User:Timothee Flutre/Notebook/Postdoc/2011/06/28

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Simple linear regression

  • Data: Let's assume that we obtained data from N individuals. We note y_1,\ldots,y_N the (quantitative) phenotypes (eg. expression level at a given gene), and g_1,\ldots,g_N the genotypes at a given SNP. We want to assess their linear relationship.
  • Model: for this we use a simple linear regression (univariate phenotype, single predictor).

\forall n \in {1,\ldots,N}, \; y_n = \mu + \beta g_n + \epsilon_n \text{ with } \epsilon_n \sim N(0,\sigma^2)

In matrix notation:

y = Xθ + ε with ε˜NN(0,σ2IN) and θT = (μ,β)

Most importantly, we want the following estimates: \hat{\beta}, se(\hat{\beta}) (its standard error) and \hat{\sigma}. In the case where we don't have access to the original data (eg. because genotypes are confidential) but only to some summary statistics (see below), it is still possible to calculate the estimates.

Here is the ordinary-least-square (OLS) estimator of θ:

\hat{\theta} = (X^T X)^{-1} X^T Y

\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}

\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}

\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}

\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}

Let's now define 4 summary statistics, very easy to compute:

\bar{y} = \frac{1}{N} \sum_{n=1}^N y_n

\bar{g} = \frac{1}{N} \sum_{n=1}^N g_n

g^T g = \sum_{n=1}^N g_n^2

g^T y = \sum_{n=1}^N g_n y_n

This allows to obtain the estimate of the effect size only by having the summary statistics available:

\hat{\beta} = \frac{g^T y - N \bar{g} \bar{y}}{g^T g - N \bar{g}^2}

The same works for the estimate of the standard deviation of the errors:

\hat{\sigma}^2 = \frac{1}{N-r}(y - X\hat{\theta})^T(y - X\hat{\theta})

We can also benefit from this for the standard error of the parameters:

V(\hat{\theta}) = \hat{\sigma}^2 (X^T X)^{-1}

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}

V(\hat{\beta}) = \frac{\hat{\sigma}^2}{g^Tg - N\bar{g}^2}

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