adding bibliography and MME

Author: filippob

bibliography/biostatistics.bib

@@ -0,0 +1,69 @@
+@article{aulchenko2007genabel,
+  title={GenABEL: an R library for genome-wide association analysis},
+  author={Aulchenko, Yurii S and Ripke, Stephan and Isaacs, Aaron and Van Duijn, Cornelia M},
+  journal={Bioinformatics},
+  volume={23},
+  number={10},
+  pages={1294--1296},
+  year={2007},
+  publisher={Oxford University Press}
+}
+@article{biscarini2016genome,
+  title={Genome-wide association study for traits related to plant and grain morphology, and root architecture in temperate rice accessions},
+  author={Biscarini, Filippo and Cozzi, Paolo and Casella, Laura and Riccardi, Paolo and Vattari, Alessandra and Orasen, Gabriele and Perrini, Rosaria and Tacconi, Gianni and Tondelli, Alessandro and Biselli, Chiara and others},
+  journal={PLoS One},
+  volume={11},
+  number={5},
+  pages={e0155425},
+  year={2016},
+  publisher={Public Library of Science}
+}
+@article{biffani2014adding,
+  title={Adding evidence for a role of the SLITRK gene family in the pathogenesis of left displacement of the abomasum in Holstein-Friesian dairy cows},
+  author={Biffani, Stefano and Morandi, Nicola and Locatelli, Valentina and Pravettoni, Davide and Boccardo, Antonio and Stella, Alessandra and Nicolazzi, Ezequiel Lu{\'\i}s and Biscarini, Filippo},
+  journal={Livestock Science},
+  volume={167},
+  pages={104--109},
+  year={2014},
+  publisher={Elsevier}
+}
+@article{henderson1950estimaton,
+  title={Estimation of genetic parameters},
+  author={Henderson, CR},
+  journal={Annals of Mathematical Statistics},
+  volume={21},
+  number={},
+  pages={309-310},
+  year={1950},
+  publisher={Institute of Mathematical Statistics}
+}
+@article{kijas2016genome,
+  title={Genome-wide association reveals the locus responsible for four-horned ruminant},
+  author={Kijas, James W and Hadfield, Tracy and Naval Sanchez, Marina and Cockett, Noelle},
+  journal={Animal genetics},
+  volume={47},
+  number={2},
+  pages={258--262},
+  year={2016},
+  publisher={Wiley Online Library}
+}
+@article{meuwissen2001prediction,
+  title={Prediction of total genetic value using genome-wide dense marker maps},
+  author={Meuwissen, THE and Hayes, BJ and Goddard},
+  journal={Genetics},
+  volume={157},
+  number={4},
+  pages={1819--1829},
+  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}
+}

r_notebooks/mixed_models.Rmd

@@ -7,6 +7,7 @@ output:
     keep_md: true
   github_document:
     pandoc_args: ['--lua-filter', 'math-github.lua']
+bibliography: ../bibliography/biostatistics.bib
 ---
 
 ```{r setup, include=FALSE}
@@ -48,7 +49,7 @@ This is a **linear additive mixed model**:
 - *additive* because variables are summed together
 - *mixed* because (again) it mixes fixed and random effects
 
-While **fixed effects** are just point estimates of an underlying *true* value, **random effects** are associated with a distribution and, importantly, a **variance component**. 
+While **fixed effects** are just point estimates of underlying *true* values, **random effects** are associated with a distribution and, importantly, a **variance component**. 
 The total variance of the observations (**y**'s) is partitioned into one component belonging to the random effect **u** and the residual variance:
 
 ```{r, echo=FALSE}
@@ -70,6 +71,7 @@ where:
 - **K**: *covariance* (relationship) *matrix* between the observations (for the specific random effect)
 - **R** is the residual covariance matrix, usually simplified to $\mathbf{I}$ (identity matrix) under the assumption of independence between residuals
 
+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:
 
 ### Mixed models: an illustration in R
 
@@ -127,4 +129,5 @@ fit <-mmer(fixed = Yield ~ Env,
 summary(fit)
 ```
 
+## References
 

r_notebooks/mixed_models.md

@@ -7,6 +7,7 @@ output:
     keep_md: true
   github_document:
     pandoc_args: ['--lua-filter', 'math-github.lua']
+bibliography: ../bibliography/biostatistics.bib
 ---
 
 
@@ -36,7 +37,7 @@ This is a **linear additive mixed model**:
 - *additive* because variables are summed together
 - *mixed* because (again) it mixes fixed and random effects
 
-While **fixed effects** are just point estimates of an underlying *true* value, **random effects** are associated with a distribution and, importantly, a **variance component**. 
+While **fixed effects** are just point estimates of underlying *true* values, **random effects** are associated with a distribution and, importantly, a **variance component**. 
 The total variance of the observations (**y**'s) is partitioned into one component belonging to the random effect **u** and the residual variance:
 
 
@@ -50,6 +51,7 @@ where:
 - **K**: *covariance* (relationship) *matrix* between the observations (for the specific random effect)
 - **R** is the residual covariance matrix, usually simplified to $\mathbf{I}$ (identity matrix) under the assumption of independence between residuals
 
+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:
 
 ### Mixed models: an illustration in R
 
@@ -153,4 +155,5 @@ summary(fit)
 ## Use the '$' sign to access results and parameters
 ```
 
+## References