A method of estimating broad-sense heritability for quantitative traits in the type 2 modified augmented design

Heritability is a basic genetic parameter for quantitative traits that may determine their selection generation and intensity as well as predict their selection response and efficiency in plant breeding. Estimation of heritability varies based on experimental design. The type 2 modified augmented design (MAD2) as an unbalanced experimental design, has been proposed for evaluating numerous unreplicated test genotypes with several replicated control genotypes to adjust for soil heterogeneity. Here, we define an inter-environment correlation ( ), that is, the mean Pearson’s correlation coefficient of trait performance for test genotypes between all pairs of environments, to approximate broad-sense heritability ( . Computer simulation and empirical results demonstrated that was consistent with estimates on a plot basis by ANOVA for non-missing data sets, and similar to those by the restricted maximum likelihood (REML)-based method for missing data sets. The method was shown to generally outperform the ANOVAand REML-based methods.


INTRODUCTION
Heritability is a basic genetic parameter for quantitative traits that may determine their selection generation and intensity as well as predict their selection response and efficiency in plant breeding.Heritability estimation varies depending on the experimental design (Holland et al., 2003).The modified augment design (MAD) has been proposed for square plots (Type 1) (Lin and Poushinsky, 1983) and specifically for rectangular plots (Type 2 -MAD2) (Lin and Poushinsky, 1985) for field evaluation of a large number of breeding lines and used in many crops such as flax (Soto-Cerda et al., 2014a;Soto-Cerda et al., 2014b;Kumar et al., 2015), wheat (Golparvar et al., *Corresponding author.E-mail: collinsagho@yahoo.com. Author(s) agree that this article remain permanently open access under the terms of the Creative Commons Attribution License 4.0 International License 2015), potato (Schaalje et al., 1987), soybean (Lin and Voldeng, 1989), and barley (May et al., 1989;May and Kozub, 1995).This design can accommodate a large number of unreplicated test genotypes with typically only three control genotypes for error control (You et al., 2013).
Our earlier study (You et al., 2013) demonstrates that soil heterogeneity can be sufficiently adjusted for traits in MAD2 trials.The adjusted observations for test and control genotypes are expected to exclude the effect of soil heterogeneity; thus, the variation among replications of each control genotypes should be solely caused by random errors.A method based on analysis of variance (ANOVA) to approximately estimate broad-sense heritability ( ) for this design has been developed (You et al., 2016b), in which genetic variance is calculated based on the total phenotypic variance estimated from test genotypes minus the error variance estimated from control genotypes.Because is always overestimated in single trials, joint analysis over multiple environments for heritability estimation was proposed in which the error variance is jointly estimated using the three replicated control genotypes (You et al., 2016b).However, a caveat to this estimation method is the requirement for the same control genotypes to be used in all environments or trials, and this limits its potential use for joint analysis of data from multi-environment trials with different control genotypes or experimental designs.
There are two units for measurement of phenotypic variances in heritability estimation: on a plot basis and on an entry (or genotype)-mean basis.Estimates of phenotypic variance on an entry-mean basis are always larger than those on a plot basis because the error and interaction variance components are divided by the corresponding numbers of observations (You et al., 2016b).From the perspective of breeding, usually our interest is the heritability estimated from a set of trials.Thus, heritability is usually estimated on an entry-mean basis, that is, the genotype × environment variance is divided by the number of environments.However, we found from empirical results that on an entry-mean basis was overestimated in multi-environment MAD2 trials (You et al., 2016a); thus, estimates on a plot basis were put forward and used in MAD2 trials of flax (You et al., 2016a) and in this study.
Herein, we define an inter-environment correlation ( ) to approximately estimate in multi-environment MAD2 trials and which does not rely on control genotype data.
estimates from two traditional methods, ANOVA and restricted maximum likelihood (REML), are compared with using computer-simulated and empirical data sets with and without missing data.
is jointly estimated based on e environments with t replicated control genotypes in each trial (You et al., 2016b).Typically, t is three in MAD2 trials.
For a typical multi-environment MAD2 trial, the data set includes observations for g unreplicated test genotypes and three control genotypes at each of e environments (representing individual trials in multiple years or/and locations).For the three control genotypes, one main plot control has m replications and each of two subplot controls are replicated n times.The values of g, m, n and e depend on the design.Thus, a total of (g + m + 2*n)*e data points will be generated from a typical multi-environment MAD2 trial.For example, an individual MAD2 trial with a grid of 10 rows by 10 columns contained 100 whole plots with each whole plot being split into 5 subplots, resulting in a total of 500 subplots.The main plot control was placed in the center subplot of each whole plot and so m = 100.Two additional subplot controls were randomly assigned to two subplots of 5 randomly selected whole plots and thus n = 5 or 2n = 10.Test genotypes were accommodated to the remaining 390 subplots (g = 390), and the ratio of m to g and that of m to all subplots were 25.6 and 20%, respectively (see Figure 1 in You et al., 2013).To simplify the process, all simulations comprised the same percentage of test genotypes for the main plot control replicates (m = 0.25 * g) and the same number of subplot control replicates (n = 5).
To assess the effect of the number of environments (e), number of test genotypes (g), and theoretical heritability of a trait on estimation, simulation data sets for a total of 180 parameter combinations of (0.1 to 0.9 with an interval of 0.1), e (2, 4, 6, 8, and 10), and g (50, 100, 200, and 300) were generated.For each parameter combination, 500 simulations were replicated, which was sufficient to represent the sampling distribution of a parameter combination based on results of different simulation runs.
Data sets with 5, 10, 15 and 20% missing data for test genotypes were constructed from each complete data set generated for parameter combinations.Missing data were distributed in all environments at random.The R sample function was applied to all simulated observations from e environments to randomly assign NULL as missing values.Consequently, a total of 450,000 data sets were generated for analysis.
According to empirical results, plot-based estimates were more accurate than entry-mean-based estimates in MAD2 trials (You et al., 2016a).Accordingly, heritability estimates on a plot basis were used in this study.Given , , and for a trait, can be estimated as = on a plot basis.Because traits may have different and , these two variances were randomly and independently generated.In addition, the environmental variance ( ) was also randomly generated.Thus, the effects of G, E, G × E, and random error can be simulated according to Equation 1 for a multi-environment MAD2 trial.Similar simulation procedures for data generation have been described previously (You et al., 2016b).The R pipeline program for simulation is available at http://probes.pw.usda.gov/bioinformatics_tools/MADPipeline/index.html.

Empirical data
Three flax biparental genetic populations, namely 243 F6-derived recombinant inbred lines (RILs) generated from a cross between CDC Bethune and Macbeth (BM), 90 F6-derived RILs from a cross between E1747 and Viking (EV), and 78 F1-derived doubled haploid lines from a cross between SP2047 and UGG5-5 (SU), plus a core collection (CC) of 391 accessions, were field evaluated at two locations in Canada (Morden, Manitoba and Kernen Farm near Saskatoon, Saskatchewan) from 2009 to 2012.The same MAD2 design was employed with the same population in all environments but the designs differed across populations.The experimental designs and phenotyping for the 22 traits in CC, 14 in BM, 19 in EV, and 11 in SU over six to eight environments have been previously described (Cloutier et al., 2010;Soto-Cerda et al., 2014b;Kumar et al., 2015;You et al., 2016b).The adjusted observations in each environment were used for estimation and evaluation.

Traditional estimation of
The simulated and empirical adjusted observations were used to estimate on a plot basis (You et al., 2016a;You et al., 2016b).was estimated as ̂ = ̂ /( ̂ + ̂ + ̂ ), where ̂ , ̂ , and ̂ are the genetic, G × E, and error variance, respectively.These variance components were estimated using the method of moments based on both ANOVA (You et al., 2016b) and REML (Holland, 2006;Piepho and Möhring, 2011).The R (https://www.rproject.org/)package lme4 (Bates et al., 2015) was used to calculate the REML-based variance components.A test for homogeneity of error variance across environments was performed before parameter estimations to satisfy the assumption of the model (Equation 1).

Inter-environment correlation as a statistic
Based on Equation 1, the inter-environment correlation of adjusted observations (y) for two environments (E1 and E2) was defined as: (2) Here, is the covariance of adjusted observations between two environments.It is equal to because G, E, and random error are hypothesized to be independent of each other and thus the covariance between them is zero, whereas = because, for a single environment, the environmental variance 0. Therefore, corresponds to an estimate.To validate this definition, independent observations were simulated in twoenvironment MAD2 trials at given values (0.1 to 0.9 with 0.1 intervals) and for a number of test genotypes (50, 100, 200 and 300).
For a multi-environment MAD2 trial, was defined as the mean inter-environment correlation coefficient of all possible pairs of environments to estimate the of a trait: Where e is the number of environments and = 0 if < 0.

Standard error of estimates
The delta method was implemented for MAD2 (You et al., 2016b) and used to estimate the standard error (SE) of ̂ by ANOVA and REML for the empirical data sets.For simulation data, the standard deviation of 500 replicates for each parameter combination was You et al. 259 calculated to represent the SE of three estimates ( ̂ or ).
Owing to the properties of as a mean Pearson's correlation coefficient, the SE of can also be approximated based on the SE formula of Pearson's correlation (Bowley, 1928;Kendall and Stuart, 1977) using: Where e is the number of environments and g is the number of test genotypes.
The bootstrap method (Efron, 1979;Xie and Mosjidis, 1997) was also used to estimate the standard error of three heritability estimates for either simulated or empirical data.Bootstrapping was performed by resampling test genotypes with replacement from the original population, and each bootstrap sample had the same size as the original population.The R sample function was used to generate the bootstrap samples.The standard deviation of in 500 bootstrap samples was calculated to represent the standard error of , denoted as ) to distinguish from and compare with SE( ) in Equation 4, ̂ , or estimated by simulation.An R pipeline program was developed to automate the simulations and, consequently, the estimation for three heritability estimation methods.A separate R program for calculating and its SE (bootstrap method) for empirical test genotype data was also created and is available at http://probes.pw.usda.gov/bioinformatics_tools/MADPipeline/index.html.

Statistical power
Statistical power (P) of three heritability estimation methods ( , ANOVA, and REML) was calculated for all parameter combinations to determine the minimum number of environments and test genotypes required for heritability estimation and for method comparison.The power of a trait heritability estimate is the probability of correctly rejecting the null hypothesis that heritability is zero when the true heritability of the trait is greater than zero.First, the Z score for the power of a heritability estimate was calculated using (Klein, 1974): Where is the Z score of the significance level α used for a onetailed test ( ; ̂ is the expected value of the heritability, here the given or parametric heritability value; and ̂ is the SE of the expected ̂ value, estimated by the standard deviation of 500 simulated samples.The one-tailed test was adopted to test whether a heritability estimate is greater than 0 because heritability should be always equal to or greater than zero.The statistical power was defined as the area under the standard normal curve from Z to plus infinity, and can be calculated using an R function: Where pnorm is the R function for calculating the cumulative density of the normal distribution.

Genetic and phenotypic variance estimation in the definition of
Simulation results showed a perfectly consistent relationship between ̂ , ) and ̂ (r = 1.000) (Figure 1A) and between ̂ ̂ and ̂ (r = 0.999) (Figure 1B) for the ANOVA-based method, confirming the definition in Equation 2.
estimates for simulated data sets without missing data estimates by , ANOVA ( ̂ (A)), and REML ( ̂ (R)) were calculated for all complete simulated data sets.was perfectly correlated with the given values (r = 1.000) independent of the number of environments (Figure 2A) or test genotypes (Figure 2C).However, standard errors of ( ) declined with increasing number of environments or test genotypes (Figure 2B  and D).For a trait with equal to or greater than 0.5, the heritability can be correctly estimated at as few as two environments with 50 test genotypes at a statistical power of > 0.999 (Table 1).For traits with low heritability, a larger number of environments and test genotypes contribute positively to more reliable estimates through increasing statistical power.For an estimate of 0.2, a statistical power in excess of 0.95 can be achieved from data sets with 300 test genotypes at two environments, 100 test genotypes at four environments, or 50 test genotypes at ≥ six environments when there was no missing data (Table 1).

estimates for simulated data sets with missing data
estimates by , ANOVA, and REML were calculated for all simulated data sets with missing data of 5 to 25%.The SEs of estimates by all three methods increased with the rate of missing data.Among the three estimation methods, REML yielded smaller SEs than the -and ANOVA-based methods for both non-missing and missing data sets (Table 3).̂ (A) had SEs consistent with for non-missing data sets (R 2 = 1.000) but larger SEs for missing data sets (Figure 3A); had more consistent REs with ̂ (R) (R 2 = 0.974) for missing data sets (Figure 3B).
The statistical power of estimates was markedly affected by missing data rates (Tables 3 and 4).Owing to the small SEs, estimates by REML had higher power than those by , which was also higher than those by ANOVA.The power of and ̂ (R) was relatively less affected by missing data.For an estimate of 0.2 in data sets with a missing data rate of ≤5%, a statistical power over 0.95 can be achieved from data sets with 300 test genotypes at two environments, 100 test genotypes at ≤ four environments, or 50 test genotypes at ≤ eight environments for all three estimation methods (Table 4).Estimates of and ̂ (R) were less affected by missing data, remaining largely constant with increasing missing data rate at different given values (Table 3).was consistent with ̂ (R) for different numbers of environments and test genotypes (Table 5) (r = 0.995-0.999)but less correlated with ̂ (A) (r = 0.955-0.996)(Table 5).Bias of ̂ (A) and ̂ (R) from their true heritability values was observed.At a given ≤ 0.3, ̂ (A) was overestimated (Table 3).However, at a given ≥ 0.4 for ̂ (A) and ≥ 0.2 for ̂ (R), ̂ values were slightly underestimated (Table 3).A negligible deviation of estimates from their true values was also observed when given ≥ 0.3 (Table 3).

estimates for empirical data
As a case study, , ̂ (A), and ̂ (R) and their SEs were estimated for 22 traits in the CC, 14 in the BM population, 19 in the EV population, and 11 in the SU population (Table S1).These traits varied greatly in estimated heritability (0.00-0.94), were phenotyped in a different number of environments ranging from two to eight, and evaluated in different populations, that is, one natural and three biparental populations of varying size (Table 6).
Here, heritability of some traits was estimated to be 0 due to large experimental errors in data and the ANOVA based model used which resulted in negative genetic variances.The negative genetic variance was treated as 0 in calculation although the real genetic variance should be greater than 0. was highly correlated to both ̂ (A) (r = 0.948-0.998)and ̂ (RP) (r = 0.974-0.998) in individual populations (Table 6).Similar relationships among , ̂ (A), and ̂ (R) were observed in different populations despite different numbers of test genotypes (Table 6).Figure 4 depicts the similar overall relationship of with ̂ (A) and ̂ (R) for all 66 data points.In addition, a strong correlation between ̂ (A) and ̂ (R) (r = 0.995-1.000),similar to the results in the simulation data sets, was also observed in the different populations    MDR, Missing data rate on an entire multi-environment trial basis; H^2 was calculated based on 10,000 data points generated from combinations of various numbers of environments (2, 4, 6, 8 and 10) and test genotypes (50, 100, 200 and 300) with 500 replicates for each combination; Statistical power was the average of power estimates from 20 parametric sets of different numbers of environments and test genotypes.

No. of environments No. of test genotypes
(Table 6).Because the empirical data had missing values in some environments (Table S1), a slightly weaker relationship among , ̂ (A) and ̂ (R), and a stronger relation between and ̂ (R) than that between and ̂ (A) (Table 6) were observed, which resembled the results obtained in missing data sets.

Standard error of
To perform a significance test for estimates, the SEs of determined by Eq. 4 ( ) and by the bootstrap method ( )) were compared to the corresponding simulated SEs ( ).Although a strong correlation of to was observed (R 2 = 0.815) (Figure 5A), a high correlation (R 2 = 0.995) was obtained for with ) (Figure 5B) indicating the bootstrap method yielded SE estimates for that were highly consistent with those obtained by simulation.The SE estimates obtained by the bootstrap method were systematically smaller (by 0.0015 on average) than those obtained by simulation.A linear regression equation, 1.0802* ) -0.0017 (Figure 5B), may be used to adjust the bootstrap estimates.

DISCUSSION
is the ratio of genetic variance to total phenotypic variance, representing the extent with which genotypes   are affected by environments and random error.also represents the repeatability of trait performance in different environments.For highly heritable traits, the performance of genotypes in one environment has a high repeatability in other environments; in other words, the trait performance of the genotypes is strongly correlated between any pair of environments.In contrast, low heritability traits display low correlations of trait performance between any two environments.As such, the inter-environment correlation ( ) of trait performance should be an indicator of trait heritability.The theoretical derivation confirms that the between two environments corresponds to (Equation 2).Both simulation and empirical results demonstrate that is an accurate and stable estimate of on a plot basis.The method has two significant advantages.First, because is a mean correlation coefficient between pairs of environments, its calculation is simple.Second, only the test genotype data is required for calculation of , eliminating the prerequisite for the use of the same control genotypes across all trials and hence permitting flexible field designs.Thus, the method allows joint estimation over multiple environments for genetic populations that may differ in their MAD2 designs and which may not necessarily include the same control genotypes.The method is equally applicable to trials with the same test genotypes but where different control genotypes are used in the different environments.A practical example was the joint estimation for three flax biparental populations: BM, EV, and SU.These three populations were evaluated in the same six to eight environments using MAD2 designs but with different Table 5. Correlation of broad-sense heritability ( ̂ ) estimated by three methods inter-environment correlation ( ), ANOVA (A), and REML (R) for simulated data sets with missing data rates of 5 to 20%.control genotypes.In each MAD2 design, CDC Bethune was used as the main plot control but the subplot controls were the two parents of each of the three biparental populations to improve error control because the two parents shared the genetic background of their offspring.

No. of test genotypes
As a consequence of the use of different control genotypes in the three population trials, neither ANOVAnor REML-based methods could estimate joint values, but this was achievable with the method using pooled data.The results will be reported separately.
In the ANOVA-based estimation, the error variance of unreplicated test genotypes is estimated by duplicated control genotypes.This is based on the assumption that control genotypes share the same random error variance with all test genotypes.Theoretically, the total mean square (MS) of the test genotypes will be greater than the error MS in a single trial or the G × E interaction variance in a multi-environment trial.As such, the genetic variance of the test genotypes can be estimated as the total MS minus the error MS of a single trial or by the total variance minus the G × E interaction MS in a multienvironment trial.However, the sampling bias caused by a limited number of control genotypes (typically three in MAD2) may occasionally result in negative genetic variance estimates and failure to correctly estimate genetic parameters, especially when the heritability of a trait is very low.In this case, the method can avoid this potential drawback because the genetic variance in corresponds to the covariance of trait performance between two environments ( )) (Equation 2and Figure 1A).If the ) is less than zero then the genetic variance can be considered zero, whereas negative or null genetic variance obtained by ANOVA or REML might be an incorrect estimate in some cases.In the core collection, for example, both oil and linolenic acid contents have estimates of zero because their genetic variances were estimated to be zero by both ANOVA-and REML-based methods.This result is  obviously incorrect because oil and linolenic acid content are traits of moderate to high heritability (You et al., 2016a).However, -based estimates for the two traits were 0.387 and 0.661, respectively, although these estimates are still smaller than expected.
The REML-based method, as an alternative to the more traditional ANOVA techniques, has been widely used for estimation of heritability and genetic correlations (Holland, 2006;Piepho and Möhring, 2011).The advantages compared to ANOVA methods are that REML estimates of variance and covariance components have known asymptotic distributional properties and can efficiently handle both unbalanced data and complex experimental designs (Meyer, 1985;Holland, 2006).REML's main drawback is that it is much more computationally intensive than ANOVA, a disadvantage that is exacerbated with large data sets.In this study, these two estimation methods were compared to the method using both simulation and empirical data of multi-environment MAD2 trials.All results corroborate the agreements between the ANOVA-and REML-based methods with r > 0.99 in the simulated data sets without missing data, especially in the empirical data of BM and SU populations where their estimates are nearly identical (Table 2).The ANOVA-based estimates are consistent with the estimates when no missing data exist in the data sets.Nevertheless, when missing values occur, the -and REML-based methods show higher power than the ANOVA-based method, confirming that the REMLbased method is efficient for tackling unbalanced data.Overall, -, ANOVA-, and REML-based methods can be used for non-missing data sets; the -and REML-based methods are suitable for missing data sets; and the method is versatile for all cases of practical data sets in multi-environment MAD2 trials.
A significance test for estimates requires SE values.Generally, SEs ( ) estimated by simulation with a large sample size (500 in this study) provide a good estimate of the sampling error.Thus, they were used as a relative standard for comparisons.Two potential methods were assessed: the approximate standard error ( ) of mean simple correlation coefficients (Equation 4) and bootstrap.The non-parametric bootstrap is an effective alternative for determining distribution of an estimator with an unknown probability density, and has been used to estimate standard errors for heritability (Xie and Mosjidis, 1997).Results demonstrate that the bootstrap method outperforms the SE formula of simple correlation coefficient (Equation 4), and provides perfectly consistent SE estimates with .However, a systematic difference between and sb(rE) was observed; adjustments may be done using the regression equation in Figure 5B, which was constructed using 900 data points generated from different numbers of environments (2, 4, 6, 8, and 10), test genotypes (50, 100, 200 and 300), and missing data rates (0, 5, 10, 15 and 20%) at given heritability values ranging from 0.1 to 0.9.Therefore, the bootstrap method is recommended to estimate the standard error of in significance tests.The bootstrap estimation of SE has been implemented in the R pipeline program (http://probes.pw.usda.gov/bioinformatics_tools/MADPipeline/index.html).
To find the effective sample size for estimating in multi-environment MAD2 trials, the statistical power of three estimation methods was calculated.As expected, the power is affected by the number of environments and test genotypes, and missing data rates.For traits with equal to or greater than 0.5, 50 test genotypes at two environments were sufficient to achieve a statistical power over 0.95 (Table 1); for traits with lower heritability (e.g., 0.2), 300 test genotypes at two environments, 100 test genotypes at four environments, or 50 test genotypes at greater than or equal to six environments are required to obtain the same statistical power (Table 1).Increasing the missing data rate decreased the statistical power (Table 4), but an increase of the number of environments and/or test genotypes markedly improved the statistical power of estimates.
Notably, heritability estimates in the simulation data sets were slightly biased from their true values when true values were greater than 0.2 (Table 3).This bias was observed in all three estimation methods, especially when the ANOVA-based method was used for missing data sets.However, the method has less deviation than other two for both missing and nonmissing data sets.This bias may be inherently owing to the MAD2's unbalanced feature.Piepho and Möhring (2007) discussed how estimation of broad-sense heritability in unbalanced trials differs from that in the case of a balanced design.The proposed here, as well as the ANOVA- (You et al., 2016b) and REML- (You et al., 2016a) based methods, provide an approximate estimate for MAD2 trials.This approximation is due to not only unbalanced data but also approximate assumption of independence for adjusted observations from an MAD2 trial.
For adjustment of observations in an MAD2 trial, there are four different cases for quantitative traits: (1) significant additive soil variation due to row or column effects (M1); (2) significant non-additive soil variation due to row × column interaction effects (M3); (3) M1+M3, and (4) no additive or non-additive soil variation (You et al., 2013).For case 4, as no data adjustment is required, their estimates of heritability will be unbiased, while for the first three cases, data adjustment are needed and the adjusted data may be correlated to some extent, resulting in biased estimates.In this study, the simulation data was completely independent of each other, but the empirical data of some traits were unnecessarily completely independent due to data adjustment.However, we found that highly similar or consistent results were obtained in both data sets, indicating small effect of defective independence assumption to heritability estimation.In our actual trials, especially those with good quality, most of the traits would not need data adjustment or have minor row or/and column or their interaction effects, the bias due to defective independence would be small and may be disregarded in breeding applications.

Conclusion
A statistic, rE, representing an inter-environment correlation of a quantitative trait, was presented for multienvironment MAD2 trials.The method provides a simple approach to approximate in any multi-

Figure 1 .
Figure 1.Covariances ( ̂ , )) of trait performance between two environments in relation to genetic variances ( ) estimated by ANOVA (A), and the product of standard deviations of trait performance from two environments ( ̂ ̂ ) in relation to phenotypic variances ( ) on a plot basis estimated by ANOVA (B).

Figure 2 .
Figure 2. Inter-environment correlations ( ) and their standard deviations ( )) in terms of heritability ( ) for different numbers of environments (A and B) and test genotypes (C and D), respectively.

Figure 3 .
Figure 3. Relationship of standard errors (SEs) of inter-environment correlation ( ) with SEs of estimates by ANOVA (A) and REML (R).(A) SEs of estimated from simulations ( )) vs. SEs of estimates by ANOVA ( ̂ ) using all data sets (missing and non-missing data) (blue) and non-missing data only (red).(B) )) vs. ̂) using all data sets.

Figure 4 .
Figure 4. Relationship of inter-environment correlations ( ) with broad-sense heritability ( ̂ ) estimated from empirical data by ANOVA and REML.Data points include estimates for and ̂ from 22 traits in the core collection, 14 in the CDC Bethune/Macbeth, 19 in the E1747/Viking, and 11 in the SP20147/UGG5-5 populations.

Figure 5 .
Figure 5. Relationship of standard error (SE) of inter-environment correlation ( ) with SE estimates by other methods.(A) ) vs. SE of calculated by Equation 4 ().(B) ) vs. SE estimated by the bootstrap method ( )).All data sets including missing and non-missing data) were used.

Table 1 .
Statistical power (P) of broad-sense heritability estimates ( ) by inter-environment correlation ( ), ANOVA and REML for α = 0.05 (one-tailed test) as a function of number of environments and test genotypes.

Table 2 .
Correlation of broad-sense heritability ( ̂ ) estimated by three methods inter-environment correlation ( ), ANOVA (A), and REML (R) for simulated data sets without missing data.

Table 3 .
Effects of missing data on estimation of broad-sense heritability ( ) and statistical power (P).
̂ (A): ̂ on a plot basis estimated by ANOVA; ̂ (R): ̂ on a plot basis estimated by REML.Simulated data includes data

Table 4 .
Statistical power (P) of broad-sense heritability estimates ( ̂ ) at a given heritability of 0.2 by inter-environment correlation ( ), ANOVA, and REML for α = 0.05 (one-tailed test) as a function of the number of environments, and test genotypes, and missing data rates.

Table 6 .
Correlation of broad-sense heritability ( ̂ ) estimated by three methods inter-environment correlation ( ), ANOVA (A), and REML (R) for empirical data.̂ on a plot basis estimated by REML.The estimate for each trait was used as a single data point to calculate the correlation among , ̂ (A), and ̂ (R).No. of traits in each population represent the number of data points used for calculation of correlation between any two methods.Total represents all 66 data points from four individual populations for correlation calculation.