In a multi-dimensional regression problem, we have a data set of l d-dimensional independent variables x_i ∈ ℝ ^d , i = 1,..., l and dependent variables y_i ∈ ℝ. In our IQS regression problem, y_i represents the true IQS and x_i denotes the input feature vector. The goal is to find a function that approximates y_i . A solution of this problem based on a kernel method is to find the function y_i ≈ f(x_i , w, b) = w. φ (x_i ) - b, where w and b ∈ ℝ ^d are parameters and φ : ℝ ^d → ℝ ^d is a mapping such that there exists a kernel function that computes the inner product φ (x_i ). φ (x_j ) = k(x_i , x_j ). Because the radial basis function (RBF) can preserve a relatively high accuracy in comparison with other kernel functions (data not shown), our choice of the kernel function is the RBF kernel 11 12 .

Many models and algorithms have been developed to search for the parameters w and b of the regression function that maximally ts the input set of data. The ε-Support Vector Regression model (ε-SVR) is one of the useful models. Its formulation is given as:

m i n i m i z e 1 2 | | w | | 2 + C ∑ i = 1 l ( ξ i + ξ i * ) s u b j e c t t o y i - w ⋅ φ ( x i ) - b ≤ ε + ξ i w ⋅ φ ( x i ) + b - y i ≤ ε + ξ i * ξ i , ξ i * ≥ 0 .

The parameter C is used to determine the complexity of model and controls the tradeoff between the training error minimization and the model complexity. If it is too small, the model may underfit the data. The parameter ε serves as the tolerance of errors of the regression. Combined with the slack variables ξ _i ξ i * we have a soft-margin approach to regression that can be flexibly adjusted 11 12 .

The ν-Support Vector Regression (ν-SVR) introduces another parameter ν in the formulation, which is proven to be easier to adjust than C. One of the reasons is that the range of ν is [0,1] while the range of C is [0, ∞) 13 14 15 .

m i n i m i z e 1 2 | | w | | 2 + C ( v ε + 1 l ∑ i = 1 l ( ξ i + ξ i * ) ) s u b j e c t t o y i - w ⋅ φ ( x i ) - b ≤ ε + ξ i w ⋅ φ ( x i ) + b - y i ≤ ε + ξ i * ξ i , ξ i * ≥ 0 , ε ≥ 0

Moreover, the parameter ν can serve as an upper bound for the fraction of margin errors, and a lower bound for the fraction of the number of support vectors. In comparison with C, to select a suitable ν would be more intuitive 13 14 . Therefore, we chose ν-SVR over ε-SVR for our IQS prediction model. This model is also known to provide high out-of-sample generalization performance.

We chose LibSVM 16 as our implementation of the ν-SVR model. The parameter γ in the radial basis function is set as 1/d. The parameter ν was searched within {0.1, 0.2, 0.3, . . . 1.0} and an optimal value of ν were selected by applying a 10-fold cross validation on the training data set. The regression model can be applied to approximate P_o and P_c as well as the IQS.

Other regression models can also be used but the key to the success is to identify a set of variables that influence the imputation quality as the input features x_i in the regression model. We intended to use all useful information related to imputation quality as features for the regression model. Under consideration of the statistical correlation analysis (data not shown), we selected the following 12 defined features of a SNP whose allele we want to impute within a given sample.

1. Chromosome position: The chromosome where the SNP located.

2. Physical position: The position of the imputed SNP in bp.

3. Minor allele frequency (MAF): Previously, 9 have shown that the minor allele frequency is an important variable correlated with the true IQS. The above three features are available in the annotation file from the genotyping platform provider.

4. B allele frequency: This is derived from the allele signal intensity measurement for each locus of each individual in the raw CEL files. The raw CEL files are available from the Hapmap samples 17 . For each imputed SNP, we used the mean of the B allele frequency of the SNP on the samples of the corresponding ethnic population.

5. MAF in the reference panel: In addition to using the available MAF provided by the annotation file, we also consider the MAF in the reference panel.

6. Ratio of genotypes AA/AB: It is used to to indicate the proportion of genotype AA for each imputed SNP in the reference panel.

7. Ratio of genotypes BB/AB: Similar to feature 6, it is used to to indicate the proportion of genotype BB for each imputed SNP in the reference panel.

8. Distance to the nearest genotyped SNP: This is to capture an indication that the imputation quality will be better if the nearest genotyped SNP in the inference panel is closer.

9. Distance to the nearest recombination hotspot: The distance to the nearest recombination hotspot also plays an important role in the quality of the imputation. We used the recombination rates and hotspots available in the release version phase II build b35 to GRCh37 from the International HapMap Project 17 .

10. The nearest recombination hotspot's recombination rate (cM/Mb, centiMorgans per megabase): This variable is important in the imputation process. The IMPUTE2 program uses it explicitly as a required input for the imputation 1 18 .

11. Posterior probability estimated by the imputation program: This variable is available from the output of the imputation program. The Beagle program provides the genotype probabilities file and the genotype dosage file. We used the mean values of the posterior probabilities estimated for all the individuals in the inference panel.

12. B-allele dosage: Given the posterior genotype probabilities for a SNP (Pr(AA), Pr(AB), and Pr(BB)), the estimated B-allele dosage for each individual is equal to 0 × Pr(AA) + 1 × Pr(AB) + 2 × Pr(BB), which is reported in the genotype dosage file. We used the mean values of the B-allele dosage values of all the individuals in the inference panel.

It is worth mentioning that the posterior probability estimated by the imputation program and the B-allele dosage are highly correlated to predicting the IQS under the statistical correlation analysis. These features will be used in the regression model for the IQS as well as the regression for the observed agreement P_o and the chance agreement P_c . We will show that these 12 features are useful to construct an adequate regression model.

We prepared three data sets to evaluate the performance of our regression models. These data sets contain genotyping results of samples chosen to cover different ethnic backgrounds collected in different disease studies. We selected recent data sets genotyped with advanced platforms that cover a large number of SNPs so that we can flexibly keep those SNPs covered by old, obsolete platforms (with less SNPs probed) and hold out the rest to impute. Meanwhile, since we have their true genotypes, we can use the true genotypes of these SNPs as the gold standard to evaluate imputation quality and regression.

The Merlion Lung Cancer Study 2 DNA 19 and Oral Squamous Cell Carcinoma samples 20 from the NCBI GEO database 21 were used for evaluating the regression model. The Merlion Lung Cancer samples consist of two ethnic populations, East-Asian (EA) and Western-European (WE). Samples were all genotyped on the Affymetrix Genome-Wide Human SNP Array 6.0 platform. This platform contains more than 906,600 SNP probes, including the historical 482,000 SNPs in the Affymetrix GeneChip Human Mapping 500K Array Set. After the preprocessing of raw CEL files, there are 763,252 SNPs reported for the EA population of Merlion Lung Cancer samples, 778,058 SNPs for the WE population of Merlion Lung Cancer samples, and 693,494 SNPs for the oral Squamous Cell Carcinoma samples.

We designed scenarios to simulate the imputation of missing SNPs in a data set genotyped using an old platform to the large set of SNPs on the Affymetrix SNP 6.0 array. These scenarios involve a training set to construct our regression model in advance. This involves holding out a set of SNPs to impute, evaluating true IQS with known alleles, using the true IQS to train the regression model. Then the trained regression model can be applied to estimate IQS of imputed SNPs in a test set, where a set of SNPs is assumed to have missing genotypes. The design of the scenarios is to create different combinations of the training and test sets and see how the regression performance is affected.

To create both training and test sets, we basically divided the SNPs on the Affymetrix SNP 6.0 array into two sets. One contains those SNPs genotyped in both an old platform and Affymetrix SNP 6.0 array. This set simulates SNPs with "known" genotypes to be used to impute other SNPs. The other contains the remaining SNPs covered only by the Affymetrix SNP 6.0 array. This set simulates "missing" SNPs to be imputed.

Table 2 and Table 3 show our design of training and test sets for four scenarios to evaluate generalization of the regression model. Scenario 1 is the simplest case, which tests the regression performance when a sample of the same ethnic and disease phenotype is used for training. We used the WE lung cancer sample to create the training set. Alleles of the randomly picked 10% of SNPs of the training set were erased, denoted as "missing." Under the Affymetrix 500k array, these "missing" SNPs were imputed using the other 90% genotyped SNPs to a full set of SNPs on the same platform. As a result, there are 41,304 SNPs of the WE lung cancer sample used for the model training. We also used the WE lung cancer sample to create the test set, which consists of 320,172 SNPs covered only by the Affymetrix SNP 6.0 array. Their genotypes were then imputed from SNPs covered by the Affymetrix mapping 500k array and our regression model was applied to assess the imputation quality.

In Scenario 2, the generalization performance of our IQS regression model was evaluated when it was trained using "known" and "missing" SNPs covered by platforms different from those to be used in testing. We used the WE lung cancer sample again but used the Illumina 550k array instead of the Affymetrix SNP 6.0 array to choose SNPs. There are 41,304 SNPs of the WE lung cancer sample on the Illumina 550k array. After the regression model is constructed, we then used the same test set created in Scenario 1.

In Scenario 3, our IQS regression model is applied to different ethnic populations. We used the EA lung cancer sample to create the training set, resulting in 37,611 SNPs of the EA lung cancer sample on the Affymetrix 500k array. The regression model constructed by the EA lung cancer samples was used to predict the IQS of SNPs of the WE lung cancer samples as in Scenario 1.

Scenario 4 tests if our regression model can be generalized across samples collected for different disease studies. We used the same training set as in the scenarios above and used the EA Oral Squamous Cell Carcinoma sample as the test set. This test set also simulates imputation from the Affymetrix mapping 500k array to the Affymetrix SNP 6.0 array and consists of 320,172 SNPs.

For all scenarios, we chose the imputation program Beagle. Beagle is based on the Hidden Markov Model (HMM) 22 . To estimate missing alleles, an EM algorithm is adopted to optimize the parameters to fit the HMM model from a given genotyped reference panel 2 7 . In terms of imputation accuracy, Beagle perform as well as other imputation programs but is known to be more efficient with regard to running time and memory space required 23 .

The 1000 Genomes Project samples (August 2010 release) served as the reference panel. As the larger reference panel has developed, researchers have become more confident to combine two studies or extend a specific study on different platforms 23 . We removed those SNPs with MAF less than 1% that usually lead to decreased imputation accuracy 9 23 . About 2% of SNPs were removed before the imputation. Notably, there are a few SNPs with inconsistent genotyped markers compared to the reference panel. These few SNPs (< 0.01%) will be excluded from the training or test set in order to focus only on the reasonable imputation results.