Supplementary Materials. We randomly divided the original dataset S into K subsets. In the k-th cross-validation, the k-th subset

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1 Supplementary Materials Supplementary Methods K-fold cross-validation procedure We randomly divided the original dataset S into K subsets. In the k-th cross-validation, the k-th subset was left out to be the validation set, denoted by S k Va, while the other K-1 subsets were used as the training set, denoted by S k Tr. We implemented the forward selection algorithm, using S k Tr to build a series of prediction models, M k 1, M k 2,..., and applied these models to the validation dataset, S k Va, to estimate the predicted AUC values. The predicted AUC values from the K-fold cross-validation were averaged at each level of model complexity, yielding a series of averaged predicted AUC values from the cross-validation, denoted by (AUC 1, AUC 2,...). The level at which the averaged AUC value stopped increasing was used as the appropriate complexity level for the final prediction model built on the original dataset S. Cross-validation procedure with bootstrap aggregating (bagging) As an alternative to the K-fold cross-validation procedure, we also propose a cross-validation procedure with bootstrap aggregating (bagging) to further improve the method s robustness and power. Bagging was first introduced [1] to reduce an estimator s variance with little cost in bias. Its basic principle is to obtain an aggregated predictor by generating multiple predictors from bootstrap replicate samples of the data. The bagging method works especially well when the prediction method is unstable (i.e., small changes in the learning set result in large changes in prediction). Recently, Petersena et al [2] introduced a cross-validated bagging procedure by integrating bagging into the cross-validation procedure. They showed that an appropriate bias-variance trade-off for the parameter 1

2 of interest can be achieved by conducting the cross-validation at the level of the bagging estimator itself (i.e., the cross-validated bagging estimator) [1]. Based on this concept, we can use cross-validation with bagging instead of K-fold cross-validation in the forward ROC method. To do this, we randomly divide the original dataset S into K subsets. In the k-th cross-validation, we draw B bootstrap samples (B k 1, B k 2,..., B k B ) from the training set S k Tr. For the b-th bootstrap sample, we implement the forward selection algorithm to build a series of prediction models, M b 1, M b 2,..., and then apply these models to the validation dataset, S k Va, to estimate the predicted AUC values. The predicted AUC values from the B bootstrap samples are then averaged at each level of model complexity, yielding a series of bagging estimators of the predicted AUC values for the k-th cross-validation, denoted by (AUC k 1, AUC k 2,...). The K results are then averaged to provide overall cross-validated bagging AUC estimators, denoted by (AUC 1, AUC 2,...). The level with the highest average AUC value is used as the complexity level for the final prediction model built on the original dataset S. Cross-validation with bagging is a computationally intensive procedure in which a number of bootstrap samples are needed for each fold of cross-validation. Assuming time t is required for the K-fold cross-validation, cross-validation with bagging needs time B t, where B is the number of bootstrap samples in each cross-validation. Despite its heavy computational requirement, cross-validation with bagging has the potential to improve the method s performance and lead to a more stable and more accurate predictive genetic test. 2

3 Supplementary table 1 Summary of the simulation settings Simulation scenarios I and II Supplementary Simulation 2 Supplementary Simulation 3 Risk Inheritance variants a mode b Odds ratios Interaction c Number of Noise loci d Setting1 10 G e : a, b, c G: 1.2, 1.3, 1.5 a e: 1.7 Setting2 15 E e A, R, D : e E: 1.4 b c: 1.4 Setting3 20 Setting1 a, b, c A, R, D 2, 2.5, 2.6 Setting2 Setting3 a, b, c, e, f, g a, b, c, e, f, g, h, i, j, k a, b, c, e, f, A, R, D, A, R, D A, R, D, A, R, D, A, R, D, R A, A, A, A, R, 1.5, 1.7, 1.8, 1.6, 1.6, , 1.9, 1.8, 1.6, 1.7, 1.9,1.6, 1.5, 1.6, , 4.0, 1.2, 1.2, 1.3 No interaction No interaction No interaction a f: 2.5 b e: 2.4 a The minor allele frequencies of risk loci were generated randomly from a uniform distribution that ranged from 0.1 to 0.5. b A, R and D represent additive, recessive and dominant modes of inheritance, respectively. c Numbers listed in the Interaction column measure the risk to individuals who carry risk alleles of interacting loci and/or environmental factors vs. all other individuals. For the a e interaction in simulation scenario I, we assumed that individuals who were exposed to the environmental risk level and carried the risk allele a (i.e., high risk individuals) had a 1.7 times higher risk of disease than all other individuals. Similarly, for the b c interaction, we assumed that individuals who carried two risk alleles at locus b and one or more risk alleles at locus c had a 1.4 times higher risk of disease than all other individuals. d The allele frequencies of the non-causal loci were generated randomly from a uniform distribution that ranged from 0.1 to 0.9. e G represents genetic risk factors and E represents environmental risk factors. 3

4 Supplementary Simulations Supplementary simulation 1 We evaluated the effect of cross-validation with bagging on the forward ROC method using simulation scenarios I and II. The details of the simulation settings are listed in supplementary table 1. For each replicate, 50 bootstrap samples were generated for the bagging cross-validation procedure. The results are summarized in supplementary table 2. In simulation I, we found that the proposed forward ROC method has an overall similar performance, whether using 10-fold cross-validation or using cross-validation with bagging. When there were a small number of loci, 10-fold cross-validation performed slightly better than cross-validation with bagging. However, with an increasing number of noise loci, cross-valuation with bagging attained higher AUC means and smaller mean square errors (MSEs) than 10-fold cross-validation. For instance, when there were 20 noise loci, cross-validation with bagging led to an AUC mean of , a 0.25% increase from 10-fold cross-validation. In terms of computational efficiency, 10-fold cross-validation and cross-validation with bagging required 1.5 minutes and 50 minutes, respectively, to complete one simulation replicate on a computer equipped with 4G memory. In simulation II we found, when the missing rate increased, that cross-validation with bagging performed better than 10-fold cross-validation. For instance, with 10% missing data, the forward ROC method using cross-validation with bagging had an AUC mean of , a 0.39% increase from using 10-fold cross-validation. 4

5 Supplementary table 2 A comparison of bagging cross-validation and 10-fold cross validation. Scenario I Forward ROC using 10-fold CV Forward ROC using Bagging CV Number MEAN a BIAS SD b MSE c MEAN BIAS SD MSE 10 Noise Loci Noise Loci Noise Loci Scenario II Forward ROC using 10-fold CV Forward ROC using Bagging CV Missing % MEAN BIAS SD MSE MEAN BIAS SD MSE a AUC estimator, b standard deviation, c mean square error Supplementary simulation 2 The random forest (RF) method is a powerful tool for high-dimensional risk prediction [3]. RFs have several unique features, such as being capable of uncovering interactions among genes and/or environmental factors with lower marginal effects [4]. We conducted a simulation to compare the proposed forward ROC method with the RF method. In the simulation, we set the proportion of disease-susceptibility loci to 1/8, 1/4 and 1/3. The detailed settings of the simulation are described in supplementary table 1. The true AUC in all three settings was set to be approximately The RF method was performed using the R package randomforest version 4.5 with a forest size of 500 trees replicates were simulated, each consisting of 1000 cases and 1000 controls. The simulation results are summarized in supplementary table 3. We found that, when the proportion of disease-susceptibility loci is small (i.e., 1/8) and the effect size is strong, the forward ROC method attained a better performance than the RF method, with an increase of 2.17% in the AUC mean. When the proportion of disease-susceptibility loci increased to 1/4, the two methods had a similar performance, with AUC means of for forward ROC and for RF. With a 5

6 further increase in the proportion of disease-susceptibility loci and a decrease in the effect size, RFs tended to capture more disease-susceptibility loci and attained higher classification accuracy than the forward ROC method. For instance, when the proportion of disease-susceptibility loci increased to 1/3, the RF method performed better than the forward ROC method, with an increase of 5.82% in the AUC mean. Supplementary table 3 Summary of results from supplementary simulation 2 Proportion of disease-loci 1/8 1/4 1/3 Mean MSE Mean MSE Mean MSE Forward ROC Random Forest (RF) Supplementary simulation 3 Some diseases may be caused by a small set of loci with large effect sizes. For instance, five SNPs have been detected as being associated with age-related macular degeneration (AMD), and were combined to explain approximately half of the classical sibling AMD risk [5]. We conducted a simulation under such a disease scenario to assess the performance of the forward ROC method, CART and the allele counting method. In this simulation, we introduced five disease-susceptibility loci and 2 two-way interactions, without including any noise loci (For details, see supplementary table 1). The true AUC was set at 0.8. We compared the forward ROC method with the CART and allele counting methods, based on 1000-replicate simulations. The predicted AUC was calculated from the evaluation sets and the results are summarized in supplementary table 4. We found that the proposed forward ROC method performed better than the other two methods, with a higher AUC mean and a lower MSE. The forward ROC method attained a predicted AUC of , which was close to the true AUC of 0.8. The CART and allele counting methods attained AUC values of and , respectively. The standard deviations of the AUCs for all three methods were relatively small: , and 6

7 for the forward ROC method, CART and the allele counting method, respectively. Supplementary table 4 Summary of results from supplementary simulation 3 MEAN a BIAS SD b MSE c Forward ROC CART Allele counting a AUC estimator. b standard deviation. c mean square error. Wellcome Trust RA GWAS Dataset RA cases from the Wellcome Trust study were recruited by the Arthritis Research Campaign Epidemiology Unit and met the standard clinical criteria for RA. One-half of the controls were chosen from the 1958 British Birth Cohort and the other half came from the UK Blood Service controls. All individuals were genotyped using the Affymetrix 500K chip. We excluded samples of low quality (e.g., low DNA quality). The final samples for analysis comprised 2938 RA cases and 1860 controls. After removing low quality single nucleotide polymorphisms (SNPs) (e.g., those with low allele frequencies), 460,547 SNPs remained for analysis. Supplementary table 5 Summary of the 35 RA associated loci reported from different studies SNP Gene Chromosome Source Reference rs IFI K [6] rs MMEL K [6] rs MMEL1-TNFRSF K [7] rs PADI4 1 Imputed [8] rs PTPN22 1 Imputed [9] rs RSBN K [6] rs TNFRSF1B 1 Imputed [8] rs CTLA K [8] rs ITGAV 2 Imputed [10] rs STAT4 2 Imputed [11] rs ANAPC K [6] 7

8 rs IL2-IL21 4 Imputed [12] rs HLA-DRB1 6 Imputed [13] rs HLA-DRB1 6 Imputed [13] rs HLA-DRB1 6 Imputed [13] rs MHC 6 500K [6] rs MHC 6 Imputed [14] rs4678 MHC:VARS2L 6 Imputed [14] rs OLIG3-TNFAIP K [6] rs OLIG3-TNFAIP3 6 Imputed [15] rs42041 CDK K [7] rs IRF5 7 Imputed [16] rs N/A 7 500K [6] rs CCL K [7] rs PHF19 9 Imputed [17] rs TRAF1 9 Imputed [18] rs TRAF1-C5 9 Imputed [19] rs IL2RA K [6] rs PRKCQ K [7] rs KIF5A-PIP4K2C K [7] rs KLF12 13 Imputed [20] rs N/A K [6] rs CD K [7] rs N/A K [6] rs C1QTNF K [6] References 1 Breiman L: Bagging predictors. Machine Learning 1996;24: Petersena ML, Molinaro AM, Sinisi SE, Van der Laan MJ: Cross-validated bagged learning. Journal of Multivariate Analysis 2007;98: Breiman L: Random Forests. Machine Learning 2001;45: Bureau A, Dupuis J, Falls K, Lunetta KL, Hayward B, Keith TP, Van EP: Identifying SNPs predictive of phenotype using random forests. Genet Epidemiol 2005;28: Maller J, George S, Purcell S, Fagerness J, Altshuler D, Daly MJ, Seddon JM: Common 8

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