‘—..w«——"’ .4 . _W <‘+K‘;‘ 5:y:§'3"" ' . . , . 7 . . .’_ ' FACTOR ANALYSIS OF 7 ~ PLANT- TYPE VARIABLES RELATED TO YIELD OF DRY BEANS (PHASEOLUS VULGARIS L ) Thesis for the Degree of Ph. D.. MICHIGAN STATE UNIVERSITY JACQUES CLAREL DENIS. : 1971 ry*~‘ lllllllillilllllll it. . ““1 7 LIBRARY I; Michigan 3 $2 a“: A ',. Univtif)‘ If. ’1 ., ’ w. 9v . 1Mmm§ This is to certify that the thesis entitled Factor Analysis of Plant-Type Variables Related to Yield of Dry Beans (Phaseolus V l ' L. _u_g§_13 Tr)esented by Jacques Clarel Denis has been accepted towards fulfillment of the requirements for Ph.D. demvein Crop and Soil Sciences Department Major professor November 12, 1971 Date ______— %%% I; M (532920219102 ABSTRACT FACTOR ANALYSIS OF PLANT-TYPE VARIABLES RELATED TO YIELD OF DRY BEANS (PHASEOLUS VU'LGARIS L.) BY Jacques Clarel Denis Negative correlations between the components of yield, namely num- ber of pods per plant (X), number of seeds per pod (Y) and seed weight (Z), are often seen as major barriers to yield improvement in beans. A question is raised as to whether these traits are part of a larger set of variables which determine collectively the pattern of production of the bean plant. In trying to answer this question, an image factor analysis was carried out on a set of twenty-two variables involved in fixing the yield potential of the plant. The material consisted of sixteen varieties, of which eight are determinate, eight indeterminate or six are light-seeded, five have me- dium seed weight and five are heavy-seeded lines. They were grown at two locations in Michigan during the summer of 1970. A randomized block design with four replications and single-row plots was used at both 10- cations. But, in order to accentuate the differences between the two environments, a standard number of sixty seeds per plot was sown at East Lansing, the first location, whereas a standard weight of fifteen grams was planted at Gratiot, the second location. The rows were alWays Jacques Clarel Denis 6 meters long and 50 centimeters apart. Observations were made at matu- rity and all analyses were performed on the logarithms of the averages over five plants chosen at random in every plot. The twenty-two traits obtained from these observations and included in this study were : to- tal number of nodes per plant, number of nodes per plant with pod-bear- ing branches, total number of branches per plant, number of racemes on the branches per plant, number of pods on the branches per plant, total number of racemes per plant, total number of pods per plant, average number of seeds per pod, average seed weight, average number of pods per productive node, average plant weight, average pod weight, average pod breadth, average pod thickness, average pod length, number of bas- sal or short internodes, number of upper or long internodes, average short-internode length, average long-internode length, average hypocotyl diameter, average short-internode diameter, average long-internode dia- meter, numbered in the same order they are listed here. A full model of factor analysis was assumed and squared multiple correlations were used as communalities. The correlation matrix with these communalities as e- lements of the leading diagonal was then adjusted and became the image covariance matrix, the input to image analysis. The analyses were per- formed on total variation, and only the factors with eigenroot greater than unity were extracted. The results obtained with the complete-set of data were checked through analyses of sub-sets of data, namely ana- lyses by location, by growth habit, by seed-weight and double-checked through the use of multiple discriminant analysis. Jacques Clarel Denis Three major factors or patterns of production were found and together they accounted for 83% of the total variation before rota- tion and about 77% after rotation. The first two were about evenly important and each extracted after rotation 31% of the variance. The first factor or pattern of production was identified as pod-weight, and characterized a plant-type that will reach a high yield through a high seed weight. This plant has long and sturdy basal internodes, no or very few long internodes and it produces very large pods. The second pattern of production identified a plant-type capable of reae ching a high yield through a high number of pods per plant. Like the pod-weight type, the pod-number type has no or very few long interno- des but it has instead many moderately sturdy short internodes. Too many nodes will provoke a reduction in the yield of either type. It has been noted that although these two types can be inproved to yield more, far better yield can be obtained with intermediate plant-types resulted from.inter%breeding between them. New varieties can thus be produced that will approximate the ideals of the plant breeder for ma- ximum.yield on a per-plant basis or an area basis, depending on the number of branches judged acceptable by assuming that the greater their number, stronger a competitor the plant might be. The third factor, a growth factor, characterized by an excessive vegetative growth coupled with a very low production, was seen as a limit to this inter-breeding° The factors leading to these conclusions are statistically uncorre- but they are not believed to be so from.a biological standpoint. So, it Jacques Clarel Denis would be very interesting to submit these factors to an oblique rotation and determine their natural association. This can be seen as an important next step, in view of its implication on the possibility of combining maximally the characteristics of the two major plant-types encountered here. FACTOR ANALYSIS OF PLANT-TYPE VARIABLES RELATED TO.YIELD OF DRY BEANS (PHASEOLUS VULGARIS L.) BY Jacques Clarel Denis A THESIS . Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Crop and Soil Sciences 1971 This thesis is dedicated to those whose labor or thoughts have contributed to the advancement of Science in general, and Agronomy in particular, but whose names, at some point in history, for one reason or another, have ceased to be remembered. ii ACKNOWLEDGEMENTS The author expresses his gratitude and appreciation to Dr. M. W. Adams, the chairman of his advisory committee, for his gui- dance and support at every stage of the realization of this study. His sincere thanks go to Drs. C. E. Cress, J. W. Hanover and W. L. Myers, as members of his guidance committee, for their much appreciated help. A Special thank to Mr. Brian L. Doyle, for the patience and care he put in helping me obtain the data. To my wife, Wilberte, my children, Nancy and Charles, for their love and understanding, I want to say : thank you and may God bless you. Finally, I wish to acknowledge the financial support I received from the Department of Crop and Soil Sciences, without which my pro- gram of study would not probably even be started. iii TABLE OF CONTENTS INTRODUCTION REVIEW OF LITERATURE MATERIALS AND METHODS RESULTS Factor Analysis of the Complete-Set of Data Factor Analysis of Individual Locations Factor Analysis of Determinate and Indeterminate Varieties as Sub-Sets Comparison between the Factor-vectors Extracted in Analyses of Determinate and Indeterminate Varieties Factor Analysis for the Three Seed-weight Groups Comparison between the Factor-Vectors Extracted from Analyses of the Seed-Weight Groups DISCUSSION Interpretation of the Results Implications of the results on the Development of an Improved-Plant-Type in Beans SUMMARY AND CONCLUSION BIBLIOGRAPHY iv 14 14 18 20 22 28 33 33 35 38 40 Table 3a 3b 5a 5b 5c 6a 6b LIST OF TABLES AND FIGURES Factor Analysis Results, Complete Set of Data Factor Analysis Results, East Lansing Set Factor Analysis Results, Gratiot Set Factor Relationships between East Lansing and Gratiot Analyses (Correlations) Relationships between Identical pairs of Variables from East Lansing and Gratiot Analyses Factor Analysis Results Determinate Lines Factor Analysis Results, Indeterminate Lines Factor Relationships between Determinate and Indeterminate Lines Analyses (Correlations) Relationships between Identical pairs of Variables from Analyses of Determinate and Indeterminate Lines Multiple Discriminant Analysis between Determinate and Indeterminate Lines Factor Analysis Results, Light-Seeded Lines Relationships between Factors of the Light and Heavy-Seed Lines (Correlations) Relationships between Identical pairs of Variables from Analyses of Light and Heavy Sets Factor Analysis Results, Medium-Weight Lines Page 15 l7 l7 l9 19 21 21 23 23 24 26 29 29 26 Table 7a 7b 8a 8b 8c Figure LIST OF TABLES AND FIGURES (Cont.) Relationships between Factors of the Light- Seeded and Medium—Weight Lines (Correlations) Relationships between Identical Pairs of Variables from Analyses of Light-Seeded and Medium-Weight Lines Factor Analysis Results in Heavy-Seeded Lines Relationships between Factors of Heavy-Seeded and Medium-Weight Lines (Correlations) Relationships between Identical Pairs of Variables from Analyses of Heavy-Seeded and Medium-weight Lines Multiple Discriminant Analysis Between Light- Seeded, Medium-Weight and Heavy-Seeded Lines Comparison of Size of Loadings for the Variables included in the Three Main Rotated Factors vi Page 31 31 29 31 31 32 15 INTRODUCTION The yield plateau of the present varieties of dry beans, Phaseolus vulgaris L}, is low compared to that of many other field crops. Although several reasons for low yields in beans can be adduced, I shall mention two problems that are seen as major barriers to yield improve- ment: one, the physical limitation of the present bean plant-types; and two, the faulty synchionization that often prevails between the demands of the plant for environmental resources and the availability of these resources to the plant over time. Bean varieties are either bush or vine. Both types can produce a relatively high number of flowers but retain a much reduced number after fertilization. Of these retained and transformed into pods, still fewer will reach the harvest stage. This explains in part the close associa- tion generally observed between the number of pods per plant (X) and the total seed yield (W). Seldom can this association be exploited in a plant yield improvement program however, since along with it there exist also negative correlations between the yield components, namely the number of pods per plant (X), the number of seeds per pod (Y) and the average seed weight (Z); here yield is taken as the product of these three components. Some of the questions that can be raised at this point are: Is it the waste of nutrients occasioned in part by the excessive production of flowers which is responsible for the negative correlations between yield components, thus implying some kind of limit on food availability together with a marked competition between the components for that food? Or, is it instead a differential through time in the amount and quality of environmental resources available and/or in the photosynthetic capacity of the plant which is the main reason for both the high correlation between X and W and the negative correlations be- tween the components? These two questions are relevant particularly to the second problem mentioned above. But, one can also ask: Are these components the only variables involved or are they only part of a much larger set of traits which together determine the pattern and level of production of the bean plant. To answer this question is to determine whether these patterns exist or not and if so, how they are associated with the different bean plant types. This is the main objective of the present work and it is seen as a first step in the understanding of the problem of physical limitations of the present bean plant types, referred to above. Factor analysis is the statistical tool that will be used to this end. REVIEW OF LITERATURE Factor analysis is a statistical technique which facilitates the interpretation of the interrelationships among many variables (Harman, 1968). It consists in the reduction of a large number of observed correlated variables to a much smaller number of hypothetical uncorre- lated variables called factors. Scientifically, these factors or con- structs are the basic influences or dimensions in the data which can be used later instead of the more numerous original variables since they explain most of the correlations among them (Cattell, 1965). There exist many methods of factor analysis. They will differ de- pending on the assumptions made or the model chosen and the specific goal being pursued. However, on the basis of their order of execution, they can be divided between direct solutions methods and derived solu- tions methods (Harman, 1967). The direct solutions methods would be those used to transform a correlation matrix into a factor matrix, i.e., to find the common-factor space of that correlation matrix. The derived solutions methods concern a reorientation or rotation of the reference axes of that common-factor space, so as to facilitate the interpretation of the results, hence to approach a "simple structure" representation. In other words, the direct or unrotated solution defines the pattern of variance in the sample data, whereas the derived or rotated solution re— veals the clusters of intercorrelations among the variables (Rummel, 1970). In the present work, a version of principal-factor analysis, image analysis, and varimax orthogonal rotation have been used as direct solutions and derived solutions methods, respectively. Principal-factor analysis as performed today on electronic compu— ters has been developed by Kelley (1935) from the method of principal components of Hotelling presented in 1933. The first computer applica- tion was made by wrigley and Neuhaus in 1952. The original method it- self has its foundation in the "method of principal axes" of Karl Pearson (1901), where he showed how a large set of data can be reduced empirically so that a maximum portion of the variance present in the data can be accounted for. Hotelling, in turn, made of this technique the principal components method. However, the father of factor ana- lysis is considered to be Charles Spearman who dedicated forty years of his life to its development (Harman, 1967, 1968). Principal components analysis, as mentioned previously, is a method by which many correlated variables which together can possess some degree of redundancy are re- duced to a few uncorrelated variables (the principal components), thus eliminating that redundancy (Bailey, 1956). There will be as many principal components or factors as there are variables in the analysis, although only a few are needed to account for a large part (90% or more) of the sample variance. All of these components should be included, how- ever, in order to reproduce the correlations among the variables. Accor- ding to Maxwell and Rao, cited individually by Murty and Arunachalam (1967), no assumption is made concerning the distribution of the varia- bles or their randomness. In principal-factor analysis, however, it is assumed that the observed variables follow a multi-variate normal dis— tribution and that the newly established factors (fewer in number than the variables) are linear and additive. The principal components analysis model is: Z. = a. F + a F + --- + a. F (j = l, 2, ---,n) where the observed variable Z , is expressed as a linear regression on n_ factors (F's). The factors age considered as random variables and they can be defined by a normal probability density function (Harman, 1967). Harman referred to this model as a "maximum variance model". By contrast, the classical factor analysis model is constructed so as to reproduce most of the correlations among the variables. It can be written as follows: Zj = ajl Fl + aj2 F2 + --- + ajm Fm + dj Uj (j = l, 2, ---,m), where each observed variable is described as a linear regression on 9; factors only (m_sma11er than 2) plus a unique factor (Uj). The common factors result from the correlations among the variables whereas the unique factor represents the remaining variance (including error) of a particular variable not accounted for in the set of correlations. A good distinction between the implications of the two models is explicit in this characterization by Cattell, who called the principal component model a closed model and the principal-factor one an open model. In practice however, the method of "principal factors" is known as an ap— plication of principal components analysis to a reduced correlation matrix, that is a matrix which has communalities in its main diagonal instead of unities (Cattell, 1965; Harman, 1967). The rest of the correlation matrix is further adjusted in image analysis (Veldman, 1967, Rummel, 1970). The communalities can be estimated in many ways. Rummel dis- cussed at some length three categories of them. He reported also the squared multiple correlation as being the most commonly used communality estimate (Rummel, 1970). As noted by Harman, Thomson (1934) was the first to use Hotelling's principal components method with the full model of classical factor analysis (Harman, 1967) Factor analysis has a long list of very diverse fields of appli- cation which includes international relations, sociology, economics, communications, taxonomy, biology, geology, meteorology and particularly psychology and the behavioral sciences (Rummel, 1970, Harman, 1967, 1968). However, most of these applications are as recent as the availa- bility of compouter programs. In the last four or five years, a few works related to plant breeding have been published. For example, in 1967, Murty and Arunachalam, using factor analysis found that the pattern of divergence in the genus Sorghum differs under natural and artificial selection (Murty and Arunachalam, 1967). Later, joined by Jain, they es- tablished that, this was the case for both cross and self-pollinated crops. They also determined that selection, be it natural or artificial, has more influence on the factor loadings than the breeding system. 0f greater importance was their finding that for both outbreeders and in— breeders, under natural or artificial selection, the variables associated with productivity were the ones with highest loadings on the first factor (Murty, Arunachalam and Jain, 1970). Morishima, Oka and Chang, working with samples from an F population of a cross between a tall and a short 3 strain of rice, applied principal component analysis to a matrix of average genetic correlations between seven morphological traits. After rotation to "simple structure", they identified two bipolar axes of variation in rice plant type, namely the panicle-number versus panicle- 1ength and the internode—length versus internode-number axes. The panicle-number and the internode-length types had higher yielding po- tential (Morishima, Oka and Chang, 1969). Using the same material, Hashigushi and Morishima performed a principal components analysis on the phenotypic correlations. They used their own method to determine from the results what they called "the genetic vectors". Interestingly, and significantly, these were comparable to the components extracted from the genetic correlations (Hashigushi and Morishima, 1969). Hegman and DeFries found different results with phenotypically standardized phenotypic, genetic and environmental covariances than with the unstan- dardized covariances (Hegman and DeFries, 1970). This shows clearly that scaling can have an effect on results from factor analysis. The differences in size of loadings in the principal components extracted from genetic and environmental correlations, in the work by Murty and Arunachalam, seem to indicate the importance of normality as a require- ment in factor analysis (ibid.). Consequently, these authors express the view that the environmental correlations are more appropriate for factor analysis. MATERIALS AND METHODS Twenty—four homozygous lines of dry edible beans were planted at two locations in Michigan during the summer of 1970. Single—row plots in a randomized block design with four replications were used. The rows were 6 meters long and 50 centimeters apart. In order to accentuate the differences between the two environments, a standard number of seeds (sixty periplot) was sown at East Lansing, the first location, and a standard weight of seeds (fifteen grams per plot) was planted at Gratiot, the second location. I believe that the light—seeded lines are usually favored by the first, more popular type of planting, so I provoked de— liberately the opposite situation in the second type of planting. Ob— servations were to be recorded at maturity, because it is the proper time to have a measurement of some of the traits in which I was in- terested. As an indication of maturity, it was taken that most of the plants in the plot would have to have almost all the pods fully developed and, in addition, 25 to 50% of them should be at the pale green or yellowish color stage. Because of frost damage to different varieties at the two locations, complete sets of data have been obtained on a smaller number of entries than anticipated. Consequently, the present work in- cludes sixteen varieties for which all data have been recorded at both locations. These varieties are listed below. Eight are bush (B) or de- terminate and eight are vine (V) or indeterminate, or six are light—seeded (L), five have medium seed weight (M) and five are heavy- seeded (H) lines. Incidentally, the light—seeded varieties correspond to navy beans, the medium seed weight to great northern beans and the heavy-seeded lines to kidney beans. The sixteen varieties were: 1. 02- Great NOrthern 1—27 (M, V) 2. 03- Algarrobo (H, B) 3. 04— Sanilac (L, B) 4. 05- Charlevois (H, B) 5. 08— Criolla (L, B) 6. 09- Dominican (L, V) 7. 10- Mich. Cranberry (H, V) 8. 12- Red Mexican (M, V) 9. 13- Michelite—62 (L, V) 10. 14- Perry Marrow (M, V) 11. 15- Seafarer (L, B) 12. 16- Pinto 114 (M, V) 13. 17- Manitou (RKBOZSS) (H, B) l4. l8— Rinson Oscuro (M, B) 15. 21- Saginaw (L, V) 16. 23- Estrada Rosada (H, B) At maturity, five plants per line were chosen at random and observa— tions were made on each plant on a node per node basis. However, all analyses were performed on logarithms of the averages over the 5 plants because of the disparity in the variances of the different variables. 10 Twenty—two traits were finally retained after elimination of complex traits like total seed yield and number of seeds per plant Which would probably make the results less interpretable. Also, some traits were not used in order to prevent singularity or too low communality in the final set of traits. The twenty-two traits or variables together de- termine yield, the coefficient of determination exceeded 98%. These variables were: 10. 11. 12. 13. l4. l5. 16. total number of nodes per plant number of nodes with pods on branches per plant total number of branches per plant number of racemes on the branches per plant number of pods on the branches per plant total number of racemes per plant total number of pods per plant (X) average number of seeds per pod (Y) average seed weight (Z) average number of pods per productive node plant fresh weight average pod fresh weight average pod breadth average pod thickness average pod length number of short-internodes* * Those immediately above the hypocotyl (5 or 6 on the average), which are followed in turn by the long—internodes, more variable in number. *0 ll 17. number of long-internodes 18. average short-internode length 19. average long-internode length 20. average hypocotyl diameter 21. average short-internode diameter 22. average long-internode diameter Principal-factor analysis comprises two main parts, namely the cal- culation of the correlation matrix (R) and the extraction of the eigen— values and eigenvectors of that correlation matrix. In factor analysis, the use of the correlation matrix as calculated implies that all the correlations result from common factors or common variance. Consequent- ly, neither specific factor of a variable nor error is accounted for. Such an assumption has been discarded in this work. So, the image covariance matrix was used instead of the correlation matrix, thus following the full model of classical factor analysis. The image covariance matrix is a transformed correlation matrix in which the unities in the main diagonal have been replaced by the communalities of the variables, here their squared multiple correlations, and then the rest of the matrix is adjusted accordingly. Then the Jacobi method was applied to the image covariance matrix to extract its eigenroots and eigenvectors. The eigenroots or eigenvalues are the variances or por- tions of the total variation extracted by the corresponding eigenvectors or principal components. Not all the components can be considered as important, since the first ones are always associated with a greater part of the total variation. So, an eigenroot of at least one was chosen as requirement for an eigenvector to be extracted. After 12 extraction, the factors were submitted to varimax orthogonal rotation. Varimax as opposed to quartimax rotation simplifies the factor in- stead of the variables, and this is done so that the orthogonality between any two factor-vectors is preserved. All analyses were made using the total variation in the sample and assuming the full model of factor analysis. The discussions of the results will concern only the common part of that variation. Beside the complete set of data, analyses Were also performed on sub-sets of the data, namely, each location separately, each growth type separately, and finally each seed-weight group separately. The factors extracted for groups within each classification were then compared. The varimax factor loadings were used as bases for these comparisons. Two types of results were obtained here, the correlations between the factor- variables in the two sets of factors being compared, and the correla- tions between all pairs of variables in the two sets (same variables in both). The diagonal elements of this last matrix are correlations of a given variable in a particular sub-set with the same variable in the other sub-set. Prior to the calculation of these correlations the second structure or set of factors is generally transformed to attain maximum overlap with the first. Finally the results of these compari- sons are double-checked through the use of multiple discriminant ana— lysis. Since this analysis can indicate which variables in a set account for differences between groups of subjects, when these differ- ences exist, it should be possible, through it, to verify the conclu- sions arrived at after finding and comparing the rotated factors. 13 The computer programs used in this work are part of a package offered by D. J. Veldman in his book "Fortran Programming for the Behavioral Sciences" (Veldman, 1967). RESULTS Factor Analysis of the Complete Set of Data The results from the complete set of data indicate that 92.70% of the total variation is common, that is, the common factors can account for that much of that variation, and the remaining 7.30% should be attributed to unique factors and errors. The trace was 20.4, 83.16% of which was extracted by only 3 roots. The first two roots accounted for more than 73% before rotation and more than 62% after rotation (Table 1, page 15). So, it can safely be said that there are mainly two fac- tors at work in the present set of data. The varimax rotation results show the third factor increasing in importance (from about 10% before rotation to about 15% after), whereas the first two become about evenly important (31% each). The variables with the highest loadings in the first factor are: average pod thickness (l4)*, average seed weight (9), average pod fresh weight (12), average pod breadth (13); next in importance are: average pod length (15), average short-internode length (18), all three diameter- variables (20—22) and plant fresh weight (11) (Table 1 and Figure l on page 15). Number of seeds per pod (8) has a moderately high and negative loading in this factor. The variables with the lowest loadings are: for the positive, number of nodes with pods on the branches (2), number of * The numbers in parentheses identify the variables as listed on pages 10-11. 14 15 Complete Set of Data Table 1. Factor Analysis Results, Unrotated Factors Rotated Factors Traits 8 C‘ 1 3 3 SC" I 2 3 1 97.61 -.6237 .3475 .6117 88.39 .3250 .2509 .8458 2 97.24 .1393 .8606 .1122 77.25.0364 .8739 .0867 3 96.26 .2890 .8062 .0306 73.43 .1394 .8437 -.0563 4 99.64 .1676 .9337 .1128 91.2’ .0499 .9508 .0788 5 99.49 .1598 .9539 -.0798 94.19 -.0473 .9653 -.0883 6 99.44 -.1439 .9241 .1992 91.50 -.1844 .8906 .2963 7 99.42 -.1965 .9184 -.O736 88.73 -.3315 .8708 .0756 8 86.24 -.3906 .4619 3.3270 47.29 -.3641 .3830 -.0891 9 96.68 .9183 -.1373 .231? 92.17 .9355 .0036 -.2158 10 91.22 .3075 .6357' -.4277 68.16 -.0249 .6691 -.4830 11 94.51 .6510 .5728 .1286 76.85 .5385 .6763 ~.1455 12 98.68 .9102 -.1082 .2403 89.79 .9245 .0508 -.2016 13 92.38 .8268 -.2223 .3254 83.90 .9080 ' -.0738 -.0948 14 96.39 .8527 -.1443 .3651 88.12 ;..9363 .0082 -.0665 15 91.60 .8407 -.0659 -.0157 71.13 ’.7408 .0755 -.3961 16 76.56 -.3571 .5201 .0074 39.81 -.3924 .4530 .1971 17 97.59 -.6746 .2596 .6119 89.69 -.3556 .1557 .8638 18 80.80 .6273 5.3283 .2595 56.86 .7200 -.2131 -.0699 19 67.71 «.3047 *.0189 .5800 42.96 —.0014 -.0575 .6529 20 95.73 .8649 .3333 -.0586 86.26 .6792 .4722 -.4223 21 96.54 .7904 .4386 .0122 81.72 .6292 .5649 -.3197 22 87.56 .8219 -.0134 -.2952 76.28 .5893 .1182 -.6337 £.Roots 8.1827 6.7969 1.9783 8 Trace 40.13 33.33 9.70 % Var. 31.28 31.03 14.78 Cum.% Trace 40.13 73.46 83.16 Cum.% Var. 31.28 62.31 77.09 Total Trace - 20.3931 1 , 92.70% of total variation is common. a Communalzty Factor 1 Factor 2 Factor 3 Traits Traits Traits 11. I I 5 I I _ 17 I II 9 I I 4 I I 1 I I 12 I I 6 I I 19 [j 13 [:3 2 I. l 22(neg.) CI 15 CZ] 7 III—I. '18 I I 3 I I 20 [j ' 11 [j 21 [:1 10 [I 22 [j 21 CI 8(neg.) [J 11 E] Figure 1. Comparison of size of loading for the variables included in the three main rotated factors. l6 racemes on the branches (4), and total number of branches (3); and for the negative, number of pods on the branches (5), average number of pods per productive node (10) and average long-internode length (19). Essentially this is a weight factor. The second factor is highly associated with number-variables like number of nodes with pods on the branches (2), total number of branches (3), total number of racemes (6) and number of pods per plant (7), par- ticularly with number of racemes on the branches (4) and number of pods on the branches (5), but also with number of pods per productive node (10) and average short-internode diameter (21) (Table l and Figure l on page 15). In general, the uppermost three or four short or basal inter— nodes support most of the productive branches of the bean plant. Seed weight (9), pod weight (12), pod thickness (14) and pod length (15) have the lowest positive loadings whereas pod breadth (13) and long-internode length have the lowest negative loadings in this factor. Consequently, this factor indentifies itself with numbers, particularly number of branches, but not with weight nor with large pods. The third and least important factor is characterized by loadings which reflect a numerous-thin-long internode relationship, that is, total number of nodes on the plant (1), number of long-internodes (l7) and average long-internode length (19) have the highest positive loadings in this factor whereas the diameter—variables (20-22) along with average number of pods per productive node (10) and average pod length (15) have the highest negative loadings (Table 1, Figure l on page 15). Number of nodes with pod—bearing branches (2), number of racemes on the branches (4) and number of pods per plant (7) have the lowest positive loadings. Total Table 2. Factor Analysis Results, East Lansing Set . Unrotated Factors ' Rotated Factors Traits 1 ca 1 2 3 4 1 c- 1 2 3 4 1 97.39 .5829 -. 427 .6629 -.2341 89.2 .1936 .1334 .9151 .0144 2 97.66 .5788 .6738 .0551 .1095 80.67 .1231 .877 .0448 .1360 3 95.95 .4990 .6885 .0225 .2026 76.46 .1107 .8397 -.o494 .2117 4 99.77 .6386 .7000 .1141 ,-.1803 94.33 .0223 .9442 .2059 -.0870 5 99.59 .6845 .7021 -.0040 —.0481» 96.39 .1485 .9658 .0877 -.o374 6 99.73 .8129 .4311 .2218 -.0322 94.83 .2436 .8554 .3954 -.0031 7 99 74 .8620 4362 .0287 .0796 94 05 .4343 8384 .2193 .0284 8 91.10 .6354 -.0‘7~ -.3160 .5184 77.83 .8043 .2670 -.1484 .1950 9 95 92 -.7878 .4-63 .1621 -.2342 92.86 -.9.89 -.0338 -.2501 0461 10 95.36 .5021 6624 -.2395 -.1261 76.42 . 923 .8351 -.1351 -.2002 11 93.41 .0378 .7870 .1314 3316 76.17 ~.244 .6574 --2308 .4632 12 8.76 -.81/5 .4147 .182 .1349 89.77 -.7780 -.1191 -.3621 .3837 13 94 59 -.8321 .2964 .2793 0452 86.03 -.8120 -.2204 -.2191 3231 14 97.35 -.7655 .4142 .2667 -.1676 85.68 -.8967 -.0760 -.1629 .1427 15 92.04 -.7o60 .4067 ~.0804 .2066 71 30 -.s790 -.0655 4.5411 .2840 16 84.90 .7913 -.oo91 -.0582 -.1007 63.97 5493 .4380 .3133 -.z186 17 96.93 .5502 -.2615 6748 - 1922 86 35 .1909 .0976 .9023 .0577 18 85.08 —.6883 .2439 .3969 .1692 71.93 -.6801 -.1896 -.0947 .4602 19 75.75 -.0983 .2380 .5455 .4970 61.10 —.1s41 .1193 .1523 .7346 20 96.28 -.2282 .8364 -.1414 ~.1634 83.21 -.5665 .5904 -.5983 -.0629 21 97.40 -.2045 .8495 -.1234 -.2615 84.76 -.5906 .6023 -.3401 -.1428 22 88.61 -.6009 .5502 -.3526 -.1600 81.37 -.6134 .1239 -.6297 -.1556 f $222: Biggfig 53323; 1'3623 1&1332 1 Var. 28.18 31.83 15.31 7.15 ' ‘ ' ' Cwn 1 Var. _28.18 60.03 75.34 82.49. Cum-i Tr. 42.42 72.44 81.94 87.53 Total Trace - 20.7331 94.24% of total variation is common. * Communality Table 3. Factor Analysis Results, Gratiot Set Unrotated Factors . - Rotated Factors Traits % C* l 2 3 8 Ci 1 2 3 1 98.86 - -.4887 .7250 .4063 92.95 -.3286 .3457 .8379 2 98.79 .4626 .7598 ' -.0133 79.15 - .1391 .8756 .0745 3 98.42 .6613 .5699 -.0485 76.45 .3209 .8041 -.1220 4 99.76 .5690 .7705 . .0237 91.80 .2390 .9260 .0579 5 99.65 .6164 .7143 -.2501 95.28 .1289 .9461 -.2025 6 99.56 .2210 .9076 .1479 89.45 .0101 .8688 .3737 7 99.37 .2052 .8667 ~.Z417 85.18 -.2204 .8944 .0576 8 1.77 -.14 .5247 -.4556 50.53 2.5244 .4732 -.0304 9 99.4 .8866 -.2781 .3175 96.42 .9373 .0969 - 2762 10 95.14 .6526 . 05 ~.6041 79.08 .1394 .40‘2 -.7803 11 98.24 .8783 .3200 .0637 87.79 .6198 .6623 ~.2347 12 99.11 .9135 -.1613 2812 93.95 .9048 .2177 -.2710 13 95.45 .8034 -.2399 . 74 84.35 8975 .0331 -.1766 14 97.68 .8239 -.1879 .4388 90.67 .9367 .1266 - 1152 15 96.86 .8790 -.1746 .0629 80.7 .7535 "’91 -.4322 16 84.51 -.0957 .5511 .2996 40 26 - 0453 .387 .5002 17 9 4 -.6014 .6539 .3801 93.38 -.4107 .2376 .8418 18 87.31 .5576 - 4545 .3038 60.98 .7264 - 7025 -..027 19 93.22 -.6039 .3272 .5004 72 22 -.2534 -.0713 .8080 20 98 44 .9540 -.0359 .0366 91.38 7576 .3888 -.4331 21 96 S4 .9201 .2024 .0185 88.7 .6567 .5858 —.3366 22 95.34 .8110 - "437 - 312 81 50 .4989 7041 .7241 E.Roots 10.0829 5.8575 2.0810 1 Trace 47.46 27.57 9.80 1 Var. 31.80 29.94 20.18 Cum 1 Trace 47.46 75.03 84.83 Cum-1 Var. 31.80 61.74 81.92 Total Trace - 21.2432 . . 96.562 of total va'LSttcn 1s conmon. e Communality 18 number of branches (3), number of pods on the branches (5), number of seeds per pod (8), average pod breadth and thickness (13, 14) and finally average short—internode length (18) have the lowest negative loadings in this factor. Therefore, this factor expresses a growth type where at the extreme an exagerate vegetative growth causes a very low production. Factor Analysis of Individual Locations The analyses by location show that 94.24% of the total variation is common at East Lansing, that is, the common factors accounted for that much variation (Table 2, page 17) compared to 96.56% at Gratiot (Table 3, page 17), indicating the presence of a slightly greater error at the first location. The results from these analyses were identical to those obtained with the complete set of data, with only one insignificant dif— ference that the signs of the loadings in the first factor at East Lansing were inverted. That should not change the interpretation of the results, however. Another finding was the extraction of a fourth factor at East Lansing, confirming the fact already mentioned that there was more variance at that location. Mbreover, the comparison between the four factors of East Lansing with the three of Gratiot gives the follow- ing correlations between the corresponding first three factors: ~.93, .99 and .93 with no large values encountered for the fourth factor at East Lansing. But, the size of the correlations of that factor with the first and third of Gratiot (.33 and .37 respectively) may explain in part the relatively smaller correlations found for the first and third sets of factors (Table 3 a-b, page 19). 19 Table 3a. Factor Relationships Between. East Lansing and Gratiot Analyses (Correlations) East G r a t i o t Lansing l 2 3 l -.9315 .1307 .0329 2 .1147 .9894 -.0656 3 -.0915 .0498 .9264 4 .3329 .0390 .3692 Table 3b. Relationships Between Identical Pairs 0f Variables From East Lansing and Gratiot Analyses Variables Correlations 1 .9476 2 .9858 3 .9268 4 .9559 5 .9571 6 .9739 7 '. 9874 8 .8213 9 .9451 10 .6710 11 .8962 12 .8686 13 4.8931 14 .9328 15 .8584 16 .6379 17 .9534 18 .9070 19 .3121 20 .9721 21 ' .9570 22 .9605 20 Factor Analysis of Determinate and Indeterminate Varieties as Sub-Sets The results of the analyses by growth type show that there were four factors with roots greater than unity for the determinate lines and five factors for the indeterminate lines. There was greater error vari- ance among the indeterminate lines than the determinate varieties as in— dicated by the amount of the total variation which is common in the two situations (92.01% compared to 95.63%). Here again, two major factors were recovered, but with some important changes in each case. For the determinate lines, there was less emphasis put on average short internode length (18) but more on number of nodes with pods on the branches (2), on number of branches per plant (3) and also on number of racemes on the branches (4) (Table 4, page 21). The loadings for the three diameter traits (20, 21, 22) were markedly higher. The second factor was almost identical to that of the complete data set, with a little less importance given to the number of nodes with pods on branches (2) and number of branches per plant (3). These two factors account for more than 76% of the total variation before rotation and for about 64% after. The third factor was also recovered, but it can not be said that it is more important than the fourth factor of the bush-type structure, a factor which seems to contrast short internode length (18) with the number of short internodes (l6) and also the number of seeds per pod (8). For the indeterminate lines, the first factor was identical to the second factor of the complete set, whereas the second resembled the first factor of the complete set of data, but with two major differences. The average long internode length (19) has gained in importance and the diameter variables (20—22) were transposed from the second to the third O 21 Table 4. Factor Analysis Results, Determinate Lines '9 Communality 92.01% of total variation is common. . Unrotated Factors Rotated Factors 1,1115% C* 1 2 3 4 8() l 2 3 4 1 98.46 -.6377 .5966 ' .3330 .1131 92.59 -.5310' .1235 .6672 -.4235 2 98.77 .5542 .7116 3518 -.0412 93.90 .4388 .6992 .2354 ~.4495 3 96.44 .6259 .6137 3004 .0166 85.89 .5000 .6819 .1883 -.3293 4 99.56 .7489 .5847 - 0515 ,. .1255 92.12 .4472 .8449 -.0481 -.0716 5 99.28 .6805 .6654 - 2089 .1578 97.44 .2943 .9357 -.1066 -.0273 6 99.15 .3063 .8690 - 1935 .1760 91.74 -.0718 .9393 .0687 -.1586 7 99.36 .1248 .8930 - 3367 .1464 94.80 -.2958 .9147 .0047 , c.1540 8 91.90 -.2809 .5979 - 1887 -.5691 79.60 -.4588 .2632 -.2496 -.6737 9 98.73 .8939 —.3502 1468 .1270 95.94 .9131 .0937 -.1$57 .3045 10 93.63 .6369 .2014 - 6406 .1700 88.54 .1862 .6725 -.SO78 .3751 11 96.62 .8139 .3825 - 0201 -.0810 81.58 .5810 .6442 -.2086 -.1408 12 99.14 .9223 -.2128 p 1326 -.1493 93.58 .9065 .1462 c.3023 .0363 13 94.17 .8489 -.2759 1671 -.0388 82.61 .8718 .0833 -.2066 .1283 14, 98.14 .9010 -.2763 0418 .0235 89.04 .8562 .1593 -.2774 .2344_ 15 96.20 .7822 -.2528 2319 -.1839 76.33 .8453 1.0147 -.2185 -.0288 16 89.42 -.1186 .5608, 3722 -.5113 72.84 -.0591 .1535 .1387 -.8259 17‘ 97.77 -.6787 .4176 3488 .3636 88.89 -.S470 .0407 .7540 -.1398 18 '89.57 .4655 -.S677 0333 .3109 63.67 .5486 -.1835 -.0590 .5465 19 77.44 -.4659 .1839 ' 3521 .5120 63.70 .3088 -.0172 .7278 .1076 20 97.53 .9538 .0600 1120 .0702 93.09 .8397 .4459 -.1382 .0888 21 98.05 .9242 .1048 1995 .0060 90.49 .8452 .4261 -.0925 -.0215 22 94.58 .8880 -.2276 0469‘ ,-.0046 84.25 .8357 1839 9.2751 .1862- E.Roots 10.6408 5.4162 1.5705 1.2988 - ‘ 4 Trace 50.58 25.74 7.46 6.17 8 Var. 38.03 26.14 11.02' 10.84 Cum 4 Tr. 50.58 76.32 83.78 89.96 Cum.% Var. 38.03 64.17 75.19 86.03 Total Trace - 21.0392 - f -. r . 95.63% of total variation is common. 4 Commnnality . ‘ . .- Table 5. Factor Ana1ysis Results, Indeterminate Lines . Unrotated Factors Rotated Factors . . Traits ~1 C* 1 2 3 4 s- 3 cs. 1 2 3 4 s 1 82 .4444 .1540 .5118 —.0241 .0219 48 .2031 -.1796 .6225 -.0914 .1219 2 98 .7608 .2408 -.4287 -.2300 .1171 89 .9087. -.0983 -.1o76 -.0575 .1921 p‘ 3 .98 .7512 ..2992 -.4011 -.2446 .1137. 89 ‘.9167 -.0478 -.0660 -.0696 .1888 *4 1100 .8735 .3463 ‘-.2523 -.1000 -.1445 ~ 98 .9636 -.1398 .1604 .0499 3.0364 5 100 .9300 .2624 ..2064 -.0319 -.0463 98 .9335 -.2449 .1833 .0960 .0733 6 ~100 .8860 .3782 -.0540 -.0387 -.2081 98 .8976' -.1641 .3624 .0710 -.0827 7 100 .9517 .2114 “.0145 .0683 -.0568 96 .8197 -.3455 .3759 .1381 .0832 8 93 .5894 -.1770 .0658 .1032 .7094 90 .2858 -.4120 .0975 .0872 .7927 9 96 -.6406 .6726 —.0011 .0185 -.1501 89 -.2196 .8730 .0864 .0862 -.2457 10 95 .8483 .1533 -.0353 .0439 -.0450 ~75 .7393 .-.3269 ..2774 .1121 .0763 11 93 .3503 .8359 .0196 v.0802 .1769 86 .6097 .5252 .3999 .0568 .2210 12 99 -.6053 .7391 -.0200 -.0998 .1780 95 -.1608 .9587 .0559 -.0122 .0803 ' 13 96 -.6969 .6297 -.0980 -.1719 .1258 94 -.2269 .9336 -.0823 -.0851 .0099 14 98 -.6404 .6956 -.0884 -.1363 -.1929 96 -.1404 .9213 .0043 -.0406 -.29S7A 15 92 -.3129 ‘ .6677 .3021 -.1291 .4627 87 -.1110 .7287 .3585 -.1072 .4272 16 86 .6300 -.1562 .3666 -.1639 -.0976~ 59 .3192 -.5030 .4281 -.2311 .0213 17 78 .1601 .1172 -.1738 .7736 -.0130 67 .0971 -.0660 .0370 .8082 .0048 ‘18 82 -.5157 .6355 ~.1651 .0150 -.1942 73 -.0644 .7975 -.0305 .1189 -.2825 19 73 -.2993 .4626 -.3857 .4402 .1281 66 .0178 .5490 v.2049 .5618 .0564 .20 90 .4020 .6649 .3036 .1979 -.1923 7S .4372 .2424 .6750 .2393 -.1112 21 94 .5611 .6091 .3483 .2264 -.0196 86 ..5033' .1194 .7194 .2575 .0859 .22 84 .1024 .2946 .6269 .0193 -.0761 SO -.OS63 .0839 .6935 -.O702 ~.0168 E.Roots 8.8850 5.1820 1.7811 1.1354 1.0615 . 5 Trace 43.90 25.60 8.80 5.61 5.24 t Var. 30.48 27.41 12.77 5.77 5.60 Cum.% Tr.43.90 69.50 73.30 83.91 89.15 Cum. Total Trace - 20.2412 8 Var. 30.48 57.89 70.66 76.43 82.03 22 factor (Table 5, page 21). Consequently, the third factor becomes an association between the number of nodes (1) and the diameter variables (20—22) instead of the number of long internodes (17) as in the com- plete-set results. It is a significant change pe£_§g, because this third factor extracts here about the same amount of variation as it does in the complete-set analysis. The fourth and fifth factors each accounted for less than 6% of the total variance. Number and length of long internodes (l7, 19) had high loadings in the fourth factor, and number of seeds per pod (8) showed a high loading in the fifth factor. These variables had also low communalities in this analysis (Table 5, page 21). The first two factors extracted a little more than 69% of the total variation prior to rotation and about 58% after rotation. Comparison Between the Factor-vectors Extracted in Analyses of Determinate and Indeterminate Varieties The factor-variables from the determinate and the indeterminate lines were compared and the results show, in addition to the transposi- tion between the first and second factors of the two sets, that factor #3 of the determinate group is similar to factor #4 of the indeterminate group and also that factor #4 of the former is the Opposite of factor #5 in the indeterminate group (Table 5a, page 23). Consequently, factor #3 of the indeterminate sub-set is unique to that group and can be seen as another expression of the third factor of the complete-set analysis. The following variables in decreasing order of importance account for most of the differences between the determinate—group and indeterminate-group results: number of nodes on the plant (1), average number of pods per productive node, (10) average long internode diameter 23 Table 5a. Factor Relationships Between Determinate and Indeterminate Lines Analyses (Correlations) B u s h Vine - 1 2 ' 3 4 .0498 .9650 -.0765 ' .0025 .8647 -.1197 -.0547 .4007 .4740 .1616 .0223 -.4834 .0087 .0945 .9744 .1908 .1579 -.1399 .2029 -.7546 m-waI—i Table 5b. Relationships Between Identical Pairs of Variables From Analyses of Determinate and Indeterminate Lines Variables Correlations 1 .1728 2 .6613 3 .6944 4 .8817 5 .8985 6 .9783 7 .9365 8' .7253 9 .8808 10 .4399 11 .9412 12 .8049 ‘ 13 .8015 14 .8442 15 .9105 16 .6602 17 _ .7778 18 .9875 19 .4639 20 ~~ .7738 21 .7376 ‘ 22 .4600' 24 Table 5a. Multiple Discriminant Analysis between Determinate and Indeterminate Lines Variables Discriminant Axis Loadings 1 . .9418 2 .1810 3‘ .0003 4 .3076 5 .1535 6 .4906 7 .2750 8 .1289 9 -.2645 10 ' -.1866 11 -.0694 12 g -.2511 13 -.2139 14 -.1413 15 -.3661 16 .3882 17 ' .8727 18 ' . -.2497 19 .3359 20 -.3193 21 - -.l638 22 -.5614 % of Total Variation : '100.00 Chi-Square Value : 317.593 Degrees of freedom .‘ 22 Significance Level : .0000 25 (22) and average long internode length (19) (Table 5b, page 23). These results were also confirmed by the use of multiple discrimi- nant analysis. Total number of nodes on the plant (1) was the main differentiating variable between determinate and indeterminate lines, followed by the number of long internodes (l7) and also their diameter (22) (Table 5c, page 24). The discriminant function had a highly sig— nificant chi—square. Factor Analysis for the Three Seed-weight Groups Attempts were also made to test the invariance of the factors when, instead of classifying by growth type, the material was divided into seed—weight sub-sets. The average seed weight before planting was used for this purpose. Three groups were established; the light—seeded, the medium—weight and the heavy-seeded lines. They corresponded approxi— mately to the navy, the great northern and the kidney beans with an average weight of 15-16 gms per hundred dry seeds, 29-30 gms and 40—45 gms, respectively. Factor analysis for the light-seeded lines reveals that 93.61% of the total variation was common, and that 89.67% was extracted by 5 roots (Table 6, page 26). The first factor accounted for 46.20% of the trace before rotation and 41.6% after rotation. It was by far the most im- portant factor, since the next two factors together extracted only some 31% of the trace. That factor resembles a combination of the first two factors extracted or identified from the complete set of data. The short internodes and hypocotyl diameters of the first factor were com— bined with the number-variables of the second factor of the complete set to form the first factor of the light-seeded lines. Three remarks should 26 Table 6. Factor Analysis Results, Light—Seeded Lines Unrotated Factors Rotated Factors Traitst C‘ 1 2 3 4 5 1c) 1 2 3 4 S ‘ 1 98 .7193 o.4807 .1565 .2863 -.2584 92 .7425 -.2362 .2849 .4619 -.1418 2 98 .8437 -.0300 ~.4062 .1763 .0639 91 .8473 .1617 -.3523 .2091 -.0526 . 3 98 .8053 .0584 -.4352 .0967 '.1139 86 .8023 .2085 -.4067 .1005 -.0308 .4 100 .9614 -.1634 -.0179 -.0063 .0480 95 .9715 -.0128 .0121 .0258 -.0943 5 99 .9585 -.1015 -.0476 -.0923 .1870 97.'.9785 .0151 -.0$42 -.1172 —.0181 6 99 .9581 -.0718 .1304 -.0417 .0662 95 .9527 .0577 .1426 -.0721 -.0987 7 99 .9323 .0285 .0497 -.1636 .2211 95 .9340 .1043 .0157 -.2526 -.0287 8 9S .3892 .5188 -.6513 .0593 -.0748 85 .2970 .5501 -.6373 .0494 -.2335‘ 9 91 '.1406 -.2468 .8573 .1942 -.1258 87 .1451 -.1359 .9037 .1082 .0365 10 93 .5943 .0450 ' .0848 -.5285 .3671 78 .6176 -.O624 °.0381 -.6242 .0086 11 95 .7473. .5557 .1233 -.0241 .1600 91 .6590 .6107 .0790 -.2977 -.0804 12 97 .2318 .8367 .2408 .3157 -.0495' 91 .0819 .9159 .2439 -.0438 -.0826 13 81 —.0125 .6510 -.1494 .5035 .1737 73 -.0828 .7888 -.1344 .1602 .2391 14 93 .2115 .2923 .7949 .2847 .0080 84 .1485 .4054 .‘38052 -.0358 .0841 - 15 90 .1643 - .7994 .0053 o.0817 -.1870 71 .0008 .7179 o.0271 -.2483 -.3604 16 93 .6887 .1100 -.1000 -.0620 -.4935 74 .5814 .1608 -.0240 .1401 -.S997 17- 98 .5692 -.6258 .1652 .3592 v.0296 87 .6546 -.3560 .2770 .4706 .1394" 18 86 -.5787 .1110 .5428 -.1551 .1087 68 -.5789 -.0365 .4662' -.3217 .1434 19 73 -.2093 -.1830 .1249 .3788 .6237 63 -.0788 -.O354 .0969 1.0798 .7760 ’20 97 .8828 .0462 .3324 -.1244 -.1075 92 .8292 .1184 .3420 -.1472 -.2806 21 '97 .9133 -.1399 .2238 v.0107 -.0656 91 .8892 -.0032 .2605 .0113 -.1778 . _22 89 .0865 .2992 .6120 -.3491 .0261 59 .0228' .1549 .5220, -.5249 -.1467 E.Roots 9.5144 3.3311 3.1773 1.3594 1.0856 8 Trace 46.20 16.17 15.43 6.60 5.27 8 Var.4l.63 14.42 14.15 7.19' 6.54 Cum.8 Tr 46.20 62.37 77.80 84.40 89.67 Cum. - . - Total Trace _ 20.5951 % Var. 41.63 56.05 70.20 77.39 83.93 . , 93.61% of total variation is common. a Communality . Table 7.. Factor Analysis Results, Medium—Weight Lines Unrotated Factors . . , Rotated Factors' . Traits t C * 1 2 . 3 4 . t-Cfi 1 _ 2 3 4 - 1 99.10 -.7367 .5662 .2964 .1021 96.16 -.9312 .0133 -.2691 .1478 _ 2' 98.98 .6149 .6719 .1374 -.1794 88.06 .1056 .9248 -.1140 ‘.0356 3 98.57 .7203 .5503 .1182 ~ -.l754 86.64 .2452 .8966 -.0367 .0316 g 4 99.94 .6111 .7291 .1424 v.2424 98.40 .0719 .9799 «.1354 -.0158 5 99.91 .7053 .6779 .1096 -.1181 98.30 .1900 .9595 -.1266 .1007 ‘ 6 99.88 .3600 .8531 .2814 ,-.1702 96.54 -.2334 .9379 -.1631 .0689 7 99.82 .5239 .7858 .2142 .0175 93.81 -.0339 .9219 -.1644 .2452 8 93.62 .4252 .4260 .1197 .6924 85.60 .1379 .4114 -.1419 .8047 9 '97.80 .6121 -.7136 .0678 ~.0413 89.02 . .7185 -.0995 ' .6016 -.0460 10 98.25" .7634 .4643 -.1518 -.0390 82.30 .4596 .7484 -.2063 .0953 11 97.89 .8982 .1093 .3252 .0807 93.09 .4758 .6913 .3831 .2827 12 98.95 .5973 -.7027 .2312 .1656 93.14 .6360 -.1056 .6932 .1877. 13 95.06, é.0395 -.7566 .5534 .0853 88.03 .0024 -.3905 .8530 .0152 14 97.59 .2492 -.7481 .3299 '-.4106 89.92 .2985 -.1706 .7916 -.3929 15 98.71 .8185' -.4243 -.0865 .2908- 94.20 .8584 .0964 .3204 .3053 16 94.56 —.2755 .7301 -.2037 .2927 73.61 -.3835 .2093 -.6736 .3024 17 99.44 -.7423 .4771 .4253 .0048 95.96 -.9712 .0104 -.1066 .0700 18 77.35 .2738 -.S415 .5018 .0189 62.03 .1778 ‘.0707 .7589 .0884 19 87.04 -.3858 .0529 .7280 .2316 74.49 -.6615 -.0453 .4196 .3594 20 97.98 .9090 -.2867 -.0392 .0597 9182 .8312 .3121 .3251 .1552 21 98.79 .9424 -.1163 «.0442 .1132 91.63 .7906 .4436 .2287 . .2052 22 98.21 .8468 -.25'9 -.ZS92 .0438 86.92 .8951 .2330 .1054 .0516 EcRoots 9.0322 7.3545 1.9931 1.1172 - ' 5 Trace 42.46 34.57 9.37 5.25 8 Var. ,31.43 32.22 18.50 6.47 £um.5 Tr. 42.46 -77.03 86.40 91.64 Cwnrfi Var. 31.43 63.65 82.15 88.62 Total Trace - 21.2745 _ 1 96.701 of total variation is common. / e Communality 27 be made here. Total number of nodes on the plant (1) was included, so was the number of long internodes (17). The pod dimensions (12-15) along with the average long internode diameter (22) vanished. Finally, the number of short and long internodes (l6, 17) had positive loadings instead of negative loadings as in the complete-set results and the opposite occurred for the average short internode length (18). So, this is a general factor pointing toward a plant type with sturdy, short internodes producing many branches full of productive units. There can be many nodes on such a plant. The second factor in the light-seeded sub-set was a pod factor characterized by long, wide and filled pods (12, 13, 15) leading to a relatively heavy plant (11). The third factor expressed a contrast between seed size and weight (14, 9) and seed number per pod (8). Such a type would probably have sturdy long-internodes (22) and also long short—internodes (18). In the analysis of the medium-weight set, there was a relatively smaller error variance, 96.70% of the total variance was common, and 91.64% was extracted by 4 roots. The first two vectors were about equally important as indicated by the amounts of variance extracted by them, 42.46% and 34.57% before rotation and 31.4% and 32.2% after rota- tion, respectively (Table 7, page 26). The second factor was identical to the second factor of the complete set of data. The first, however, was apparently a completely new factor. It is bipolar and has large positive loadings on the diameter-variables (20-22), the length and the weight of the pod (15, 12) and seed weight (9), and negative loadings particularly on the total number of nodes (1) and on the number and the length of the long internodes (17, 19). Number of pods per productive 28 node (10) and average plant fresh weight (11) have positive loadings but intermediate in size. The third factor contrasts the number of short internodes (16) with seed weight (9), the pod dimensions (13—15) and the average short internode length (18). The number of short in- ternodes (16) had negative loading. This factor extracted 9.37% of variance before rotation and 18.5% after rotation. It was the most im- portant third factor encountered in this study (Table 7, page 26 ). The analysis of the heavy-seeded sub-set shows that 95.64% of the total variation was common and 89.0% was extracted by 5 roots. Only the first three deserve consideration, however (Table 8, page 29 ). The first factor was essentially the same as the second factor of the com— plete set analysis and the second was comparable to the third of those factors, but with some important differences. The negative loadings of the diameter-variables (20-22) and number of pods per productive node (10) were significantly higher in this factor. Finally, the third factor is identifiable with a contrast of plant weight (11), pod weight (12), pod breadth (l3) and pod thickness (14). There were no high posi— tive loadings on this factor. The three factors—variables of this set accounted for 35.59%, 26.89% and 11.15% of the total variation before rotation and 26.36%, 24.52% and 14.17% after rotation, respectively. Comparisons Between the Factor-Vectors Extracted from Analyses of the Seed-weight Groups The three sets of factors from the three seed-weight groups were compared two by two. Through these comparisons, it can be seen that the second factor of the complete-set factors was found in the results of all three seed-weight groups, although with some modification in the n-— I.- Table 8. Unrotatcd Factors 29 Factor Analysis Results, Heavy-Seeded Lines Rotated Factors Traits 8 C* 1 2 3 4 5 8 Ch 1 2 3 4 5 1 99 -.1809 .9505 -.1302 .0722 .1396 98 .2478 .9286 .0711 .1438 .1685 2 98 .5940 .6112 .2779 .0329 .1036 82 .6792. .1556 -.0855 '.2401 .5047 3 93_ .6702 .4152 .4546 -.1939 .0390 87 .6616 -.0839 .0101 .0649 .6482 '4 100 .7870 .4801 -.1473 -.0697 -.3118 97 .9638 .0671 -.1614 -.1037 .0583 5 100 .9126 .2500 -.0802 .0142 -.2782 98 .9374 -.1$S9 -.2491 -.0176 .0507 6 100 .6619 .6499 -.0885 .0264 -.2457 93 .9193 .2510~ -.0928 .0326 .1080 7 100 .8265 .3880 -.0074 .1363 -.1823 89 .8945 -.0458 ’.2176 .1512 .1143 8 90 .1331 -.0343 .1308 .8614 '.1178 79 .0566 -.0930 -.1463 .8471 -.2022 9 94 .5676 -.4943 -.0668 v.4653 .0509 79 .1632 -.6100 -.3758 -.4678 .1764 10 98 .7239 -.3536 -.2165 .0559 -.3527 82 .5537 -.5752 -.2954 -.1183 -.2914 11 _ 96 .8308 .1840 .0637 .2443 ..3429 91 ’.5649 -.1381 -.5634 .3612 .3453 12 95 .6464 -.2511 -.1481 .2155 .5577 86 .1308 '.3067 -.8082 .2478 .1857 13 89 .5730 -.3017 -.4987 —.0379 .3172 77 .1638 -.2458 -.8043 -.1674 -.0895 14 96 .6105 ’.0649 -.6634 ‘ .0320 .2511 88 .3400 -.0202 -.8397 -.1521 -.1916 15 94 -.0058 -.4777 .7690 -.0037 .1534 84 -.3209 -.6301 .2635 .2607 .4536‘ 16 94 -.0102 .4424 .4712 .5224 -.0966 70 .2449 .1924 .3638 .6718 .1392 17 99 -.2385 .8805 -.0719 -.1066 .2748 92 .1019 .8973 .0478 .0232 .3252 18 89 .2699 .2419 .3271 -.6046 .1522 63 -.2164 -.0039 .0573 -.3553 .6714 19 93 -.0498 .7532 .1722 -.2780 .3674 81 .1298 .6463 .0337 -.0290 .6126 20 '97 .7767 5.3495 .3779 -.1769 -.0765 91 .4427 -.7512 -.1377 -.0479 .3517 21 97 .7345 -.4315 .2986 .0765 .0398 82 .3309 -.7481m -.2753 .1641 ‘ .2248 22 94 .3627 -.8042 .1951 .0490 .1470 84 -.1706 -.8479 -.2909 ‘ .0705 .0525 E.Roots 7.4877 5.6581 2.3454 1.9070 ’1.3286 8 Trace 35.59 26.89 11.15 9.06 6.31 % Var.26.36 24.52 14.17 9.06 11.02 Cum.% Tr 35.59 62.48 73.63 82.69 89.01 Cun- 26.36 50.88 55.05 64.11 75.12 4 Var. ' Total Trace e 21.0401 . Communality ' Table 6a. Relationships between Factors of the Light and Heavy-seeded lines 95(648 of total variation is common. (Correlations.) Light-seeded Heavy-seeded - 1 2 3 4 5 l .9570 .0551 -.1064 .1785 .1946 2 .1801 -.2833 -.7882 .5115 .0660 3 , -.0311 -.2918 --.2381 .6274 .6808 4 .2109 ..7353 .0184 - .2869 .5764 5 .0784 .5394 -.5571 -.4801 -.4025 ‘Table 66. Relationships between identical pairs of variables from.Analyses of Light and Heavy Sets. I Traits : 1 Correlations .5767 2 .8381 3 .5557 4 -9201 5 '9515 6 .9067 7 3873 8 3895 9 5562‘ 10 3114 ' i; 3166 13 3183 14 3250 15 6684, 16 7459 17 3276 18 1040 19 2181 20 . 8379 21 5640 22 8233 30 light—seeded results (Table 6 a—b, page 29). This confirms the finding that the first factor of the complete—set analysis is indeed a weight factor, since in none of the seed-weight sub-set analyses was it possible to recover it. These comparisons also indicate that the fourth factors of the medium-weight and heavy—seeded groups are almost the same. Furthermore, combinations of factors in one set may probably have the same effect of another factor or combination of factors in another set, an exception being made of the factors similar to the second factor of the complete set results (Table 7 a—b, 8 a-b, page 31), These results were verified through multiple discriminant analysis. Of the two discriminant functions necessary to separate the three groups, both with very significant chi—squares, the first was the most important. It accounted for 88.60% of the variance (Table 8c, page 32). Seed weight (9), and the pod-dimension variables had the highest positive correlations with that function. This function is almost identical to the first factor-variable found through factor analysis of the complete set of data. The second function, on the other hand, resembles more the third factor from that structure. It is also similar, in pattern, but not in size of the loadings, to the discriminant function between vine and bush types, that is, it has high correlations with number of nodes per plant (1) and number of long internodes (17). These results show also that the three seed-weight groups can include branchy types of plants. E31 coda. Nwom. “nNm. xmxm. meme. Ammo. Ncco. cmqo. wmwm. emmm. wan. no_m. - moms. __:o. mean. o_na. macs. c_ro. ammo. wcxm. scam. . .; ammo. . o—dqu-MONQO‘OHN dH—‘HHHHHHHNNN fiNnemohwo occuue~ouuoo . ooanouuu> was“; unmask -E:_ooz ac< vacuom-a>oo= mo coma—cc< Eben ao—pouuo> uo ”swam uoouuzocn cognac: na_:nco«uouom .Am u~pea nmwn. ngmq. wand. hoo~.. nnww. on~¢.- Noofi. unmo. wmm0.a moms.u «noo.c o-~.u «woo. Nauo. unoo.a oqwm.u. ~nn~.c unwo.. ouwa. oooc.u w-CNF'DQV‘ a n u . a ugwsuz-eassoz vuvoom gh>uo= “accuuoaouuoov was“; azuuozussuvoz vc< cocoomua>no= we auouonu coerce: neuzocouuouoz .uo oases aunn. NnNm. «fiao. cow“. cwoe. nmon. mama. onwm. non». Nmmm. Onhw. Mono. ooem. ance. ouno. aoem. owoa. owma. ammo. cums. swam. «new. MN NN MNMQWONQO‘O o-‘HP‘H'dFd—CHHN w-‘NflQWOh-QO‘O .-a asoque~ouuoo use“; sauna: -asavo: can cocoom-u;m.3 no nonauoc< soup oounuuua> we «each usuuucova :uozuon onusacoquuuoz oodaoauu> b .an oupuh cano.a anew. m-o.- moms. ammo. nu me no am so on. Race. no. «050.- cc. case. an. on—c.- oo.c nnoo. ocmm.c $550.- menu. couc. unna.a .4854an n .u “swag:-esseuz vovooa -uamuq “accuse—9.6603 coca; sauna: .asuoaz we. cocoom.u=waa no enouunu sooauon amu=uoouuc~ou ..a was.“ 32 Table 8c. Multiple Discriminant Analysis between Light-seeded, Medium-weight and Heavy-seeded Lines Variables Discriminant Axes Loadings 1 2 _ l -.2864 .5705 2 .0493 -.O356 3 .1351 -.1261 ’4 .0439 .0145 5 -.0634 -.0462 6 -.2134 .1899 7 -.4005 .1214 8 -.5265 .2964 9 .9115 —.1776 10 -.0127 -.1501 11 .4259 ' .0305 12 .8874 -.0830 13 .9159 -.l644 14 .8987 -.1240 15 .7612 .0784 16 -.2884 .1546 17 -.3580 . .5227 18 .6435 -.3102 19 ~11154 . .1257 20 .5787 -.l801 21 .5693' -.0167 22 $5543 -.1267 % 0f Total Variance : 88.60 11 .40 Chi-Square value : 292.655 105.486 Degrees of Freedom : 23 21 Significance Level : .0000 .0000 . DISCUSSION Interpretation of the results Factor analysis of the complete-set of data reveals that there are two major factors or patterns of production in beans and a much less important factor more related to the morphology of the plant than to its productivity. Thus, the relation of this third factor to production is indirect. The three factors have been tentatively called pod-weight, pod-number and growth,respectiVe1y. All three factors remained invariant from.one location to another despite the different types of planting which were used to accentuate the diffe— rences between the two environments. It is to be noted, however, that that the means of many of the variables were found to be significantly different between locations. Factor analyses with the growth types show that these two major factors exist among both determinate and indeterminate lines and fur- thermore, that whereas the first factor is the most important for the bush type, the second is probably the main factor for the vine type. The third factor was also recovered in both cases, but with a diffe- rent configuration among the indeterminate lines. Consequently, it can be concluded that these patterns are consistent across locations and growth types. Multiple discriminant analysis, both between growth types and between seed-weight groups, establishes beyond any doubt that the third factor of the complete-set results is indeed a growth factor. It was 33 34 responsible for about 10% of the total variation before rotation and 15% after rotation in the factor analysis of the complete-set of data. This indicate that growth habit has a relatively low impact on bean productivity. The factor of growth can be expressed in terms of the total number of nodes and the number of long internodes on the plant. Probably, the greater the number of internodes of the plant (presuma- bly indeterminate), the thinner the stem.and the pods and also the lighter in weight will be the whole plant. In an extreme case, the plant will certainly produce too many racemes, many undeveloped pods giving way to an unbalanced set of yield components, and consequently a very low yield. The results of the factor analyses of seed-weight groups data are in accord with those of the complete-set. The first factor in the latter was in fact a weight factor, since it was not recovered in any of the seed-weight groups analyses. But,the second factor, or number-factor was identified in all of them. This indicates that the grouping of bean va- rieties by seed weight is very appropriate and its implication on pro- ductive potential is great, given the importance of the weight—factor in this study. As a multiple discriminant function, the weight factor accounted for about 89% of the total variance. In the results from factor analy- sis of the complete—set of data, it was as important as the number- factor. This number-factor was recovered in all seed-weight group a- nalyses and as in the indeterminate sub-set, it was the most important factor in the light-seeded sub-set results. 35 Implications of the results on the Development of an Improved Plant-type in beans The results indicate that in order to reach a high yield through high seed weight, a plant has to have a sturdy stem and relatively few but long basal internodes. Such a plant can not therefore also have many nodes and certainly will have no or very few long internodes. It will be more feasible to increase the number of pods on that plant than to augment its number of seeds per pod. Its production will be mainly or uniquely located on the stem. The height of the plant is due mostly to the length of its basal internodes. By contrast to this pod-weight type, in order to attain a high yield per plant through a high pod number, a bean plant needs many basal internodes which should be moderately sturdy. The number of nodes on the plant can be higher than in the preceding type. Its pod—bearing potential is higher on the branches than on the stem. This plant—type can have considerably more seeds per pod, but the seed-weight should be kept at a minimum. Interestingly, this pod— number type can weigh more than the pod—weight type and is also more efficient since its number of pods per productive node is higher. The height of the plant is due primarily to the number of basal inter- nodes. The factor related to growth shows that a bean plant with numerous, long and thin internodes will be an extremely poor yielder. This also will be a strong competitor. Consequently, in improving yield, either on a per—plant basis or on an area basis, such a plant-type should be avoided. However, this result can also be seen as an invitation for inter-breeding between the pod—weight and the pod-number 36 types, so long as it is possible to maintain a favorable balance be- tween the number of nodes and the diameter of the stem. Whenever this can be done and at the same time the number of upper-internodes can be kept at a minimum, the yield of the plant, whatever its original plant-type will be improved considerably. It is interesting to note that although most of the unadapted, south-american varieties were not included in the factor analyses, their yields and morphological characteristics strongly confirm the result that too many nodes or internodes on the plant, particularly the upper ones, will provoke a reduction in its yield, It seems therefore feasible to attempt to develop an improved bean plant-type by increasing the god-bearing capacity of the stem of a heavy-seeded line. This can be done by augmenting the number of its short or basal internodes and also maintaining the sturdiness of its stem. If the seed size can be kept at a level commercially accep- table, without increasing the number of branches, this superior plant- type will fit well to the objective of improving yield on an area basis, because inter-plant competition will be thus minimum. However, the pod- bearing capacity of the plant as a whole can still be enhanced by aug- menting the number of its branches. This can be realized through back- crossing with a pod-number type parent. This may not appeal to most plant breeders in View of the fact that the inter-plant competition might be consequently increased. But, it still may be wise to do it, at least as an intermediate step, since this is likely to produce a reduction in seed size, something which in itself may be economically beneficial. Moreover, the pod-number type parent may also be used as 37 as the carrier of other useful traits. But, if the objective is to improve yield on a per-plant basis, the number of branches on the plant should always be a matter of concern, since the higher their number, the higher the yield. In fact, everything else being compa- rable, any bean plant will yield more with than without branches. So, keeping in mind the limit imposed by the third factor, the growth factor, several improved bean plant-types can be obtained through repeated inter-breeding between the two major plant-types, namely the pod-weight and the pod-number types, Shifting emphasis, through increasing the number of backcrosses toward one type or the other, permits the plant breeder to enhance the differences between those improved plant—types, that is, to separate genetically the ones approaching his ideal of a good variety on a plant-yield basis from those getting close to his"ideotype" on an area—yield basis. SUMMARY AND CONCLUSION There results of this study indicate that there three uncorre- lated factors or patterns of production in beans and that only the pod of the plant can be used to distinguish between the first two. The first most important factor is pod-weight, the weight of the pod as expressed through the size of both the seed and the pod. The second most important factor is pod-number and is related more with the number of racemes on the plant than with the number of pods per productive node. It is interesting to note that the number of branches on the plant has only a very low weight in the obtaining of this fac- tor. Surprisingly, number of seeds per pod does not have a high posi- tive loading on any of these two major factors. On the contrary, that variable has a negative and moderately high loading only in the first factor. The third and least important factor is a growth factor with emphasis on the general size of the plant. Incidentally, this factor will not necessarily discriminate between determinate and indetermina- te lines. Its resemblance with the discriminant function calculated for that purpose indicates rather that the indeterminate lines are more likely to have many nodes and long internodes as compared to the determinate lines. It was found that it is possible to make progress in yield im- provement by breeding within any of the two major types, that is, increasing the weight of the seed of a pod-weight type variety or the number of pods of a pod-number type variety will often result in better yields. In the first case, the short or basal internodes 38 39 should be made longer and sturdier whereas in the second they should become’more numerous while remaining sturdy. However, greater advance is obtainable through interbreeding between the two types. For exam: ple, the number of short internodes can be increased in a pod-weight type line so that it can produce more pods. In the same manner, the weight of the pod and the seed can be improved in a pod-number type variety by making its short internodes longer and sturdier. However, two variables should be kept under control in either case, the number of long internodes and the number of branches. The first will prevent the plant from.approximating its maximum yield potential, and the se- cond will make the plant a strong competitor, thus limiting its use to production on a per-plant basis. Obviously, since beans are categorized for commercial usage lar- gely on the basis of seed size, the plant breeder therefore must place greater emphasis upon obtaining a plant-type which, on an area basis, maximizes the pod number. B IBL IO GRAPHY BIBLIOGRAPHY Bailey, D. W. 1956. A comparison of genetic and environmental prin- cipal components of morphogenesis in mice. Growth 20:63-74. Cattell, R. B. 1965. Factor analysis: An introduction to essentials. I. The purpose and underlying models. Biometrics 21:190-210. 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