Theory Development on Optimal Replication

This protocol is extracted from research article:

Estimation of the Optimal Number of Replicates in Crop Variety Trials

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Front Plant Sci**,
Jan 13, 2021;
DOI:
10.3389/fpls.2020.590762

Estimation of the Optimal Number of Replicates in Crop Variety Trials

Procedure

The achieved heritability within a single trial (*H*_{ST}) is defined by Falconer (1989); DeLacy et al. (1996):

where ${\mathrm{\sigma}}_{G}^{2}$ is the genotypic variance and ${\mathrm{\sigma}}_{\epsilon}^{2}$ the experimental error variance estimated on a single trial basis, and *r* is the number of replicates in the trial. This equation indicates that for a given set of genotypes, natural conditions, and management, the trial heritability can be improved only by increasing the number of replicates.

From Eq. 1, we have

Observing the curvilinear relationship between *r* and *H*_{ST} revealed that heritability improves nearly linearly with the increase in *r* when the heritability is lower than a certain level, say, 0.75, and the effect of increasing *r* gradually diminishes after that (Yan et al., 2015). Therefore, a trial may be regarded as optimally replicated when the achieved heritability is equal to 0.75, and the number of replicates required to achieve this level of accuracy can be estimated by Yan et al. (2015):

The so-estimated number of replicates may be referred as the number of replicates for optimal replication. Equation 3 was adopted to estimate the required number of replicates for China national cotton (Xu et al., 2016) and wheat (Zhang et al., 2020) variety trials, the required number of replicates for wheat and cotton variety trials in the Mediterranean regions (Baxevanos et al., 2017a, b), the required number of replicates in soybean variety trials in Brazil (Woyann et al., 2020), and the required number of replicates for winter wheat in California (George and Lundy, 2019).

The heritability on the basis of single-year, multi-location test is determined by

where ${\mathrm{\sigma}}_{G,ML}^{2}$ is the genotypic variance and ${\mathrm{\sigma}}_{\epsilon ,ML}^{2}$ the experimental error variance estimated on the single-year, multi-location trial basis; ${\mathrm{\sigma}}_{GL}^{2}$ is the variance for genotype by location interaction (GL) and *l* is the number of locations. From Eq. 4, the number of replicates required to achieve a target level of heritability on the multi-location basis is determined by:

where *H*_{MML} is the maximum achievable across-location heritability and is determined by:

Equation 6 is a special case of Eq. 4, i.e., the cross-location heritability with 0 experimental error variance or with an infinite number of replicates. The target level of cross-location heritability must be smaller than the maximum possible heritability, i.e., *H*_{ML}¡*H*_{MML}, for Eq. 5 to be meaningful. Therefore, the target cross-location heritability should not be a specific level of *H*_{ML}; rather, it should be a certain level of *H*_{ML}/*H*_{MML}, which may be called “relative cross-location heritability.” The relative cross-location heritability is the measure for adequate replication in the multi-location trial framework. If the target cross-location heritability is set such that *H*_{ML}/*H*_{MML}=0.75, i.e., *H*_{ML}=0.75*H*_{MML}, then Eq. 5 becomes

Because the error variance across locations, ${\mathrm{\sigma}}_{\epsilon ,ML}^{2}$, is the accumulative error variance of individual locations, σ^{2}, the value of $\frac{{\mathrm{\sigma}}_{\epsilon ,ML}^{2}}{l{\mathrm{\sigma}}_{G,ML}^{2}}$ in Eq. 7 should be similar to that of $\frac{{\mathrm{\sigma}}_{\epsilon}^{2}}{{\mathrm{\sigma}}_{G}^{2}}$ in Eq. 3; the optimal number of replicates estimated based on Eq. 7, therefore, shrinks with *H*_{MML}, relative to that based on Eq. 3. The smaller the *H*_{MML}, the greater the shrinkage is, and the fewer replicates will be needed. This may appear counter intuitive. However, it means that when *H*_{MML} is low, the key to improve the cross-location heritability is not to increase the number of replicates; rather, it is to increase the number of locations or to divide the region into meaningful mega-environments.

From Eq. 6 and analogous to Eq. 3, the required number of locations for adequate testing, i.e., to achieve *H*_{MML}=0.75 in a mega-environment, can be estimated by Yan et al. (2015)

More locations would be needed if *H*_{MML}¡0.75, and fewer locations would be needed if *H*_{MML}¿0.75, relative to the actual number of locations used in the test.

The discussion above can be extended to multi-year, multi-location tests. The heritability under the multi-year, multi-location framework, *H*_{MLY}, is determined by

Where ${\mathrm{\sigma}}_{G,MLY}^{2}$ is the genotypic variance and ${\mathrm{\sigma}}_{\epsilon ,MLY}^{2}$ the experimental error variance estimated on the multi-location, multi-year trial basis; ${\mathrm{\sigma}}_{GY}^{2}$ and ${\mathrm{\sigma}}_{GLY}^{2}$ are variances for genotype by year interaction (GY) and genotype by location by year three-way interaction (GLY), respectively, and *y* is the number of years the test is conducted.

From Eq. 9 the number of replicates required to achieve a certain level of heritability is determined by:

where

*H*_{MMLY} is the maximum possible heritability on the multi-year, multi-location basis. Equation 11 is a special case of Eq. 9, i.e., the cross location and year heritability, assuming 0 experimental error variance or infinite number of replicates. If the target heritability is set to *H*_{MLY}=0.75*H*_{MMLY}, then the number of replicates for optimal replication can be determined by

The implications discussed above regarding Eq. 7 can also be extended to Eq. 12. Furthermore, the definition of heritability in Eq. 9 is for variety trial systems in which a single crop is grown each year and is under a single management. Factors such as season and management should be added when multiple crops are grown in the same year (e.g., Swallow and Wehner, 1989) or in agronomic experiments in which multiple managements are involved.

Note that the definition of the heritability at various levels is consistent with the concept of mixed effect models (DeLacy et al., 1996). The genotypic main effect (G), the genotype by environment effects (GL, GY, GLY), and the experimental errors (ε) are treated as random effects as they appear in the formulas of heritability (Eqs 1, 4, and 9). Effects not included in the heritability formulas, such as the main effects of block, location, and year, are treated as fixed effects.

The yield data from the 2015 to 2019 ORDC oat registration test were used as an example in this study (the raw data, which belong to AAFC, are available upon request). Each year the test was conducted at several locations within Ontario as well as in other provinces, including Quebec and Prince Edward Island in eastern Canada and Manitoba and Alberta in western Canada, as listed in Table 1. The trials were conducted with four replicates at the Ontario locations, as required by the Ontario Cereal Crops Committee, and three at locations in other provinces. Each year the same set of 36 oat genotypes were tested at all locations, and the set of genotypes varied each year. All trials were conducted based on randomized complete blocks designs and in rain-fed conditions. Not all locations listed in Table 1 were used in all years. Locations in Ontario have generally lower latitudes (<45.5°N), except for the northern Ontario location New Liskeard (Table 1).

Test locations involved in the 2015–2019 oat registration test and their geographical coordinates (sorted by latitude).

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