9 Multiple Trait Selection and Selection Index

Learning Objectives

By the end of this chapter, you will be able to:

Distinguish between breeding goal (H) and selection index (I)
Explain why economic weights are necessary for multi-trait selection
Describe how genetic correlations influence selection index weights
Calculate a simple selection index
Interpret index values and rank animals for selection

9.1 Introduction

[Content to be developed: Most breeding programs have multiple objectives. Selection index theory provides the optimal method for combining information to maximize genetic merit.]

9.2 The Need for Multi-Trait Selection

[Content to be developed: Why we can’t select for a single trait in isolation.]

9.2.1 Multiple Objectives

[Content to be developed: Breeding goals typically include production, efficiency, health, reproduction, and quality traits.]

9.2.2 Genetic Correlations

[Content to be developed: Traits are genetically correlated (Chapter 8), so selecting for one affects others.]

9.2.3 Economic Trade-offs

[Content to be developed: Traits have different economic values. A 1% improvement in trait A may be worth more or less than 1% in trait B.]

9.3 Breeding Goal (H) vs. Selection Index (I)

[Content to be developed: Key distinction.]

9.3.1 Breeding Goal (H)

[Content to be developed:]

The traits we want to improve, weighted by their economic importance
Aggregate genotype
Not directly observable

\[ H = v_1 BV_1 + v_2 BV_2 + \ldots + v_n BV_n \]

Where v_i are economic weights.

9.3.2 Selection Index (I)

[Content to be developed:]

The information we use to predict H and rank animals
Linear combination of phenotypes or EBVs
Observable

\[ I = b_1 X_1 + b_2 X_2 + \ldots + b_m X_m \]

Where b_i are index weights, X_i are information sources (phenotypes, EBVs).

9.3.3 Why H and I May Differ

[Content to be developed:]

Some traits in H are hard or expensive to measure (not in I)
Correlated traits can provide indirect information (in I but not H)
Example: Select on backfat ultrasound (in I) to improve carcass leanness (in H)

9.4 Economic Weights

[Content to be developed: How much is each trait worth economically?]

9.4.1 Definition

[Content to be developed: Economic value (v) is the change in profit per unit change in the trait, holding all other traits constant.]

Units: $/unit (e.g., $/kg, $/egg, $/day)

9.4.2 Deriving Economic Weights

[Content to be developed:]

Define profit function (revenue - costs)
Take partial derivatives with respect to each trait
Evaluate at current population means

9.4.3 Example: Dairy Cattle

[Content to be developed:]

Traits and economic values (simplified):

Milk yield: +$0.30/kg
Fat yield: +$4.00/kg
Protein yield: +$6.00/kg
Fertility (days open): -$3.00/day
Longevity (lactations): +$200/lactation

9.4.4 Relative vs. Absolute Economic Weights

[Content to be developed: Relative weights (ratios) are sufficient for ranking animals. Absolute weights needed for economic simulations.]

9.5 Selection Index Theory

[Content to be developed: How to calculate optimal index weights.]

9.5.1 Objective

[Content to be developed: Choose index weights (b) to maximize the correlation between I and H.]

\[ \max \ \text{cor}(I, H) \]

9.5.2 Index Equation

[Content to be developed:]

\[ \mathbf{b} = \mathbf{P}^{-1} \mathbf{G} \mathbf{v} \]

Where:

b = vector of index weights
P = phenotypic (co)variance matrix
G = genetic (co)variance matrix (additive)
v = vector of economic weights

9.5.3 Interpretation of Index Weights

[Content to be developed:]

Index weights (b) differ from economic weights (v) because:
1. They account for heritabilities (traits with higher h² get more weight)
2. They account for genetic correlations (to avoid double-counting)
3. They reflect measurement error
Sign of b can differ from sign of v when traits are genetically correlated

9.6 Calculating a Simple Selection Index

[Content to be developed: Two-trait example.]

9.6.1 Example: Swine Growth and Backfat

[Content to be developed:]

Breeding goal traits:

Average daily gain (ADG): v_ADG = $2.00/kg per day
Backfat thickness (BF): v_BF = -$10.00/mm (negative because thinner is better)

Genetic parameters:

h²_ADG = 0.30, h²_BF = 0.50
r_A(ADG, BF) = 0.30 (positive, unfavorable)
σ_P(ADG) = 0.10 kg/day, σ_P(BF) = 3 mm

Calculate index weights:

[Content to be developed: Step-by-step calculation using the index equation]

9.6.2 Ranking Animals

[Content to be developed: Calculate index value for each animal and rank. Animals with highest I are selected.]

9.7 Examples of Selection Indices in Practice

[Content to be developed: Real-world indices used in livestock breeding.]

9.7.1 Dairy: Net Merit Index

[Content to be developed:]

Includes milk, fat, protein, fertility, health, longevity, calving ease
Published by CDCB (Council on Dairy Cattle Breeding)
Widely used in US dairy industry

9.7.2 Beef: Terminal Index vs. Maternal Index

[Content to be developed:]

Terminal index: Emphasizes growth, feed efficiency, carcass quality
Maternal index: Emphasizes reproduction, maternal ability, longevity

9.7.3 Swine: Sow Productivity Index vs. Growth Index

[Content to be developed:]

Sow index: Litter size, piglet survival, sow longevity
Growth/carcass index: ADG, feed efficiency, backfat, loin depth

9.7.4 Layers: Egg Production Index

[Content to be developed:]

Egg production, egg weight, shell quality, feed efficiency, mortality

9.8 R Demonstration: Calculating Selection Index

[Content to be developed:]

# Two-trait selection index example

# Economic weights
v <- c(2.0, -10.0)  # ADG ($/kg per day), BF ($/ mm, negative)

# Genetic parameters
h2 <- c(0.30, 0.50)
r_A <- 0.30
sigma_P <- c(0.10, 3.0)

# Calculate genetic and phenotypic (co)variances
sigma2_A <- h2 * sigma_P^2
sigma_A <- sqrt(sigma2_A)
cov_A <- r_A * sigma_A[1] * sigma_A[2]
sigma2_P <- sigma_P^2

# Genetic (co)variance matrix G
G <- matrix(c(sigma2_A[1], cov_A,
              cov_A, sigma2_A[2]), nrow = 2)

# Phenotypic (co)variance matrix P (assuming r_P = r_A for simplicity)
cov_P <- r_A * sigma_P[1] * sigma_P[2]
P <- matrix(c(sigma2_P[1], cov_P,
              cov_P, sigma2_P[2]), nrow = 2)

# Calculate index weights
b <- solve(P) %*% G %*% v
cat("Index weights:\n")
cat("  ADG:", round(b[1], 3), "\n")
cat("  Backfat:", round(b[2], 3), "\n")

# Rank animals
# Example animals (phenotypes)
animals <- data.frame(
  ID = 1:5,
  ADG = c(0.85, 0.90, 0.88, 0.82, 0.92),
  BF = c(12, 10, 11, 14, 9)
)

# Calculate index values
animals$Index <- b[1] * animals$ADG + b[2] * animals$BF

# Rank
animals <- animals %>% arrange(desc(Index))
print(animals)

9.9 Restrictions on Index Weights

[Content to be developed: Sometimes we want to constrain selection.]

9.9.1 Zero Economic Weight (Maintain Current Level)

[Content to be developed: Set economic weight to zero for traits we don’t want to change.]

9.9.2 Desired Gains Index

[Content to be developed: Specify desired genetic change for each trait, and index weights are calculated to achieve those changes.]

9.10 Summary

[Content to be developed.]

9.10.1 Key Points

Multi-trait selection requires balancing improvements across traits
Breeding goal (H): traits we want to improve, weighted by economic value
Selection index (I): information we use to predict H
Economic weights (v) reflect the value of each trait
Index weights (b) are calculated optimally: b = P⁻¹ G v
Selection indices are widely used in commercial breeding programs

9.11 Practice Problems

[Problems to be developed]

Explain the difference between the breeding goal (H) and the selection index (I). Why might they include different traits?
Given economic weights v_1 = $5 and v_2 = $3, explain why the index weights b_1 and b_2 are not necessarily in a 5:3 ratio.
A dairy breeding program selects only for milk yield, ignoring fertility and health. After 10 years, profitability has not improved as expected. Explain why and propose a solution.

9.12 Further Reading

[References to be added]

Hazel (1943): The genetic basis for constructing selection indexes
Modern selection indices: Net Merit (dairy), $W (beef)

# Multiple Trait Selection and Selection Index {#sec-selection-index} <div class="learning-objectives"> ### Learning Objectives {.unnumbered} By the end of this chapter, you will be able to: 1. Distinguish between breeding goal (H) and selection index (I) 2. Explain why economic weights are necessary for multi-trait selection 3. Describe how genetic correlations influence selection index weights 4. Calculate a simple selection index 5. Interpret index values and rank animals for selection </div> ```{r} #| echo: false #| message: false #| warning: false # Load required packages library(tidyverse) library(knitr) # Set default theme for plots theme_set(theme_minimal(base_size = 12)) ``` ## Introduction [Content to be developed: Most breeding programs have multiple objectives. Selection index theory provides the optimal method for combining information to maximize genetic merit.] ## The Need for Multi-Trait Selection [Content to be developed: Why we can't select for a single trait in isolation.] ### Multiple Objectives [Content to be developed: Breeding goals typically include production, efficiency, health, reproduction, and quality traits.] ### Genetic Correlations [Content to be developed: Traits are genetically correlated (Chapter 8), so selecting for one affects others.] ### Economic Trade-offs [Content to be developed: Traits have different economic values. A 1% improvement in trait A may be worth more or less than 1% in trait B.] ## Breeding Goal (H) vs. Selection Index (I) [Content to be developed: Key distinction.] ### Breeding Goal (H) [Content to be developed:] - The **traits we want to improve**, weighted by their economic importance - Aggregate genotype - Not directly observable $$ H = v_1 BV_1 + v_2 BV_2 + \ldots + v_n BV_n $$ Where v_i are economic weights. ### Selection Index (I) [Content to be developed:] - The **information we use** to predict H and rank animals - Linear combination of phenotypes or EBVs - Observable $$ I = b_1 X_1 + b_2 X_2 + \ldots + b_m X_m $$ Where b_i are index weights, X_i are information sources (phenotypes, EBVs). ### Why H and I May Differ [Content to be developed:] - Some traits in H are hard or expensive to measure (not in I) - Correlated traits can provide indirect information (in I but not H) - Example: Select on backfat ultrasound (in I) to improve carcass leanness (in H) ## Economic Weights [Content to be developed: How much is each trait worth economically?] ### Definition [Content to be developed: Economic value (v) is the change in profit per unit change in the trait, holding all other traits constant.] **Units**: $/unit (e.g., $/kg, $/egg, $/day) ### Deriving Economic Weights [Content to be developed:] 1. Define profit function (revenue - costs) 2. Take partial derivatives with respect to each trait 3. Evaluate at current population means ### Example: Dairy Cattle [Content to be developed:] **Traits and economic values** (simplified): - Milk yield: +$0.30/kg - Fat yield: +$4.00/kg - Protein yield: +$6.00/kg - Fertility (days open): -$3.00/day - Longevity (lactations): +$200/lactation ### Relative vs. Absolute Economic Weights [Content to be developed: Relative weights (ratios) are sufficient for ranking animals. Absolute weights needed for economic simulations.] ## Selection Index Theory [Content to be developed: How to calculate optimal index weights.] ### Objective [Content to be developed: Choose index weights (b) to maximize the correlation between I and H.] $$ \max \ \text{cor}(I, H) $$ ### Index Equation [Content to be developed:] $$ \mathbf{b} = \mathbf{P}^{-1} \mathbf{G} \mathbf{v} $$ Where: - **b** = vector of index weights - **P** = phenotypic (co)variance matrix - **G** = genetic (co)variance matrix (additive) - **v** = vector of economic weights ### Interpretation of Index Weights [Content to be developed:] - Index weights (b) differ from economic weights (v) because: 1. They account for heritabilities (traits with higher h² get more weight) 2. They account for genetic correlations (to avoid double-counting) 3. They reflect measurement error - **Sign of b can differ from sign of v** when traits are genetically correlated ## Calculating a Simple Selection Index [Content to be developed: Two-trait example.] ### Example: Swine Growth and Backfat [Content to be developed:] **Breeding goal traits**: - Average daily gain (ADG): v_ADG = $2.00/kg per day - Backfat thickness (BF): v_BF = -$10.00/mm (negative because thinner is better) **Genetic parameters**: - h²_ADG = 0.30, h²_BF = 0.50 - r_A(ADG, BF) = 0.30 (positive, unfavorable) - σ_P(ADG) = 0.10 kg/day, σ_P(BF) = 3 mm **Calculate index weights**: [Content to be developed: Step-by-step calculation using the index equation] ### Ranking Animals [Content to be developed: Calculate index value for each animal and rank. Animals with highest I are selected.] ## Examples of Selection Indices in Practice [Content to be developed: Real-world indices used in livestock breeding.] ### Dairy: Net Merit Index [Content to be developed:] - Includes milk, fat, protein, fertility, health, longevity, calving ease - Published by CDCB (Council on Dairy Cattle Breeding) - Widely used in US dairy industry ### Beef: Terminal Index vs. Maternal Index [Content to be developed:] - **Terminal index**: Emphasizes growth, feed efficiency, carcass quality - **Maternal index**: Emphasizes reproduction, maternal ability, longevity ### Swine: Sow Productivity Index vs. Growth Index [Content to be developed:] - **Sow index**: Litter size, piglet survival, sow longevity - **Growth/carcass index**: ADG, feed efficiency, backfat, loin depth ### Layers: Egg Production Index [Content to be developed:] - Egg production, egg weight, shell quality, feed efficiency, mortality ## R Demonstration: Calculating Selection Index [Content to be developed:] ```{r} #| echo: true #| eval: false # Two-trait selection index example # Economic weights v <- c(2.0, -10.0) # ADG ($/kg per day), BF ($/ mm, negative) # Genetic parameters h2 <- c(0.30, 0.50) r_A <- 0.30 sigma_P <- c(0.10, 3.0) # Calculate genetic and phenotypic (co)variances sigma2_A <- h2 * sigma_P^2 sigma_A <- sqrt(sigma2_A) cov_A <- r_A * sigma_A[1] * sigma_A[2] sigma2_P <- sigma_P^2 # Genetic (co)variance matrix G G <- matrix(c(sigma2_A[1], cov_A, cov_A, sigma2_A[2]), nrow = 2) # Phenotypic (co)variance matrix P (assuming r_P = r_A for simplicity) cov_P <- r_A * sigma_P[1] * sigma_P[2] P <- matrix(c(sigma2_P[1], cov_P, cov_P, sigma2_P[2]), nrow = 2) # Calculate index weights b <- solve(P) %*% G %*% v cat("Index weights:\n") cat(" ADG:", round(b[1], 3), "\n") cat(" Backfat:", round(b[2], 3), "\n") # Rank animals # Example animals (phenotypes) animals <- data.frame( ID = 1:5, ADG = c(0.85, 0.90, 0.88, 0.82, 0.92), BF = c(12, 10, 11, 14, 9) ) # Calculate index values animals$Index <- b[1] * animals$ADG + b[2] * animals$BF # Rank animals <- animals %>% arrange(desc(Index)) print(animals) ``` ## Restrictions on Index Weights [Content to be developed: Sometimes we want to constrain selection.] ### Zero Economic Weight (Maintain Current Level) [Content to be developed: Set economic weight to zero for traits we don't want to change.] ### Desired Gains Index [Content to be developed: Specify desired genetic change for each trait, and index weights are calculated to achieve those changes.] ## Summary [Content to be developed.] ### Key Points - Multi-trait selection requires balancing improvements across traits - Breeding goal (H): traits we want to improve, weighted by economic value - Selection index (I): information we use to predict H - Economic weights (v) reflect the value of each trait - Index weights (b) are calculated optimally: b = P⁻¹ G v - Selection indices are widely used in commercial breeding programs ## Practice Problems [Problems to be developed] 1. Explain the difference between the breeding goal (H) and the selection index (I). Why might they include different traits? 2. Given economic weights v_1 = $5 and v_2 = $3, explain why the index weights b_1 and b_2 are not necessarily in a 5:3 ratio. 3. A dairy breeding program selects only for milk yield, ignoring fertility and health. After 10 years, profitability has not improved as expected. Explain why and propose a solution. ## Further Reading [References to be added] - Hazel (1943): The genetic basis for constructing selection indexes - Modern selection indices: Net Merit (dairy), $W (beef)