User:Timothee Flutre/Notebook/Postdoc/2011/06/28
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Simple linear regression
[math]\displaystyle{ \forall n \in {1,\ldots,N}, \; y_n = \mu + \beta g_n + \epsilon_n \text{ with } \epsilon_n \sim N(0,\sigma^2) }[/math] In matrix notation: [math]\displaystyle{ y = X \theta + \epsilon }[/math] with [math]\displaystyle{ \epsilon \sim N_N(0,\sigma^2 I_N) }[/math] and [math]\displaystyle{ \theta^T = (\mu, \beta) }[/math]
Here is the ordinary-least-square (OLS) estimator of [math]\displaystyle{ \theta }[/math]: [math]\displaystyle{ \hat{\theta} = (X^T X)^{-1} X^T Y }[/math] [math]\displaystyle{ \begin{bmatrix} \hat{\mu} \\ \hat{\beta} \end{bmatrix} = \left( \begin{bmatrix} 1 & \ldots & 1 \\ g_1 & \ldots & g_N \end{bmatrix} \begin{bmatrix} 1 & g_1 \\ \vdots & \vdots \\ 1 & g_N \end{bmatrix} \right)^{-1} \begin{bmatrix} 1 & \ldots & 1 \\ g_1 & \ldots & g_N \end{bmatrix} \begin{bmatrix} y_1 \\ \vdots \\ y_N \end{bmatrix} }[/math] [math]\displaystyle{ \begin{bmatrix} \hat{\mu} \\ \hat{\beta} \end{bmatrix} = \begin{bmatrix} N & \sum_n g_n \\ \sum_n g_n & \sum_n g_n^2 \end{bmatrix}^{-1} \begin{bmatrix} \sum_n y_n \\ \sum_n g_n y_n \end{bmatrix} }[/math] [math]\displaystyle{ \begin{bmatrix} \hat{\mu} \\ \hat{\beta} \end{bmatrix} = \frac{1}{N \sum_n g_n^2 - (\sum_n g_n)^2} \begin{bmatrix} \sum_n g_n^2 & - \sum_n g_n \\ - \sum_n g_n & N \end{bmatrix} \begin{bmatrix} \sum_n y_n \\ \sum_n g_n y_n \end{bmatrix} }[/math] [math]\displaystyle{ \begin{bmatrix} \hat{\mu} \\ \hat{\beta} \end{bmatrix} = \frac{1}{N \sum_n g_n^2 - (\sum_n g_n)^2} \begin{bmatrix} \sum_n g_n^2 \sum_n y_n - \sum_n g_n \sum_n g_n y_n \\ - \sum_n g_n \sum_n y_n + N \sum_n g_n y_n \end{bmatrix} }[/math] Let's now define 4 summary statistics, very easy to compute: [math]\displaystyle{ \bar{y} = \frac{1}{N} \sum_{n=1}^N y_n }[/math] [math]\displaystyle{ \bar{g} = \frac{1}{N} \sum_{n=1}^N g_n }[/math] [math]\displaystyle{ g^T g = \sum_{n=1}^N g_n^2 }[/math] [math]\displaystyle{ g^T y = \sum_{n=1}^N g_n y_n }[/math] This allows to obtain the estimate of the effect size only by having the summary statistics available: [math]\displaystyle{ \hat{\beta} = \frac{g^T y - N \bar{g} \bar{y}}{g^T g - N \bar{g}^2} }[/math] The same works for the estimate of the standard deviation of the errors: [math]\displaystyle{ \hat{\sigma}^2 = \frac{1}{N-r}(y - X\hat{\theta})^T(y - X\hat{\theta}) }[/math] We can also benefit from this for the standard error of the parameters: [math]\displaystyle{ V(\hat{\theta}) = \hat{\sigma}^2 (X^T X)^{-1} }[/math] [math]\displaystyle{ V(\hat{\theta}) = \hat{\sigma}^2 \frac{1}{N g^T g - N^2 \bar{g}^2} \begin{bmatrix} g^Tg & -N\bar{g} \\ -N\bar{g} & N \end{bmatrix} }[/math] [math]\displaystyle{ V(\hat{\beta}) = \frac{\hat{\sigma}^2}{g^Tg - N\bar{g}^2} }[/math]
[math]\displaystyle{ PVE = \frac{V(\beta g)}{V(y)} = \frac{V(\beta g)}{V(\beta g) + V(\epsilon)} }[/math] with [math]\displaystyle{ V(\beta g) = \frac{1}{N}\sum_{n=1}^N (\beta g_n - \bar{\beta g})^2 }[/math] and [math]\displaystyle{ V(\epsilon) = \sigma^2 }[/math] This way, by also fixing [math]\displaystyle{ \beta }[/math], it is easy to calculate the corresponding [math]\displaystyle{ \sigma }[/math]: [math]\displaystyle{ \sigma = \sqrt{\frac{1}{N}\sum_{n=1}^N (\beta g_n - \bar{\beta g})^2 \frac{1 - PVE}{PVE}} }[/math] Here is some R code implementing this: set.seed(1859) N <- 100 mu <- 5 g <- sample(x=0:2, size=N, replace=TRUE, prob=c(0.5, 0.3, 0.2)) # MAF=0.2 beta <- 0.5 pve <- 0.8 beta.g.bar <- mean(beta * g) sigma <- sqrt((1/N) * sum((beta * g - beta.g.bar)^2) * (1-pve) / pve) # 0.18 y <- mu + beta * g + rnorm(n=N, mean=0, sd=sigma) plot(x=0, type="n", xlim=range(g), ylim=range(y), xlab="genotypes (allele dose)", ylab="phenotypes", main="Simple linear regression") for(i in unique(g)) points(x=jitter(g[g == i]), y=y[g == i], col=i+1, pch=19) ols <- lm(y ~ g) summary(ols) # muhat=5.01, betahat=0.46, R2=0.779 abline(a=coefficients(ols)[1], b=coefficients(ols)[2])
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