Broad-sense heritability in plant breeding

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):

It is plugged into the breeder’s Equation to predict the response to selection.
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 (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

Table 1: 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
            , fixed.model = "0 + (1|bloque) + geno"
            , random.model = "1 + (1|bloque) + (1|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 components

hr$tabsmr %>% kable(caption = "Variance component table")

Table 2: Variance component table
trait	rep	geno	env	year	mean	std	min	max	V.g	V.e	V.p	repeatability	H2.s	H2.p	H2.c
stemdw	5	15	1	1	12.59867	4.749994	2.818	22.302	19.96002	9.410932	21.84221	0.913828	0.913828	0.9502395	0.9533473

Best Linear Unbiased Estimators (BLUEs)

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

Table 3: 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

Best Linear Unbiased Predictors (BLUPs)

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

Table 4: 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

Outliers

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

Table 5: 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

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

Table 6: 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

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

Maria Belen Kistner & Flavio Lozano-Isla

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

Breeder´s equation

Usual Problems

How calculate the Heritability?

1) Two stages approach

Model

Phenotypic variance

2) One stage approach

Phenotypic variance

Differentes heritability calculations

Heritability function in the package

Load packages

H2cal function

Model information

Variance components

Best Linear Unbiased Estimators (BLUEs)

Best Linear Unbiased Predictors (BLUPs)

Outliers

Comparison: H2cal and asreml