# Methods for Mixed Linear Model AnalysisΒΆ

## Overview

The mixed linear model analysis tools are powerful utilities to not only perform a regression analysis on genotypic data while correcting for cryptic relatedness and pedigree structure, but also provide an estimation of random effects related to genotypic data.

These tools include the following utilities:

- Genomic Best Linear Unbiased Predictors (GBLUP) [VanRaden2008], [Taylor2013]
(
**Genotype > Compute Genomic BLUP (GBLUP)**) - AI REML GBLUP and gene by environemnt interactions using algorithms from GCTA [Yang2011]
- Bivariate GREML method from GCTA [LeeSH2012]
- Bayes C and C (also called C-pi) [Habier2011], [Fernando2009a],
[Fernando2009b], [Sorensen2002] (
**Genotype > Bayesian Genomic Prediction**) - GWAS Mixed Linear Model Analysis (
**Genotype > Mixed Linear Model Analysis**and**Genotype > Mixed Linear Model Analysis with Interactions**) [Segura2012], [Vilhjalmsson2012], [Kang2010], [Kang2008]

Each of these tools creates and finds a solution to, or at least an approximate solution to, one or more sets of mixed linear model (MLM) equations.

Additionally, there is a supplemental utility that may be used with any mixed-model tool:

- Compute the GBLUP Genomic Relationship Matrix (
**Genotype > Quality Assurance > GBLUP Genomic Relationship Matrix**)

Please see *Mixed Linear Model Analysis*,
*Genomic Best Linear Unbiased Predictors Analysis*, and *Separately Computing the Genomic Relationship Matrix*
below for further details on these tools.

For cross validation of genomic prediction methods, K-Fold Cross Validation
is available. See *K-Fold Cross Validation*.

This chapter contains the following sections:

*Overview of Mixed Linear Models**Large Kinship Matrices or Large Numbers of Samples**Mixed Linear Model Analysis**Mixed Linear Model Analysis with Interactions**Genomic Best Linear Unbiased Predictors Analysis**Genomic Best Linear Unbiased Predictors Analysis Using Bins**Genetic Correlation of Two Traits using GBLUP**Bayes C and C-pi Genomic Prediction Analysis**K-Fold Cross Validation**Predict Phenotypes From Existing Results*

- Overview of Mixed Linear Models
- The Mixed Model Equation
- Finding the Variance Components
- Finding the Variance Components Using EMMA
- Finding the Variance Components Using the Average Information (AI) Technique
- Estimating the Variance of Heritability for One Random Effect
- Solving the Mixed Model Equation Using EMMA
- Using the Mixed Model for Association Studies
- The Mixed Model Equation for Gene-by-Environment Interactions
- The Multi-Locus Mixed Model (MLMM)
- Gender Correction for EMMAX and MLMM
- Genomic Best Linear Unbiased Predictors (GBLUP)
- Correcting GBLUP for Gender
- Genomic Prediction
- Bayes C and C-pi

- Large Kinship Matrices or Large Numbers of Samples
- Mixed Linear Model Analysis
- Mixed Linear Model Analysis with Interactions
- Genomic Best Linear Unbiased Predictors Analysis
- Genomic Best Linear Unbiased Predictors Analysis Using Bins
- Genetic Correlation of Two Traits using GBLUP
- Bayes C and C-pi Genomic Prediction Analysis
- K-Fold Cross Validation
- Predict Phenotypes From Existing Results