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I am reading a paper on "multi-SNP" GWAS data analysis method, The author fitted a linear model $y = \beta_g g + \epsilon$, where $g$ is a genotype vector for a single gene, $\beta_g$ is the true regression coefficient, $\epsilon$ is the error, assuming the regression line goes through the origin. Let $\hat{y}$, $b_g$, $s^2(g)$ and $s^2(y)$ be the estimate of $y$, $\beta_g$, $\sigma^2(g)$ and $\sigma^2(y)$ respectively, since the intercept $b_{g0} = \overline{y} - b_g \overline{x} = 0$, we have $\overline{y} = b_g \overline{x}$, it follows that

\begin{align*} SSR &= \sum (\hat{y} - \overline{y})^2 \\ &= \sum (b_g g_i - \overline{y})^2 \\ &= \sum(b_g g_i - b_g \overline{g})^2 \\ &= b_g^2 \sum (g_i - \overline{g})^2 \\ &= b_g^2 s^2(g)(n-1) \end{align*} \begin{align} r^2 &= \frac{SSR}{SSR + SSE} \notag\\ &= \frac{b_g^2 s^2(g)(n-1)}{b_g^2 s^2(g)(n-1) + s^2(y)(n-1)} \notag\\ &= \frac{b_g^2 s^2(g)}{b_g^2 s^2(g) + s^2(y)} \end{align}.

The equation above is quite similar to what the author offers in his first equation, but they essentially different unless the author has been confused about true and estimated quantities. The second equation about $\hat{r^2}_{locus}$ totally escapes my comprehension, any help will be appreciated.

The original article can be found here

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