analysis of genotype-environment interactions based on the method of path coefficient analysis
TRANSCRIPT
ANALYSIS OF GENOTYPE-ENVIRONMENT INTERACTIONS BASED ON THE METHOD OF BATH
COEFFICIENT ANALYSIS
G. C. C. TAI Agriculture Canada Research Station, Fredericton, Nerv Brlinswick
A method of investigating genotype-environment interactions of field crops is presented. Based on the concept that yield components are determined sequentially at different stages in the ontology of plants and the hypothesis that the environmental resources can be separated into independent groups with each contributing to the development of a component trait, a causal relationship between environmental resources, component traits and yield is established (Fig. I ) . The GE interaction effect is then represented by three multiplicative terms which are composed of three genotypic and three environmental components. These components are estimated using the method of path coefficient analysis based on the postulated causal relationship. Data of marketable yield and yield components of seven potato cultivars collected from two series of trials are examined by the proposed method.
Introduction Recently much attention has been paid to the statistical analysis of genotype-
environment (GE) interactions. An excellent review of this subject was given by Freeman (1973). In his review, suggestions are made on the use of the various methods of multivariate analysis in the study of GE interactions especially when the simpler methods such as the joint regression analysis do not give clear answers. One example of the use of multivariate methods was the principle component analysis of carrot, data reported by Freeman and Dowker (1973).
In a series of papers, Grafius (1969), Grafius and Thomas (1971) and Thomas et erl. (1971 a, b, c) presented the concept of a sequential develcdpment process of yield compcments. The components of yield in their studies were X, the number of heads per plant; Y, the number of seeds per head; and Z, the average seed weight. The chronological developmental sequence of the components is X to Y to Z. Yield, W, is a multiplicative product of the components, i.e., W = XYZ. A transformation technique was given by Thomas et nl. (1 97 1 a) to remove the part of variation of a component trait which is contributed by the trait(s) which appear earlier in the development sequence.
Rasmusson and Cannell (1970) pointed out that yield components of cereal crops are determined at different stages in the ontogeny of plants and thus are differentially affected by variation in environment. They suggested that the three yield components in cereals are affected by independent environmental factors during the same or different periods of plant development.
The aim of the present paper is to suggest the use of the method of path coefficient analysis to study GE interactions based on the concept of sequential development of yield components. Data of marketable yield and yield components of potato (Solanuin tuberosurn L.) collected from two series of trials in New Brunswick are then examined by the proposed method.
A New Mathematical Model for Yield The fact that yield components are formed in sequence results in a different
relationship between a component trait and the environmental resources. The develop-
Manuscript received November 1 I , 1974.
Can. J. Genet. Cytol. 17: 141-149, 1975.
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ment of the first component trait is entirely supported and hence solely determined by the resources available during the early stage of growth. A component trait whose development is subsequent to others, however, is not only influenced by the resources available during its formation, but by the development and conformation of its predecessors. The mechanism for controlling the formation s f a yield component is thus increasingly complicated along with the timing of its ontsgeny in the chronological development sequence.
We assume that the environmental resources can be hypothetically separated into independent groups. These resources are available to the process of plant development at sequential stages in which the yield components are formed. Using the WXYZ system, a cause-effect diagram can be constructed as shown in Fig. 1 . The three. independent groups of environmental resources are represented as W,, W p and W3 respectively in Fig. 1 . The relations of the R's to the component traits, X, Y and Z, and yield, W, are self-explanatory in the diagram.
The method of path coefficient analysis of Wright (1921, 1934) is applied to the causation diagram in Fig. 1 to determine the relation between the three groups of resources and yield. Let p,,. p,,, p,,, p,,, p,,, and p,, be the correlation coefficients between yield components and yield and a,, . . ., as be the corresponding path coefficients, we have
Fig. 1 . A causation diagram showing the developmental relationship between yield (W) and yield components: number of sterns/plot (X), number of tubers/stern (Y) , and average tuber weight (Z). The lower case Betters on the various single-anow paths are path coefficients. R,, R, and R3 indicate three independent groups o f environmental resources.
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PATH ANALYSIS OF GE INTERACTIONS 143
The six path coefficients can be obtained by solving the above simultaneous equations, i.e.,
A = A-lp (2) where
A' = (al a2 as a, a, as) P = (PXY Pxs PYZ Pxw PYW pzw)
and 1 0 0 0 0 0
A =t kXY rxy : : : ) 0 0 Q 1 PXY Pxz 0 0 0 PXY 1 Pyz 0 0 0 Pxz pyz 1
The path coefficients from R,, R2 and R, to X, Y and Z are, respectively, u l = k l u2 = iz(l-a3l/2 u3 = i ~ ( l - a ~ p ~ ~ - a ~ p ~ $ ~ h (3)
These coefficients can be either positive or negative depending on the scaling system we wish to use to account for the effects of per unit inputs of the three independent groups of environmental resources on the yield components. The positive values will be used throughout the paper.
Let w, r,, r, and r3 represent yield and the three independent groups of environmental resources measured in standard deviation units, the following relation can be obtained:
w = vir, + v,Pr2 + vArg + e' (4) in which vi , v,P, and vi are the path coefficients from R,, R, and R, to yield, W, and e" the residual effect. The three path coefficients can be derived as follows:
vi = u1 (a4 + alas + a2a6 + a1a3a6) = ulpxw v i = up (asas + a,) ( 5 ) v; = u3a6 If a group of m genotypes is tested over n environments, the yield of the ith
genotype in the jrh environment can be expressed as:
wi j ' Ldwi + vlirlj + v2ir21 + V3ir31 + eid @I where vgi = v",iaWi for g = 1,2 and 3 and ~ 2 , . ~ is the variance of yields of the ith genotype. This formula represents a new mathematical model for the observed yield, Wii. It is composed by a mean genotypic effect, pWb three multiplicative terns of the genotype-environment interaction effects formed by three genotypic components, vli, v2i and v3i, and three environmental components, rlj, r2j and r3j, and an error deviate, eij. The three genotypic components each represent the efficiency s f a genotype to utilize a standard deviation unit input in one of the three environmental components during the succeeding stages of plant development for the formation of final yield.
Materials and Methods Data for seven potato cultivars derived from two series of trials were pooled f ~ r
this study. The seven cultivars were Avon, Fundy, Hunter, Katahdin, Kennebec, Netted Gem and Sable. In the first series the seven cultivars were each planted as a 100-hill row over three sites in 197 1, 1972 and 1973. Each row was divided into 18-hill sections and harvested consecutively at a bi-weekly interval from early July to early October. Only
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one set of data from plants harvested in September was taken from each site for the present study. All cultivars except Netted Gem had 25.4 cm spacing between adjacent plants. A 45.7 cm spacing was used for the latter. The spacing between adjacent rows was 9 1.4 cm.
In the second series of trials the seven cultivars were tested in two replicates over three sites in 1973. Single row plots of 10 hills each with 25.4 cm spacing between adjacent hills were used for all cultivars. The spacing between adjacent rows was 91.4 cm.
Fifteen sets of data were available for analysis in the present study. These were taken as representing fifteen "environments'. More information on the fifteen environ- ments can be found in Table I.
Data of marketable tuber yield in kg/plot (W, one plot = 23.2 m y and three yield components: number of stems/plst (X), number of tubers/stem (Y), and average tuber weight in kg/tuber (%) were derived from the data of number and weight of marketable tubers, number and weight of all tubers and number of steins in a plot. It is noted that the multiplicative product of the three yield components fornas total yield. Marketable yield rather than total yield was used in this study because only the former is of economic importance.
Simple correlation coefficients between marketable yield and yield components were estimated for each of the seven cultivars. The six correlations of a cultivar were used in (2) to obtain estimates of the six path coefficients a, to as which, in turn, were used in (3) for estimating u,, u, and u$ values. The three genotypic components of the GE interactions for a cultivar were finally estimated using (5 ) .
Using sample estimates of mean yield and genotypic components of the GE interactions for each of the seven cultivars in (61, the three environmental components rIj , rZj and r,i, of the GE interactions of each of the fifteen environments were estin~ated in standard units by the method of least squares.
TABLE I Descriptions of the Fifteen environments used for potato trials
Planting Harvesting Environment date in date in
no. Site Year Replicate May September
I Benton* 1971 - 18 17 2 Centrevilk 197 1 - 19 17 3 Grand Falls 197 1 - 20 17 4 Grand Halls 1972 - 25 7 5 Benton 1972 - 10 7 6 CentrevilIe I972 - 1 1 11 7 Benton 1973 - 14 5 8 Centreville 1973 - 15 5 9 Fredericton 1973 - I8 4
10 Fredericton 1973 1 B 8 26 I I Fredericton 1973 2 18 26 12 Benton 1973 B 14 2 4 13 Benton I973 2 14 24 14 CentreviHe 1973 I 15 2 5 1 5 GentrevilIe 1973 2 15 25
----A -- *A11 sites are in New Brunswick
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PATH ANALYSIS OF GE INTERACTIONS 145
Shukla's stability variance (Shukla, 1972b) was estimated for marketable yields of each of the seven cultivars. The yield stability information was then used to compare the results of the genetic and environmental component estimates of the GE interactions.
Results Correlation coefficients between yield and yield components for each of the seven
cultivars are shown in Table 11. The correlation structures of the cultivars were quite different from one another. The correlation between nunzber of stems per plot and average tuber weight, for example, was negative for all cultivars but Fundy. The latter had a highly significant positive correlation. Genotypic and environmental components of the GE interactions for the seven cultivars were estimated using the correlations in Table 11. The results are given in Table 111 and Table IV.
Mean marketable yields and estimates of the three genotypic and environmental components of the GE interactions (Table I11 and Table IV) were used to obtain fitted yields for the seven cultivars in the fiftken environments. The correlation coefficient between the observed and fitted yield data was high, r = 0.77. The distributions of the fitted yields with the observed ones are shown in Fig. 2.
All cultivars showed positive and large estimates of the v, component. The estimates of the v, component were positive for six of the seven cultivars except Avon. The ?, estimates were about 30 to 80% of the sizes of the corresponding ?, values. With the exception of Katahdin which had a distinctly negative value, the ?, estimates of the other six cultivars were small or positive. The estimates were also considerably smaller than the corresponding ?,values.
Correlation coefficients were calculated between mean yields and the three genotypic components of the GE interactions. The correlation analyses were carried out both with or without Katahdin. The results are shown in Table V. An interesting result was the highly positive correlation between the ?, and ?, estimates when Katahdin was excluded from the analysis. This suggested that for six of the tested cultivars the genotypic responses to the first and third groups of environmental resources were probably controlled by one genic system.
The cultivars Kennebec and Fundy were the highest yielders with the largest positive estimates of 9 , and ?, components (Table 111). Kennebec also had the largest ?, estimate. The ?, estimate of Fundy was positive but the smallest in absolute value among the seven cultivars. Avon and Sable had slightly lower mean yields than Kennebec and Fundy but the lowest ?, and ?, estimates. Avon was the only cultivar
Correlation coefficients between yield and yield components for seven cultivars grown over 15 environments
---- - - .- -
Avon Fundy Hunter Katahdin Kennebec N . Gem Sable - - -- - -- -
Yield and No. stemslplot -.1M .46 .31 -.30 .44 .23 -.02 No. tuberslstelma .49 - .29 .47 .59* .38 . 1 1 .41 Av. tuber wt .25 .77** .42 .56* .35 .58* .24
No. stemslplot and No. tuberslstem .15 - . $ I * * .21 -.46 - .02 -.64** -.66** Av. tuber wt. -.71** .65*".52* -.46 - . I7 - .32 -.56*
No. tuberslstem and Av. tuber wt. -.56* -.70** -.32 -.09 - . - 54* - .04 -.01
- - - *Significantly different from r = 0 at the 5 % level .
**Significantly different from r = 0 at the I B' level .
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showing a negative but moderate .G2 estimate. The 6, estin~ate of Sable was positive and moderately high. Hunter and Netted Gem had low mean yields and nloderately positive estimates of the three genotypic components. Katahdin, as indicated before, had a distinctly negative .GI estimate. This cultivar and Kennebec were the only two genotypes showing a high Bevel of responses to all three environmental components of the GE interactions. However, their reactions to the first environmental component were in entirely opposite directions.
Estimates of Shukla's stability variance (Shukla, 1972b) indicated the following order of yield stability: Sable, Avon, Katahdin, Netted Gem, Hunter, Kennebec and Fundy (see 3 in Table 111). Tests of equality of any pair sf the stability variance estimates were cmied out according to Shukla (1972b). Significant differences were obtained between the stability variance estimates of Fundy and Sable, Fundy and Avon, and Kennebec and Sable. It appeared that the sizes o f 3 increased with increasing G I
TABLE III Estimates of mean yields, genotypic components of GE interactions and
stability variances of seven potato varieties -- --
Genotypic components of GE interactions Mean Mark. Stability
Variety yield (kgjplot) 0, 0 2 C, variances, &f
Avon 7.09 Fhendy 8.32 Hunter 6.38 Katakdin 6.58 Kennebec 8.26 Netted Gem 5.50 Sable 7.76 Mean 7.12
Estimates of mean yields amd environmental components sf the GE interactions of fifteen environments
-- - Environmental components of GE interactions
Env. Mean no. yield 7 I r2 r3
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PATH ANALYSIS OF GE INTERACTIONS 14'7
and 8, values and decreased with increasing G2 values. Fundy, for example, had high 8, and G, and low C2 estimates. Sable. on the other hand, had low C1 and G, and a moderately high G, estimates. Kennebec had 0, and G3 estimates higher than those of Fundy but a smaller estimate of stability variance, apparently because the former had a highly positive G, estimate. The relationship between 3 and G's for other cultivars can be similarly interpreted. A multiple correlation coefficient was calculated between 8 and the three genotypic component estimates which was highly significant, W = 0.93.
OBSERVED, KGIPLOT
Fig. 2. Distributions of observed and fitted marketable yields of seven potato varieties tested in fifteen environments.
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TABLE V Correlation coefficients betueen mean yield and three genotypic components of GE interactions with or
without the cultivar Katahdin in the analysis - -- P -
Gtv& e l Gw & 0, $, & $3 e l & 0, e l & C3 $,& c, -
With Katahdin .45 - .03 .30 .08 .63 .68 Without Katahdin .38 .13 .37 .55 .Y8* * .69 -- .- -- -
**Sign~ficant at the 1 9 level.
A high multiple correlation coefficient was also obtained between the mean yields of environments and the three environmental component estimates (Table IV), R = 0.91. The least squares estimates of the three hypothetically independent environmental components thus represented a satisfactory and reasonable separation of the environ- mental effects. The third environmental component was apparently the most influential one on yield formation which alone had a highly significant correlation with the mean yields of the fifteen environments, r = 0.8 1 . There was a general trend of low yield for the trials in 1973 (Table IV). Inspection of the results of the environmental component estimates in Table IV reveals that r̂, and r̂, estimates were either low or negative in 1973. No clear relation, however. can be found of the ?, estimates with sites or years.
Discussion The method presented in this paper is an application of factor analysis. Factor
analysis, as pointed out by Lawley and Maxwell (1963)' attempts to account for a matrix of covariances or correlations of a group of variates by a small number of hypothetical 'factors'. The basic assumption of the present analysis was that there were three independent 'common factors', i.e., the three environmental components r,, r, and r, in (4) and (51, which accounted for most of the variation of any of the tested genotypes which were grown over a series of environments. The assumption of three common factors and the determination of the three factor loadings of a genotype, which were actually the path coefficients v,', v,' and v,' in (4), was based on the concept of sequential development of the three yield components. No rotation of the factor axes was attempted as is usually done in most factor analyses since the positions of the three factors were decided through the postulated causal relationship of yield and yield components. The success of the method depends on the validity of the causation scheme and whether the three environmental components of the GE interactions were in fact conamon among the tested genotypes. An overall assessment of the situation would be to investigate the relationship between the observed and estimated yields. A satisfactory linear relationship was obtained from the potato data used in this study. Thus some conclusions can be drawn with respect to the GE interactions based on the new method of analysis.
There appeared to be a joint genetic control for six of the seven cultivars in their responses to the changes of the first and third environmental components. A cultivar such as Fundy which is inclined to vary substantially in its number of stems in response to the changes of the environmental conditions before and during the period of stem emergence and growth, also reacted sensitively to the environmental changes during the period of tuber bulking. However, the fact that Katahdin did not join others in showing such an association proved the existence of another genetic strategy or strategies for controlling the sequential developmental processes of stem emergence and growth. tuberization, and tuber bulking. Undoubtedly, similar studies involving more genotypes
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PATH ANALYSIS OF GE INTERACTIONS 149
would be required to reveal more about the genetics of the developmental dynamics of potato plants when grown over a range of environments.
The process of tuberization which was reflected by the second yield component appeared to hold the key to the stability of the marketable yield. Over the range of cultivars studied, a cultivar which was able to produce a flexible number of tubers per stem according to the environmental conditions in that period gave more stable yield performance than did less flexible others grown in the same set of environments. This was especially so for those genotypes showing strong responses to the other two environmental components as well. Nine of the 15 environments used were in the same year (1 973) when both the second and third environmental components were poor. That is to say, a majority of the environments in which the trials were conducted gave a poor supply of resources during the later stages of the potato growth including the developmental process of tuberization and tuber bulking. Yield stability was achieved by a genotype which was capable of controlling its number of tubers per stem and hence allowed a high proportion of tubers to reach marketable size in those environments.
All cultivars showed the highest response to the final environmental component. This suggested that the availability of the resources during the period of tuber bulking was more critical and variable than those supporting the two previous stages of potato growth. Conversely, the relative importance of the first and second environmental components on the marketable yield was not as pronounced and it depended on the responses of individual cultivars.
Acknowledgments I wish to thank Drs. Perry Jui and C. S . Lin for going through the manuscript and
making some valuable suggestions; to Drs. G. M. Weaver, G. C. Misener, H. De Jong and D. A. Young for their critical reading and correction of the manuscript.
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