2 - Mixed models

This is an introduction to some basic properties of mixed linear models

The transition from linear models to mixed models is for solving real-world problems

Mixed models were developed at Cornell University, why here?

First expanding the linear model (y = Wb + e), W → [X Z]

modeling y (for E(y) and var(y))

is a combination of Selective Indexing (SI) and Generalized Linearization (GLS)

** y = Xβ + Zu + e **

where: y is the observation vector (known); β is the fixed effects (unknown); u is the random effects (unknown); e is the residuals (unknown); X and Z are the matrices associating y with β and u (known)

Based on the modeling of y, the derived objectives are: β, u,

** y = Xβ + Zu + e **

such as X = [1 0 24

0 1 34

1 0 23

1 0 27]

Represented as: 4 observations in 2 cow pastures (first 2 columns), with age at calving in column 3;

Mixed modeling has what can be called a system of Mixed model equations (MME)

Or abbreviated Cs = r

Need to know the (co)variances of each unknown variable:

Estimation of (co)variances (R and G), some of the mainstream algorithms:

1. REML (DF-, EM-, AI-...) is based on Maximum Likelihood. All based on Maximum Likelihood)

2. MCMC ("Gibbs Sampling")

3. Others as Method R (based on BLUP properties) (I haven't used it personally)

variance components estimation(VCF) methods on their own => special class

The mainstream algorithms for the above 3 will be described in more detail later

Back to the topic, let's move on to the MME: y = Xβ + Zu + e,

if abbreviated as: y = Wb +e (similar to OLS)

Then both of the above equations are solved:

Note that here W = [ X Z]

There are two major shortcomings in this estimator of u:

How do you compare u OLS and u SI ?

It is necessary to compare (Z'Z) -1 Z' with Cov(u,y)(Var(y)) -1

because:

Cov(u,y) = GZ' = AZ'(σ g ) 2 ;

Var(y) = V = ZGZ' + R = ZAZ'(σ g ) 2 + I (σ e ) 2

So bring it to the following Equation:

Then:

The u of SI can be converted to:

Two unrelated bulls S and T, both with three daughters, and their six daughters' dams are also unrelated

We want to calculate the genetic contribution of these two bulls to the phenotypes of their respective sons and daughters

Two methods of calculation are used:

The data are as follows:

Given the following definition:

Then the variance structure of the phenotypic value y:

The covariance structure of y:

The variance-covariance matrix of y:

According to the previous formula: Var(y) = ZZ'(σ g ) 2 + I (σ e ) 2

The variance-covariance matrix of Cov(u,y):

The variance-covariance matrix of Cov(u,y):

The variance-covariance matrix of Cov(u,y):

The variance-covariance matrix of Cov(u,y) is given by the SI. Solve using SI:

Choose the weights of the indices:

According to

SI: Minimize the error variance of the prediction Var(T-I) , and also maximize the correlation between T and I

OLS: Minimize the error variance, which is ultimately weighted by the variance of the observations (the residuals)

GLS: Minimize the error variance of the weights (least squares) MME: mixed models, simultaneously minimizing error variance and random effectde prediction error variance

BLUP evolved from SI