Background
Some geneticists have predicted a genetic revolution in healthcare: involving a future in which individuals take a battery of genetic tests, at birth or later in life, to determine their individual 'genetic susceptibility' to disease
However, there are also many critics of this strategy, who argue that it is likely to be of limited benefit to health
There are two possible approaches to considering this issue. The 'bottom-up' approach seeks to identify individual genetic and environmental risk factors and their interactions and quantify the risks. However, this approach is limited by the difficulties in establishing the statistical validity of genetic association studies and of quantifying gene-gene and gene-environment interactions: see, for example,
A 'top-down' approach instead considers risks at the population level using twin and family studies and data on the importance of environmental factors in determining a trait. However, analysis of twin data is usually limited by the assumptions made in the classical twin study
The four-category model of population risks developed by Khoury and others
The aim of this paper is to combine the four-category model with population level data from twin, family and environmental studies, without adopting the classical twin model assumptions. This model of gene-gene and gene-environment interactions is then used to implement a 'top-down' approach to quantifying the utility of genetic 'prediction and prevention'.
Method
The four-category model
Consider a population divided into genotypic or environmental risk categories for a given trait (Figure
The four-category model
The four-category model. A population divided into: (a) high and low genotypic risk categories (rg and rog); (b) high and low environmental risk categories (re and roe); (c) four categories based on combined genotypic and environmental risk.
or by:
The same population can alternatively be divided into four categories, making a four-category model (Figure
The four category model: risks and cases for a population of size N.
Category
Risk of being in category
Number of people in category
Number of cases in category
ge (high-risk genotype/high-risk exposure)
Rge
γεN
γε RgeN
go (high-risk genotype/low-risk exposure)
Rgo
γ (1-ε)N
γ (1-ε)RgoN
oe (low-risk genotype/high-risk exposure)
Roe
ε (1-γ)N
ε (1-γ)RoeN
oo (low-risk genotype/low-risk exposure)
Roo
(1-ε) (1-γ)N
(1-ε) (1-γ)RooN
Total
N
rtN
The risks are related to the previous definitions by:
The category risks R remain constant in different populations (i.e. as ε and γ vary), provided there are no confounders. This assumption restricts the model to special cases of gene-gene and gene-environment interaction. Note that for a single genetic variant, rg corresponds to the penetrance of the variant, and that in general (provided Rge ≠ Rgo) this varies with the proportion of the population in the high exposure group, ε, as has been observed
The total risk for the given trait is given by:
The subpopulation of cases has different characteristics from the general population: for example, it contains a higher proportion of people from the 'ge' subgroup. The relative risk for a person drawn randomly from a subpopulation with the same genotypic and environmental characteristics as the cases, RRcases, is given by the sum of the relative risks for each category shown in Table
Similarly, the relative risk for a person drawn randomly from a subpopulation with the same genotypic characteristics as the cases (but with the environmental characteristics of the general population) is:
The relative risk for a person drawn randomly from a subpopulation with the same environmental characteristics as the cases (but with the genotypic characteristics of the general population) is:
Population attributable fractions
Provided there are no confounders, the population attributable fraction (PAFEe) due to the presence of the high exposure (E) in the high exposure population subgroup (e) may be defined as:
If the trait is a disease, PAFEe is the proportion of cases that could be avoided if an environmental intervention (such as a lifestyle change or reduction in exposure) succeeds in moving everyone in the 'high environmental risk group' to the 'low environmental risk' category, as shown in Figure
The targeted population attributable fraction (PAFEge) may be defined as the proportion of cases that could be avoided by targeting the same environmental intervention at the 'high genotypic + high environmental risk' subgroup only (the 'ge' subgroup), as shown in Figure
Note that PAFEge differs from PAFge as defined by Khoury & Wagener
Although in practice it is not possible to change the genotype of the population, the parameter PAFGg is nevertheless useful in the calculations that follow.
Measures of utility
Khoury et al.
PI is one possible measure of the usefulness of targeting the environmental intervention (E) at the 'ge' subgroup. It measures the proportion of cases avoided by targeting the 'high genotypic + high environmental risk' subgroup (the 'ge' subgroup), compared to the proportion avoided by applying the environmental intervention to the whole 'high environmental risk' group. PI has the property:
0 ≤
and has its maximum value when PAFEge = PAFEe. However, as a measure of the utility of genotyping, PI has the disadvantage that it takes no account of the proportion of the population γ in the high genotypic risk group. This means PI = 1 when γ = 1 simply because the whole population is then in the high genotypic risk group, although using genotyping to target environmental interventions is more likely to be useful if PI = 1 and γ is also small.
Therefore, consider an alternative utility parameter Uge, defined by:
which has the property
-
Uge tends to 1 only if PI = 1 and γ is also small. It is a measure of the utility of using genotyping to target the environmental intervention at the 'ge' subgroup, compared to randomly selecting the same proportion γ of the population to receive the intervention. Uge is positive if those at high genotypic risk have
Note that even if genotyping is better than random selection, other types of test that are more useful may be available
Limits on parameters
Consider only populations where rg ≥ rog and re ≥ roe for all values of ε and γ. Then the risks in the four box model must be ordered such that:
1 ≥
and
Using the known relationships (Equations (11), (13) and (16)) between PAFEe, PAFGg, Uge and the risks Roo, Rgo, Roe and Rge, leads to the limits on the utility parameter Uge shown in Table
Constraints on model parameters
Condition
Limits on U ge
Limits on γ
Limits on p DZ g
Limits on f ge
The twin and familial risks model
Data from studies of monozygotic and dizygotic twins are commonly used to estimate the genetic and environmental variances Vg and Ve of a trait. Here, the aim is to use twin and other data to estimate the possible magnitudes of the population attributable fractions and measures of utility defined above. To do this it is necessary to estimate Vg, Ve and the variance due to gene-environment interaction, Vge. The standard methodology for twin data analysis is inappropriate because it assumes Vge = 0.
First note that we are interested in the extent to which relatives share
For monozygotic (MZ) twins, assumed to share their entire genome, pMZg = 1. For dizygotic (DZ) twins and other siblings, who share half their genome, pDZg = psibg = 1/2 for a single allele model (dominant Mendelian disorder) or an additive polygenic model. For a two allele model (recessive Mendelian disorder) or the dominance term of a polygenic model (in which multiple pairs of alleles interact), pDZg = psibg = 1/4. Here, allowing for the possibility of multiple gene-gene interactions (epistasis), require only that:
The meaning of pDZg and its relationship to the polygenic risk model first adopted by Ronald Fisher in 1918 is discussed further below.
Similarly, define prele as the correlation in environmental risk category (e) between relatives of type "rel", requiring only that:
Assume that prelg and prele are independent (so that there is no genotype-environment correlation) and that risks within a category are randomly distributed. The relative risk for a relative of type "rel" may then be written:
Substituting for the relative risks RRcasesgen, RRcasesenv and RRcases using Equations (8), (9) and (10) leads (after some algebra) to:
where
Note that if the G-E interaction component of the variance, Vge, is zero, the utility of targeting the environmental intervention by genotype, Uge, is also zero (Equation (26)), because those at high genotypic risk have no more to gain from the intervention than those at low genotypic risk (Rge-Rgo = Roe-Roo).
Equation (23) can also be derived more formally using matrix methods (Appendix A).
The gene-environment interaction factor and remaining inequalities
Without loss of generality, define the gene-environment interaction factor fge such that:
and choose its sign so that (combining Equations (24), (25) and (26)):
Uge is zero if fge = 0 (i.e. for an additive G-E model, with no G-E interaction), but for a given γ and Vg, Uge increases with increasing gene-environment interaction factor, fge. For a fixed fge and genetic variance component Vg, Uge is maximum when γ = 1/2, i.e. when half the population is in the high genotypic risk group, provided solutions with γ = 1/2 exist (see also below:
Using the definitions of Ve, Vg and Vge (Equations (24), (25) and (26)) and the remaining inequalities, Rge ≤ 1 and Roo ≥ 0, two limits can be derived on the proportion of the population in the 'high genotypic risk' group, γ (see Table
Scoping studies
The general system of equations represented by Equation (23) may be simplified where data exist from monozygotic twins, dizygotic twins and other siblings, such that λDZ > λsib. This implies that environmental risks are more strongly correlated in dizygotic twins than in other siblings, peDZ > pesib. Remembering that pMZg = 1 and psibg = pDZg, three independent equations for the relative risk in monozygotic, dizygotic twins and siblings may then be written:
To solve, assume the recurrence risks λ are known (see Appendix B and
with
and
0 ≤
Note that if RSD = 1, Equations (30) and (31) are identical, peDZ = pesib, and more relatives are needed to obtain solutions, except in the special case where there is no environmental variance (see below:
In addition, define the variable parameters (assumed unknown):
with
and
0 ≤
For λDZ > 1 and RSD < 1 the simultaneous Equations (29), (30) and (31) can then be solved to give:
provided
For situations in which a targeted intervention is under consideration, the population attributable fraction PAFEe and exposure ε are likely to be known, allowing Ve to be treated as an input variable. However, pDZe is usually unknown, since environmental correlations are often difficult to measure. Therefore, it is useful to eliminate pDZe from Equations (41) and (42), leading to:
where
and
Equations (27), (40) and (43) allow the gene-environment interaction factor fge to be written as:
The parameter pDZg, which defines the form of the genetic model, is then given by:
For known RMD, RSD and λDZ a solution space can now be mapped, which includes all possible variances consistent with the data and with the inequalities derived above.
Requiring the variances to be positive leads to the additional conditions on pDZg and cSD shown in Table
Further constraints on model parameters
Condition
Limits on p DZ g
Limits on c SD
If
The limits on Uge shown in Table
Noting that fge = 0 corresponds to pDZg = pDZgmin (Equation (64)), this implies that, for Uge ≥ 0, the solution space may be defined by:
where pDZgmax is given by Equation (47) with fge = 1/PAFEe.
For Uge ≤ 0, the solution space may be defined by:
where pDZgneg is given by Equation (47) with fge = -ε/(1-ε)PAFEe.
The remaining limits on Uge lead to the additional conditions on the range of γ values (the proportion of the population in the high risk group) shown in Table
where (noting that γmaxge = γo when fge = 1):
and (noting that γminge = γneg when fge = -rt/(1-rt)):
Two transition lines can therefore be defined such that pDZg = pDZgt when fge = 1 and pDZg = pDZgnegt when fge = -rt/(1-rt). The values of pDZgt and pDZgnegt may be calculated using Equation (47).
The full range of gene-environment interaction models specified by fge (within the limits given by Equation (48)) and the corresponding range of γ values are summarized in Table
Limits on the gene-environment interaction factor (fge) and the proportion of the population in the high-genotypic risk group (γ).
Gene-environment interaction model
Interaction factor f ge
Risk distribution
Utility U ge
Fraction of population at high genotypic risk
Maximum γ max
Minimum γ min
Genetic effect in high-exposure group only
1/PAFEe
R 00
R ge
Positive
γmaxge (where PAFEge = PAFEe; PI = 1; and Uge = 1-γ).
γminge (where Rge = 1).
R 00
R 0e
Multiplicative
1
R g0
R g0 R 0e /R 00
γmaxge = γ0 (where PAFEge = PAFEe; R00 = 0; and PAFGg = 1).
R 00
R 0e
Additive
0
R g0
R g0 +R 0e -R 00
Zero
γ0 (where R00 = 0).
R 00
R 0e
Reverse multiplicative
-rt/(1-rt)
R g0
(1-Rg0) (1- R0e)/ (1-R 00 )
Negative
γneg = γminge (where PAFEge = 0 and Rge = 1)
R 00
R 0e
Genetic effect in low-exposure group only
-ε/(1-ε)PAFEe
R g0
R 0e
γneg (where PAFEge = 0 and PI = 0).
R 00
R 0e
One additional condition is necessary for solutions to exist, namely:
This condition is always met if
where
and F1 and F2 are given by:
However, if λMD is greater than this, the requirement γmax ≥ γmin further restricts the values of cSD that lie within the solution space (Table
If Ve and ε are known, a solution space can be now be mapped for pDZg and fge with known input data from twin and sibling studies (λMZ, λDZ and λsib), for a given cMD and all values of cSD within the assumed range. The boundaries of the solution space are determined by the limits on fge given by Equation (48), the condition γmax ≥ γmin (Equation (54)), and the requirement that pDZg is less than or equal to 1/2 (Equation (20)) – no other condition on the genetic model is specified a priori. For each genetic risk model and gene-environment interaction model in the solution space, defined by pDZg and fge respectively, the variances Vg and Vge can then be calculated, as can γmax and γmin. For a chosen γ value in the allowed range, Uge can then be calculated from Equation (28).
The model code is available as [
Gene-gene and gene-environment interaction model. Contains the Visual Basic macro (Twincal), input and output datasheets and charts used to calculate the solutions described in the text. The program is run by entering parameters in the 'Inputs' sheet and clicking on the 'Run' button. Note that for the final chart ('fe') the number of categories on the horizontal axis changes depending on the environmental input parameters ε and PAFEe. If these parameters are changed it is therefore necessary to delete the lower part of the output sheet prior to running the model and, after the run, to redraw the chart using the source data option from the chart. All other charts are drawn automatically. The line γmax = γmin is calculated exactly for the chart 'fe' but is approximated in the charts 'pgdz' and 'pgdzneg' using Newton's method and an initial guess for fge (f0) and step (fet). For some input parameters it may be necessary to change these values by editing the Visual Basic code (Twincal) to obtain a valid solution.
Click here for file
Note that the condition on pDZg ≤ 1/2 may also be rewritten using Equation (47), so that:
which is always met if
Before mapping the solution space, first consider some special cases and a comparison of the model with the classical twin studies approach.
Special cases
1. No genetic variance
If Vg = 0, Equation (27) implies that Vge = 0 also. Equations (29), (30) and (31) then give:
and
Under the usual assumption that cMD = 1 (the 'equal environments' assumption), this is the well-known result that genetic variance can be zero only when the concordance in monozygotic and dizygotic twins is the same (leading to RMD = 1). However, if the equal environments assumption is not met (cMD > 1), values of RMD greater than 1 do not necessarily imply that a genetic component to the variance exists (see, for example,
2. No environmental variance
If Ve = 0, Equation (27) implies that Vge = 0 also. Equations (29), (30) and (31) then give:
and
For a purely genetic model with no environmental variance, Equation (64) implies that if RMD > 2, pDZg < 1/2. This is consistent with Risch's finding
3. Classical twin study assumptions
Assuming no gene-environment interaction (Vge = 0); an additive genetic risk model (pDZg = 1/2); and the 'equal environments' assumption (cMD = 1) in Equations (29), (30) and (31) gives:
This is the classical twin study result, assuming the dominance term of the genetic variance is negligible. Note that, if RMD = 2, the classical solution implies that the environmental variance terms in Equations (29) to (31) are zero and shared sibling risk is due to entirely to shared genes.
4. No correlation in genotypic risk in siblings (pDZg = 0)
Equation (20) allows pDZg to tend to zero. Substituting pDZg = 0 in Equations (29), (30) and (31) and using the definition of the gene-environment interaction factor (Equation (28)) gives:
and
Note that, from Equations (30) and (31), pDZg = 0 corresponds to a purely environmental explanation for shared sibling risks (although there may remain a genetic component to shared risks in monozygotic twins, from Equation (29)). The solution pDZg = 0 may not exist in reality; however, the solution at this limit is of interest because low values of pDZg are plausible.
Also, note that if fge = 0 (no gene-environment interaction) and cMD = 1 (the 'equal environments' assumption), the genetic variance Vg given by Equation (67) is half the classical twin study result (Equation (65)).
5. Cases where γmax = γmin
If the line γmax = γmin exists within the solution space, some special cases may arise with risk distributions of particular interest (including, for example, a solution with Rge = 1 and all other risks zero). These special cases and the conditions that they meet are shown in Table
Special cases with γmax = γminfor Uge ≥ 0
Special cases with γ max = γ min
Special cases with γ max = γ min and specific G-E interaction models
Special cases with γ max = γ min and all risks all 0 or 1
Risk distribution
Conditions
Population impact and Utility
Risk distribution
Conditions
Population impact and Utility
Risk distribution
Conditions
Population impact and Utility
1
1
rt = 1 PAFe = 0
Undefined (PAFge = 0)
R00
1
γminge = γmaxge (Rge = 1 and PAFge = PAFe) fge = 1/PAFe
PI = 1 Uge = 1-γ
1
1
Rg0
1
γminge = γmaxge (Rge = 1 and PAFge = PAFe) fge ≥ 1
PI = 1 Uge = 1-γ
R00
R00
0
1
rt = γε PAFe = 1
PI = 1 Uge = 1-γ
R00
R00
Rg0
1
γminge = γ0 = γmaxge (Rge = 1; R00 = 0; PAFge = PAFe) fge = 1
PI = 1 Uge = 1-γ
0
0
Rg0
1
γminge = γ0 (Rge = 1; R00 = 0) 0 ≤ fge ≤ 1
0 = PI = 1 Uge = PI-γ
0
0
1
1
rt = γ PAFe = 0
Undefined (PAFge = 0)
0
R0e
1-R0e
1
γminge = γ0 (Rge = 1; R00 = 0) fge = 0
PI = γ Uge = 0
0
0
0
R0e
0
1
rt = ε PAFe = 1
PI = γ Uge = 0
0
1
6. Cases where γmaxge < 1/2
Equation (27) shows that for a fixed gene-environment interaction factor fge and genetic variance component Vg, the utility Uge is maximum when γ = 1/2, i.e. when half the population is in the high genotypic risk group, provided this solution exists. However, if γmax < 1/2, utility is maximum when γ = γmax. As a smaller proportion of the population is then targeted, these solutions are of particular interest. Because solutions with population impact PI = 1 may exist when 1 ≤ fge ≤ 1/PAFEe if γ = γmaxge (Table
where pDZgx is given by:
solving for pDZgx allows the region of the solution space where γmaxge < 1/2 to be defined.
7. Cases where the 'equal environments' assumption holds (cMD = 1)
In the special case where the 'equal environments' assumption holds (cMD = 1, and hence pDZgtop = 1/RMD), Equation (63) simplifies to give RMD ≥ 2. Equation (62) also simplifies to give:
where
Meeting the condition pDZg ≤ 1/2 at cSD = 0 then requires:
It follows that if cMD = 1, solutions with pDZg = 1/2 (an additive genetic model) and positive utility exist only when the following condition holds for RMD:
Further, all three classical twin study assumptions (cMD = 1, pDZg = 1/2 and fge = 0) can be met only for values of RMD that are low enough to satisfy:
1 +
If RMD lies within this range, the classical twin study gives one possible solution; however, other solutions also exist. All alternative solutions favour a less 'genetic' and more 'environmental' explanation for shared sibling risks (i.e. they have higher values of cSD). If RMD is greater than 1+RSD, all three assumptions of the classical twin study cannot be met simultaneously.
Comparison with the classical twins approach
Table
Comparison with classical twin study
Classical twin study
Twins + siblings model
Genetic model
Additive and dominance terms only: VDZg = 1/2VA+1/4VD
Variable: VDZg = pDZgVg with 0 < = pDZg < = 1/2
Shared twin environments
Equal environments assumption: cMD = 1
Variable: 1 < = cMD < = RMD cMD = RMD implies Vg = 0
Shared sibling environments
Siblings not included.
Variable: 0 < = cSD < = RSD Familial aggregation may be due to genes (cSD = 0) or environment (cSD = RSD).
Gene-environment interactions
None
Variable: Vge = f2ge· Vg· Ve/r2t -ε/(1-ε)PAFe < = fge < = 1/PAFe
Gene-environment correlations
None
None
Method
Total phenoptypic variance given by: VP = Vg+Ve VP is input and a single solution for Ve and Vg calculated. Heritabilities are given by: H2 = Vg/VP h2 = VA/VP
Ve and ε are input and Vg and Vge calculated, for a chosen cMD and all possible values of fge and pDZg. Method is not valid if RSD = 1.
A central feature of the model is that it abandons Fisher's assumption
Fisher saw his polygenic model as "
The alternative model adopted here is based on correlations in
The classical twin study assumptions (see above) allow a single solution to be calculated from the under-determined system of simultaneous Equations (29), (30) and (31). However, in the absence of prior knowledge about the form of the genetic model, the presence or absence of gene-environment interactions, and the validity of the 'equal environments' assumption, the approach adopted here is more rigorous.
Results
General model solutions
First consider the behaviour of the model when the 'equal environments' assumption holds and hence cMD = 1 (as described above).
Figures
Example model solution space with RMD < 1+RSD and Uge ≥ 0
Example model solution space with RMD < 1+RSD and Uge ≥ 0. Input parameters: λMZ = 3.4, λDZ = 3, λsib = 2, ε = 0.2, PAFEe = 0.5, cMD = 1, rt = 0.1. Hence RMD = 1.2, RSD = 0.5.
Example model solution space with RMD < 1+RSD and Uge ≤ 0
Example model solution space with RMD < 1+RSD and Uge ≤ 0. Input parameters as for Figure 2.
Example full model solution space with RMD < 1+RSD
Example full model solution space with RMD < 1+RSD. Input parameters as for Figure 2, with the solution space transformed so that fge is on the horizontal axis.
For lower values of RMD, the curves defining the solution space are shifted downwards [see Additional files
Supplementary Figure 1: Example model solution space with RMD = 1.7 and Uge ≥ 0. Model solution space with Uge ≥ 0 for the same input parameters as Figure 2, apart from λMZ = 4.4.
Click here for file
Supplementary Figure 2: Example model solution space with RMD = 1.8 and Uge ≥ 0. Model solution space with Uge ≥ 0 for the same input parameters as Figure 2, apart from λMZ = 4.6.
Click here for file
Supplementary Figure 3: Example model solution space with RMD = 1.95 and Uge ≥ 0. Model solution space with Uge ≥ 0 for the same input parameters as Figure 2, apart from λMZ = 4.9.
Click here for file
Supplementary Figure 4: Example model solution space with RMD = 2.1 and Uge ≥ 0. Model solution space with Uge ≥ 0 for the same input parameters as Figure 2, apart from λMZ = 5.2.
Click here for file
Supplementary Figure 5: Example full solution space with RMD = 1.7. Full model solution space for the same input parameters as Figure 5, transformed so that fge is on the horizontal axis.
Click here for file
Supplementary Figure 6: Example full solution space with RMD = 1.8. Full model solution space for the same input parameters as Figure 6, transformed so that fge is on the horizontal axis.
Click here for file
Supplementary Figure 7: Example full solution space with RMD = 1.95. Full model solution space for the same input parameters as Figure 7, transformed so that fge is on the horizontal axis.
Click here for file
Supplementary Figure 8: Example full solution space with RMD = 2.1. Full model solution space for the same input parameters as Figure 8, transformed so that fge is on the horizontal axis.
Click here for file
When cMD > 1, lines of constant fge no longer decrease monotonically to zero, and are also shifted upwards, so that solutions with strong G-E interactions are no longer possible [see Additional files
Supplementary Figure 9: Example model solution with cMD > 1 and Uge ≥ 0. Input parameters: λMZ = 5.2, λDZ = 3, λsib = 2, ε = 0.2, PAFEe = 0.5, cMD = 2, rt = 0.1.
Click here for file
Supplementary Figure 10: Example model solution with cMD > 1 and Uge ≥ 0. Input parameters as for Figure 13.
Click here for file
Supplementary Figure 11: Example full solution space with cMD > 1. Full model solution space for the same parameters as Figure 13, transformed so that fge is on the horizontal axis.
Click here for file
Example applications using twin, sibling and environmental data
Input values
Consider example applications of the model for male lung cancer, female breast cancer and schizophrenia. The model input variables used are shown in Table
The recurrence risks, λ, and total risks, rt, for breast and lung cancer are those calculated by Risch
The recurrence risks λ, and total risk, rt, for schizophrenia are those used by Risch
Detailed results for the three diseases are shown in [Additional file
Supplementary Figure 12: Breast cancer solution space with Uge ≥ 0. Input parameters are as shown in Table
Click here for file
Supplementary Figure 13: Breast cancer variances with fge = 0. Input parameters as for Figure 16. Additive model of G-E interaction (fge = 0). Variance components are genetic (Vg) or environmental (Ve).
Click here for file
Supplementary Figure 14: Breast cancer variances with fge = 1. Input parameters as for Figure 16. Multiplicative G-E interaction model (fge = 1). Variance components are genetic (Vg), environmental (Ve) or due to gene-environment interaction (Vge).
Click here for file
Supplementary Figure 15: Breast cancer variances with fge = 1/PAFEe. Input parameters as for Figure 16. Maximum G-E interaction model (fge = 1/PAFEe). Variance components are genetic (Vg), environmental (Ve) or due to gene-environment interaction (Vge).
Click here for file
Supplementary Figure 16: Breast cancer γ values with fge = 0. Input parameters as for Figure 16. The proportion of the population in the 'high genotypic risk' group, γ, may take any value in the shaded area. γmin occurs when Rge = 1, i.e. when the Positive Predictive Value (PPV) of being in the 'ge' subgroup is 100%. γmax occurs when Roo = 1 for an additive G-E model and solutions with a Population Impact of 100% (PI = 1) cannot exist.
Click here for file
Supplementary Figure 17: Breast cancer γ values with fge = 1. Input parameters as for Figure 16. The proportion of the population in the 'high genotypic risk' group, γ, may take any value in the shaded area. A solution with a Population Impact of 100% (PI = 1) may exist if γ = γmax.
Click here for file
Supplementary Figure 18: Breast cancer γ values with fge = 1/PAFEe. Input parameters as for Figure 16. The proportion of the population in the 'high genotypic risk' group, γ, may take any value in the shaded area. A solution with a Population Impact of 100% (PI = 1) may exist if γ = γmax.
Click here for file
Supplementary Figure 19: Breast cancer solution space with Uge ≤ 0. Input parameters are as for Figure 16. The solution space is shown for negative fge (where the utility of targeting environmental interventions at the high genotypic risk group is negative, Uge ≤ 0). Solutions exist only in the shaded area where γmax ≥ γmin.
Click here for file
Supplementary Figure 20: Breast cancer: full solution space. Input parameters are as for Figure 16. The same solution space as Figures 16 and 23 is shown (shaded), transformed so that the G-E interaction factor is plotted on the horizontal axis. Again, each point in the shaded solution space represents a genetic model defined by pDZg and a G-E interaction model defined by fge. The area of solutions with γmaxge < 1/2 is highlighted with darker shading. The classical twin study solution lies on the vertical axis (fge = 0) at the point pDZg = 1/2, and is slightly outside the solution space.
Click here for file
Supplementary Figure 21: Lung cancer solution space with Uge ≥ 0. Input parameters are as shown in Table
Click here for file
Supplementary Figure 22: Lung cancer variances with fge = 0. Input parameters as for Figure 25. Note that the horizontal axis has been expanded to show high values of cSD/RSD only.
Click here for file
Supplementary Figure 23: Lung cancer variances with fge = 1. Input parameters as for Figure 25. Note that the horizontal axis has been expanded to show high values of cSD/RSD only.
Click here for file
Supplementary Figure 24: Lung cancer variances with fge = 1/PAFEe. Input parameters as for Figure 25. Note that the horizontal axis has been expanded to show high values of cSD/RSD only.
Click here for file
Supplementary Figure 25: Lung cancer γ values for fge = 1. Input parameters as for Figure 25. The proportion of the population in the 'high genotypic risk' group, γ, may take any value in the shaded area.
Click here for file
Supplementary Figure 26: Lung cancer γ values for fge = 1/PAFEe. Input parameters as for Figure 25. The proportion of the population in the 'high genotypic risk' group, γ, may take any value in the shaded area.
Click here for file
Supplementary Figure 27: Lung cancer Uge values for fge = 1. Input parameters as for Figure 25. The utility parameter, Uge, may take any value in the shaded area, but is maximum when γ = 1/2.
Click here for file
Supplementary Figure 28: Lung cancer Uge values for fge = 1/PAFEe. Input parameters as for Figure 25. The utility parameter, Uge, may take any value in the shaded area, but is maximum when γ = 1/2.
Click here for file
Supplementary Figure 29: Lung cancer: full solution space. Input parameters as for Figure 25.
Click here for file
Supplementary Figure 30: Schizophrenia Uge ≥ 0, small environmental variance and cMD ≥ 1. Input parameters are as shown in Table
Click here for file
Supplementary Figure 31: Schizophrenia Uge ≥ 0, small environmental variance and cMD > 1. Input parameters are as shown in Table
Click here for file
Supplementary Figure 32: Schizophrenia Uge ≥ 0, large environmental variance and cMD = 1. Input parameters are as shown in Table
Click here for file
Breast cancer results
For breast cancer, the PAFEe associated with known environmental factors is low. The value of the model is therefore less in calculating the utility of targeted environmental interventions than in exploring the solution space for a complex disease with RMD close to 2.
Although strictly speaking the classical twin study solution (with an additive genetic model, pDZg = 1/2, and an additive G-E model, fge = 0) does not exist as a solution, it might lie within the margin of error of the data. However, an infinite number of other models also could also fit the data. The classical twin model result always overestimates the genetic component of the variance, which reduces as the gene-environment interaction factor fge increases, and also as pDZg decreases (i.e. as epistatic terms begin to dominate the genetic model). These alternative models imply that shared environmental factors may partially explain familial aggregation of breast cancer. This contrasts with the classical twin method result (see earlier), which for RMD = 2 leads to the inevitable conclusion that shared sibling risk must be due solely to shared genes
In theory, a model with pDZg = 0 (where shared sibling risk is due entirely to shared environmental factors) could fit the data. However, for breast cancer the existence of known mutations that significantly increase risk (particularly mutations in the BRCA1 and BRCA2 genes, which are relatively common) rules out this solution. Although it is not possible to subtract out the effect of these mutations from the model, it is possible to show that they could be sufficient to explain the twin data if a G-E interaction also exists. For example, one possible solution consistent with the data could involve one or more dominant genes (pDZg = 1/2), a strong G-E interaction (fge = 1/PAFE e), but a largely environmental explanation for shared sibling risk (say cSD/RSD = 0.9). This solution implies that the genetic component of the variance is less than a fifth of the classical twin study result, which could be low enough to be explained by mutations in the BRCA1 and BRCA2 genes alone
The line γmax = γmin does not occur within the solution space for breast cancer; however, in some circumstances the lines γmax and γmin may be rather close together. This suggests that, although as expected there is always a trade-off between selecting a small proportion (γmin) of the population with a high Positive Predictive Value (PPV), or a larger proportion of the population (γmax) with a higher Population Impact (PI)
Lung cancer results
For lung cancer, all the possible solutions imply that shared sibling risk is largely due to shared environmental factors (smoking) because solutions occur only when cSD/RSD is close to 1. Unlike for breast cancer, the line γmax = γmin lies outside the solution space, even for negative fge, as does the area of solutions with γmaxge < 1/2. However, the classical twin study solution, with fge = 0 and pDZg = 1/2, clearly lies within the solution space.
Although the classical twin model again provides an upper limit to the genetic component of the variance, even the classical result indicates that the risk of lung cancer is dominated by smoking in this population and the variance has at most a small genetic component.
Unlike the breast cancer example, γmax and γmin are always far apart, suggesting a strong trade off between high Postive Predictive Value (Rge) for a genotypic test and a high Population Impact (PI) for a targeted intervention. This means that a genotypic test that predicts which smokers will get lung cancer cannot exist. To predict all cases of lung cancer in smokers (i.e. to obtain PI = 1), 95% or more of the population would have to be in the high genotypic risk group, and the predictive value of such a test would be very low.
Because the genetic component of the variance is so small, it follows that the utility of genetic 'prediction and prevention' (measured by Uge) is also small (from Equation (28)). Utility is maximum when γ = 1/2, but even then values are low. The maximum utility of genotyping occurs when about 60% of cases could be prevented by targeting the 50% of smokers at high genotypic risk. However, other possible solutions have zero or negative utility.
Schizophrenia results
For schizophrenia, the classical twin study solution (with fge = 0 and pDZg = 1/2 and cMD = 1) cannot not fit the data. If the 'equal environments' assumption holds, neither a single dominant gene (pDZg = 1/2), nor additive polygenic model (also with pDZg = 1/2), nor single recessive gene (pDZg = 1/4) can explain the twin and family data, consistent with Risch's 1990 findings
The possible solution spaces include purely genetic explanations for shared sibling risk (at cSD/RSD = 0), or purely environmental ones (at cSD/RSD = 1, applicable if pDZg = 0).
Assuming a small environmental component to the variance, there is no region of the solution space for which γmaxge < 1/2, suggesting that the utility of targeted environmental interventions under these assumptions is likely to be low. However, if the environmental component of the variance is assumed to be much larger, the available solution space changes dramatically, because the line γmax = γmin now constrains the solution space to a much smaller area, which excludes solutions with no G-E interaction (fge = 0). Special solutions may exist along the line γmax = γmin, as shown in Table
Because prenatal development is thought to be important in schizophrenia, it is plausible that monozygotic twins are more likely to share environmental risk factors than dizygotic twins are. Breaking the 'equal environments' assumption changes the shape of the solution space significantly, and, assuming a small environmental component to the variance, only limited G-E interactions are now possible (the multiplicative G-E model, fge = 1, lies largely outside the solution space). The utility of targeting environmental interventions by genotype is then likely to be low. However, in these circumstances it is possible that an additive genetic model (pDZg = 1/2) with some G-E interaction, or a recessive gene (pDZg = 1/4) with no G-E interaction, could explain the data.
Discussion
If Fisher's polygenic model
Abandoning Fisher's assumption that the polygenic model is necessarily dominated by an additive term can be justified by the growing evidence that phenotypic effects can result from the synergistic action of alleles in many genes
The model also allows the impact of the much criticised 'equal environments' assumption to be explored.
A number of conclusions can be drawn about the merits of the classical twin study and the utility of genetic 'prediction and prevention'.
Firstly, the model confirms that the classical twin study solution is not always valid and gives at best an upper limit to the genetic component of the variance of a trait. The importance of the 'equal environments' assumption and of gene-environment interactions have previously been recognised
Secondly, the model confirms that the potential for reducing the incidence of common diseases using environmental/lifestyle interventions targeted by genotype may be limited
(i) the low importance of genetic differences in determining the risk of some conditions (for example, lung cancer);
(ii) the complexity of gene-gene and gene-environment interactions and/or lack of knowledge of environmental factors (for example, schizophrenia).
Targeting environmental/lifestyle interventions at those at 'high genotypic risk' can be of high utility only in specific circumstances. The utility of targeting environmental interactions by genotype (compared to randomly selecting the same number of people from the population) is zero if there is no gene-environment interaction. Utility can also be negative in the presence of a negative interaction (i.e. if the people at high genotypic risk have
The lung cancer example is apparently trivial but also of critical importance. The RMD value for lung cancer is close to 1, and neither the Scandinavian data used here
Finally, the model illustrates the argument of Terwilliger and Weiss
The model has several limitations. Measurements of shared sibling risk (λsib) are needed from the same population as twin data, and the scoping studies are only valid for λDZ > λsib, implying that environmental risks are more strongly correlated in dizygotic twins than other siblings. Some data exist to support this assumption for smoking
Treating exposure and environmental variance (or population attributable fraction) as input data is also problematic when the effects of environmental factors on risk are often unknown. Further, the simple nature of the model (with one environmental axis) cannot adequately represent the complexity of environmental (including socio-economic) causes of disease. However, if targeting environmental interventions by genotype is to be considered, this implies that at least something is known (or expected to be learned) about environmental factors, such as particular exposures, that are amenable to intervention.
The assumption of no gene-environment correlation will often hold (for example it is rather implausible that the same genes strongly influence both lung cancer risk and nicotine addiction), but is not necessarily always true. Adult lactose intolerance is an example of a condition with a strong gene-environment interaction where targeted intervention to avoid drinking milk may be of high utility. However, the model is invalid for lactose intolerance unless exposures are applied equally to the population studied because, in general, people who are lactose intolerant may be less likely to drink milk (a gene-environment correlation) owing to the unpleasant symptoms.
A more fundamental problem is caused by the assumptions that: (i) the risks Roo, Rgo, Roe and Rge are inherent properties of a given trait within a given population (with a given γ and ε) and that there are therefore no confounders; and (ii) risks are randomly distributed within these categories.
These assumptions, although often made, are implausible in many situations. The assumption of no confounders means that the model can only represent a subset of the potential models of gene-gene and gene-environment interaction described by more complex models (for example
The second assumption neglects the fact that for most exposures there is a gradient in risk, with higher exposure meaning higher risk, and that the same may also be true of genetic factors. This means that increasing the number of categories in the model will increase Ve (see
More broadly, these assumptions make the model, like the classical twin model, essentially deterministic: it assumes that all the factors contributing to correlations in risk between relatives are perfectly known and are either environmental or genetic. Retention of these assumptions here may be problematic and could limit the applicability of the results. Nevertheless, all the other questionable assumptions of the classical twin model have been simultaneously removed.
Conclusion
The model shows that the potential for reducing the incidence of common diseases using environmental interventions targeted by genotype may be limited, except in special cases. The model also confirms that the importance of an individual's genotype in determining their risk of complex diseases tends to be exaggerated by the classical twin studies method, owing to the 'equal environments' assumption and the assumption of no gene-environment interaction. In addition, if phenotypes are genetically robust, because of epistasis, a largely environmental explanation for shared sibling risk is plausible, even if the classical heritability is high. The model therefore highlights the possibility – previously rejected on the basis of twin study results – that inherited genetic variants are important in determining risk only for the relatively rare familial forms of diseases such as breast cancer. If so, genetic models of familial aggregation may be incorrect and the hunt for additional susceptibility genes could be largely fruitless.
Competing interests
The author(s) declare that they have no competing interests.
Appendix A: formal derivation of equation (31)
Equation (23) may be derived more formally by extending the matrix method of Li and Sacks
Define the probability that an affected proband is in genotypic risk category z and environmental risk category w as Pzw and assume that risks are randomly distributed within categories. Using the definitions of the four category model given in Table
A risk vector R may also be defined:
Now define Gxy as the conditional probability P(relative is in genotypic risk category y|proband is in genotypic risk category x). Similarly, define Exy as the conditional probability P(relative is in environmental risk category y|proband is in environmental risk category x). Using the definitions of prelg and prele given in Section 2.5, matrices G and E may be written such that:
Finally, define Xab-cd as the conditional probability P(relative is in risk category cd|proband is in risk category ab), where the risk categories are as defined in Table
Then the risk in a relative of the proband is given by:
After some algebra, this yields equation (23).
Appendix B: calculating recurrence risks for twins
The sibling recurrence risk λsib is often available directly from familial studies. For twins the recurrence risks, if not reported, may be calculated from the case-wise concordance (Cc):
where, if there is complete ascertainment of all affected twins in a population,
and C is the number of concordant and D the number of discordant pairs
Input variables
Condition
λ MZ
λ DZ
λ sib
ε
PAF E e
r t
Breast cancer
4.09
2.51
2.01
0.62
0.15
0.036
Lung cancer
6.27
6.14
3.16
0.15
0.86
0.017
Schizophrenia
52.1
14.2
8.6
0.62
0.15
0.01
0.15
0.86