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Power analysis for GWAS

Type I, type II errors and Statistical power

This table shows the relationship between the null hypothesis \(H_0\) and the results of a statistical test (whether or not to reject the null hypothesis \(H_0\) ).

H0 is True H0 is False
Do Not Reject True negative : \(1 - \alpha\) Type II error (false negative) : \(\beta\)
Reject Type I error (false positive) : \(\alpha\) True positive : \(1 - \beta\)

\(\alpha\) : significance level

By definition, the statistical power of a test refers to the probability that the test will correctly reject the null hypothesis, namely the True positive rate in the table above.

\(Power = Pr ( Reject\ | H_0\ is\ False) = 1 - \beta\)

Power

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Factors affecting power

  • Total sample size
  • Case and control ratio
  • Effect size of the variant
  • Risk allele frequency
  • Significance threshold

Non-centrality parameter

NCP describes the degree of difference between the alternative hypothesis \(H_1\) and the null hypothesis \(H_0\) values.

Consider a simple linear regression model:

\[y = \mu +\beta x + \epsilon\]

The variance of the error term:

\[\sigma^2 = Var(y) - Var(x)\beta^2\]

Usually, the phenotypic variance that a single SNP could explain is very limited, so we can approximate \(\sigma^2\) by:

\[ \sigma^2 \thickapprox Var(y)\]

Under Hardy-Weinberg equilibrium, we can get:

\[Var(x) = 2f(1-f)\]
  • \(f\) : the allele frequency for this variant

So the Non-centrality parameter(NCP) \(\lambda\) for \(\chi^2\) distribution with degree of freedom 1:

\[ \lambda = ({{\beta}\over{SE_{\beta}}})^2\]

Power for quantitative traits

\[ \lambda = ({{\beta}\over{SE_{\beta}}})^2 \thickapprox N \times {{Var(x)\beta^2}\over{\sigma^2}} \thickapprox N \times {{2f(1-f) \beta^2 }\over {Var(y)}} \]

Significance threshold: \(C = CDF_{\chi^2}^{-1}(1 - \alpha,df=1)\)

  • \(CDF_{\chi^2}^{-1}(x)\) : is the inverse of the cumulative distribution function for \(\chi^2\) distribution.
\[ Power = Pr(\lambda > C ) = 1 - CDF_{\chi^2}(C, ncp = \lambda,df=1) \]
  • \(CDF_{\chi^2}(x, ncp= \lambda)\) : is the cumulative distribution function for non-central \(\chi^2\) distribution with non-centrality parameter \(\lambda\).

Power for large-scale case-control genome-wide association studies

Denote :

  • \(P_{case}\) : Risk allele frequency in cases
  • \(N_{case}\) : Number of cases. The total allele count for cases is then \(2N_{case}\).
  • \(P_{control}\) : Risk allele frequency in controls
  • \(N_{control}\) : Number of control. The total allele count for control is then \(2N_{control}\).

Null hypothesis : \(P_{case} = P_{control}\)

To test whether one proportion \(P_{case}\) equals the other proportion \(P_{control}\), the test statistic is:

\[z = {{P_{case} - P_{control}}\over {\sqrt{ {{P_{case}(1 - P_{case})}\over{2N_{case}}} + {{P_{control}(1 - P_{control})}\over{2N_{control}}} }}}\]

Significance threshold: \(C = \Phi^{-1}(1 - \alpha / 2 )\)

\[ Power = Pr(|Z|>C) = 1 - \Phi(-C-z) + \Phi(C-z)\]

GAS power calculator

GAS power calculator implemented this method, and you can easily calculate the power using their website

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Reference:

  • Skol, A. D., Scott, L. J., Abecasis, G. R., & Boehnke, M. (2006). Joint analysis is more efficient than replication-based analysis for two-stage genome-wide association studies. Nature genetics, 38(2), 209-213.
  • Johnson, J. L., & Abecasis, G. R. (2017). GAS Power Calculator: web-based power calculator for genetic association studies. BioRxiv, 164343.
  • Sham, P. C., & Purcell, S. M. (2014). Statistical power and significance testing in large-scale genetic studies. Nature Reviews Genetics, 15(5), 335-346.