User:Timothee Flutre/Notebook/Postdoc/2011/06/28: Difference between revisions
(Autocreate 2011/06/28 Entry for User:Timothee_Flutre/Notebook/Postdoc) |
(→Entry title: first version) |
||
| Line 6: | Line 6: | ||
| colspan="2"| | | colspan="2"| | ||
<!-- ##### DO NOT edit above this line unless you know what you are doing. ##### --> | <!-- ##### DO NOT edit above this line unless you know what you are doing. ##### --> | ||
== | ==Calculate OLS estimates with summary statistics for simple linear regression== | ||
We obtained data from <math>n</math> individuals. Let be <math>y_1,\ldots,y_n</math> the (quantitative) phenotypes (eg. expression level at a given gene), and <math>g_1,\ldots,g_n</math> the genotypes at a given SNP. | |||
We want to assess the linear relationship between phenotype and genotype. For this with use a simple linear regression: | |||
<math>y_i = \mu + \beta x_i + \epsilon_i</math> with <math>\epsilon_i \rightarrow N(0,\sigma^2)</math> and for <math>i \in {1,\ldots,n}</math> | |||
In vector-matrix notation: | |||
<math>y = X \theta + \epsilon</math> with <math>\epsilon \rightarrow N_n(0,\sigma^2 I)</math> and <math>\theta^T = (\mu, \beta)</math> | |||
Here is the ordinary-least-square (OLS) estimator of <math>\theta</math>: | |||
<math>\hat{\theta} = (X^T X)^{-1} X^T Y</math> | |||
<math>\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>\begin{bmatrix} \hat{\mu} \\ \hat{\beta} \end{bmatrix} = | |||
\begin{bmatrix} n & \sum_i g_i \\ \sum_i g_i & \sum_i g_i^2 \end{bmatrix}^{-1} | |||
\begin{bmatrix} \sum_i y_i \\ \sum_i g_i y_i \end{bmatrix} | |||
</math> | |||
<math>\begin{bmatrix} \hat{\mu} \\ \hat{\beta} \end{bmatrix} = | |||
\frac{1}{n \sum_i g_i^2 - (\sum_i g_i)^2} | |||
\begin{bmatrix} \sum_i g_i^2 & - \sum_i g_i \\ - \sum_i g_i & n \end{bmatrix} | |||
\begin{bmatrix} \sum_i y_i \\ \sum_i g_i y_i \end{bmatrix} | |||
</math> | |||
<math>\begin{bmatrix} \hat{\mu} \\ \hat{\beta} \end{bmatrix} = | |||
\frac{1}{n \sum_i g_i^2 - (\sum_i g_i)^2} | |||
\begin{bmatrix} \sum_i g_i^2 \sum_i y_i - \sum_i g_i \sum_i g_i y_i \\ - \sum_i g_i \sum_i y_i + n \sum_i g_i y_i \end{bmatrix} | |||
</math> | |||
Let's now define 4 summary statistics: | |||
<math>\bar{y} = \frac{1}{n} \sum_{i=1}^n y_i</math> | |||
<math>\bar{g} = \frac{1}{n} \sum_{i=1}^n g_i</math> | |||
<math>g^T g = \sum_{i=1}^n g_i^2</math> | |||
<math>g^T y = \sum_{i=1}^n g_i y_i</math> | |||
<math>\hat{\beta} = \frac{g^T y - n \bar{g} \bar{y}}{g^T g - n \bar{g}^2}</math> | |||
<!-- ##### DO NOT edit below this line unless you know what you are doing. ##### --> | <!-- ##### DO NOT edit below this line unless you know what you are doing. ##### --> | ||
Revision as of 14:27, 28 March 2012
 Project name
|
<html><img src="/images/9/94/Report.png" border="0" /></html> Main project page Next entry<html><img src="/images/5/5c/Resultset_next.png" border="0" /></html> |
Calculate OLS estimates with summary statistics for simple linear regressionWe obtained data from [math]\displaystyle{ n }[/math] individuals. Let be [math]\displaystyle{ y_1,\ldots,y_n }[/math] the (quantitative) phenotypes (eg. expression level at a given gene), and [math]\displaystyle{ g_1,\ldots,g_n }[/math] the genotypes at a given SNP. We want to assess the linear relationship between phenotype and genotype. For this with use a simple linear regression: [math]\displaystyle{ y_i = \mu + \beta x_i + \epsilon_i }[/math] with [math]\displaystyle{ \epsilon_i \rightarrow N(0,\sigma^2) }[/math] and for [math]\displaystyle{ i \in {1,\ldots,n} }[/math] In vector-matrix notation: [math]\displaystyle{ y = X \theta + \epsilon }[/math] with [math]\displaystyle{ \epsilon \rightarrow N_n(0,\sigma^2 I) }[/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_i g_i \\ \sum_i g_i & \sum_i g_i^2 \end{bmatrix}^{-1} \begin{bmatrix} \sum_i y_i \\ \sum_i g_i y_i \end{bmatrix} }[/math] [math]\displaystyle{ \begin{bmatrix} \hat{\mu} \\ \hat{\beta} \end{bmatrix} = \frac{1}{n \sum_i g_i^2 - (\sum_i g_i)^2} \begin{bmatrix} \sum_i g_i^2 & - \sum_i g_i \\ - \sum_i g_i & n \end{bmatrix} \begin{bmatrix} \sum_i y_i \\ \sum_i g_i y_i \end{bmatrix} }[/math] [math]\displaystyle{ \begin{bmatrix} \hat{\mu} \\ \hat{\beta} \end{bmatrix} = \frac{1}{n \sum_i g_i^2 - (\sum_i g_i)^2} \begin{bmatrix} \sum_i g_i^2 \sum_i y_i - \sum_i g_i \sum_i g_i y_i \\ - \sum_i g_i \sum_i y_i + n \sum_i g_i y_i \end{bmatrix} }[/math] Let's now define 4 summary statistics: [math]\displaystyle{ \bar{y} = \frac{1}{n} \sum_{i=1}^n y_i }[/math] [math]\displaystyle{ \bar{g} = \frac{1}{n} \sum_{i=1}^n g_i }[/math] [math]\displaystyle{ g^T g = \sum_{i=1}^n g_i^2 }[/math] [math]\displaystyle{ g^T y = \sum_{i=1}^n g_i y_i }[/math] [math]\displaystyle{ \hat{\beta} = \frac{g^T y - n \bar{g} \bar{y}}{g^T g - n \bar{g}^2} }[/math] | |
