A model of gene-gene and gene-environment interactions and its implications for targeting environmental interventions by genotype

1742-4682-3-35 1742-4682 Research A model of gene-gene and gene-environment interactions and its implications for targeting environmental interventions by genotype Wallace M Helen helen.wallace@genewatch.org

GeneWatch UK, The Mill House, Tideswell, Buxton, Derbyshire, SK17 8LN, UK

Theoretical Biology and Medical Modelling 1742-4682 2006 3 1 35 http://www.tbiomed.com/content/3/1/35 17029623 10.1186/1742-4682-3-35 13 4 2006 09 10 2006 09 10 2006 2006 Wallace; licensee BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Background

The potential public health benefits of targeting environmental interventions by genotype depend on the environmental and genetic contributions to the variance of common diseases, and the magnitude of any gene-environment interaction. In the absence of prior knowledge of all risk factors, twin, family and environmental data may help to define the potential limits of these benefits in a given population. However, a general methodology to analyze twin data is required because of the potential importance of gene-gene interactions (epistasis), gene-environment interactions, and conditions that break the 'equal environments' assumption for monozygotic and dizygotic twins.

Method

A new model for gene-gene and gene-environment interactions is developed that abandons the assumptions of the classical twin study, including Fisher's (1918) assumption that genes act as risk factors for common traits in a manner necessarily dominated by an additive polygenic term. Provided there are no confounders, the model can be used to implement a top-down approach to quantifying the potential utility of genetic prediction and prevention, using twin, family and environmental data. The results describe a solution space for each disease or trait, which may or may not include the classical twin study result. Each point in the solution space corresponds to a different model of genotypic risk and gene-environment interaction.

Conclusion

The results show 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 results therefore highlight 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.

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 12. In theory, once the risk of particular combinations of genotype and environmental exposure is known, medical interventions (including lifestyle advice, screening or medication) could then be targeted at high-risk groups or individuals, with the aim of preventing disease 3.

However, there are also many critics of this strategy, who argue that it is likely to be of limited benefit to health 45678. One area of debate concerns the proportion of cases of a given common disease that might be avoided by targeting environmental or lifestyle interventions to those at high genotypic risk. Known genetic risk factors have to date shown limited utility in this respect 9. However, some argue that combinations of multiple genetic risk factors may prove more useful in the future 10.

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, 11121314.

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 15, including that: (i) there are no gene-gene interactions (epistasis); (ii) there are no gene-environment interactions; (iii) the effects of environmental factors shared by twins are independent of zygosity (the 'equal environments' assumption). These assumptions have all been individually explored and shown to be important in influencing the conclusions drawn from twin and family data 161718. In addition, the magnitude of any gene-environment interaction is critically important in determining the utility of targeting environmental interventions by genotype 19. Although a general methodology to analyze twin data without making these assumptions has been developed, the algebra becomes intractable once multiple loci are involved 17. This is problematic because, for common diseases, the impacts of multiple genetic variants, and potentially the whole genetic sequence, on disease susceptibility (here called 'genotypic risk') may be important.

The four-category model of population risks developed by Khoury and others 19 is a useful starting point for a top-down analysis of genetic prediction and prevention. It allows the merits of a targeted intervention strategy (which seeks to reduce the exposure of the high-risk genotype group only) to be explored, and can readily be extended to include more than four risk categories 10. However, this model's use to date has been limited to bottom-up consideration of single genetic variants or to studying hypothetical examples of multiple variants. The four-category model is limited by the assumption of no confounders, which means it is applicable to only a subset of possible models of gene-gene and gene-environment interaction. However, situations where the 'no confounders' assumption is valid are arguably most likely to be of relevance to public health.

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 1a and 1b). The fraction of the population in the 'high environmental risk group' (designated by subscript e) is ε, and this subpopulation is at risk r_e. The remainder of the population is at risk r_oe. The fraction of the population in the 'high genotypic risk' group (designated by the subscript g) is γ, and this subpopulation is at risk r_g, with the remainder of the population at risk r_og. The total risk r_tfor this trait in this population is then given by:

Figure 1

The four-category model

The four-category model. A population divided into: (a) high and low genotypic risk categories (r_gand r_og); (b) high and low environmental risk categories (r_eand r_oe); (c) four categories based on combined genotypic and environmental risk.

r_t= γr_g+ (1-γ)r_og (1)

or by:

r_t= εr_e+ (1-ε)r_oe (2)

The same population can alternatively be divided into four categories, making a four-category model (Figure 1c)) with risks R_oo, R_oe, R_goand R_ge. Table 1 shows the risk categories in this model.

Table 1

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)

R_ge

γεN

γε R_geN

go (high-risk genotype/low-risk exposure)

R_go

γ (1-ε)N

γ (1-ε)R_goN

oe (low-risk genotype/high-risk exposure)

R_oe

ε (1-γ)N

ε (1-γ)R_oeN

oo (low-risk genotype/low-risk exposure)

R_oo

(1-ε) (1-γ)N

(1-ε) (1-γ)R_ooN

Total

r_tN

The risks are related to the previous definitions by:

r_g= εR_ge+ (1-ε) R_go (3)

r_og= εR_oe+ (1-ε) R_oo (4)

r_e= γR_ge+ (1-γ) R_oe (5)

r_oe= γR_og+ (1-γ) R_oo (6)

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, r_gcorresponds to the penetrance of the variant, and that in general (provided R_ge≠ R_go) this varies with the proportion of the population in the high exposure group, ε, as has been observed 2021.

The total risk for the given trait is given by:

r_t= γεR_ge+ γ(1-ε)R_go+ ε(1-γ)R_oe+ (1-ε)(1-γ)R_oo (7)

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, RR^cases, is given by the sum of the relative risks for each category shown in Table 1:

R R c a s e s = γ ε R g e 2 + γ ( 1 − ε ) R g o 2 + ε ( 1 − γ ) R o e 2 + ( 1 − ε ) ( 1 − γ ) R o o 2 r t 2 ( 8 ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=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@7152@

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:

R R g e n c a s e s = γ r g 2 + ( 1 − γ ) r o g 2 r t 2 ( 9 ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGsbGucqWGsbGudaqhaaWcbaGaem4zaCMaemyzauMaemOBa4gabaGaem4yamMaemyyaeMaem4CamNaemyzauMaem4CamhaaOGaeyypa0ZaaSaaaeaaiiGacqWFZoWzcqWGYbGCdaqhaaWcbaGaem4zaCgabaGaeGOmaidaaOGaey4kaSIaeiikaGIaeGymaeJaeyOeI0Iae83SdCMaeiykaKIaemOCai3aa0baaSqaaiabd+gaVjabdEgaNbqaaiabikdaYaaaaOqaaiabdkhaYnaaDaaaleaacqWG0baDaeaacqaIYaGmaaaaaOGaaCzcaiaaxMaadaqadaqaaiabiMda5aGaayjkaiaawMcaaaaa@5403@

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:

R R e n v c a s e s = ε r e 2 + ( 1 − ε ) r o e 2 r t 2 ( 10 ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGsbGucqWGsbGudaqhaaWcbaGaemyzauMaemOBa4MaemODayhabaGaem4yamMaemyyaeMaem4CamNaemyzauMaem4CamhaaOGaeyypa0ZaaSaaaeaaiiGacqWF1oqzcqWGYbGCdaqhaaWcbaGaemyzaugabaGaeGOmaidaaOGaey4kaSIaeiikaGIaeGymaeJaeyOeI0Iae8xTduMaeiykaKIaemOCai3aa0baaSqaaiabd+gaVjabdwgaLbqaaiabikdaYaaaaOqaaiabdkhaYnaaDaaaleaacqWG0baDaeaacqaIYaGmaaaaaOGaaCzcaiaaxMaadaqadaqaaiabigdaXiabicdaWaGaayjkaiaawMcaaaaa@54F7@

Population attributable fractions

Provided there are no confounders, the population attributable fraction (PAF^E_e) due to the presence of the high exposure (E) in the high exposure population subgroup (e) may be defined as:

P A F e E = ε ( r e − r o e ) r t = ε { γ ( R g e − R g o ) + ( 1 − γ ) ( R o e − R o o ) } / r t ( 11 ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGqbaucqWGbbqqcqWGgbGrdaqhaaWcbaGaemyzaugabaGaemyraueaaOGaeyypa0ZaaSaaaeaaiiGacqWF1oqzcqGGOaakcqWGYbGCdaWgaaWcbaGaemyzaugakeqaaiabgkHiTiabdkhaYnaaBaaaleaacqWGVbWBcqWGLbqzaOqabaGaeiykaKcabaGaemOCai3aaSbaaSqaaiabdsha0bGcbeaaaaGaeyypa0ZaaSGbaeaacqWF1oqzdaGadaqaaiab=n7aNjabcIcaOiabdkfasnaaBaaaleaacqWGNbWzcqWGLbqzaOqabaGaeyOeI0IaemOuai1aaSbaaSqaaiabdEgaNjabd+gaVbGcbeaacqGGPaqkcqGHRaWkcqGGOaakcqaIXaqmcqGHsislcqWFZoWzcqGGPaqkcqGGOaakcqWGsbGudaWgaaWcbaGaem4Ba8MaemyzaugakeqaaiabgkHiTiabdkfasnaaBaaaleaacqWGVbWBcqWGVbWBaOqabaGaeiykaKcacaGL7bGaayzFaaaabaGaemOCai3aaSbaaSqaaiabdsha0bGcbeaaaaGaaCzcaiaaxMaadaqadaqaaiabigdaXiabigdaXaGaayjkaiaawMcaaaaa@6C7B@

If the trait is a disease, PAF^E_eis 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 1b.

The targeted population attributable fraction (PAF^E_ge) 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 1c. Again assuming no confounders, it is given by:

P A F g e E = ε γ ( R g e − R g o ) / r t ( 12 ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGqbaucqWGbbqqcqWGgbGrdaqhaaWcbaGaem4zaCMaemyzaugabaGaemyraueaaOGaeyypa0dcciGae8xTduMae83SdCMaeiikaGIaemOuai1aaSbaaSqaaiabdEgaNjabdwgaLbGcbeaacqGHsislcqWGsbGudaWgaaWcbaGaem4zaCMaem4Ba8gakeqaaiabcMcaPiabc+caViabdkhaYnaaBaaaleaacqWG0baDaOqabaGaaCzcaiaaxMaadaqadaqaaiabigdaXiabikdaYaGaayjkaiaawMcaaaaa@4BB5@

Note that PAF^E_gediffers from PAF_geas defined by Khoury & Wagener 19. The latter implicitly assumes that both environmental and genetic risk factors are reduced and thus is inappropriate for assessing the merits of a targeted environmental intervention. PAF^E_geas defined here is instead equivalent to the targeted attributable fraction (AF_T) defined by Khoury et al. 10. To avoid confusion, the notation adopted here specifies both the nature of the intervention (environmental, denoted by superscript E) and the target subpopulation (the 'ge' subgroup, at both high genotypic and high environmental risk). Thus, the proportion of cases that would be avoided were it possible to move the 'high genotypic risk' subgroup to 'low genotypic risk' (as shown in Figure 1a) is written as PAF^G_g, given by:

P A F g G = γ ( r g − r o g ) r t = γ { ε ( R g e − R o e ) + ( 1 − ε ) ( R g o − R o o ) } / r t ( 13 ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGqbaucqWGbbqqcqWGgbGrlmaaDaaabaGaem4zaCgabaGaem4raCeaaOGaeyypa0ZaaSaaaeaaiiGacqWFZoWzcqGGOaakcqWGYbGCdaWgaaWcbaGaem4zaCgakeqaaiabgkHiTiabdkhaYnaaBaaaleaacqWGVbWBcqWGNbWzaOqabaGaeiykaKcabaGaemOCai3cdaWgaaqaaiabdsha0bqabaaaaOWaaSGbaeaacqGH9aqpcqWFZoWzdaGadaqaaiab=v7aLjabcIcaOiabdkfasnaaBaaaleaacqWGNbWzcqWGLbqzaOqabaGaeyOeI0IaemOuai1cdaWgaaqaaiabd+gaVjabdwgaLbqabaGccqGGPaqkcqGHRaWkcqGGOaakcqaIXaqmcqGHsislcqWF1oqzcqGGPaqkcqGGOaakcqWGsbGudaWgaaWcbaGaem4zaCMaem4Ba8gakeqaaiabgkHiTiabdkfasnaaBaaaleaacqWGVbWBcqWGVbWBaOqabaGaeiykaKcacaGL7bGaayzFaaaabaGaemOCai3cdaWgaaqaaiabdsha0bqabaaaaOGaaCzcaiaaxMaadaqadaqaaiabigdaXiabiodaZaGaayjkaiaawMcaaaaa@6C8F@

Although in practice it is not possible to change the genotype of the population, the parameter PAF^G_gis nevertheless useful in the calculations that follow.

Measures of utility

Khoury et al. 10 define the Population Impact (PI) as:

P I = P A F g e E / P A F e E ( 14 ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGqbaucqWGjbqscqGH9aqpdaWcgaqaaiabdcfaqjabdgeabjabdAeagnaaDaaaleaacqWGNbWzcqWGLbqzaeaacqWGfbqraaaakeaacqWGqbaucqWGbbqqcqWGgbGrlmaaDaaabaGaemyzaugabaGaemyraueaaaaakiaaxMaacaWLjaWaaeWaaeaacqaIXaqmcqaI0aanaiaawIcacaGLPaaaaaa@41E2@

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 ≤ PI ≤ 1 (15)

and has its maximum value when PAF^E_ge= PAF^E_e. 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 U_ge, defined by:

U g e = P A F g e E P A F e E − γ = γ ( 1 − γ ) [ ( R g e − R g o ) − ( R o e − R o o ) ] [ γ ( R g e − R g o ) + ( 1 − γ ) ( R o e − R o o ) ] ( 16 ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGvbqvlmaaBaaabaGaem4zaCMaemyzaugabeaakiabg2da9maalaaabaGaemiuaaLaemyqaeKaemOray0cdaqhaaqaaiabdEgaNjabdwgaLbqaaiabdweafbaaaOqaaiabdcfaqjabdgeabjabdAeagTWaa0baaeaacqWGLbqzaeaacqWGfbqraaaaaOGaeyOeI0ccciGae83SdCMaeyypa0ZaaSaaaeaacqWFZoWzcqGGOaakcqaIXaqmcqGHsislcqWFZoWzcqGGPaqkdaWadaqaaiabcIcaOiabdkfasnaaBaaaleaacqWGNbWzcqWGLbqzaOqabaGaeyOeI0IaemOuai1aaSbaaSqaaiabdEgaNjabd+gaVbGcbeaacqGGPaqkcqGHsislcqGGOaakcqWGsbGudaWgaaWcbaGaem4Ba8MaemyzaugakeqaaiabgkHiTiabdkfasnaaBaaaleaacqWGVbWBcqWGVbWBaOqabaGaeiykaKcacaGLBbGaayzxaaaabaWaamWaaeaacqWFZoWzcqGGOaakcqWGsbGudaWgaaWcbaGaem4zaCMaemyzaugakeqaaiabgkHiTiabdkfasnaaBaaaleaacqWGNbWzcqWGVbWBaOqabaGaeiykaKIaey4kaSIaeiikaGIaeGymaeJaeyOeI0Iae83SdCMaeiykaKIaeiikaGIaemOuai1cdaWgaaqaaiabd+gaVjabdwgaLbqabaGccqGHsislcqWGsbGulmaaBaaabaGaem4Ba8Maem4Ba8gabeaakiabcMcaPaGaay5waiaaw2faaaaacaWLjaGaaCzcamaabmaabaGaeGymaeJaeGOnaydacaGLOaGaayzkaaaaaa@862D@

which has the property

-γ ≤ U_ge≤ (1-γ) (17)

U_getends 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. U_geis positive if those at high genotypic risk have more to gain than those at low genotypic risk from the intervention ((R_ge-R_go) ≥ (R_oe-R_oo)) and negative if they have less to gain from the intervention. This reflects the fact that targeting those who have least to gain through an intervention is worse than using random selection in terms of its impact on population health.

Note that even if genotyping is better than random selection, other types of test that are more useful may be available 22; a population-based approach still has the potential to reduce more cases of disease 91923; and such targeting also has broader psychological and social implications. Therefore a positive U_gedoes not necessarily imply that genotyping is the best means of selecting a subpopulation to target, or that a targeted approach is necessarily effective or socially acceptable. Note also that the measure U_geapplies only to interventions that are considered applicable to the whole population (such as smoking cessation) and neglects other relevant issues such as cost-effectiveness and the burden of disease 24. In addition, it is necessary to consider the magnitude of the Population Attributable Fraction, PAF^E_ebefore proposing this approach. This is because both PI and U_gemay tend to unity even if only a small proportion of cases can be avoided by means of environmental interventions.

Limits on parameters

Consider only populations where r_g≥ r_ogand r_e≥ r_oefor all values of ε and γ. Then the risks in the four box model must be ordered such that:

1 ≥ R_ge≥ R_oe≥ R_oo≥ 0 (18)

and

R_ge≥ R_go≥ R_oo (19)

Using the known relationships (Equations (11), (13) and (16)) between PAF^E_e, PAF^G_g, U_geand the risks R_oo, R_go, R_oeand R_ge, leads to the limits on the utility parameter U_geshown in Table 2. These conditions also ensure that PAF^E_e, PAF^G_gand PAF^E_geare all positive. The two remaining inequalities (R_ge≤ 1 and R_oo≥ 0) are considered later, where they are used to derive limits on the proportion of the population in the 'high genotypic risk' group, γ. This step is not possible at this stage because PAF^E_e, PAF^G_gand PAF^E_geare themselves dependent on γ.

Table 2

Constraints on model parameters

Condition

Limits on U _ge

Limits on γ

Limits on p ^DZ _g

Limits on f _ge

R_oe≥ R_oo

U_ge≤ (1 - γ)

γ ≤ γ_{max ge}where γmax⁡ge=11+VgeVe MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaaiiGacqWFZoWzdaWgaaWcbaGagiyBa0MaeiyyaeMaeiiEaGNaem4zaCMaemyzaugabeaakiabg2da9maalaaabaGaeGymaedabaGaeGymaeJaey4kaSYaaSaaaeaacqWGwbGvdaWgaaWcbaGaem4zaCMaemyzaugabeaaaOqaaiabdAfawnaaBaaaleaacqWGLbqzaeqaaaaaaaaaaa@4011@

R_go≥ R_oo

U g e ≤ ( 1 − γ ) P A F g G P A F e E MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGvbqvdaWgaaWcbaGaem4zaCMaemyzaugabeaakiabgsMiJkabcIcaOiabigdaXiabgkHiTGGaciab=n7aNjabcMcaPmaalaaabaGaemiuaaLaemyqaeKaemOray0aa0baaSqaaiabdEgaNbqaaiabdEeahbaaaOqaaiabdcfaqjabdgeabjabdAeagnaaDaaaleaacqWGLbqzaeaacqWGfbqraaaaaaaa@438B@

p g D Z ≤ p g max ⁡ D Z MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGWbaCdaqhaaWcbaGaem4zaCgabaGaemiraqKaemOwaOfaaOGaeyizImQaemiCaa3aa0baaSqaaiabdEgaNjGbc2gaTjabcggaHjabcIha4bqaaiabdseaejabdQfaAbaaaaa@3D07@

f g e ≤ 1 P A F e E MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGMbGzdaWgaaWcbaGaem4zaCMaemyzaugabeaakiabgsMiJoaalaaabaGaeGymaedabaGaemiuaaLaemyqaeKaemOray0aa0baaSqaaiabdwgaLbqaaiabdweafbaaaaaaaa@3972@

R_ge≥ R_go

U_ge≥ -γ

γ ≥ γ_negwhere γneg=11+VeVge MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaaiiGacqWFZoWzdaWgaaWcbaGaemOBa4MaemyzauMaem4zaCgabeaakiabg2da9maalaaabaGaeGymaedabaGaeGymaeJaey4kaSYaaSaaaeaacqWGwbGvdaWgaaWcbaGaemyzaugabeaaaOqaaiabdAfawnaaBaaaleaacqWGNbWzcqWGLbqzaeqaaaaaaaaaaa@3D50@

R_ge≥ R_oe

U g e ≥ − ( 1 − γ ) ε P A F g G ( 1 − ε ) P A F e E MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGvbqvdaWgaaWcbaGaem4zaCMaemyzaugabeaakiabgwMiZkabgkHiTiabcIcaOiabigdaXiabgkHiTGGaciab=n7aNjabcMcaPmaalaaabaGae8xTduMaemiuaaLaemyqaeKaemOray0aa0baaSqaaiabdEgaNbqaaiabdEeahbaaaOqaaiabcIcaOiabigdaXiabgkHiTiab=v7aLjabcMcaPiabdcfaqjabdgeabjabdAeagnaaDaaaleaacqWGLbqzaeaacqWGfbqraaaaaaaa@4B5C@

p g D Z ≤ p g n e g D Z MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGWbaCdaqhaaWcbaGaem4zaCgabaGaemiraqKaemOwaOfaaOGaeyizImQaemiCaa3aa0baaSqaaiabdEgaNjabd6gaUjabdwgaLjabdEgaNbqaaiabdseaejabdQfaAbaaaaa@3CF0@

f g e ≥ − ε ( 1 − ε ) P A F e E MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGMbGzdaWgaaWcbaGaem4zaCMaemyzaugabeaakiabgwMiZkabgkHiTmaalaaabaacciGae8xTdugabaGaeiikaGIaeGymaeJaeyOeI0Iae8xTduMaeiykaKIaemiuaaLaemyqaeKaemOray0aa0baaSqaaiabdwgaLbqaaiabdweafbaaaaaaaa@405F@

R_ge≤ 1

γ ≥ γ_{min ge}where γmin⁡ge=11+F12(Vg/rt2) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaaiiGacqWFZoWzdaWgaaWcbaGagiyBa0MaeiyAaKMaeiOBa4Maem4zaCMaemyzaugabeaakiabg2da9maalaaabaGaeGymaedabaGaeGymaeJaey4kaSYaaSaaaeaacqWGgbGrdaqhaaWcbaGaeGymaedabaGaeGOmaidaaaGcbaWaaSGbaeaacqGGOaakcqWGwbGvdaWgaaWcbaGaem4zaCgabeaaaOqaaiabdkhaYnaaDaaaleaacqWG0baDaeaacqaIYaGmaaGccqGGPaqkaaaaaaaaaaa@4503@

R_oo≥ 0

γ ≤ γ_owhere γo=11+F22(Vg/rt2) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaaiiGacqWFZoWzdaWgaaWcbaGaem4Ba8gabeaakiabg2da9maalaaabaGaeGymaedabaGaeGymaeJaey4kaSIaemOray0aa0baaSqaaiabikdaYaqaaiabikdaYaaakiabcIcaOmaalyaabaGaemOvay1aaSbaaSqaaiabdEgaNbqabaaakeaacqWGYbGCdaqhaaWcbaGaemiDaqhabaGaeGOmaidaaaaakiabcMcaPaaaaaa@3F90@

The twin and familial risks model

Data from studies of monozygotic and dizygotic twins are commonly used to estimate the genetic and environmental variances V_gand V_eof 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 V_g, V_eand the variance due to gene-environment interaction, V_ge. The standard methodology for twin data analysis is inappropriate because it assumes V_ge= 0.

First note that we are interested in the extent to which relatives share risk categories (which may be either environmental or genotypic, or both), rather than a particular genetic variant. The probability that a relative of a proband is also a case depends on the extent to which their environmental and genotypic risks are correlated with those of the proband. Rather than adopting a specific form for the genetic model, define p^rel_gas the correlation in genotypic risk category (g) between relatives of type denoted by the superscript 'rel'. The parameter p^rel_gis the probability that the genotypic risk category (high or low) is identical by descent.

For monozygotic (MZ) twins, assumed to share their entire genome, p^MZ_g= 1. For dizygotic (DZ) twins and other siblings, who share half their genome, p^DZ_g= p^sib_g= 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), p^DZ_g= p^sib_g= 1/4. Here, allowing for the possibility of multiple gene-gene interactions (epistasis), require only that:

1 / 2 ≥ p g D Z ≥ 0 ( 20 ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaadaWcgaqaaiabigdaXaqaaiabikdaYiabgwMiZkabdchaWnaaDaaaleaacqWGNbWzaeaacqWGebarcqWGAbGwaaaaaOGaeyyzImRaeGimaaJaaCzcaiaaxMaadaqadaqaaiabikdaYiabicdaWaGaayjkaiaawMcaaaaa@3D10@

The meaning of p^DZ_gand its relationship to the polygenic risk model first adopted by Ronald Fisher in 1918 is discussed further below.

Similarly, define p^rel_eas the correlation in environmental risk category (e) between relatives of type "rel", requiring only that:

1 ≥ p e r e l ≥ 0 ( 21 ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqaIXaqmcqGHLjYScqWGWbaCdaqhaaWcbaGaemyzaugabaGaemOCaiNaemyzauMaemiBaWgaaOGaeyyzImRaeGimaaJaaCzcaiaaxMaadaqadaqaaiabikdaYiabigdaXaGaayjkaiaawMcaaaaa@3DD9@

Assume that p^rel_gand p^rel_eare 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:

λ r e l = ( 1 − p g r e l ) ( 1 − p e r e l ) + p g r e l ( 1 − p e r e l ) R R g e n c a s e s + ( 1 − p g r e l ) p e r e l R R e n v c a s e s + p g r e l p e r e l R R c a s e s ( 22 ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=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@A657@

Substituting for the relative risks RR^cases_gen, RR^cases_envand RR^casesusing Equations (8), (9) and (10) leads (after some algebra) to:

λ r e l − 1 = p g r e l V g r t 2 + p e r e l V e r t 2 + p g r e l p e r e l V g e r t 2 ( 23 ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=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@6E4A@

where

V e r t 2 = ( 1 − ε ) ε [ P A F e E ] 2 ( 24 ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaadaWcaaqaaiabdAfawnaaBaaaleaacqWGLbqzaeqaaaGcbaGaemOCai3aa0baaSqaaiabdsha0bqaaiabikdaYaaaaaGccqGH9aqpdaWcaaqaaiabcIcaOiabigdaXiabgkHiTGGaciab=v7aLjabcMcaPaqaaiab=v7aLbaadaWadaqaaiabdcfaqjabdgeabjabdAeagnaaDaaaleaacqWGLbqzaeaacqWGfbqraaaakiaawUfacaGLDbaadaahaaWcbeqaaiabikdaYaaakiaaxMaacaWLjaWaaeWaaeaacqaIYaGmcqaI0aanaiaawIcacaGLPaaaaaa@492C@

V g r t 2 = ( 1 − γ ) γ [ P A F g G ] 2 ( 25 ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaadaWcaaqaaiabdAfawnaaBaaaleaacqWGNbWzaeqaaaGcbaGaemOCai3aa0baaSqaaiabdsha0bqaaiabikdaYaaaaaGccqGH9aqpdaWcaaqaaiabcIcaOiabigdaXiabgkHiTGGaciab=n7aNjabcMcaPaqaaiab=n7aNbaadaWadaqaaiabdcfaqjabdgeabjabdAeagnaaDaaaleaacqWGNbWzaeaacqWGhbWraaaakiaawUfacaGLDbaadaahaaWcbeqaaiabikdaYaaakiaaxMaacaWLjaWaaeWaaeaacqaIYaGmcqaI1aqnaiaawIcacaGLPaaaaaa@493A@

V g e r t 2 = ( 1 − ε ) ε γ ( 1 − γ ) [ U g e P A F e E ] 2 ( 26 ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaadaWcaaqaaiabdAfawnaaBaaaleaacqWGNbWzcqWGLbqzaeqaaaGcbaGaemOCai3aa0baaSqaaiabdsha0bqaaiabikdaYaaaaaGccqGH9aqpdaWcaaqaaiabcIcaOiabigdaXiabgkHiTGGaciab=v7aLjabcMcaPaqaaiab=v7aLjab=n7aNjabcIcaOiabigdaXiabgkHiTiab=n7aNjabcMcaPaaadaWadaqaaiabdwfavnaaBaaaleaacqWGNbWzcqWGLbqzaeqaaOGaemiuaaLaemyqaeKaemOray0aa0baaSqaaiabdwgaLbqaaiabdweafbaaaOGaay5waiaaw2faamaaCaaaleqabaGaeGOmaidaaOGaaCzcaiaaxMaadaqadaqaaiabikdaYiabiAda2aGaayjkaiaawMcaaaaa@556D@

Note that if the G-E interaction component of the variance, V_ge, is zero, the utility of targeting the environmental intervention by genotype, U_ge, 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 (R_ge-R_go= R_oe-R_oo).

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 f_gesuch that:

V g e r t 2 = f g e 2 V g r t 2 . V e r t 2 ( 27 ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaadaWcaaqaaiabdAfawnaaBaaaleaacqWGNbWzcqWGLbqzaeqaaaGcbaGaemOCai3aa0baaSqaaiabdsha0bqaaiabikdaYaaaaaGccqGH9aqpcqWGMbGzdaqhaaWcbaGaem4zaCMaemyzaugabaGaeGOmaidaaOWaaSaaaeaacqWGwbGvdaWgaaWcbaGaem4zaCgabeaaaOqaaiabdkhaYnaaDaaaleaacqWG0baDaeaacqaIYaGmaaaaaOGaeiOla4YaaSaaaeaacqWGwbGvdaWgaaWcbaGaemyzaugabeaaaOqaaiabdkhaYnaaDaaaleaacqWG0baDaeaacqaIYaGmaaaaaOGaaCzcaiaaxMaadaqadaqaaiabikdaYiabiEda3aGaayjkaiaawMcaaaaa@4E53@

and choose its sign so that (combining Equations (24), (25) and (26)):

U g e = f g e γ ( 1 − γ ) V g r t 2 ( 28 ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGvbqvdaWgaaWcbaGaem4zaCMaemyzaugabeaakiabg2da9iabdAgaMnaaBaaaleaacqWGNbWzcqWGLbqzaeqaaOWaaOaaaeaaiiGacqWFZoWzcqGGOaakcqaIXaqmcqGHsislcqWFZoWzcqGGPaqkdaWcaaqaaiabdAfawnaaBaaaleaacqWGNbWzaeqaaaGcbaGaemOCai3aa0baaSqaaiabdsha0bqaaiabikdaYaaaaaaabeaakiaaxMaacaWLjaWaaeWaaeaacqaIYaGmcqaI4aaoaiaawIcacaGLPaaaaaa@487F@

U_geis zero if f_ge= 0 (i.e. for an additive G-E model, with no G-E interaction), but for a given γ and V_g, U_geincreases with increasing gene-environment interaction factor, f_ge. For a fixed f_geand genetic variance component V_g, U_geis 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: cases where γ_maxge < 1/2).

Using the definitions of V_e, V_gand V_ge(Equations (24), (25) and (26)) and the remaining inequalities, R_ge≤ 1 and R_oo≥ 0, two limits can be derived on the proportion of the population in the 'high genotypic risk' group, γ (see Table 2).

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 λ_DZ> λ_sib. This implies that environmental risks are more strongly correlated in dizygotic twins than in other siblings, p^e_DZ> p^e_sib. Remembering that p^MZ_g= 1 and p^sib_g= p^DZ_g, three independent equations for the relative risk in monozygotic, dizygotic twins and siblings may then be written:

λ M Z − 1 =