writing out the MMEs

Author: filippob

bibliography/biostatistics.bib

@@ -57,13 +57,9 @@ @article{meuwissen2001prediction
   year={2001},
   publisher={Genetics Soc America}
 }
-@article{wolf2015genome,
-  title={Genome-wide association studies in dogs and humans identify ADAMTS20 as a risk variant for cleft lip and palate},
-  author={Wolf, Zena T and Brand, Harrison A and Shaffer, John R and Leslie, Elizabeth J and Arzi, Boaz and Willet, Cali E and Cox, Timothy C and McHenry, Toby and Narayan, Nicole and Feingold, Eleanor and others},
-  journal={PLoS genetics},
-  volume={11},
-  number={3},
-  pages={e1005059},
-  year={2015},
-  publisher={Public Library of Science}
+@book{mrode2014linear,
+  title={Linear models for the prediction of animal breeding values},
+  author={Mrode, Raphael A},
+  year={2014},
+  publisher={Cabi}
 }

r_notebooks/mixed_models.Rmd

@@ -73,6 +73,29 @@ where:
 
 Estimates for the parameters in the mixed model above are obtained by solving the **mixed model equations** (MME, @henderson1950estimaton). MME are set up by expanding the equation $\left(\mathbf{X'+Z'}\right) \cdot \left(\mathbf{Xb}+ \mathbf{Zu} \right) = \left( \mathbf{X'+Z'} \right) \cdot \mathbf{y}$, and then adding to the random term **u** the corresponding covariance matrix:
 
+$$
+\left[
+\begin{array}{cc}
+\mathbf{X'X} & \mathbf{X'Z} \\
+\mathbf{Z'X} & \mathbf{Z'Z} + \mathbf{K^{-1}\cdot \frac{\sigma_e^2}{\sigma_u^2}}
+\end{array}
+\right] \cdot
+\left[ 
+\begin{array}{c}
+\mathbf{b}\\
+\mathbf{u}
+\end{array}
+\right]
+= \left[
+\begin{array}{c}
+\mathbf{X'y}\\
+\mathbf{Z'y}
+\end{array}
+\right]
+$$
+
+**MMEs** are a very powerful statistical/mathematical tool, that can be extended in a very flexible way to a very large number of different (and increasingly) models (see for instance @mrode2014linear)
+
 ### Mixed models: an illustration in R
 
 We take data on the yield of 41 potato lines (`DT_example` dataset) from different environments (combination of location and year).

r_notebooks/mixed_models.md

@@ -53,6 +53,29 @@ where:
 
 Estimates for the parameters in the mixed model above are obtained by solving the **mixed model equations** (MME, @henderson1950estimaton). MME are set up by expanding the equation $\left(\mathbf{X'+Z'}\right) \cdot \left(\mathbf{Xb}+ \mathbf{Zu} \right) = \left( \mathbf{X'+Z'} \right) \cdot \mathbf{y}$, and then adding to the random term **u** the corresponding covariance matrix:
 
+$$
+\left[
+\begin{array}{cc}
+\mathbf{X'X} & \mathbf{X'Z} \\
+\mathbf{Z'X} & \mathbf{Z'Z} + \mathbf{K^{-1}\cdot \frac{\sigma_e^2}{\sigma_u^2}}
+\end{array}
+\right] \cdot
+\left[ 
+\begin{array}{c}
+\mathbf{b}\\
+\mathbf{u}
+\end{array}
+\right]
+= \left[
+\begin{array}{c}
+\mathbf{X'y}\\
+\mathbf{Z'y}
+\end{array}
+\right]
+$$
+
+**MMEs** are a very powerful statistical/mathematical tool, that can be extended in a very flexible way to a very large number of different (and increasingly) models (see for instance @mrode2014linear)
+
 ### Mixed models: an illustration in R
 
 We take data on the yield of 41 potato lines (`DT_example` dataset) from different environments (combination of location and year).