shows the disease risks for genotype scores of up to 20 for the two models when relative risks are 1.1, 1.2 and 1.5, and genotype frequency is 0.1 and 0.3. We considered genotype scores of up to 20 because anyone in the population having a genotype score of more than 20 is almost zero for these genotype frequencies. The disease risk for the additive model increases linearly with increasing risk genotypes. When genotype frequency is 10%, the disease risk for the additive model is higher than the disease risk for the multiplicative model for up to a genotype score of 5, and when genotype frequency is 30%, the disease risk for the additive model is higher than the disease risk for the multiplicative model for up to a genotype score of 13 for relative risk 1.2 and up to a genotype score of 14 for relative risk 1.5. When the genotype score is 4 in the genomic profile, with relative risk 1.2 and genotype frequency 10%, the increase in disease risk for the additive model is only 0.006; when relative risk is increased to 1.5, the increase is 0.028; when genotype frequency is 30% and relative risk 1.2, the increase is 0.033; and when genotype frequency is 30% and relative risk 1.5, the increase is 0.041. Theoretically, multiplicative models give higher disease risk compared with additive models when there are a large number of risk genotypes.

The vertical lines in denote the distribution of risk genotypes in the population. When the genotype frequency is 0.1, only 8 out of 10

000 in the population can have more than 10 risk genotypes, whereas when the genotype frequency is 0.3, more than 75% of the population can have more than 10 risk genotypes. For more than 79% of the population, the risk under the additive model is higher than the risk under the multiplicative model. However, when relative risk is around 1.1, the difference in risk between the two models is almost negligible. For the population for which the multiplicative model gives higher risk than the additive model, the difference in risk is very large. This situation is to be expected, as both models assume identical prevalence of disease in the population.

We also considered 40 risk genotypes with genotype frequency 0.1 and relative risk 1.2, and another risk genotype, A, with the same genotype frequency and relative risk of 3.5 in a genomic profile. The risks for the two models for the populations with and without risk genotype A are shown in . For the sub-population that has risk genotype A, almost 77% shows a higher risk under the multiplicative model compared with the additive model. However, the proportion of the sub-population that has the risk genotype is only 10% of the overall population.

shows the ROC curves for the different combinations of relative risks and genotype frequencies studied. The AUC for the multiplicative model (AUC_{M}) and the AUC for the additive model (AUC_{A}) increase with increasing values of relative risks. When relative risk is 1.1 and genotype frequency is 10%, the AUC_{M} and AUC_{A} are 0.56 and 0.53, respectively. When the relative risk is 1.5, AUC_{M} increases to 0.74 and AUC_{A} increases to only 0.60. This shows that higher relative risks result in larger differences between AUC_{M} and AUC_{A}. When genotype frequency is increased from 0.1 to 0.4, keeping relative risk at 1.5, the AUC_{M} increases from 0.74 to 0.85, but AUC_{A} declines from 0.6 to 0.55. As seen in , there is not much difference between AUC_{M} and AUC_{A} for lower values of relative risks (around 1.1) and for lower values of genotype frequencies (less than or equal 10%). For example, when relative risk is 1.1 and genotype frequency is 0.1, the difference between AUC_{M} and AUC_{A} is only 0.02. However, when relative risk is 1.5 and genotype frequency is 0.4, the difference between AUC_{M} and AUC_{A} is 0.30. AUC_{M} increases steeply with increasing relative risks and genotype frequency, whereas the increase in AUC_{A} remains small.

To explore this phenomenon further, we plotted the AUC against genotype frequency for relative risks of 1.2 and 1.5 (). The AUC for the multiplicative model increases with increasing genotype frequency up to 50% and then declines. The AUC for the additive model increases with increasing genotype frequency for rare genotypes of up to 5–7% range and then declines. and show that, for rare genotypes (less than 10%) and low relative risks (les than 1.2), there is not much difference between the AUC for multiplicative models and that for additive models. When relative risks are around 1.2–1.5, the largest difference in AUC for the two models is achieved when genotype frequency is around 50%. The AUCs for the 41 genetic variants illustrated in are 0.635 and 0.590 for the multiplicative and additive models, respectively. When the genotype frequency is increased to 0.3 for all the 41 risk genotypes, the AUCs for the multiplicative and additive models are 0.690 and 0.588, respectively.

The population attributable fractions (PAF) of the 40 risk genotypes for the additive and multiplicative models are also different.^{14} For example, when *R*=1.2 and *G*=0.1, the PAF for the additive model is 0.444, whereas the PAF for the multiplicative model is 0.615. When the genotype frequency is increased to 0.2, the PAF values for additive and multiplicative models are 0.547 and 0.792, respectively.

Zheng *et al*^{15} studied the genetic predisposition to prostate cancer by examining the association between prostate cancer and five SNPs that map to the three 8q24 loci, to 17q12, and to 17q24.3. The genotype frequencies of the five SNPs were 0.3, 0.25, 0.07, 0.77, and 0.6, and the relative risks were 1.38, 1.28, 1.53, 1.37, and 1.22. The prevalence of diagnosed prostate cancer in the US adult population is about 1.6%, based on estimates from the National Health Interview Survey. As the true prevalence of prostate cancer is unknown, we assumed an upper bound of 3.2% for the prevalence of prostate cancer. We used the same methods developed in this study for identical risk genotypes to calculate AUC for these five SNPs after adjusting the probabilities calculated in the equations for sensitivity and specificity based on different genotype frequencies and relative risks. The AUCs for the multiplicative model and additive model were 0.569 and 0.541, respectively.