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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1 Answer 1


This question has been posted here for a long time now and has not been answered even with a 50-point bounty. After a second thought, it looks like the second equation is using the adjusted version of $r^2$: $$ r^2 = 1 - \frac{SSE/df_e}{SST/df_t} $$

In this case, $\frac{SST}{df_t} = \frac{\sum(y_i - \overline{y})^2}{n-1} = \frac{(n-1)s^2(y)}{n-1} = 1$, since everything is standardized, and $\frac{SSE}{df_e} = \frac{SSE}{n-m}$, because there is no constant term in the model, finally we have: $$ r^2 = 1 - \frac{SSE}{n-m} $$

I hope I have not made a mistake.


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