planning rice breeding programs for impact models, means, variances, lsd’s and heritability

Planning rice breeding programs for impact

Models, means, variances, LSD’s and Heritability

IRRI: Planning breeding Programs for Impact

Learning objectives

1. Review the linear model for plot measurements in variety trials and nurseries, and the derived statistics

2. Understand the purpose of replication in breeding programs

3. Model the relationship between replication, the standard error of a cultivar mean (SEM), and the least significant difference (LSD) between the means of 2 cultivars


Introduction

• Measurements made on field plots contain both genotypic effects (G) and plot residuals (e)

• Purpose of experimental design and statistical analysis is to separate genotypic “signal” from “noise” of plot residuals.


2.0 2.3 1.9 2.2 0.8 1.1 2.0 2.5 2.6 2.3 2.8 3.2 3.83.5

2.5 2.6 2.3 2.4 1.0 0.6 1.8 3.1 3.2 2.9 3.3 3.5 4.13.9

2.7 2.8 2.4 2.6 2.7 1.3 0.5 3.1 3.4 3.5 3.3 3.7 4.44.0


Linear model for plot measurements

For a completely randomized design (CRD):

Where:

• Yij = a plot measurement

• μ = the mean of all plots

• Gi = the effect of the ith genotype

• ej = the “residual” effect of the jth plot

G’s and e’s sum to 0

Yij = μ + Gi + ej [4.1]


• As r increases, e approaches 0, Y approaches μ + Gi

Breeders replicate to reduce effect of e!

• But even if r is 3 or 4, e’s have big effect on estimates of G

Yi. = μ + Gi + e [4.2]

E(e) =0


Yi. = μ + Gi + ej [4.1]

Thus, for measurements on a single plot, G and e are confounded

• Because of the confounding, Y is an unreliable estimator of G

• In replicated trials, the mean of Y over several plots is a better estimator of G, because e’s tend to cancel each other out


Variance of a mean

The variance of a genotype mean is an important measure of the precision of a trial:

σ2Y = σ2

e/r [4.3]

σ2e is the error mean square from the ANOVA

Standard error of a mean (SEM)

SEM = σ2Y

Variances, standard errors and LSD’s


Variance of a difference between 2 means

σ2D = 2σ2

e/r [4.4]

Standard error of a difference (SED)

SED = √(2σ2e/r ) [4.5]


Least significant difference (LSD)

LSD = tα/2,edf x SED

= tα/2,edf x √(2 σ2e /r) [4.6]

tα/2,edf roughly equals 2, so LSD = 3 SEM

SEM, SED, and LSD are important measures of the precision of a trial

Precision is determined mainly by replication


Repeatability

• H integrates information on genetic variation and environmental “noise” into a measure of repeatability

• H is closely related to selection response (R)

• H can be used to model effect of changes to breeding program organization on R


Cultivar mean:

Yi. = m + Gi + Σeij

Variance AMONG cultivar means:

σ2P = σ2

G + (σ2e /r)

The phenotypic variance: single trial model


σ2G

σ2P

=H

σ2G

σ2G + (σ2

e /r)

=

Broad-sense heritability for single trial


What does H tell us, and what is it useful for?

• Proportion of phenotypic variation in genotype means that is due to genotypic differences (“signal:noise” ratio)

• Repeatability of a trial, or the expected correlation between 2 identical variety trials conducted in the same field

• It tells us how reliable the results of an experiment are

• It can be used to examine the effect of increasing or decreasing replicate number on repeatability of the experiment


What does H NOT tell us?

• Mendelian transmissability

• Anything about genetic control of a trait

• Note that H is not a constant! It is affected by the level of replication of the selection unit


Source MS EMS

Genotypic MSG σ2e + rσ2

G

Plot residuals MSe σ2e

Estimating H for the single-trial model

Variance components (including σ²G) are estimated from ANOVA table (for balanced trials) or REML software


Example: a 40-entry micro plot trial

40 upland varieties were evaluated in single-row micro plots at IRRI

SourceMean square

(g/plot)²EMS

Replicates

Genotypes (G) 6891 σ2e + rσ2

G

Plot residuals 1544 σ2e


σ2G = (6891 – 1544) / 3

= 1782

Table 8.3. Predicted H for yield in micro plots with 1- 4 replicates

Replicates H

1

2

3

4

σ2G / [σ2G + (σ2e /r)] = 1782/[1782 + (1544/1)] = 0.54

σ2G / [σ2G + (σ2e /r)] = 1782/[1782 + (1544/2)] = 0.70

σ2G / [σ2G + (σ2e /r)] = 1782/[1782 + (1544/3)] = 0.78

σ2G / [σ2G + (σ2e /r)] = 1782/[1782 + (1544/4)] = 0.82


H for the single trial model

• H is not a constant; it approaches 1.0 with increased r

• Single-trial H estimates are biased upward by GEI

• Estimates apply only to TPE and genetic population from which they were derived


Can anyone briefly explain:

the purpose of replications?

heritability?


Conclusion 1

• In field trials & nurseries, genotype & plot effects are confounded

• Purpose of replication in breeding programs = reduce this confounding, increasing our ability to identify superior genotypes

• Error mean square from representative experiments = used to predict LSD value we obtain from given level of replication


Conclusion 2

• H = a measure of repeatability of variety trials

• Genotype and error variances estimated from replicated trials used to model H

• Gains in precision and repeatability from increasing replication diminish quickly for trials with > 4 reps

planning rice breeding programs for impact models, means, variances, lsd’s and heritability

Documents

single plot

genotypic effects g

better estimator of

single trial model cultivar

noise of plot residuals

plot measurementsfor

single trial modelh

e r2gplot residuals15442e2g