I very much appreciate the thoughtful comments made by Drs. Brian and John
Healy and, in particular, I am grateful for the time they took to read my paper carefully.
However, while I agree with their theoretical construction of the various conditional
probabilities in their letter, it seems from their comments that, possibly, I was not
sufficiently clear in the paper about the fact that my construction was based less on pure
theoretical considerations and more upon conclusions drawn from the available
epidemiological evidence. In part, this may have been the result of dividing my logic
between the text and the Appendix of the paper. In the following paragraphs, therefore, I
have attempted to clarify my arguments and to address, hopefully, the concerns raised.
The starting point of my analysis was that, in the general population, it should be
possible to divide MS cases into two broad categories – those cases that developed MS
through “genetic” pathways and those that developed MS through “non-genetic”
pathways. In this context, the term “genetic” MS is used to indicate that the development
of MS has occurred through a pathway requiring both a susceptible genotype (G) and
specific environmental events (E). In this formulation, the terms (PG) and (PE) are
conceptualized very broadly. Thus, (PG) refers to the probability of possessing any
genotype that could possibly develop MS through the “genetic route” under some set of
environmental exposures. Similarly, (PE) refers to the probability of experiencing any
environmental exposure that could possibly produce MS under some selected set of
genetic preconditions. The life-time probability that an individual will develop MS by
the genetic route is termed (PMSG), whereas the corresponding probability for non-genetic
MS is termed (PMSE). This pathway to non-genetic MS may include cases caused by
either special environmental or purely stochastic events. Thus, my starting equation is:
PMS = PMSG + PMSE – (PMSG)(PMSE) = 0.0015
(1a)
where (0.0015) was chosen because it is the mid-point of the estimated prevalence range
for MS in Canada (i.e., 0.1-0.2%). Because both (PMSE) and (PMSG) must be (≤ 0.0015),
the cross-product term (the probability of getting MS through both routes) is negligible
compared to the other two terms and can be ignored.
If we define the proportion (p) as:
PMSG = (p)(PMS)
then Equation (1a) can be rewritten as:
PMS = PMSG + PMSE = (p)(PMS) + (1-p)(PMS) = 0.0015
(2a)
This equation is equivalent to Healy Equation (5), in which their first term represents
(PMSG) and the sum of the last two terms represents (PMSE). However, I believe that only
their assumption that P(MS|E1,G2)=0} is necessary for my construction. Their two further
assumptions that {P(MS|E2,G1)=1 and P(MS|E2,G2)=1}seem (to me) unnecessary.
My logic at this point in the analysis was to use actual epidemiological
observations to estimate the value of (p). Two independent observations were used for
this purpose. In considering the implications of these observations it is important to
recognize that that essentially every case of concordant MS in monozygotic-twins (and
also cases of concordant dizygotic-twins and siblings), represent individuals who are
genetically susceptible. For example, because, as noted above, (PMSE < 0.0015), and
because (PMSE │S ≈ PMSE) for the same reasons that lead to Equation (9a) below, the
probability that a sibling (S) of an MS proband will get MS can be expressed as:
PMS│S = 0.029
Therefore:
= PMSG│S + PMSE│S
< PMSG│S + 0.0015
PMSG│S > (0.95) (PMS│S)
(3a)
This percentage increases to much more than (95%) when either a more realistic estimate
for (PMSE) is used or when considering the same circumstance for monozygotic (MZ) and
dizygotic (DZ) twins where the observed concordance rates are considerably higher (0.25
and 0.054 respectively). Consequently, the conclusion that essentially all cases of
concordant MS in monozygotic twins represent “genetic” MS is justified.
In this circumstance, we can define (PMS*) as the probability of getting MS in the
susceptible population (assuming, for the moment, that either the genetic profile of
patients with clinical MS is similar to that of the susceptible population or that the
average probability of getting MS is the same for the two groups), Equation (2a) can be
written as:
PMS│MZ = CRMZ = PMSG│MZ = (p)(PMS*)
so that:
PMS* = CRMZ / p
(4a)
The first epidemiological observation used to estimate (p) is the difference (Δ) in
concordance rates for monozygotic twin-pairs, split according to whether or not they
carry the HLA DRB1*1501 allele (Table). In the MS population of Canada (and in
northern Europe), 55% of individual carry at least one copy of this allele compared to
24% in the general Canadian population (D. Sadovnick, personal communication).
Because this particular allele has been clearly established as an MS-susceptibility allele,
its excess in the MS population must be the result of “genetic” MS. Therefore this excess
indicates that 31% (i.e., 55% – 24%) must represent a minimum value for (p) and, thus,
that these individuals are genetically susceptible due, in part, to carrying the HLA
DRB1*1501 allele. Presumably, some of the remaining 69% of MS cases are also
genetically susceptible either due, in part, to this allele or for other reasons. Indeed, by
reasoning similar to that leading to Equation (3a) above, the 28% proband-wise
monozygotic-twin concordance rate for twin-pairs who lack this allele entirely (Table),
clearly indicates the presence of other important genetic susceptibility factors.
We will let (v) represent the proportion of MS patients who are not susceptible to
MS due to possessing this allele, but who are still genetically susceptible for other
reasons (0 ≤ v ≤ 1); we will let (r1) represent the probability of an individual possessing at
least one copy of the HLA DRB1*1501 allele in the general population; and we will let
(r2) represent the percent increase above the general population in the likelihood of an
MS patient possessing this allele. Making the conservative assumption that HLA
DRB1*1501 allele contributes to MS susceptibility only in those (r2) genotypes overrepresented in the MS population, leads to an estimate for (p) of:
p = [(1 – r1 – r2)(v)] + [r2 + (r1)(v)]
= [r2 + (1 – r2)(v)]
(5a)
Moreover, by Equation (4a), because all concordant MS in monozygotic twins is
“genetic”, and assuming equal penetrance of different genotypes and using adjusted
concordance rates (CRMZ(S)) as in Equation (14a) below, then:
CRMZ(S)(HLA+) = [1/(r1 + r2)][r2 + (r1)(v)][CRMZ(S) / p]
and:
CRMZ(S)(HLA–) = [1/(1 – r1 – r2][ 1 – r1 – r2][v][CRMZ(S) / p] = [v][CRMZ(S) / p]
Subtracting these two equations and substituting for (p) from Equation (5a) yields:
Δ = [r2/(r1 + r2)][1– v][CRMZ(S)] / [r2 + (1 – r2)(v)]
or:
v = [(r2/{r1 + r2})(CRMZ(S)) – Δ(r2)] / [(r2/{r1 + r2})(CRMZ(S)) + Δ(1 – r2)]
(6a)
Because this is a linear system of two equations in two unknowns (v and p), it can
be solved uniquely. As discussed below in the arguments leading to Equation (14a), we
need to use adjusted values for CRMZ and Δ (i.e., CRMZ(S) = 0.156 and Δ = 0.01), and
substitute these together with the observed values of (r1 = 0.24), and (r2 = 0.31) into
Equation (6a), to get the point estimate for (v) of (v = 0.89). Substituting this value back
into Equation (5a) gives the point estimate for (p) of:
p = 0.93
This analysis considers only the possibility that the HLA DRB1*1501 allele
contributes to MS susceptibility to the extent that persons with this allele are overrepresented in the MS population (i.e., in 31% of MS cases). If this allele, when present,
contributes more often than this, the estimated of the proportion (p) will increase.
The second method used to estimate the proportion (p) of genetic MS was based
on a population-based survey of monozygotic twins in Finland. Using an unadjusted
25% concordance rate for monozygotic twins of an MS proband, together with the
knowledge that essentially all twin pairs concordant for MS have “genetic” MS (see
Equation 3a), permits the estimation of PMSG as four times the prevalence of concordant
twin-pairs in the cohort. Although this analysis is based on only a small sample, using the
data from this cohort, the estimated prevalence of “genetic” MS is (817/105 population).
As pointed out in the original paper, this number greatly exceeds the reported prevalence
of MS in the general population of Finland. It also exceeds the total estimated prevalence
of MS (298/105 population) taken directly from this cohort.
Consequently, both of these analyses of existing data suggest that the large
majority of MS (perhaps all) occurs by the route of genetic susceptibility together with an
appropriate environmental exposure and, therefore, suggest that the assumption (p ≈ 1) is
a reasonable approximation.
It was on this basis (i.e., assuming that: p ≈ 1) that my Equation (1) was
developed and formulated. In this circumstance, the joint probability P(MS,G,E) should
be equal to the marginal probability P(MS) and, thus, under these conditions, my
Equation (1) and Healy Equations (1) and (5) are all the same. Moreover, even if (p < 1),
PMSG would be smaller than I have assumed and, thus, the estimated proportion of the
population who are genetically susceptible will be even less than that proposed.
I agree with Drs. Healy, that my Equation (3) requires adjustment. However,
contrary their assertion, I did provide the experimental data necessary to make this
adjustment. Thus, as I pointed out in the paper, the 5.4% recurrence risk for MS in
dizygotic twins (CRDZ) of an MS proband is higher than the 2.9% recurrence risk in nontwin siblings (CRS). Moreover, I argued that, because several experimental studies have
failed to identify any differences in MS risk among adopted individuals, conjugal
couples, brothers and sisters of different birth order, and in siblings and half-siblings
raised together or apart, this difference between dizygotic twins and siblings must reflect
the impact of a shared intra-uterine or early post-natal environment on MS risk.
Consequently, my Equation (2) should have been expressed as:
(PG│MZ)(PE│G, MZ)(PMS│G, MZ, E) = PMS│MZ
which can be simplified to:
(PE│MZ)(PMS│MZ, E) = CRMZ
(7a)
where (PE│MZ) is the conditional probability of exposure given that the individual is a
monozygotic (MZ) twin of an MS proband and (PMS│MZ, E) is the conditional
probability of developing MS given that the person is a monozygotic twin and has
received a sufficient environmental exposure. The relationship between
(PE│MZ)(PMS│MZ, E) and (PE│G)(PMS│G, E) needs to be determined from existing
data regarding the impact of a shared intrauterine or early postnatal environment on the
likelihood of MS. Fortunately, this impact can be estimated from:
CRDZ = (CRDZ/CRS)(CRS) = (5.4 / 2.9) (CRS) = 1.86 (CRS)
Because (PG) is the same for both twin and non-twin siblings, then:
CRDZ = (PG)(PE│DZ)(PMS│DZ, E) = 1.86 (PG)(PE│S)(PMS│S, E)
(8a)
Where the different conditional probabilities are defined for the dizygotic (DZ) and
sibling (S) cases in the same manner as terms of Equation (7a) were defined for the MZ
case. Moreover, as discussed above, there seems to be no change in the risk of
environmental exposure due to siblings sharing their childhood environment with the MS
proband compared to the same risk in siblings growing up in an environment experienced
by unrelated individuals in the general population. Similarly, there seems to be no change
in the risk of environmental exposure due to an unrelated individual sharing their
childhood environment with an MS proband compared to their risk growing up in the
general population. Thus, it seems that the observed difference in MS risk between nontwin siblings and members of the general population is related to their genetic make-up
(i.e., the PG term) and not their environmental exposure terms (other than the shared
intrauterine effect noted above). Consequently, it seems reasonable to conclude that:
(PE│S) = (PE│G)
(9a)
In my paper, I made the further assumption that:
(PMS│S, E) ≈ (PMS│G, E)
(10a)
and therefore that:
CRDZ = (PG)(PE│DZ)(PMS│DZ, E) ≈ 1.86 (PG)(PE│G)(PMS│G, E)
(11a)
Dividing out the common, but unknown, (PG) term from Equation (11a) yields:
(PE│DZ)(PMS│DZ, E) ≈ 1.86 (PE│G)(PMS│G, E)
(12a)
Using this same estimate for the impact of a shared intra-uterine and early post-natal
environment in monozygotic twins, yields:
(PE│MZ)(PMS│MZ, E) = CRMZ ≈ 1.86 (PE│G)(PMS│G, E)
(13a)
Using Equation (13a), we can define an adjusted concordance rate (CRMZ(S)), removing
the intrauterine/early postnatal effect, as:
CRMZ(S) = CRMZ / 1.86 = 0.25 / 1.86 = 0.134
(14a)
Using Equations (13a) and (14a) above, Equation (3) from the original paper should
properly have been expressed as an adjusted rate of:
PG ≈ (PMS / CRMZ(S)) = 0.0015 / 0.134 = 1.1%
(15a)
Nevertheless, I agree that my assumption in Equation (10a) requires justification.
Specifically, the concern is that, given two different susceptible genetic profiles (G1 and
G2), although the epidemiological observations leading to Equation (9a) suggest that:
(PE│ G1) = (PE│ G2)
they may not necessarily also suggest that:
(PMS│ G1, E) = (PMS│ G2, E)
In essence, I believe that this is same concern expressed by Drs Healy in their
discussion surrounding the development of their equation Healy (3), in which they
defined the set G as representing “the set of genetic profiles for the monozygotic twins.”
Unlike the situation for dizygotic twins, however, the tendency to have monozygotic
twins is not inherited, so that monozygotic twins should have the same distribution of
genotypes (i.e., the same genetic profile) as the general population. Therefore, I suspect
that Drs. Healy meant to define the set G as “the set of genetic profiles for individuals
known to have a monozygotic twin with MS.” In this circumstance (i.e., if the penetrance
term, [PMS│G, E], is different for different genotypes), then the genetic profile of such
individuals will be different from that of the general population.
To consider the impact of this possibility, I will let (x) represent the number of
susceptibility loci (i.e., loci that harbor susceptibility alleles) in addition to the known
HLA DRB1 locus. I will let (X) be the total number of possible combinations of alleles at
the (x + 1) susceptibility loci, P(i) be the probability of the ith combination in the general
population, and (PMSi*) be the conditional probability of developing MS given that the
person has the ith genetic combination (Gi). Thus:
PMSi* = (PE│Gi)(PMS│Gi, E)
If each locus combines uniquely with other loci, then:
x+1
X
=
Σ (x + 1)! / (x + 1 – i)! (i)!
i=1
If, however, different susceptible states at one locus can combine with different
susceptible states at other loci, then the number of possible combinations will be much
greater. Regardless, letting (X) be the total number of possible combinations yields:
X
PG
Σ [(P(i)] | (PMSi* > 0)
=
i=1
X
PMS | G
=
(Σ [PMSi*][(P(i)] ) / (PG)
(16a)
i=1
Thus, (PMS | G) is a weighted average of the different conditional non-zero (PMSi) terms.
Clearly, if all non-zero (PMSi) terms are approximately equal, then:
PMS | G
=
PMSi*
and estimate of (1.1%) from Equation (15a) will be approximately correct. However, if
the different conditional (PMSi) terms are not equal, this may affect the estimated
prevalence of genetic susceptibility. For example, consider the circumstance in which the
different genotypes are grouped into two classes of susceptibility genotypes – the first
(G1) being those genotypes with an expected penetrance higher than average and the
second (G2) being those with an expected penetrance lower than average. For the purpose
of this example, we will assume that the penetrance within each class is approximately
the same for all genotypes. Again assuming that essentially all MS is “genetic”, we will
define (PG1) and (PG2) as the probabilities of these two different classes of genotype in the
general population so that:
PMS ≈ PMSG1 + PMSG2
where:
PMSG1 = (PG1)(PE│G1)(PMS│G1, E)
and:
PMSG2 = (PG2)(PE│G2)(PMS│G2, E)
For simplicity, we will also define (PMSG1*) and (PMSG2*) respectively as the
probabilities either that an individual in the G1 subset will develop MS when (PG1 = 1) or
that an individual in the G2 subset will develop MS when (PG2 = 1). Thus:
PMSG1* = (PE│G1)(PMS│G1, E)
and:
PMSG2* = (PE│G2)(PMS│G2, E)
In the circumstance of a monozygotic twin, where (PG1) and (PG2) are the
respective probabilities of class-membership in the general population, we will define
(PG1app) and (PG2app) as these (apparent) probabilities in the population of monozygotic-
twins in which the proband is known to have MS. Because (PMS ≈ PMSG) it is apparent
that, under any circumstance, including those of Equation (3a), essentially all
monozygotic twins of an MS proband are genetically susceptible (i.e., PGapp ≈ 1) and,
therefore, that:
PGapp = PG1app + PG2app = 1
and also that:
PG1app = [(PMSG1*) (PG1)] / [(PMSG1*) (PG1) + (PMSG2*) (PG2)]
PG2app = [(PMSG2*) (PG2)] / [(PMSG1*) (PG1) + (PMSG2*) (PG2)]
or:
PG1app = [(PMSG1*) (PG1)] / PMS
(17a)
PG2app = [(PMSG2*) (PG2)] / PMS
From Equation (16a):
PMS | G = [(PMSG1*)( PG1app) + (PMSG2*)( PG2app)] / (1) = CRMZ(S)
where:
(18a)
1 ≥ PMSG1* ≥ CRMZ(S) ≥ PMSG2* > 0
This leads to the linear system of two equations in two unknowns:
(PMSG1*)(PG1app) + (PMSG2*)(PG2app) = 0.134
(19a)
PG1app + PG2app = 1
In the case of two classes of susceptibility loci, this system can be solved uniquely as:
and:
PG1app = (0.134 – PMSG2*) / (PMSG1* – PMSG2*)
(20a)
PG2app = 1 – PG1app = (PMSG1* – 0.134) / (PMSG1* – PMSG2*)
(21a)
In the general population, where the estimated prevalence of MS is 0.15%, then:
(PMSG1*)(PG1) + (PMSG2*)(PG2) = 0.0015
or:
PG1 = [0.0015 – (PMSG2*)(PG2)] / PMSG1*
(22a)
and:
PG2 = [0.0015 – (PMSG1*)(PG1)] / PMSG2*
(23a)
Equating the RHS of Equations (17a) and (20a), using Equation (23a) to
substitute for (PG2), and solving for (PG1) yields:
PG1 = [(0.0015)(0.134 – PMSG2*)] / [(PMSG1*)(PMSG1* – PMSG2*)]
(24a)
Equation (24a) cannot be solved uniquely but can still provide an estimate of how
the spread of the expected penetrance between the two classes of susceptible genotypes
might affect the total percentage of the population that is capable of developing MS (i.e.,
PG). For example, if the penetrance of either class is equal to CRMZ(S) (i.e., 0.134) the
percentage of the population who are members of the other class is (0%) and the estimate
of (PG = 1.1%) is accurate. When (PMSG1*) and (PMSG2*) are both removed from CRMZ(S),
and also are somewhat removed from each other (e.g. PMSG1* = 0.2; PMS2 = 0.07), the
estimated percentage of susceptible individuals is barely altered (i.e., PG = 1.5%). Even
when they are removed by one order of magnitude (e.g., PMSG1* = 0.3; PMS2 = 0.03), the
estimated percentage of susceptible individuals is still quite small (i.e., PG = 3.3%). It is
only when (PMSG2*) approaches zero that the estimate for (PG) begins to increase more
markedly. Thus, when (PMSG1*) and (PMSG2*) are removed from CRMZ(S) by two orders of
magnitude (e.g., PMSG1* = 0.3; PMSG2* = 0.003), the estimated percentage of susceptible
individuals becomes (PG = 28.2%).
Nevertheless, there are also other important constraints on this system. Thus,
because, by definition, PG1 and PG2 are mutually exclusive, and because (PMS ≈ PMSG),
Equation (1a) becomes:
PMS ≈ PMSG = (PG1)(PMSG1*) + (PG2)(PMSG2*) =
PMSG1 + PMSG2
(25a)
We will define (p) to be the proportion of susceptible individuals in the general
population who belong to the G1 subset (0 ≤ p ≤ 1).
Thus:
PG = PG1 + PG2 = (p)(PG) + (1 – p)(PG)
(26a)
In the case of identical genotypes outside the IU environment, by Equations (17a), (25a),
and (26a), Equation (18a) can be rewritten as the quadratic equation:
(p)(PMSG1*)2(PG)/PMS + (1 – p)(PMSG2*)2 (PG)/PMS = CRMZ(S) = 0.134
(27a)
The proportion of concordant monozygous twin-pairs that belong to the (G1) subset
(PMZG1) is:
PMZG1 = (p)(PMSG1*)2 / [(p)(PMSG1*)2 + (1 – p)(PMSG2*)2]
(28a)
Multiplying Equation (27a) by (PMZG1) yields:
(p)(PMSG1)2(PG)/(PMS) = (PMZG1)(CRMZ(S)) = (PMZG1)(0.134)
(29a)
Also, as before, we can make use of the fact that the proband-wise monozygotictwin concordance-rates for probands who carry the HLA DRB1*1501 allele
(CRMZ(S)[HLA+]) and for probands who do not carry this allele (CRMZ(S)[HLA–]) are
substantially the same (0.16 and 0.15 respectively) and that the overall (i.e., combined)
estimated proband-wise monozygotic-twin concordance-rate from this HLA-typed cohort
is (CRMZ(S) = 0.156). Moreover, because HLA DRB1*1501 is the best-established and
most consistently identified susceptibility allele in MS, because it is present in 55% of
MS patients, and finally because genotypes with this allele have a penetrance at least as
large as the average penetrance in the susceptible population (Table), those genotypes, in
which this allele contributes to susceptibility, presumably belong to the G1 subset. Again,
we will make use of the difference (Δ) in penetrance between genotypes with or with out
the HLA DRB1*1501 allele; we will let (v) represent the proportion of MS patients who
are not in the G1 subset due to having this allele, but who are, nonetheless, still in the G1
subset (0 ≤ v ≤ 1); we will let (r1) represent the probability of an individual possessing at
least one copy of the HLA DRB1*1501 allele in the general population; and we will let
(r2) represent the percent increase above the general population in the likelihood of an
MS patient possessing this allele. Also, as before in Equation (5a):
p = [(1 – r1 – r2)(v)] + [r2 + (r1)(v)]
= [r2 + (1 – r2)(v)]
(30a)
Because, in the Canadian population, (r2 = 0.31), this requires that (p ≥ 0.31). Putting this
constraint into Equation (28a) yields
PMZG1 ≥ PMZG1min = (0.31)(PMSG1*)2 / [(0.31)(PMSG1*)2 + (0.69)(PMSG2*)2]
where (PMZG1min) is the minimum value that (PMZG1) can take. Therefore, Equation (29a)
can be rewritten as:
or:
(p)(PMSG1*)2(PG)/PMS ≥ (PMZG1min)(CRMZ(S))
(31a)
PMSG1* ≥ [(PMZG1min)(CRMZ(S))(PMS) / (p)(PG)]1/2
(32a)
Moreover, assuming approximately equal penetrance of the different genotypes within
each of the two subsets, then:
CRMZ(S)(HLA+) = [1/(r1 + r2)][r2 + (r1)(v)][ PMSG1*] + [r1 / (r1+ r2)][1–v][PMSG2*]
and:
CRMZ(S)(HLA–) = [1/(1 – r1 – r2][ 1 – r1 – r2][v][ PMSG1*] + [1-v][PMSG2*]
= [v][ PMSG1*] + [1-v][PMSG2*]
Subtracting these two equations yields:
Δ = [r2/(r1 + r2)][1– v][PMSG1* – PMSG2*]
(33a)
As shown earlier using Equation (24a), if (PMSG1*) and (PMSG2*) are even close to
each other in magnitude, the estimate for (PG) remains very small. It is only when
(PMSG1* >> PMSG2*) that this estimate increases substantially. Nevertheless, in this
circumstance, Equation 33a becomes:
Δ ≈ [r2/(r1 + r2)][1– v][PMSG1*]
Letting:
R = r2/(r1 + r2)
and:
X = (PMZG1min)CRMZ(S))(PMS / PG) so that:
(34a)
PMSG1* = [X / p]1/2
and substituting both the value of (p) from Equation (30a) and the minimum value for
PMSG1* from Equation (32a), Equation (34a) can be rearranged into the quadratic
equation:
[(R2)(1/ Δ2)(X)] v2 – [(1 – r2) + (2)(R2)(1/ Δ2)(X)] v – (r2) + [R2][1/ Δ2] [X] = 0
(35a)
Solving Equation (34a), using the assigned values of (PMSG1* and PMSG2*), and
substituting the observed values of (Δ = 0.01), (CRMZ(S) = 0.156), (r1 = 0.24), (r2 = 0.31),
and (PMS = 0.0015), together with the value of (PG) from Equations (23a) and (24a)
yields a minimum estimated (v) and, after substituting this back into Equation (30a), a
minimum estimated (p) for any two pairs of penetrance values (PMSG1* and PMSG2*).
This derived minimum expected value of (p) can then be compared to the actual value of
(p) determined from Equations (23a and 24a) to assess whether the proposed pair of
values is plausible. For example, in circumstances considered earlier (i.e., PMSG1* = 0.3
and PMSG2* = 0.03), the approximation of Equation (35a) holds, and Equations (30a) and
(35a) yield an estimated minimum value for (p) of (p = 0.86) compared to the actual
value of (p = 0.06) for this pair of penetrance values. This is obviously an implausible
circumstance. In fact, for (PMSG1* ≥ 0.2), there are no values for (PMSG2*) over its entire
possible range of (0 < PMSG2* ≤ 0.134) for which the actual value of (p) even reaches its
minimum value of (p = 0.31). In this circumstance (i.e., PMSG1* = 0.2), the best fit is at
(PMSG2* = 0.07), although, even here, the actual (p) is only 25%, which is still lower than
its minimum value of (p ≥ 0.31). It is only when (PMSG1*) approaches CRMZ(S) that the
two values of (p) approximate each other and, even then, only when the magnitude of
(PMSG2*) is quite small (e.g., PMSG1* = 0.135 and PMSG2* = 0.01; p > 0.9 for both
estimates,; and PG = 1.2%). Moreover, in all of these circumstances, the estimated (PG)
from Equations (23a) and (24a) for the optimal fit is never exceeds (1.5%).
Consequently, in any situation, either Equation (35a) approximately holds or
(PMSG1*) and (PMSG2*) are close enough in magnitude so that that the estimate of (PG) is
quite stable. Therefore, the approximation of Equation (10a) is justified and the
conclusion that only a very small fraction of the general population is genetically
susceptible to MS is secure. Moreover, this percentage can be approximated as:
PG ≈ PMS / CRMZ(S)
Also, presuming the impact of a shared intrauterine/early postnatal environment on twin
concordance is similar in different locations, the prevalence of MS susceptibility in
different regions of the world can be compared using unadjusted concordance rates:
PMS / CRMZ
Table. Concordance rates for MS in Monozygotic Twins of HLA DRB1*1501-positive
(ZH+) and HLA DRB1*1501-negative (ZH-) Probands *.
Monozygotic Twins of MS Probands
HLA DRB1*1501
HLA DRB1*1501
Positive
Negative
Concordant (C)
9
11
20
Discordant (D)
31
42
73
Totals
40
53
93
ZH+ = (9/40) = 23%
ZH- = (11/53) = 21%
ZH+ = 30%
ZH- = 28%
ZH+ = 16%
ZH- = 15%
Pair-wise
Totals
Concordance†
Proband-wise
Concordance††
Adjusted Probandwise Concordance†††

Data derived from:
Willer et al., Proc Natl Acad Sci (USA) 2003;100:12877-12882.
†
Pair-wise rates calculated as (Z = C/(C + D).
††
Proband-wise concordance rates calculated as (Z = 2C/(2C + D)
adjusted for the overall probability of doubly ascertaining an
affected twin (70%) in the Willer, et al., 2003 study.
†††
See Text, Equation (14a)