Supplementary Material for: A Novel Kernel for Correcting Size Bias in the Logistic Kernel Machine Test with an Application to Rheumatoid Arthritis

<b><i>Objectives:</i></b> The logistic kernel machine test (LKMT) is a testing procedure tailored towards high-dimensional genetic data. Its use in pathway analyses of case-control genome-wide association studies results from its computational efficiency and flexibility in incorporating additional information via the kernel. The kernel can be any positive definite function; unfortunately, its form strongly influences the test's power and bias. Most authors have recommended the use of a simple linear kernel. We demonstrate via a simulation that the probability of rejecting the null hypothesis of no association just by chance increases with the number of SNPs or genes in the pathway when applying a simple linear kernel. <b><i>Methods:</i></b> We propose a novel kernel that includes an appropriate standardization in order to protect against any inflation of false positive results. Moreover, our novel kernel contains information on gene membership of SNPs in the pathway. <b><i>Results:</i></b> When applying the novel kernel to data from the North American Rheumatoid Arthritis Consortium, we find that even this basic genomic structure can improve the ability of the LKMT to identify meaningful associations. We also demonstrate that the standardization effectively eliminates problems of size bias. <b><i>Conclusion:</i></b> We recommend the use of our standardized kernel and urge caution when using non-adjusted kernels in the LKMT to conduct pathway analyses.

<b><i>研究目的：</i></b> 逻辑核机器检验（logistic kernel machine test，LKMT）是一种专为高维遗传数据设计的检验方法。因其计算效率优异，且可通过核函数灵活整合额外信息，故而被应用于病例对照全基因组关联研究的通路分析中。核函数可为任意正定函数，但其形式会显著影响检验的效能与偏倚。多数研究者建议使用简单的线性核函数。本研究通过模拟实验证实，当采用简单线性核函数时，仅因随机因素而拒绝“无关联”原假设的概率，会随通路中单核苷酸多态性（single nucleotide polymorphism，SNPs）或基因的数量增加而升高。<b><i>研究方法：</i></b> 本研究提出一种新型核函数，其内置了恰当的标准化流程，可有效防范假阳性结果膨胀问题；此外，该新型核函数还整合了通路内单核苷酸多态性的基因归属信息。<b><i>研究结果：</i></b> 将该新型核函数应用于北美类风湿关节炎联盟（North American Rheumatoid Arthritis Consortium）的数据集时，研究发现即便仅基于这一基础基因组结构，也可提升LKMT识别有意义关联的能力。同时本研究证实，该标准化流程可有效消除规模偏倚问题。<b><i>研究结论：</i></b> 本研究推荐使用本文提出的标准化核函数，并提醒研究者在使用LKMT进行通路分析时，若采用未校正的核函数需格外谨慎。