Power analysis for GWAS

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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.

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

!!! info "Power"

!!! tip "Factors affecting power"

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

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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:

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

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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.

$$ Power = Pr(X > C ) = 1 - CDF_{\chi^2}(C, ncp = \lambda,df=1) $$

where $X \sim \chi^2(df=1, ncp=\lambda)$ is the test statistic under the alternative hypothesis.

• $CDF_{\chi^2}(x, ncp= \lambda)$ : is the cumulative distribution function for non-central $\chi^2$ distribution with non-centrality parameter $\lambda$.

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

Under the alternative hypothesis, the test statistic $Z$ follows a normal distribution with mean $\mu = \frac{P_{case} - P_{control}}{\sqrt{ \frac{P_{case}(1 - P_{case})}{2N_{case}} + \frac{P_{control}(1 - P_{control})}{2N_{control}} }}$ and variance 1.

$$ Power = Pr(|Z|>C) = 1 - \Phi(C-\mu) + \Phi(-C-\mu)$$

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

![image](https://user-images.githubusercontent.com/40289485/218300614-cc36e850-e5ee-4ec8-aa41-b75d5002518a.png)

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References

• 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.