Preface The father of genetics, Gregor Johann Mendel proposed the laws of inheritance based on his experiments in pea crop. The important laws are: 1) Law of segregation or purity of gametes and 2) Law of independent assortment. Mendel had two important assumptions viz., 1) a heterozygote produces two types of gametes in equal proportion and 2) gametes fuse together randomly. The two laws are valid till date. Interestingly, the segregation ratio obtained by Mendel in his experiments are similar to the ratios obtained using probability principles. Thus, Mendel recognized the statistical nature of the genetical variables (Singh and Chaudhary, 1977). Later, Fisher, Sewall Wright and Haldane developed the generalized theories and a separate discipline namely Quantitative Genetics bloomed. In Quantitative Genetics, the measurable characters are considered. Continuous variation is the characteristic feature of a quantitative trait. Quantitative genetics perceive genes based on phenotypic expression. Johannsen (1909) proposed the concept of genotype and phenotype. Multiple factor hypothesis was proposed by Nilson-Ehle (1909); Emerson and East (1913) and Davenport (1913). The work of Mather and his coworkers, Comstock, Robinson Cockerham, Kempthorne, Hayman, Jinks etc. in plants and Lush and his co-workers in animals gave further impetus to the study of quantitative genetics. Quantitative Genetics provides the foundation for breeding methodologies. Several books are available in Quantitative or biometrical genetics. However, the present work is unique in that sense it gives formulae along with actual data (analyzed) for the easy understanding. This book is mainly meant for post graduate and research scholars in Quantitative Genetics. A careful perusal of the book will give clear cut idea about the interpretation of the data and formulation of breeding strategies. However, the authors ready to accept ideas and suggestions to improve the quality of the book. The ultimate aim of the authors is to make the scholars to write dissertation (thesis), project reports and articles without any hurdle. The authors wish all the best for the readers.

Any crop improvement programme primarily depends on the amount of genetic variability available and the extent to which the economic traits are heritable. A wide survey of genetic variability and a thorough understanding of the genetic makeup of the crop with the biometrial tools is very much indispensable for initiating an effective breeding programme. Effectiveness in breeding programme depends upon three main factors viz., (i) the extent of genetic variation available, (ii) heritabiity of the variation and (iii) intensity of the selection pressure that can be applied. As most of the characters of economic importance and related characters which influence the performance of the economic character in a given set of environment are highly influenced by the environmental conditions, the heritable (genotypic) variation usually masked by non-heritable (environmental) variation and thereby creates difficulty in exercising the selection. Hence, it becomes necessary to partition the overall variability into heritable and non-heritable components to enable the breeders to plan for proper breeding programme. The estimates of mean, variance and standard error are worked out by adopting the standard methods of Panse and Sukhatme (1961). Usually, randomized block design is the choice of the statistical design. Various genetic parameters are worked out from the mean squares. Assessment of genetic variation is usually made through the computation of genotypic variance and genotypic coefficient of variation. The proportion of the total genetic variance expressed as percentage of total phenotypic variance has been used as a measure of genetic variability of heritability in broad sense.

Seed yield is the foremost important attribute and is the end product of many complex component characters which singly or jointly influence the seed yield. Seed yield does not possess gene per se as such. It is the interaction phenotype of the harmonious understanding, mutual adjustment and manifestation of its component characters. As most of the component characters are polygenic, they are greatly influenced by the environment. So, the selection of genotypes based on yield as such is not likely to be effective. The information on the strength and direction of association of the component characters with seed yield and also among themselves will be very useful in formulating an effective breeding programme for seed yield improvement in a given time. Correlation studies provide about the magnitude and direction of genetic association between the seed yield and its component characters. Correlation coefficients provide the measure of association between two characters. But, they do not provide the causal basis of such association. Further, correlation between seed yield and its component characters are often misleading due to the inter-relationship that exist among the component characters and the operation of size symmetry and component compensation that thwart the breeding progress. The direct contribution of each component to the seed yield and the indirect effects it has through its association with other components cannot be differentiated from mere correlation studies. This is how, it became necessary for a plant breeder to have information on the direct and indirect effects of these components on the seed yield to identify the key characters for improvement.

One of the important studies in the crop improvement programme is the discriminant function analysis. This is a method of exercising selection in plants, which show the extent of which character is genetically related to yield (Goulden, 1959). The technique of discriminant function was developed by Fisher (1936). But, the application of this method for plant selection was first made by Smith (1936) in wheat. He developed an index design for the selection of plant lines in wheat using the concept of discriminant function to derive a linear equation based on observable characteristics as the best available guide to the genetic value of each line. Stephens (1942) reported about the interaction of yield components within the plant and within the population, which serves as an index for yielding ability. According to Hazel and Lush (1943), selection for a total score or net desirability is more efficient that selection for one trait at a time. The formulae presented by them for selection index give proper weight to each trait that are more efficient than selection for one trait at a time or for several traits with an independent culling level for each trait. According to Darlington and Mather (1949) discriminant function has been defined as a linear compound of series of varieties, obtained by giving the different varieties with individual coefficients which will minimize the difference between classes relative to variation within classes of objects of which the variates are measurements. Mather (1949) reported that these functions afford to the best available means of discriminating the classes and evaluating the genotype of a plant in terms of the observed values of various characters. Pundir and Raj (1971) reported that selection index based on three characters combination i.e., seeds per siliqua, 1000 seed weight and yield per plant was efficient in toria.

A successfully developed new cultivar should have stable performance and broad adaptation over a range of environments, seasons and locations, in addition to high yielding potential. Evaluating performance stability and range of adaptation is become increasingly important in breeding programmes. Since, stability and adaptability are important selection criteria in breeding programmes, in depth research on stability is needed for a better estimate of crop performance and adaptability. Genotype × season interaction is of major performance to the plant breeder in developing stable but high yielding varieties of crop plants. When new varieties are tested over a series of seasons, the relative ranking of the varieties for any given attribute is rarely the same at each season. This increases the difficulty of identifying superior stable genotypes. Comstock and Moll (1963) have shown statistically the effect of large genotype × environment interaction in reducing the progress from selection. In order to minimize the genotype × season interaction, stratification of the seasons can be practiced. The implication in stratification is that suitable selections could be made for unfavourable and favourable seasons encountered within the sampling units proposed. Stratification of the environments has been practiced for a long time to identify suitable selection that could be made for the narrower range of environments encountered within the sampling units proposed. However, even with this refinement of technique, the interactions of genotypes with locations in a sub-region with environments encountered at the same location in different years, frequently remain too large (Allard and Bradshaw, 1964). Little is known about the environmental factors contributing to such interactions. Sprague (1966) suggested that even if such information were available, the possibility of materially reducing the genotype × environment interaction in field experiments would remain questionable.

Genetic architecture of a population is the result of prolonged natural selection. The populations which exist in diverse environments might have been strongly diversified genetically. So, it is necessary to understand the extent and rates of genetic divergence existing between the diversified forms or types. Genetic diversity may be available in improved germplasm or it may exist only in genetically inferior stocks. Genetic divergence among the parents is important in self pollinated crops because a cross involving genetically diverse parents is likely to produce high heterozygous as well as heterotic effect. In such a cross, more variability could be expected in the segregating generations due to the accumulation of the genes of both the parents. Genetic divergence, as one of the criteria of selection of parents is considered in plant breeding as early as 1962 (Murthy et al., 1962; Timothy, 1963). Later, Joshi and Dhawan (1966), Murthy and Arunachalam (1966), Arunachalam et al. (1984) and Lefort-Buson (1987) also stated that for exploiting heterosis as a means of increasing production, it is necessary to have parents of maximum divergence. The more diverse the parents, more is the chance of pronounced heterotic effects and increased spectrum of variability in the segregating generations. The availability of statistical tools to quantitatively measure the genetic divergence between two or more populations and the relative contribution of individual characters to the total divergence have permitted to trace the evolutionary patterns in some crops such as rice and tobacco and in choosing the parents for hybridization in crop plants (Rao, 1958; Blackith, 1960; Morishima and Oka, 1960; Murthy et al., 1962). Among the several statistical methods developed for measuring the divergence between the populations, multivariate analysis (D2 statistic) developed by Mahalanobis in 1936, has been found to be a potent tool. Mahalanbobis’s D2 statistic is an effective tool in quantifying the degree of divergence at genetic level and provides a quantitative measure of the association between geographic and genetic diversity based on generalized distance. It is now known that the estimates of genetic divergence are also influenced by environments and seasons. Therefore, if the divergence analysis is conducted in a number of seasons and parents are selected on the basis of consistent divergence over the seasons, the problem of influence of seasons on divergence analysis can be largely overcome.

6.1. Line × tester analysis A successful approach of crop breeding programme depends on the proper choice of best parents for hybridization and the ideal selection adopted in the early generation. But, the selection of parents for hybridization programme is relatively tough in the case of complex traits like yield and their components as they governed by large number of quantitative genes which are influenced by environments and seasons. Thus, breeders often meet with difficulty to fix in advance the desirable parents which will give superior progenies. In such a situation, knowledge on the nature of gene action on such complex quantitative traits of economic importance is necessary to plan and adopt appropriate selection techniques (Simmonds, 1979) and breeding methodology. Combining ability analysis gives useful information regarding the selection of parents in terms of the performance of the hybrids. This analysis elucidates the nature and magnitude of various types of gene action involved in the expression of quantitative traits (Dhillon, 1975). Thus, the knowledge of combining ability serves as an useful tool for the selection of plants for hybridization and further exploitation. Combining ability consists of general and specific combining ability. Sprague and Tatum (1942) defined general combining ability (gca) as the “average performance of a line in hybrid combinations” and specific combining ability (sca) “as those cases in which certain combinations do relatively better of worse than would be expected on the basis of the average performance of the lines involved. The general combining ability provides an indication of the importance of genes of largely additive in nature, while specific combining ability indicates the importance of non-additive gene effects. Several biometrical techniques are available to estimate the general combining ability of parents and specific combining ability of hybrids. Among them, line × tester analysis is extensively used both in self and cross pollinated crops. The advantage of line × tester analysis is that a large number of inbred lines can be evaluated than diallel analysis. Besides, it provides additional information on the performance of the specific hybrids on gene action in contrast to top cross or polycross. The line × tester analysis is an efficient biometrical approach for assessing the combining ability of parents and hybrids (Kempthorne, 1957).

Eswaran (2007) made genetic analysis of six parents and their 30 hybrids of Okra with the method of analysis proposed by Hayman (1954b). He estimated the various genetic parameters viz., D?, F?, H?1, H?2 and Ê as well as the various genetic ratio. The results are presented in Tables 69 and 70. The estimates of D? were significant for seven out of ten characters studied. This indicated that the component of variation due to additive genetic variance was important for days to seedling emergence, number of first fruiting node, height of first fruiting node, plant height, days to first picking, number of fruits per plant and fruit weight. These characters could be improved by resorting to simple selection. The estimates of H? 1and H? 2were positive and significant for almost all the characters studied. It indicated that there were unequal frequency of alleles i.e., u 1 v at all the loci, in this context, ‘u’ refers to the frequency of alleles which increase the mean expression of the character and are situated at loci which exhibited dominance. On the other hand, ‘v’ corresponds to the frequency of alleles at loci that decreases the expression of the character. Further proof for the unequal distribution of the alleles over loci was obtained

Eswaran (2007) made graphic analysis with the model proposed by Jinks and Hayman (1953). He evolved 30 hybrids from six parents in a diallel fashion. The results are presented in Tables 71-74

The narrow sense heritability estimates were high for days to first flowering, plant height at first flowering, number of leaves at first flowering, plant height at maturity, productive tillers per plant, boot leaf area, 100 grain weight, grain length and grain L/B ratio. These characters may be controlled by additive genetic variance. This indicated that the individual genotype can be evaluated readily from their phenotypic expression. Simple selection would more effective in the sets of materials exhibiting greater additive genetic variability and desirable mean performance. Thus, it merit selection in the next generation. On the other hand, the remaining traits may largely be controlled by non-additive genetic variance or the number of genes controlling these traits may be more, or these traits may largely be influenced by a large number of modifiers or might be largely influenced by environment. As the effects of the environmental variance were generally low, the observed low heritability for the above traits might be due other factors other than environment. These traits are hard to improve by selection. It necessitates progeny testing.

The relative magnitude of different gene effects and an understanding of the mode of inheritance of complex quantitative traits have a direct bearing on the method of hybridization and selection which should be adopted in a specific breeding programme. Useful biometrical methods have been developed by the Birmingham group of scientists, to analyse the genetic architecture in organisms, through the use of first and second degree statistics obtainable from the true breeding inbreds, their first generation crosses and the different types of segregating progenies (Mather, 1949; Hayman, 1958; Gamble, 1962; Mather and Jinks, 1971). Since then this method has been widely used in the genetic analysis of various crops.

Among the different biometrical methods available, triple test cross analysis is one of the most efficient designs currently available for investigating the genetic structure of populations. It provides a test of epistasis and in its absence it gives independent and equally precise estimates of additive and dominance genetic components (Mather and Jinks, 1982). 11.1. Statistical analysis Triple test cross Jinks and Perkins (1970) method was applied to detect epistasis and estimate additive and dominance components of genetic variance.

Generation of new genetic variability through recombination breeding followed by appropriate selection procedure is one of the key factor for crop improvement. Gain in selection is restricted by correlated response of agronomically desirable as well as undesirable characters. There is a host of examples in various crop species where genes for desirable agronomic traits have been found to be closely linked with those governing undesirable characters (Murthy, 1971; Clegg et al., 1972). Such close linkages delay the realization of the full recombination potential in many crops. The biparental mating or the disruptive mating scheme, finds its importance in the improvement of autgamous speice,s because it increases the probabilities of obtaining useful recombinants as compared to selection without intermating (Hanson, 1959; Jensen, 1970). In contrast to classical pedigree breeding, random mating population could produce large amounts of variability. This variability can be maintained in the population by the use of recurrent intermating, while the population is improved by selection. According to the standard breeding methodology in self-pollinated crops selection for quantitative characters are generally taken up in early segregating generations. For characters like yield, selection is continued till the material becomes homozygous. Such characters are controlled by large number of genes and a very large population has to be raised to make selection effective. Since most plant breeders could never grow the theoretically required size of population, the recombination is highly restricted. It is likely that many of the valuable genes may be practically lost in the process of advancing the material by continued selfing the generations. Andrus (1963) suggested crossing of selected sibs in early generation for reassembling the genes capable of functioning in a balanced polygenic system.

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