THE AREA UNDER THE FUNCTION: AN INDEX FOR SELECTING DESIRABLE GENOTYPES 8 9 c. M. Hernandez, J. Crossa, A. castillo 0 8 9 0 Universidad de Colima, Mexico. International Maize and Wheat Improvement Center (CIMMYT), Apdo. Postal -, Mexico D. F., Mexico. Centro de Estadistica y Calculo, Colegio de Postgraduados 0, Chapingo, Mexico.
\ SUMMARY The linear regression approach has been widely used for selecting high yielding and stable genotypes targeted to 8 9 0 8 9 0 several environments. The genotype mean yield and the regression coefficient of a genotypef/!al performance on an.. ( index of environmental productivity are" 'two main stability parameters. Often, using both can complicate the breeder's decision when comparing high yielding, less stable genotypes with low yielding, stable genotypes. This study proposed to combine the mean yield and regression coefficient into a unified desirability index (Di). Di is defined as the area under the linear regression function divided by the.'\\\v'~/ V\ I difference between the two extreme environmental indexes.adi is equal to the mean of the ith genotype across all environments plus its slope multiplied by the mean of the X: environmental indepes of the two extreme environments (symmetry). Desirable genotypes are those with a large Di. For symmetric trials the desirability index depends largely on the mean yield of the genotype and for asymmetric trials the slope has an important influence on the desirability index. The use of Di was illustrated by a 0 environments maize yield trial and a environments wheat yield trial. ' Three maize genotypes out of nine showed Di that were significantly larger than a hypothetical, stable genotype. These were considered desirable, eventhough two of them had slopes significantly greater than.0.... _..
. ' Key Words: Genotype x environment interaction, Adaptation, stability, Desirability Index. ',
INTRODUCTION Differential genotypic responses to varied environmental conditions are known, in the classical sense, as genotype- S environment interactions (GEI). Their most important effect in plant breeding is to complicate the identification of superior genotypes for general adaptation and for specific 8 environments. 9 The most widely used method for selecting high yielding 0 and stable genotypes is the linear regression fitted by least squares. It was first proposed by Yates and Cochran (98) for analyzing a barley yield trial and used by Finlay and Wilkinson (9) to examine the adaptation of several hundred barley genotypes. A genotypelt stability is shown by its proportional response to the environmental index (regression coefficient). It is considered to be relatively stable if it has a regression line whose slope is near to 8.0. In general, this is a good criterion for selecting a 9 genotype or a group of genotypes targeted to several 0 environments. Eberhart and Russell (9) used the linear regression approach for assessing genotypic yield stability. They proposed pooling the sum of squares for environments and GEI and subdividing it into a linear effects between ' environments, a linear effect for genotype-environment, and a deviation from re~ression. The deviation from the regression line of each genotype is considered another
'. 8 9 0 8 9 0 stability parame' er. A stable genotype has slope equal to one J. a small deviation from regression. However, deviation from regression has been criticized because biological and algebraic interdependencies exist between the slopes and the sum of squares due to deviations from regression (Hardwick and Wood, 9; Westcott, 98). Several authors have pointed out the statistical and biological limitations of regression analysis (Freeman, 9; Hill, 9; Westcott, 98; Lin et al., 98; Crossa, 990). Although a genotype(ljs' slope is the primary stability parameter, not all stable genotypes are desirable, since mean yield is also an important parameter. Often these two parameters complicate a breeder,~ decision when comparing high yielding, less stable genotypes with low yielding, stable genotypes (Crossa, 988). In general, the regression model is used to partition the overall response pattern of a genotype into two components: yield performance and stability. However, an analytical method for examining the total behavior of a genotype across the tested environments should consider both of these components simultaneously. The main objective of this study was to present a simple statistical method for selecting genotypes. The procedure is called the "area under the function" and combinei the two principal parameters (regression coefficient and mean yield) which describe genotype performance into a unified desirability index. The method was illustrated with examples
from an international maize (Zea mays ~.) trial and an international wheat trial (Triticum aestivum L. em Thell.) distributed by the International Maize and Wheat Improvement Center (CIMMYT). MBT~ODOLOGY \... - ' l ~\ FOR DEVELOPING A GENOTYPIC DESIRABILITY INDEX 8 From a breeding perspective, a practical problem of the 9 regression approach is how to reconcile and utilize the two 0 most important parameters, slope and mean yield. Often a high yielding genotype is not stable (~significantly different from.0) or a genotype with ~-l~:may show poor yield response. Relationship between slope and mean yield production of the ith genotype and a standard genotype 8 In this section we examine different situations where the 9 response pattern of a specific genotype, given by its slope 0 and mean yield, is co~pared with the response of a stable genotype.,}) t''' Consider an experiment where "g" genotypes (i l,,,g) are evaluat&d in "e" environments (j l,,..,e). The function that estimates the yield of the ith genotype is where fi(i) ui + ~ii; ui is the mean yield of the ith genotype over all environments and is estimated by Y. ; ~..
is the regression coefficient of the i th genotype on the 8 environmental index (Ij) and is estimated by bi. The enviromental index is the mean of an environment minus the grand mean (Y. - Y ) J Let the function that estimates.the yield of a standard genotype "s" be defined by Ys fs(i) where fs(i) u + I; u is the mean of all genotypes in all 9 the environments and is estimated by the overall mean (Y ). 0 By definition, the standard genotype is stable because ~s l. The relationships between the two parameters, slope and mean yield production, under the models Yi and Ys can be displayed in a diagram where phenotypic values of the ith genotype and the standard genotype are regressed on the environmental indexes (Ia being the lowest yielding environment and Ib the highest yielding environment). Figure displays six hypothetical cases. In case the ith 8 genotype is unstable (bi greater than.0); however it can 9 be considered a desirable genotype because its yield 0 production is better than that of the standard genotype in all environments (Fig. A). In case the ith genotype is unstable (bi less than.0) and its yield is lo~er than the standard ~enotype in all environments (Fig. lb). Note thai in both cases GEI exists but without change in ge~otypic performance across environments. In cases and the ith genotype is unstable (bi is greater than.0 in Fig. lc and less than.0 in Fig. ld) and yield is higher than that of
8 the standard genotype in some high and low yielding environments, respectively. Both cases show genotypic rank change across environments. In case the ith genotype is stable (bi l.o) and more productive than the standard one (Fig. le), whereas in case the ith genotype is stable (bi l.o) and yield is lower than that of the standard genotype (Fig. lr). When both genotypes are stable (~.0; 8 case and case ), it is clear.that the ith genotype would 9 be desirable only if it shows better yield performance than 0 the standard one (Case ). Clearly the response of the ith genotype in Fig. la is much more desirable than the response in rig. le. It represents a response to favorable environments and also superiority in stress environments. For the cases where the ith genotype is unstable but more productive than the standard genotype in all environments (Case ) or unstable and more productive than the standard genotype in only some environments (Case and Case ) the 8 decision as to whether ~r not the ith genotype should be 9 selected is not clear. For these cases it would be useful to 0 obtain an index that includes both stability parameters. The desirability index of the ith genotypes expressed as the area under the linear function Desirability index
9 The area (A ) under the linear fun~tion gives a 8 9 0 8 9 0 combined estimate of the influence oi Y. and b.. i.. Because Ai is an area, it should be converted to a unit of measurement that is easier to interpret. By the fundamental theorem of calculus dividing Ai by the distance between the two extreme environment indexes (Ia and Ib) we obtain the desirability index (Di): Di Ai/(Ib-Ia). This value represents the mean expected yield of the ith genotype expressed in performance units when tested in any of the environments between Ia and Ib. The area under the linear function (A ) is the integral of fi(i) over the range of the tested environments (Ia,Ib) such that Ib Oi [J f (I) d(i)]/(ib-ia) [J (Yi. + b I) d(i)]/(ib-ia) Ia Ib [ ( Y i. ) I I + ( b i ) ( I )/ I I ( I b-i a) - ra [(Yi.)(Ib-Ia) + (bi){(ib ~Ia )/}]/(Ib-Ia). Therefore, Di Yi. + (b )c where c (Ib+Ia)/ is the mean of the two extreme environmental indexes and defines the asymmetry ot the Ia distribution of environmental indexes around zero (c -o for complete symmetry). The contribution of the slope to the desirability index increases with asymmetry. Ib Ia Ib
0 Similarly, the desirability index (0 8 ) of the standard genotype is defined as the area under its linear function (A ) divided by the difference between the two extreme s environmental indexes (Ib-Ia): Ds As/(Ib-Ia) where As is the integral of fs(i) over the range of environments such that 8 9 Ib Ib 0 Ds (J fs(i) d(i)]/(ib-ia) [J (Y +I) d(i)]/(ib-ia) Ia Ia ((Y.. ) (Ib-Ia) + (Ib -Ia )/]/(Ib-Ia) Y.. + c where c is the same as above. According to these results, the desirability index of a genotype depends on its mean yield, the slope of the regre'ssion line, and the asymmetry (C ) of the experiment. 8 When the two extreme environmental indexes approach complete 9 0 symmetry,. ~yon Y. and the overall mean (Y ), respectively. In I l. this case, the slopes of the genotypes~ regression lines do not influence the desirability index, thus the desirable genotypes are those with higher mean yields (higher o ). However, when the two extreme environmental indexes are asymmetric (c tends to be different than 0), two different cases are possible: ) when \r ~> ~~ and c <0, then Di and c are inversely related. For two genotypes with the same mean
yield the regression line of the desirable one (with higher Di) will have a smaller slope (bi); ) Ia<Ib and c >0; then Di and c are directly related. For two genotypes with the same mean yield, the one with greater slope will have a 8 9 0 greater Di. The desirability index of a genotype with quadratic or cubic responses can be calculated. For a quadratic response ~, the corresponding coefficient ~calculated as (Ib -Ia )/, whereas for a cubic pattern its coefficient is {Ib -Ia )/. Comparing the area under the function of the ith genotype with a constant: Hypothesis testing 8 9 0 A linear model testing methodology (Draper and Smith, 9) is used to test hypotheses about the parameter o and its comparison with D 8 Let the vector C be (C, c ) and the vector b be {Yi., b.); then D. can be written as D. C' b and the variance of Di is estimated by s D. (C' (I'I)- C) (MSE/r) where MSE is the pooled error, r is the number of replicates, and I is a matrix containing ones in the first column and the environmental indexes in the second column. To test the null hypothesis Ho:D Ds' we use the following F test
where MSHo (C'bi- o 8 ) /S Di" The value Fe is compared with the F value from the table with one degreej of freedom (, df error). As pointed out in the previous section, those genotypes with values of Di that are significantly larger than Ds are the most desirable. 8 Trial l 9 0 8 9 0 A maize experimental variety trial (EVT ) provides a, simple illustration of the use of the area under the model for characterizing desirable genotypes. The trial had 9 genotypes tested in 0 international environments in a randomized complete block design with replicates. The response variable was grain yield (kg ha- ). The highest yielding genotypes are,,, and 9 (Table ). Of these, genotypes and 9 are stable (siopes not significantly different from.0), whereas genotypes and were unstable (slopes significantly greater than.0). The asymmetry of this trial is c - (Ia -,0 kg ha-l and - Ib,8 kg ha ). Therefore, the desirability index for the ith genotype is D. Yi. + (b. )(). The overall mean of the, I. - " ~~> experiment was,88 kg ha /aad' the desirability index for the standard genotype was Ds S08 kg ha- Genotypes,,,, and 8 have values of Di that were significantly different from those of the standard genotype (Table ). Of these, genotypes,, and were superior because they had
values of Di that were significantly larger ~han the standard genotype. In contrast, genotype 8 had the lowest mean yield response and the lowest Di. The response pattern of the 9 genotypes given by their regression coefficients and mean yields was compared with the response of the hypothetical standard genotype (b.0; Figs. to 0). Deviations from the average response can 8 also be seen from these graphs. Genotypes and represent 9 refer~nce case, but only the latter had values of Di 0 significantly smaller than Ds. Genotype corresponded to case, but Di is not different from Ds (Table ). Genotypes and belong to Case ; they had slopes greater than one (Table ) and a Di that was significantly larger than Ds. Case is typified by genotype, which showed the highest mean yield and therefore the largest desirability index. Genotypes and 8 fall into case ; however, the latter had a slope that was less than.0 and a Di significantly 8 smaller than D (Table ). Genotype 8 not only had a low Di 9 but has large deviations from regression (Fig. 9). Although 0 genotype 9 belongs to case, its Di does not differ significantly from Ds. For this trial, mean yields were the predominant factor contributing to Di and were highly correlated to the desirability index. However, for trials with higher asymmetry than that of EVT, the slope of the regression line will have a greater influence on D. and thus the
results of ranking genotypes on mean yield should differ from a ranking based on Di. Trial This data come from a elite spring wheat trial (ESWYT 8) that included 8 genotypes planted in international 8 environments in a randomized complete with replicates. 9 Grain yield is the response variable (kg ha- ). 0 Genotype has the highest mean yield but it is unstable because has a slope significantly different from one (bi l.lo}. Genotype has the highest Di ( kg ha- ) followed by genotypes 9 and with Di's of and 80 kg h - a, r~spectively. Genotype has a relatively high mean yield ( 0 kg ha- ) but according to its slope (b -.) it is unstable. The asymmetry of this trial is c -8 (Ia - kg ha-l and Ib 98 kg ha- ), that is, trial is about 8 five times more asymmetric than trail. The desirability 9 index for the ith genotype is Di Yi + (bi) (8), whereas 0 os-80 kg ha-. For this trial, both parameters mean yields and slope were important factors contributing to o Results of ranking genotypes on mean yield differ from a ranking based ' on o For example genotype 9 was ranked second based on mean yield and third based on Di; genotype ranked third based on mean yield and second based on Di; genotype ranked loth based on mean yield and th based on Di.
Conclusions The desirability index, expressed as the area under the regression function, attempts to quantify what most plant breeders actually do, that is, to use both the mean yield and the regression coefficient for determining the 8 desirability of a genotype. This index can facilitate the 9 breeder's decision when selecting superior genotypes 0 especially when high yielding genotypes have slope greater than one (i.e., genotypes and on the maize trial). For trials where the environmental indexes are fairly symmetric (trial ), the genotypic desirability index is highly dependent on the mean yield. In this case, superior genotypes selected by their mean yield or by the desirability indexes are the same. However, for trials with a more asymmetric such as trial, distribution of 8 environmental indexes, the slope of the regression line is 9 an important factor, and therefore selecting on the basis of 0 mean yield will identify different genotypes from those selected using the desirability index. ACKNOWLEDGEMENTS The authors thank Drs. c.o. Gardner and c. Qualset for their helpful comments on the manuscript.
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- Table. Mean yield (Yi) (kg ha ), slope of the regression line (bi) and desirability index (Di) in kg ha-l of 9 maize genotypes (gi) at 0 environments (trial ). Genotype Yield 8 9 0 8 9 0 * + 8 0.98 80 0 0.9 8+ 8 0.9 08 8.* ++.* ++ 0.0 0++ 0.8 8 8 8 0.** 0++ 9 90. 8 ** Significant at the % and % probability level, respectively. ++ Significantly different from Ds at the % and % probability level, respectively.