1. Planning rice breeding programs for impact Models, means, variances, LSD’s and Heritability

2. 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

3. IRRI: Planning breeding Programs for Impact 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.

4. IRRI: Planning breeding Programs for Impact 2.02.31.92.20.81.12.02.52.62.32.83.23.83.5 2.52.62.32.41.00.61.83.13.22.93.33.54.13.9 2.72.82.42.62.71.30.53.13.43.53.33.74.44.0

5. IRRI: Planning breeding Programs for Impact Linear model for plot measurements  For a completely randomized design (CRD): Where: Y ij = a plot measurement μ = the mean of all plots G i = the effect of the ith genotype ej = the “residual” effect of the jth plot G’s and e’s sum to 0 Y ij = μ + G i + e j [4.1]

6. IRRI: Planning breeding Programs for Impact As r increases, e approaches 0, Y approaches μ + G i  Breeders replicate to reduce effect of e! But even if r is 3 or 4, e’s have big effect on estimates of G Y i. = μ + G i + e [4.2] E(e) =0

7. IRRI: Planning breeding Programs for Impact Y i. = μ + G i + e j [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

8. IRRI: Planning breeding Programs for Impact Variance of a mean The variance of a genotype mean is an important measure of the precision of a trial: σ 2 Y = σ 2 e /r [4.3] σ 2 e is the error mean square from the ANOVA Standard error of a mean (SEM) SEM = σ 2 Y Variances, standard errors and LSD’s

9. IRRI: Planning breeding Programs for Impact Variance of a difference between 2 means σ 2 D = 2σ 2 e /r [4.4] Standard error of a difference (SED) SED = √(2σ 2 e /r )[4.5]

10. IRRI: Planning breeding Programs for Impact Least significant difference (LSD) LSD = t α/2,edf x SED = t α/2,edf x √(2 σ 2 e /r) [4.6] t α/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

11. IRRI: Planning breeding Programs for Impact 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

12. IRRI: Planning breeding Programs for Impact Cultivar mean: Y i. = m + G i + Σe ij Variance AMONG cultivar means: σ 2 P = σ 2 G + (σ 2 e /r) The phenotypic variance: single trial model

13. IRRI: Planning breeding Programs for Impact σ 2 G σ2Pσ2P = H σ 2 G + (σ 2 e /r) = Broad-sense heritability for single trial

14. IRRI: Planning breeding Programs for Impact 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

15. IRRI: Planning breeding Programs for Impact 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

16. IRRI: Planning breeding Programs for Impact SourceMSEMS GenotypicMS G σ 2 e + rσ 2 G Plot residualsMS e σ2eσ2e Estimating H for the single-trial model Variance components (including σ²G) are estimated from ANOVA table (for balanced trials) or REML software

17. IRRI: Planning breeding Programs for Impact Example: a 40-entry micro plot trial  40 upland varieties were evaluated in single-row micro plots at IRRI Source Mean square (g/plot)² EMS Replicates Genotypes (G)6891σ 2 e + rσ 2 G Plot residuals1544σ2eσ2e

18. IRRI: Planning breeding Programs for Impact σ 2 G = (6891 – 1544) / 3 = 1782 Table 8.3. Predicted H for yield in micro plots with 1- 4 replicates ReplicatesH 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

19. IRRI: Planning breeding Programs for Impact 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

20. IRRI: Planning breeding Programs for Impact Can anyone briefly explain:  the purpose of replications?  heritability?

21. IRRI: Planning breeding Programs for Impact 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

22. IRRI: Planning breeding Programs for Impact 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