1. Sequence Kernel Association Tests (SKAT) for the Combined Effect of Rare and Common Variants
統計論文
奈良原

2. The American journal of Human Genetics (2013)
SKAT
Wu, M.C., Lee, S., Cai, T., Li, Y., Boehnke, M., and Lin, X. (2011). Rare-variant association testing for sequencing data with the sequence kernel association test. Am. J. Hum. Genet. 89, 82–93.
Developed for rare-variant analysis

3. Background of rare variant analysis
Classic: burden tests
Collapsing method: rare variant +/- in a region
Counts of rare alleles
Combined multivariate and collapsing (CMC) method
rare variants: collapsed, common variants: each forms a separate group
--> Combined by Hotelling's T2 statistic
Weighted sum
Non-burden tests
C-alpha test
Sequence kernel association test (SKAT)
Problem of Burden tests
Burden tests assume that all rare variants influence the phenotype in the same direction with the same magnitude of effects (after weighting).
methods that are robust to different direction and magnitude of effects

4. Development of SKAT SKAT (2011) A kernel regression approach
non-parametric non-linear regression
flexible weighting function
weights based on minor allele frequency
based on SNP functional annotation
Wide range of application
binary/continuous traits
adjustment for covariates
both rare and common variants (up-weighting rare variants)
Efficient computation
Score test for variance-component in linear mixed model

5. Development of SKAT (2) SKAT-O: Optimal unified approach (AJHG, 2012)
Combination of a burden test and SKAT
Burden test ... optimal when most variants in a region are causal and the effects are in the same direction
SKAT ... optimal when a large fraction of the variants in a regions are non-causal or the effects of causal variants are in different directions
Extension of SKAT-O to testing a combined effect of rare and common variants (AJHG, 2013)

6. methods

7. Linear mixed model
Genetic effect: random effect

8. Variance component score test
Choice of a kernel function
weighted linear
weighted quadratic
weighted IBS
= kernel function
Genetic similarity between subjects
(weighted)
Choice of weights
Typical parameter:
a1=1, a2=25
P value given by the Davies method
Approximation of Q statistic

9. Optimal unified approach, SKAT-O
Next, they unified burden test and SKAT to optimize the rare variant analysis.
Burden test
More powerful than SKAT when most variants in a region are causal and the effects are in the same direction
SKAT
More powerful than burden test when a large fraction of the variants in a regions are non-causal or the effects of causal variants are in different directions

10. Weighted burden test statistic
SKAT statistic
aggregates the variants before regression
first regresses and aggregates the individual variant statistics

11. Unifying two test statistics
Optimal value of ρ is determined by grid search.
Qρ is equivalently calculated by the formula of score test statistic
ρ: correlation between different βj's
ρ=0: regression coefficients are not correlated to each other --> SKAT
ρ=1: regression coefficients are perfectly correlated --> Burden test

12. Rare and common variants together in the SKAT-O framework
Different weighting functions are defined for rare and common variants.
The effects of rare and common variants are fitted together using separate random effect terms.

13. Model

14. Statistic
Weighted sum of statistics of rare and common variants

15. Predefined parameters
Weights
Rare variants
Common variants
Contribution, φ
Equal contribution
or searching the optimal value of φ
Beta(1, 25)
Beta(0.5, 0.5)
MAF

16. Appendix

17. Kernel in statistics In Bayesian statistics
The kernel of a probability density function or probability mass function is the form of PDF or PMF in which any factors that are not functions of any of the variables in the domain are omitted (normalization factor).
Ex. kernel of a normal distribution
PDF:
Kernel:

18. Kernel in statistics (2)
In non-parametric statistics
A kernel is a weighting function
Usage
Kernel density estimation to estimate random variables' density functions
In kernel regression to estimate the conditional expectation of a random variable
In time-series to estimate the spectral density
Estimation of a time-varying intensity for a point process
Definition
A kernel is a non-negative real-valued integrable function K satisfying the following two requirements:
If K is a kernel, then so is the function K*.
K*(u) = λK(λu), where λ > 0
--> A kernel is a PDF.
--> A kernel is symmetric about u=0.

19. Kernel regression
The kernel regression is a non-parametric approach to find a non-linear relation between a pair of random variables X and Y.
The goal is to estimate a function m that gives conditional expectation of a variable Y relative to a variable X:
A kernel is used to estimate a function m.

20. Kernel trick
A kernel trick is a method to project data into a higher-dimensional space so that non-linear data can be separated by a hyperplane.
non-linear --> linear
Kernel function
K(x, z) = <Φ(x), Φ(z)>
Φ(・): a function to project data into higher-dimensional space
<・, ・> : inner product

21. Application of kernel method
Kernel PCA (non-linear PCA)
Kernel CCA
Support vector machine

22. Support vector machine
SVM is a machine learning approach that utilizes the kernel function to project data in a higher-dimensional space that can separate the data by a hyperplane.
SVM is a non-linear classifier.

23. Variance component score test
Lin, X. (1997). Variance component testing in generalised linear models with random effects. Biometrika84, 309–326.
Variance component tests in linear mixed model
Likelihood ratio test
Score statistic
Computationally efficient
Wald statistic