# 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:

- \(\Delta G\) is the genetic gain
- \(S\) is the mean phenotypic value of the selected genotypes, expressed as a deviation from the population mean.

# 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.

### 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)
    library(agridat)

## H2cal function

     dt <- john.alpha
     hr <- H2cal(data = dt
                , trait = "yield"
                , gen.name = "gen"
                , rep.n = 3
                , fix.model = "rep + (1|rep:block) + gen"
                , ran.model = "rep + (1|rep:block) + (1|gen)"
                , emmeans = TRUE
                , plot_diag = TRUE
                , plot_dots = "rep"
                , outliers.rm = TRUE
                )

## Model information

    hr$model %>% summary()
    #> Linear mixed model fit by REML ['lmerMod']
    #> Formula: yield ~ rep + (1 | rep:block) + (1 | gen)
    #>    Data: dt.rm
    #> Weights: weights
    #>
    #> REML criterion at convergence: 78
    #>
    #> Scaled residuals:
    #>      Min       1Q   Median       3Q      Max
    #> -1.96200 -0.28318  0.04692  0.42242  2.20560
    #>
    #> Random effects:
    #>  Groups    Name        Variance Std.Dev.
    #>  gen       (Intercept) 0.14584  0.3819
    #>  rep:block (Intercept) 0.07895  0.2810
    #>  Residual              0.05334  0.2310
    #> Number of obs: 71, groups:  gen, 24; rep:block, 18
    #>
    #> Fixed effects:
    #>             Estimate Std. Error t value
    #> (Intercept)   4.4597     0.1471  30.318
    #> repR2         0.3564     0.1759   2.026
    #> repR3        -0.3555     0.1759  -2.021
    #>
    #> Correlation of Fixed Effects:
    #>       (Intr) repR2
    #> repR2 -0.601
    #> repR3 -0.601  0.503

## 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
--- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | ---
yield | 3 | 24 | 1 | 1 | 4.4318 | 0.419937 | 3.437591 | 5.035595 | 0.1458434 | 0 | 0 | 0.0533381 | 0.8913393 | 0.8486282 | 0.9117696

## Best Linear Unbiased Predictors (BLUPs)

    hr$blups %>% kable(caption = "BLUPs")

BLUPs
gen | yield
--- | ---
G01 | 4.967734
G02 | 4.473221
G03 | 3.620271
G04 | 4.584676
G05 | 4.571117
G06 | 4.495029
G07 | 4.103546
G08 | 4.566418
G09 | 3.555795
G10 | 4.334937
G11 | 4.346410
G12 | 4.687456
G13 | 4.712533
G14 | 4.775001
G15 | 4.920382
G16 | 4.693132
G17 | 4.547410
G18 | 4.362023
G19 | 4.781144
G20 | 4.114060
G21 | 4.748910
G22 | 4.579263
G23 | 4.293791
G24 | 4.198864

## Best Linear Unbiased Estimators (BLUEs)

    hr$blues %>% kable(caption = "BLUEs")

BLUEs
gen | yield | SE | df | lower.CL | upper.CL
--- | --- | --- | --- | --- | ---
G01 | 5.035595 | 0.1287046 | 42.59093 | 4.775965 | 5.295224
G02 | 4.426944 | 0.1286734 | 42.58668 | 4.167376 | 4.686511
G03 | 3.534301 | 0.1308346 | 42.66688 | 3.270388 | 3.798214
G04 | 4.591766 | 0.1305419 | 42.66130 | 4.328443 | 4.855089
G05 | 4.545263 | 0.1539083 | 40.56170 | 4.234337 | 4.856189
G06 | 4.550213 | 0.1297772 | 42.61679 | 4.288424 | 4.812002
G07 | 4.050188 | 0.1288641 | 42.60147 | 3.790239 | 4.310138
G08 | 4.252017 | 0.1534515 | 40.47391 | 3.941993 | 4.562041
G09 | 3.437591 | 0.1308101 | 42.66424 | 3.173727 | 3.701455
G10 | 4.304469 | 0.1288617 | 42.60145 | 4.044525 | 4.564414
G11 | 4.359846 | 0.1300335 | 42.63849 | 4.097544 | 4.622148
G12 | 4.766961 | 0.1298095 | 42.62006 | 4.505108 | 5.028815
G13 | 4.814309 | 0.1297867 | 42.61849 | 4.552501 | 5.076117
G14 | 4.820699 | 0.1284878 | 42.57709 | 4.561505 | 5.079894
G15 | 4.906773 | 0.1288035 | 42.58312 | 4.646943 | 5.166604
G16 | 4.705367 | 0.1286877 | 42.58950 | 4.445772 | 4.964963
G17 | 4.564521 | 0.1282290 | 42.57519 | 4.305848 | 4.823194
G18 | 4.354190 | 0.1283802 | 42.58577 | 4.095214 | 4.613166
G19 | 4.912830 | 0.1298183 | 42.62142 | 4.650959 | 5.174701
G20 | 4.030450 | 0.1289288 | 42.58961 | 3.770368 | 4.290533
G21 | 4.751738 | 0.1284928 | 42.57747 | 4.492533 | 5.010943
G22 | 4.568926 | 0.1305487 | 42.66215 | 4.305589 | 4.832263
G23 | 4.250782 | 0.1285805 | 42.58354 | 3.991401 | 4.510162
G24 | 3.827448 | 0.1534577 | 40.48016 | 3.517413 | 4.137483

## Outliers

    hr$outliers$random %>% kable(caption = "Outliers random model")

Outliers random model
| rep | block | gen | yield | resi | res_MAD | rawp.BHStud | index | adjp | bholm | out_flag
--- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | ---
3 | R1 | B1 | G05 | 5.8757 | 0.7593638 | 4.833477 | 0.0000013 | 3 | 0.000001341688 | 0.0000966 | OUTLIER

    hr$outliers$fixed %>% kable(caption = "Outliers fixed model")

Outliers fixed model
| rep | block | gen | yield | resi | res_MAD | rawp.BHStud | index | adjp | bholm | out_flag
--- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | ---
3 | R1 | B1 | G05 | 5.8757 | 0.7022176 | 4.929350 | 0.0000008 | 3 | 0.0000008250342 | 0.0000594 | OUTLIER
16 | R1 | B4 | G08 | 4.9989 | 0.5179535 | 3.635874 | 0.0002770 | 16 | 0.0002770401283 | 0.0193928 | OUTLIER
23 | R1 | B6 | G24 | 4.9577 | 0.5385472 | 3.780435 | 0.0001566 | 23 | 0.0001565546183 | 0.0111154 | OUTLIER

# References

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.
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