User:Timothee Flutre/Notebook/Postdoc/2011/06/28
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Calculate OLS estimates with summary statistics for simple linear regressionWe obtained data from n individuals. Let be the (quantitative) phenotypes (eg. expression level at a given gene), and 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: y_{i} = μ + βx_{i} + ε_{i} with and for In vectormatrix notation: y = Xθ + ε with and θ^{T} = (μ,β) Here is the ordinaryleastsquare (OLS) estimator of θ:
Let's now define 4 summary statistics, very easy to compute:
This allows to obtain the estimate of the effect size only by having the summary statistics available:
The same works for the estimate of the standard deviation of the errors:
We can also benefit from this for the standard error of the parameters:
