Broad-sense heritability in plant breeding

Maria Belen Kistner & Flavio Lozano-Isla

Broad-sense heritability (\(H^2\))

Broad-sense heritability (\(H^2\)) is defined as the proportion of phenotypic variance that is attributable to an overall genetic variance for the genotype (Schmidt et al., 2019b). There are usually additional interpretations associated with \(H^2\): (i) It is equivalent to the coefficient of determination of a linear regression of the unobservable genotypic value on the observed phenotype; (ii) It is also the squared correlation between predicted phenotypic value and genotypic value; and (iii) It represents the proportion of the selection differential (\(S\)) that can be realized as the response to selection (\(R\)) (Falconer and Mackay, 2005).

There are two main reasons why heritability on an entry-mean basis is of interest in plant breeding (Schmidt et al., 2019a):

  1. It is plugged into the breeder’s Equation to predict the response to selection.
  2. It is a descriptive measure used to assess the usefulness and precision of results from cultivar evaluation trials.

Breeder´s equation

\[\Delta G=H^2S\] Where:

Usual Problems

In practice, most trials are conducted in a multienvironment trial (MET) presente unbalanced data as not all cultivars are tested at each environment or simply when plot data is lost or when the number of replicates at each location varies between genotypes (Schmidt et al., 2019b). However, the standard method for estimating heritability implicitly assumes balanced data, independent genotype effects, and homogeneous variances.

How calculate the Heritability?

According Schmidt et al. (2019a), the variance components could be calculated in two ways:

1) Two stages approach

For the two stage approach, in the first stage each experiment is analyzed individually according their experiment design (Lattice, CRBD, etc) (Zystro et al., 2018). And for the second stage environments are denotes a year-by-location interaction. This approach assumes a single variance for genotype-by-environment interactions (GxE), even when multiple locations were tested across multiple years (Buntaran et al., 2020).

Model

\[y_{ikt}=\mu\ +\ G_i+E_t+GxE_{it}+\varepsilon_{ikt}\]

Phenotypic variance

\[\sigma_p^2=\sigma_g^2+\frac{\sigma_{g\cdot e}^2}{n_e}+\frac{\sigma_{\varepsilon}^2}{n_e\cdot n_r}\]

2) One stage approach

For the one stage approach only one model is used for the MET analysis. The environmental effects are included via separate year, and location main interaction effects.

\[y_{ikt}=\mu+G_i+Y_m+E_q+YxE_{mq}+GxY_{im}+GxE_{iq}+GxYxE_{imq}+\varepsilon_{ikmq}\]

Phenotypic variance

\[\sigma_p^2=\sigma_g^2+\frac{\sigma_{g\cdot e}^2}{n_e}+\frac{\sigma_{g\cdot y}^2}{n_y}+\frac{\sigma_{g\cdot y\cdot e}^2}{n_y\cdot n_e}+\ \frac{\sigma_{\epsilon}^2}{n_e\cdot n_y\cdot n_r}\]

Differentes heritability calculations

Differentes heritability calculation
Standart Cullis Piepho
\(H^2=\frac{\sigma_g^2}{\sigma_p^2}=\frac{\Delta G}{S}\) \(H_{Cullis}^2=1-\frac{\overline{V}_{\Delta..}^{^{BLUP}}}{2\cdot\sigma_g^2}\) \(H_{Piepho}^2=\frac{\sigma_g^2}{\sigma_g^2+\frac{\overline{V}_{\Delta..}^{BLUE}}{2}}\)

Heritability function in the package

For calculate the standard heritability in MET experiments the number of location and replication should be include manually in the function H2cal(). In the case of difference number of replication in each experiments, take the maximum value (often done in practice) (Schmidt et al., 2019b).

For remove the outliers the function implemented is the Method 4 used for Bernal-Vasquez et al. (2016): Bonferroni-Holm using re-scaled MAD for standardizing residuals (BH-MADR).

Load packages

library(inti)

H2cal function

 dt <- potato
 hr <- H2cal(data = dt
            , trait = "stemdw"
            , gen.name = "geno"
            , rep.n = 5
            , ran.model = "1 + (1|bloque) + (1|geno)"
            , fix.model = "0 + (1|bloque) + geno"
            , emmeans = TRUE
            , plot_diag = TRUE
            , outliers.rm = TRUE
            )

Model information

hr$model %>% summary()
## Linear mixed model fit by REML ['lmerMod']
## Formula: stemdw ~ 1 + (1 | bloque) + (1 | geno)
##    Data: dt.rm
## Weights: weights
## 
## REML criterion at convergence: 796.1
## 
## Scaled residuals: 
##      Min       1Q   Median       3Q      Max 
## -2.38440 -0.64247 -0.08589  0.57452  2.84508 
## 
## Random effects:
##  Groups   Name        Variance Std.Dev.
##  geno     (Intercept) 19.960   4.4677  
##  bloque   (Intercept)  0.110   0.3316  
##  Residual              9.411   3.0677  
## Number of obs: 148, groups:  geno, 15; bloque, 5
## 
## Fixed effects:
##             Estimate Std. Error t value
## (Intercept)    12.51       1.19   10.51

Variance component

hr$tabsmr %>% kable(caption = "Variance component table")
Variance component table
variable rep geno env year mean std min max V.g V.gxl V.gxy V.e h2.s h2.c h2.p
stemdw 5 15 1 1 12.59867 4.749994 2.818 22.302 19.96002 0 0 9.410932 0.913828 0.9533473 0.9502395

Best Linear Unbiased Predictors (BLUPs)

hr$blups %>% kable(caption = "BLUPs")
BLUPs
geno stemdw
G01 15.587018
G02 10.228658
G03 9.821839
G04 15.057007
G05 12.843686
G06 20.631268
G07 3.254483
G08 10.517060
G09 15.525899
G10 9.389236
G11 6.699074
G12 15.948953
G13 14.533681
G14 16.126578
G15 11.515963

Best Linear Unbiased Estimators (BLUEs)

hr$blues %>% kable(caption = "BLUEs")
BLUEs
geno stemdw SE df lower.CL upper.CL
G01 15.73200 1.030325 119.7830 13.6919903 17.77201
G02 10.12100 1.030325 119.7830 8.0809903 12.16101
G03 9.69500 1.030325 119.7830 7.6549903 11.73501
G04 15.17700 1.030325 119.7830 13.1369903 17.21701
G05 12.87106 1.086433 122.5189 10.7204483 15.02167
G06 22.30200 1.030325 119.7830 20.2619903 24.34201
G07 2.81800 1.030325 119.7830 0.7779903 4.85801
G08 10.42300 1.030325 119.7830 8.3829903 12.46301
G09 15.66800 1.030325 119.7830 13.6279903 17.70801
G10 9.24200 1.030325 119.7830 7.2019903 11.28201
G11 6.42500 1.030325 119.7830 4.3849903 8.46501
G12 16.11100 1.030325 119.7830 14.0709903 18.15101
G13 14.62900 1.030325 119.7830 12.5889903 16.66901
G14 16.29700 1.030325 119.7830 14.2569903 18.33701
G15 11.46900 1.030325 119.7830 9.4289903 13.50901

Outliers

hr$outliers$random %>% kable(caption = "Outliers random model")
Outliers random model
bloque geno stemdw resi res_MAD rawp.BHStud index adjp bholm out_flag
68 IV G05 80.65 61.39925 18.886677 0.0000000 68 0.0000000000 0.0000000 OUTLIER
100 IV G06 33.52 12.02340 3.698449 0.0002169 100 0.0002169207 0.0323212 OUTLIER
hr$outliers$fixed %>% kable(caption = "Outliers fixed model")
Outliers fixed model
bloque geno stemdw resi res_MAD rawp.BHStud index adjp bholm out_flag
68 IV G05 80.65 60.36709 18.84505 0 68 0 0 OUTLIER

Comparison: H2cal and asreml

https://inkaverse.com/articles/extra/stagewise.html

Bernal-Vasquez, A.-M., H.-F. Utz, and H.-P. Piepho. 2016. Outlier detection methods for generalized lattices: A case study on the transition from ANOVA to REML. Theoretical and Applied Genetics 129(4): 787–804. doi: 10.1007/s00122-016-2666-6.
Bolker, B. 2021. Mean variance of a difference of BLUEs or BLUPs in lme4. Stack Overflow. https://stackoverflow.com/questions/38697477/mean-variance-of-a-difference-of-blues-or-blups-in-lme4 (accessed 21 May 2021).
Buntaran, H., H.-P. Piepho, P. Schmidt, J. Rydén, M. Halling, et al. 2020. Cross-validation of stagewise mixed-model analysis of Swedish variety trials with winter wheat and spring barley. Crop Science 60(5): 2221–2240. doi: 10.1002/csc2.20177.
Falconer, D.S., and T.F. Mackay. 2005. Introduction to quantitative genetics (Pearson Prentice Hall, editor). Fourth.
Schmidt, P., J. Hartung, J. Bennewitz, and H.-P. Piepho. 2019a. Heritability in Plant Breeding on a Genotype-Difference Basis. Genetics 212(4): 991–1008. doi: 10.1534/genetics.119.302134.
Schmidt, P., J. Hartung, J. Rath, and H.-P. Piepho. 2019b. Estimating Broad-Sense Heritability with Unbalanced Data from Agricultural Cultivar Trials. Crop Science 59(2): 525–536. doi: 10.2135/cropsci2018.06.0376.
Zystro, J., M. Colley, and J. Dawson. 2018. Alternative Experimental Designs for Plant Breeding. Plant Breeding Reviews. John Wiley & Sons, Ltd. p. 87–117