title: Determination of sample size for a multi-class classifier based on single-nucleotide polymorphisms: a volume under the surface approach.
abstract: BACKGROUND
  Data on single-nucleotide polymorphisms (SNPs) have been found to be useful in predicting phenotypes ranging from an individual's class membership to his/her risk of developing a disease. In multi-class classification scenarios, clinical samples are often limited due to cost constraints, making it necessary to determine the sample size needed to build an accurate classifier based on SNPs. The performance of such classifiers can be assessed using the Area Under the Receiver Operating Characteristic (ROC) Curve (AUC) for two classes and the Volume Under the ROC hyper-Surface (VUS) for three or more classes. Sample size determination based on AUC or VUS would not only guarantee an overall correct classification rate, but also make studies more cost-effective.
  
  
  RESULTS
  For coded SNP data from D(≥2) classes, we derive an optimal Bayes classifier and a linear classifier, and obtain a normal approximation to the probability of correct classification for each classifier. These approximations are then used to evaluate the associated AUCs or VUSs, whose accuracies are validated using Monte Carlo simulations. We give a sample size determination method, which ensures that the difference between the two approximate AUCs (or VUSs) is below a pre-specified threshold. The performance of our sample size determination method is then illustrated via simulations. For the HapMap data with three and four populations, a linear classifier is built using 92 independent SNPs and the required total sample sizes are determined for a continuum of threshold values. In all, four different sample size determination studies are conducted with the HapMap data, covering cases involving well-separated populations to poorly-separated ones.
  
  
  CONCLUSIONS
  For multi-classes, we have developed a sample size determination methodology and illustrated its usefulness in obtaining a required sample size from the estimated learning curve. For classification scenarios, this methodology will help scientists determine whether a sample at hand is adequate or more samples are required to achieve a pre-specified accuracy. A PDF manual for R package "SampleSizeSNP" is given in Additional file 1, and a ZIP file of the R package "SampleSizeSNP" is given in Additional file 2.
authors.forename: Xinyu
authors.initials: X
authors.lastname: Liu
authors.forename: Yupeng
authors.initials: Y
authors.lastname: Wang
authors.affiliation: Department of Statistics, University of Georgia, Athens, GA 30602, USA. tn@uga.edu.
authors.forename: T N
authors.initials: TN
authors.lastname: Sriram
mesh_terms: D001499:Bayes Theorem; D006801:Humans; D009010:Monte Carlo Method; D020641:Polymorphism, Single Nucleotide; D012372:ROC Curve; D018401:Sample Size