With the advent of GWAS we are gaining a clearer understanding of the genetic architecture of common complex diseases. Empirical evidence suggests an architecture of many genetic loci with many variants of small effect. Interest in genomic profiling, the use of a genome-wide markers to predict genetic disease risk, is growing (for example, [19, 20]), as is the establishment of companies offering profiling services. The prediction of disease risk from many risk loci or markers requires a model that combines the effects of these loci and the choice of this model is the topic of this paper.

### Total variance of risk loci is the driving force

We chose two parameters that are directly measurable in real populations for benchmarking models: disease prevalence (that is, *K*) and the effect size of a single risk allele (that is, *τ*). We recognized that many combinations of the number of loci (that is, *n*) allele frequency (that is, *p*) and *τ* were consistent with the same heritability on the underlying scale in the Probit model (that is,
) and that the predictions of all the models were insensitive to the exact combination of *n*, *p* and *τ* provided
was held constant. Therefore, we have compared the models while holding constant *K* and
. In Figures 1 and 2 we present results for *n* = 1,000 and *p* = 0.3, to provide some comparison to empirical estimates of *τ*. Since the distribution of genetic risk of disease in a population is driven by total genetic variance rather than the variance contributed by each locus, it is unlikely that relaxing the restriction of equal allele frequencies and effect sizes will impact the results; this is consistent with the results of other studies [4, 10, 21].

Although we show that the unconstrained Risch model is not a practical model, its mathematical tractability can still provide valuable insight into our understanding of the factors influencing genetic risk. We show (Additional file 4) that the scaled contribution to the genetic variance on the risk scale by each risk allele (*v*) is a function of *p* and *τ*, *v* = *p*(1 - *p*)(*τ* - 1)^{2}/[1 + *p*(*τ* - 1)]^{2} and the total genetic variance on this scale is proportional to *nv*. For small values of *τ* (that is, *τ*; → 1), *nv* ≈ *np*(1 - *p*)(*τ* - 1)^{2}, which can be used to derive the proportion of genetic variance explained by one locus.

### Rejection of simple additive and simple multiplicative models on the risk scale

Risch [3], using schizophrenia as an example, was the first to show that recurrence risk to relatives in complex diseases is better explained by a multiplicative than an additive model of gene action on the risk scale because (*λ*
_{
MZ
}- 1)/(*λ*
_{
sib
}- 1) >2 as shown in Table 1. In preliminary simulations (not reported) we confirmed that additivity on the risk scale of all risk loci simply could not produce the steep rise in probability of disease (Figure 1) necessary to achieve the disease prevalences and recurrence risks to relatives typical of complex diseases. In contrast, Slatkin [13], under his thesis of exchangeable models, demonstrated that an additive model on the risk scale could explain complex disease. However, to achieve the steep rise in disease risk, he imposed stringent constraints, so that the additive effect of risk alleles only occurred in the (very narrow) range of the number of risk alleles associated with the steep rise in probability of disease. Outside this range probability of disease was either zero or 1. In this way, the shape of the risk function is similar to the models that are multiplicative on the risk scale.

Other theoretical studies have used the Risch model [2, 13], the CRisch model [13], the Odds model [4] and the Probit model [22]. Although there is a generally accepted dogma that these models are similar, in trying to compare studies it is important to know if any differences are a function of the choice of risk model. In a previous study [10] we made derivations under the Risch model and for the parameter combinations considered the probability of disease being greater than 1 was rare. However, in this study, where we have considered the full range of parameters, we have recognized that under the unconstrained Risch model, individuals for whom probability of disease is greater than 1 (*g*
_{
x
}>1) make a huge contribution to the genetic variances.

Risch [3] investigating schizophrenia and Brown *et al*. [6] studying ankylosing spondilitis recognized that the observed ratio
was less than one, whereas this ratio is expected to be 1 under the Risch model [3]. The sampling variance on estimates of recurrence rates is high and so the greater consistency with multiplicative rather than additive models (risk scale) was their main conclusion. However, by looking at a range of complex diseases (Table 1) there is consistent evidence that
is less than 1, particularly for low prevalence diseases. These observed ratios are consistent with our simulation results, which show that under the CRisch, Odds and Probit models, the ratio
only as *K* → 0.5 and
→ 0, but under parameters typical of common complex genetic diseases
, particularly as *K* → 0 and
→ 1. The mathematical tractability of the Risch model has often made it the method of choice in theoretical studies and the equality
has been used to underpin predictions (for example, see the Supplement of Clayton [23]); in the mathematical expressions the impact of not constraining the probability of disease to be less than 1 is not obvious, but it is because of this important constraint that equality
is often much less than 1.

Therefore, we conclude that the unconstrained Risch model is simply not realistic, particularly for parameters typical of human complex disease (K < 0.1 and
> 0.5), so here we have made comparisons on the more realistic constrained (CRisch) model.

### Differences between the models unlikely to be detectable in practice

Since we reject the additive and Risch models, we concentrate on the comparison of the CRisch, Odds and Probit models. We chose to compare models with two fixed benchmarks, disease prevalence and effect size of an individual risk allele, taken at the average number of risk alleles (that is, *τ*). Under this benchmarking, the probability of disease associated with carrying the minimum number of alleles in the population differs between models, but in all models this will be very close to zero given the number or risk loci now expected to contribute to complex genetic disease. Although we assume that each risk locus has the same individual effect size, the models differ in the way that the effect sizes combine. For example, a given risk locus with observed *τ* and *p* explains a smaller proportion of the risk to relatives under a Probit model than under a CRisch model However, we conclude that for all operational purposes, in the foreseeable future, it is unlikely that we will be able to distinguish between the models either on the basis of recurrence risks to relatives or on the basis of estimates of effect sizes of risk loci. Slatkin [13] also compared the CRisch and Probit models and benchmarked on a range of parameters. Our results are complementary to, and consistent with, his, although direct comparison is prevented by his models distinguishing between heterozygotes and homozygotes at each locus, so that the multiplicativity of risk alleles was only between loci and not within loci. Inability to distinguish between multi-locus risk models on the basis of recurrence risks is perhaps not surprising given that Smith [24] was unable to distinguish between more extreme models on this basis. Ability to distinguish between the models is only possible in the very tail of the risk curve and would only be achievable if genomic profiles could be constructed using measured variants that accounted for the totality of the genetic variance. If this were possible, sets of individuals could be identified with high predicted risk and the proportion succumbing to disease could be measured and compared to the proportion expected under different models. Such hypothetical scenarios at present seem unattainable.

### Each individual carries a unique portfolio of risk loci

From Figure 1 it becomes clear that when there are many risk loci contributing to disease each of small effect, that all individuals in the population necessarily carry a large number of risk alleles. For example, when 1,000 loci with risk alleles of frequency 0.1 underlie a complex disease, all individuals in the population carry at least 150 risk alleles, an average individual carries 200 risk alleles and, when disease prevalence is low and heritability is high, most of those with disease carry 230 to 250 risk alleles. Since, in this example, there is a total of 2,000 risk alleles, each individual will carry their own unique portfolio, which could underlie the phenotypic heterogeneity typical of many complex diseases.

### Large amounts of epistasis on the risk scale despite additivity on underlying scales

Our results show that additivity of individual genetic variants on some underlying scale can convert to, sometimes considerable, non-additive genetic variance on the risk scale, particularly when the disease prevalence is low. These results are not new and were presented by Dempster and Lerner [14], but are sometimes overlooked. Human diseases usually have prevalences of less than 0.1, in which case the majority of the genetic variance on the risk scale is epistatic. These results imply that the models underpinning GWAS already account for one type of gene-gene interaction, if each *τ* could be estimated without error. Likewise, our usual models also imply genotype-environment interaction on the risk scale because the effect of an environmental factor is greater in people with higher genetic risk. Our definition of epistasis is one of statistical interaction; the extent to which statistical interaction relates to biological or functional interaction has been much debated (see [25] for a review) and will not become clear until more of the genetic variance can be explained by identified genomic variants.

### True versus estimated *τ*

We set out to benchmark models on the basis of two observable parameters, disease prevalence (that is, *K*) and the effect size of a single risk allele (that is, *τ*). In building the models we have assumed that the true *τ* is known and have defined it as the effect of a single risk locus in the background of the average number of risk loci. However, the estimates of *τ* made from experimental data may be quite different to these true values. If the genotypes at all risk loci were known and a complete model was fitted to the data, then the correct estimate of *τ* would be obtained (within experimental sampling error). In practice, however, usually only the effect of a single risk locus is included in the statistical model and under these circumstances we will estimate the effect of an extra risk allele averaged across all background genotypes rather than the effect at the mean background genotype. The effect of this may be dependent on the true way in which loci combine to influence risk of disease, which, of course, is unknown. Under the CRisch model of Figure 1a, all individuals with >650 risk alleles get the disease, so above 650 risk alleles there is no effect of an extra risk allele. Conversely, below 650 risk alleles each extra risk allele increases the probability of disease by *τ*. The experimental estimate will be a weighted average of these two estimates (zero and *τ*). In practice, therefore, variants detected with small relative risk may reflect greater biological importance than might otherwise be inferred. Under the Probit model the *τ* calculated at the average number or risk loci is
whereas the *τ* estimated when a single risk locus is in the statistical model is Φ(a-t)/Φ(-*t*) because then all other risk loci are part of the residual variance in liability and so the residual variance approaches the phenotypic variance, which is 1.0.

Comparison of the models in practice is difficult and distinguishing between them may be impossible, especially if the true *n* is large and the true *τ* is small. Since we have demonstrated that the models are difficult to differentiate, the use of the Probit model, which has mathematical tractability and a known relationship between the estimates of *τ* in different genetic backgrounds, is likely to be the model of choice. The estimated variance on the liability scale explained by a locus with estimated effect size
is
[26], so that the estimated effect on the liability scale is
, where *i* is the mean liability of the diseased group, *i = z/K*, where *z* is the height of the normal curve at the threshold *t*.

### Limitations

The true genetic architecture (in terms of number, frequency and effect size of risk variants and the way in which they combine) is unknown and may be quite different for the different diseases listed in Table 1. For simplicity, we have described disease in terms of affected/unaffected, ignoring time-dependent onset, and we have ignored phenotypic heterogeneity (which may reflect genetic heterogeneity) in the definition of disease status and other real-life complications. In principle, our approach could reflect any definition of disease if the genetic epidemiology and genetic risk variants can be defined - for example, early and late onset disease may be considered as different diseases - but despite this any simple model is likely to be a poor representation of disease. None of the models we have considered are likely to be the true model, but since they can all generate recurrence risks consistent with complex genetic diseases (given the right combination of parameters), they can give useful insight until empirical data provide evidence for them to be rejected. These simple models provide some boundaries, demonstrating some properties that must be upheld by the true genetic architecture in order to be consistent with observed data.