probands, then π = 1.0, C2 = 0, and equation (4) reduces to the
probandwise rate in equation (2).
Curiously, testing the significance of the difference between MZ and DZ
concordance rates uses only the pairwise rates. First compute Amz and Adz—the pairwise
concordance rates for MZ and DZ twins. Next, compute A, the pairwise concordance for all
twins, ignoring zygosity:
©1991, 2000 Gregory Carey Twin Concordance - 3
A
C C
C C D D
mz dz
mz dz mz dz
=
+
+ + +
.
Finally, compute the likelihood ratio χ2:
χ 22 1 1
1
= + − + + −
− − −
[ log( ) log( ) log( ) log( )
log( ) log( )]
C A D A C A D A
C A D A
mz mz mz mz dz dz dz dz (5)
This will be a χ2 with one degree of freedom. Note that the logarithm taken here is the
natural or Naperian logarithm.
A numerical example
Gottesman & Shield’s (1972) twin data on schizophrenia will be used to illustrate
the procedure. The data used here consist of a consensus diagnosis of schizophrenia or
probable/questionable schizophrenia derived from the independent evaluation of case
histories by six clinicians (see Gottesman & Shields, 1972, Appendix C) and are given in
Table 1.
[Insert Table 1 about here]
Using both MZ and DZ twins, the estimate of the ascertainment probability, π, is
given from equation (3) as
π =
+
+ + +
=
2 4 1
2 4 1 7 2
526
( )
( )
. .
The concordance rate for identical twins is found by entering the quantities in Table 1 into
equation (4), or
θmz =
+ −
=
2 11
2 11 2 526 11
576
( )
( ) ( . )
. .
Similarly, the concordance for DZ twins is
θdz =
+ −
=
2 3
2 3 2 526 30
119
( )
( ) ( . )
. .
©1991, 2000 Gregory Carey Twin Concordance - 4
To test for a significant difference in concordance, we compute the pairwise rates for
the MZ twins, the DZ twins, and all twins regardless of zygosity. Thus Amz = 11/22 = .50,
Adz = 3/33 = .091, and A = 14/55 = .255. Using these quantities and equation (5), the
likelihood ratio χ2 is
χ 2 2 11 5 11 5 3 091 30 909
14 255 41 745 11 80
= + + +
− − =
[ log(.) log(.) log(. ) log(. )
log(.) log(.)] .
.
The value of χ2 exceeds the critical value of 6.64 at the .01 level. Hence, there is clear
evidence that concordance for schizophrenia is significantly greater in MZ than in DZ twins.
Proof
Let p denote the prevalence of a trait in the general population. Under the
assumption that the trait has the same prevalence in twins as in the general population the
distribution of twins in the general population will be given in the simple two by two
contingency table illustrated in Table 2.
[Insert Table 2 here]
Note that when twin pairs are randomly sampled from the general population, regardless of
their trait status, then an estimate of θ may be derived by simply double-entering the pairs
into the table and dividing the proportion of concordant pairs by p.
When twins are ascertained through pairs where are least one twin has the trait, a
correction for ascertainment is required to arrive at an unbiased estimate of θ and it is
necessary to estimate π. A critical assumption of the following derivation is that
ascertainment is independent in members of concordant pairs. That is, it is assumed that the
probability of independently ascertaining twin 2 in a concordant pair, given that twin 1 has
already been ascertained, is π.
The probability of ascertaining twinships where neither partner has the trait is
obviously 0. For discordant pairs, the probability of ascertaining the pair is simply π. For
©1991, 2000 Gregory Carey Twin Concordance - 5
concordant pairs, however, the ascertainment probability is more complicated. There are
three ways to ascertain concordant pairs. First, twin 1 may be ascertained but twin 2 is not
ascertained. This will occur with frequency π(1 - π). Second, twin two may be ascertained
while twin 1 remains unascertained. This also occurs with frequency π(1 - π). Finally, both
twins may be independently ascertained. This should occur with frequency π2.
Using these quantities, one can then construct a table of the frequency with which
twins should occur in the general population and in an ascertained sample. This table is
given below in Table 3.
[Insert Table 3 here]
The quantity λ in the last column of table is the proportion of twin pairs in the
general population that is ascertained. It may be found by summing the quantities in the
sixth column (Frequency in the General Population by Ascertainment Status) of the table
for all those pairs that are ascertained, or,
λ =θpπ2+2θpπ(1−π)+2(1−θ)pπ=(2−θπ)pπ
Inspection of the last column in the table reveals that there are only three categories
of twins in the ascertained twin sample. The first of these consists of concordant pairs in
which both members are independently ascertained. Let C1 denote the number of these
pairs in the sample and note that the probability of observing pairs of this type of twin pair
in the sample is θpπ2λ-1. The second type are concordant pairs in which only one twin is
ascertained. Let C2 denote these pairs. Their probability is 2θpπ(1 - π)λ-1. The final type
are discordant pairs. Their probability is 2(1 - θ)pπ λ-1, and their number will be denoted
by D.
Consequently, the log likelihood of the sample becomes
l og(L)=Clog( p )+Clog[ p ( − ) ] +Dlog[ ( − )p ] − − −
1
2 1
2
θ π λ 2θ π1 π λ1 21 θ πλ1
which reduces to
©1991, 2000 Gregory Carey Twin Concordance - 6
log( ) log( ) log( ) log( ) log( )
log( ) log( )
L cons C C C C
D N
= + + + + − +
− − −
1 1 2 2 1
1 2
θ π θ π
θ θπ
(6)
where cons is a constant and N is the total number of twin pairs (C1 + C2 + D). Thus, the
log likelihood is a function of two parameters, θ and π. Differentiating the log likelihood
with respect to each of these parameters and performing tedious algebra gives the maximum
likelihood estimates of π and θ as
ˆπ =
+
2
2
1
1 2
C
C C
and
√ ( )
( ) ( ) ( )
θ
π π
=
+
+ + −
=
+ −
2
2 2
2
2 2
1 2
1 2
C C
C C D
C
C D
.
Note that these are equations (3) and (4) given in the first section of this paper.
The likelihood ratio χ2 is twice the difference in log likelihoods between a general
model that fits three parameters to the data (π, θmz, and θdz) and a constrained model that fits
π but constrains θmz to equal θdz. By subscripting the quantities in equation (6) to reflect
MZ twins and DZ twins and then algebraically reducing the formula for χ2, one arrives at
equation (5) given earlier. It is remarkable that all terms involving π cancel in the process
and one is left with comparisons of the MZ, DZ, and whole-sample concordance rates.
©1991, 2000 Gregory Carey Twin Concordance - 7
References
Gottesman,I.I. & Shields,J. (1972). Schizophrenia and Genetics: A twin study vantage
point. New York: Academic Press.
©1991, 2000 Gregory Carey Twin Concordance - 8
Table 1. Twin concordance for schizophrenia.
Zygosity
Type: MZ DZ All
C1 4 1 5
C2 7 2 9
D 11 30 41
Source: Gottesman & Shields (1972).
©1991, 2000 Gregory Carey Twin Concordance - 9
Table 2. Frequency of twins with (+) and without (-) a trait in the general
population
Twin 2 + Twin 2 - Total
Twin 1 +: θp (1 - θ)p p
Twin 1 -: (1 - θ)p q - (1 - θ)p q
Total p Q 1.0
©1991, 2000 Gregory Carey Twin Concordance - 10
Table 3. Distribution of Twin Types in the General Population and in an Ascertained Sample
Twin Type
Twin
1
Twin
2
Frequency in
the General
Population
Ascertainment
Status
Probability of
Ascertainment
given Twin Type
Frequency in the General
Population by
Ascertainment Status
Frequency in the
Ascertained Sample
both ascertained π2 θpπ2 θpπ2λ-1
twin 1 ascertained π(1 - π) θpπ(1 - π) θpπ(1 - π)λ-1
twin 2 ascertained π(1 - π) θpπ(1 - π) θpπ(1 - π)+ + θp λ-1
neither 1 - 2π(1 - π) - π2 θp[1 - 2π(1 - π) - π2] 0
ascertained π (1 - θ)pπ (1 - θ)pπ λ+ - (1 - θ)p -1
not ascertained 1 - π (1 - θ)p(1 - π) 0
ascertained π (1 - θ)pπ (1 - θ)pπλ- + (1 - θ)p -1
not ascertained 1 - π (1 - θ)p(1 - π) 0
- - q - (1 - θ)p ascertained 0 0 0
not ascertained 1 q - (1 - θ)p 0