Correlation Studies on Yield and its Components in Inter Specific Cotton Hybrids (G. hirsutumxG. barbadense) for Developing Heterotic Box  

Yanal Alkuddsi , S.S. Patil , S.M. Manjula , B.C. Patil
Agricultural Research Station, Dharwad Farm, University of Agricultural Sciences, Dharwad, Karnataka, 580005, India
Author    Correspondence author
Molecular Plant Breeding, 2013, Vol. 4, No. 28   doi: 10.5376/mpb.2013.04.0028
Received: 09 Sep., 2013    Accepted: 11 Sep., 2013    Published: 16 Sep., 2013
© 2013 BioPublisher Publishing Platform
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Preferred citation for this article:

Alkuddsi et al., Correlation Studies on Yield and its Components in Inter Specific Cotton Hybrids (G.hirsutum×G.barbadense) for Developing Heterotic Box, Molecular Plant Breeding, Vol.4, No.28 228-237 (doi: 10.5376/mpb.2013.04.0028)

Abstract

This present study was conducted with 92 F1cotton hybrids to provide information on interrelationships of seed cotton yield with some characters (plant height (cm), number of monopodial per plant, number of sympodia per plant, number of bolls per plant, mean boll weight (g), reproductive points on sympodia, sympodial length at 50 % plant height (cm), inter branch distance (cm), seed index (g),  ginning outturn (%) and lint index (g)  and to partition the observed correlations into their direct and indirect effects. In this experiment, 92 inter specific crosses (G.hirsutumxG.barbadense) along with three checks (MRC 6918 Bt check, RAHB 87 and DCH 32 non Bt checks) were subjected to LinexTester analysis (23 hirsutum testers and 4 barbadense lines). This experiment was laid out in Randomized Block Design (RBD) with two replications at University of Agricultural Sciences, Dharwad, Indis. Seed cotton yield exhibited significant positive correlation with mean boll weight (0.1123), ginning outturn (0.1339) and lint index (0.1020). Among these, seed cotton yield recorded significant positive strong correlation with ginning outturn.The direct effect on seed cotton yield was positive in plant height (0.0274), number of monopodia per plant (0.0480), number of sympodia per plant (0.0103), mean boll weight (0.1241), sympodial length at 50% plant height (0.0601) and lint index (0.6884), while the direct effect on seed cotton yield was negative in number of bolls per plant (-0.0024),  reproductive points on sympodia (-0.0687), inter branch distance (-0.0571), seed index (-0.4620) and ginning outturn (-0.4280).

Keywords
Seed cotton yield; Correlation co-efficient; Path co-efficient

Estimates of genotypic and phenotypic correlations among characters are useful in planning and evaluating breeding programs.Knowledge of the correlation that exists between important characteristics may facilitate the interpretation of results and provide the basis for planning more efficient programs.

Correlation co-efficient determines simple relations among the traits, so it doesn’t determine always decisive results about determination of plant selection criteria (Cakmakci et al., 1998). Path co-efficient analysis as to correlation co-efficient gives more detailed information on the relations so it is commonly used by researches in plant breeding to determine seed cotton yield and seed cotton yield criteria relations (Williams et al., 1990; Kang et al., 1993; Board et al., 1997).
Muthuswamy and Vivekanandan (2004) worked out correlation involving 48 crosses of 12 lines tester combination and reported that seed cotton yield was significant and positively correlated with GOT. Boll weight had positive significant association with lint index and GOT. Seed index had positive significant association with lint index.
The plant height showed positive correlation with seed cotton yield in E2, while negative correlation was recorded in E3. Length of sympodia was positively correlated with seed cotton yield in E3, while non-significant correlation was observed in E1 and E2 (Ashok et al., 2004).
Muhammad Iqbal (2006) reported positive significant correlation of node of first fruiting branch, monopodial branches/plant, boll number and boll weight with yield.
Sun-Junling et al (2006) reported in correlation analysis indicated that the correlation co-efficients of phenotypic characters and quality characters, including lint percentage boll weight, plant height, fibre length fibre strength and micronaire, between M4 and M5 were highly significant, suggesting that the genetic variation in the irradiated progenies was heritable.
Verma et al (2006) reported that seed cotton yield per plant had positive and highly significant correlation with number of bolls per plant, number of sympodia and ginning outturn.
The association of yield with monopodia was negative (Griffee et al., 1929; Patil, 1974). Contrary to this, Venkataraman and Santhanam (1962) reported positive association of monopodia with yield but Singh et al (1968) noticed non significant positive correlation with yield. Significant positive correlations between number of monopodia per plant and seed cotton yield per plant were observed by Vijendradas (1981).
The sympodia per plant was represented to be significantly and positively correlated with seed cotton yield by many workers (Miller et al., 1958; Singh et al., 1968; Gill and Singh, 1981; Singh and Singh, 1981; Musande et al., 1981; Bhatade, 1982; Shaikh and Upadhyay, 1982; Gawand et al., 1984; Basu and Bhat, 1987; Choudhary et al., 1988; Arshad et al., 1993 and Killi, 1995).
Annapurve et al (2007) studied the phenotypic and genotypic correlation in American cotton and reported that both phenotypic and genotypic correlation were significant for most of the characters like number of bolls per plant, boll weight, boll bursting and yield per plant.
Neelimaand Chenga Reddy (2008) conducted the correlation co-efficient studies involving 4 lines, 10 testers and their 40 F1 crosses in G. hirsutum and revealed that number of sympodia per plant, number of bolls per plant, boll weight, seed index and micronaire value showed significant positive association with seed cotton yield both at genotypic and phenotypic level indicating that these characters can be simultaneously improved.
The method of path co-efficient analysis provides an effective means of finding out direct and indirect causes of association of various component characters. The method was developed by Wright (1921) and was first applied to plant breeding by Dewey and Lu (1959).
Verma et al (2006) in a path co-efficient analysis study involving 51 genotypes having different cytoplasm sources developed through successive back crossing and reported that selection for high seed cotton yield seemed to be positive through number of bolls per plant, number of sympodia, ginning out turn as these exerted positive direct effects and exhibited positive significant association with seed cotton yield.
Gururaj (2006) revealed that compact genotypes showed differences for per cent deviation from the population mean. However, the mean per cent deviations were positive for plant height (6.37%), number of monopodia per plant (72.38 %), inter boll distance (1.54%), number of bolls per plant (24.46%), boll weight (8.12%), seed index (8.31%) and GOT (6.1%). Negative per cent deviations were recorded by number of sympodia per plant, sympodia length at fifty per cent height (-6.40%), diameter of plant (-0.57%) and halo length (-4.11%). However, potential genotype had the some difference in their path of productivity as obtained in the genotypes RAH-216 (number of sympodia per plant), RAH-205-91 (for diameter of plant and inter boll distance) and 433 0503AYC (for halo length).
Neelima and Chenga Reddy (2008) studied the path co-efficient analysis in 4 lines, 10 testers and their 40 F2s of G. hirsutum indicated that bolls per plant through boll weight, lint index and boll weight through lint index exerted high positive indirect effects on seed cotton yield.
Hanamaraddi (2010) revealed that top performing genotypes have positive value for seed cotton yield (40.53%), number of bolls (40.21%), boll weight (11.11%), plant height (7.09%), halo length (7.21%), number of nodes (6.55%), seed index (4.85%), lint index (4.04%), ginning out turn (2.48%) number of seed per boll (2.38%) and number of seed (1.92%). And also show negative per cent deviations for monopodia (-1.96%) and inter boll distance (-2.25%). These negative deviations for number of monopodia and inter boll distance are highly desirable.
This present study was conducted with 92 F1 cotton hybrids to provide information on interrelationships of seed cotton yield with some characters, plant height (cm), number of monopodial per plant, number of sympodia per plant, number of bolls per plant, mean boll weight (g), reproductive points on sympodia, sympodial length at 50% plant height (cm), inter branch distance (cm), seed index (g), ginning outturn (%) and lint index (g) and to partition the observed correlations into their direct and indirect effects.
1 Results and Discussion
Yield is a complex polygenically inherited character resulting from multiplicative interaction of its contributing characters. It is highly influenced by the environment, hence selection based on yield alone may limit the improvement.Whereas, the component characters of yield are less complex in inheritance and are influenced by the environment to a lesser extent. Thus, effective improvement in yield may be brought about through selection for yield component characters.
Yield component characters show association among themselves and also with yield. Favorable associations between desirable attributes will help improvement in a joint manner. Whereas, unfavorable associations between the desirable attributes under selection may limit genetic advance. Hence, knowledge of associations between the yield components and also among themselves are essential for planning a sound breeding programme.
Grafius (1959) reported that there may not be genes for yield as such, but operate only through its components. So, correlation analysis provides the information on nature and magnitude of the association of different components characters with seed yield, which is regarded as highly complex trait in which the breeder is ultimately interested. So, it is a matter of great importance to the plant breeders to find out as to which of the characters are correlated with yield and also how they are associated among themselves, if negative association between characters is due to pleiotropic effects it would be very difficult to obtain the desired combinations while if linkage is involved, special breeding programmes are needed to break these linkage blocks.
Phenotypic correlation co-efficient in inter specific crosses: The phenotypic correlation co-efficients among all characters related to seed cotton yield per plant were estimated and the results are presented in Table 1 & Figure 1. Seed cotton yield (kg/ha): Seed cotton yield exhibited significant positive correlation with mean boll weight (0.1123), ginning outturn (0.1339) and lint index (0.1020). Among these, seed cotton yield recorded significant positive strong correlation with ginning outturn. Similar results of association of these traits with seed cotton yield were reported by Bhatade (1982), Gulamov et al. (1974), Manivasakam (1985), Nazir and Khan (1974), Verma et al. (2006) and Muthu et al. (2004).


Table 1 Phenotypic correlation co-efficient of seed cotton yield per plant and different quantitative characters in Line×Tester inter specific crosses


Figure 1 Phenotypic correlation co-efficient of seed cotton yield per plant and different quantitative characters in Line×Tester inter specific crosses
 
Number of monopodia per plant: Number of monopodia per plant recorded non significant negative correlation with plant height (-0.1146). Number of sympodia per plant: Number of sympodia per plant exhibited highly significant positive correlation with plant height (0.5141). Number of bolls per plant: Number of bolls per plant recorded highly significant positive correlation with number of sympodia per plant (0.2654). Mean boll weight (g): Mean boll weight exhibited non significant correlation with other characters. Reproductive points on sympodia: Reproductive points on sympodia showed highly significant positive correlation with plant height (0.2229) and number of sympodia per plant (0.1880). Among these, reproductive points recorded highly significant positive strong correlation with plant height. Sympodial length at 50% plant height (cm): Sympodial length at 50% plant height exhibited highly significant positive correlation with plant height (0.5214), number of sympodia per plant (0.3137) and reproductive points on sympodia (0.6418). Among these, sympodial length at 50% plant height showed highly significant positive strong correlation with reproductive points on sympodia, while sympodial length at 50 per cent plant height showed highly significant negative correlation with number of monopodia per plant (-0.1895). Inter branch distance (cm): Inter branch distance exhibited highly significant positive correlation with plant height (0.4779), number of sympodia per plant (0.2179) and sympodial length at 50% plant height (0.3424). Among these, inter branch distance had highly significant positive strong correlation with plant height, while exhibited significant positive correlation with reproductive points on sympodia (0.1441). Seed index (g): Seed index recorded highly significant positive correlation with number of bolls per plant (0.1939) and highly significant negative correlation with inter branch distance (-0.1896). Ginning outturn (%): Ginning outturn showed non significant correlation with other characters.
Lint index (g)
Lint index exhibited highly significant positive association with number of bolls per plant (0.2316), seed index (0.5508) and ginning outturn (0.7942). Among these, lint index recorded highly significant positive strong correlation with ginning outturn, while lint index recorded significant positive correlation with number of sympodia per plant (0.1510). The same result of significant positive correlation between seed index and lint index confirmed by Sambamurthy et al. (1994).
Direct and indirect phenotypic effects of component characters on seed cotton yield in inter specific crosses
The phenotypic path co-efficient analysis among all characters related to seed cotton yield per plant were estimated and the results are presented in Table 2 & Figure 2.


Table 2 Direct and indirect phenotypic effects of seed cotton yield and different quantitative characters in Line ×Tester inter specific crosses


Figure 2 Direct and indirect phenotypic effects of seed cotton yield and different quantitative characters in Line×Tester inter specific crosses (YHB)
Plant height (cm)
The direct effect of this character on seed cotton yield was positive (0.0274), although plant height exhibited non significant positive association with seed cotton yield. The contribution of other characters indirectly on seed cotton yield through plant height appeared to be positive value in respect of number of sympodia per plant (0.0141), number of bolls per plant (0.0008), reproductive points on sympodia (0.0061), sympodial length at 50 % plant height (0.0143), inter branch distance (0.0131), seed index (0.0015), ginning outturn (0.000) and lint index (0.0006). The contribution had negative indirect effects via number of monopodia per plant (-0.0031) and mean boll weight (-0.0009).

Plant height (cm)
The direct effect of this character on seed cotton yield was positive (0.0274), although plant height exhibited non significant positive association with seed cotton yield. The contribution of other characters indirectly on seed cotton yield through plant height appeared to be positive value in respect of number of sympodia per plant (0.0141), number of bolls per plant (0.0008), reproductive points on sympodia (0.0061), sympodial length at 50 % plant height (0.0143), inter branch distance (0.0131), seed index (0.0015), ginning outturn (0.000) and lint index (0.0006). The contribution had negative indirect effects via number of monopodia per plant (-0.0031) and mean boll weight (-0.0009).
Number of monopodia per plant
It was observed that this character appeared to influence seed cotton yield directly as positive value (0.0480). Balakotaiah (1973), Gill and Singh (1981), Vijendaradradas (1981) represented the same result in their study. Although number of monopodia per plant recorded non significant positive correlation with seed cotton yield. The contribution of other characters indirectly on seed cotton yield through number of monopodia per plant appeared to be positive value in respect of number of bolls per plant (0.0066), ginning outturn (0.0013) and lint index (0.0007). The association recorded negative indirect effects via plant height (-0.0055), number of sympodia per plant (-0.0021), mean boll weight (-0.0018), reproductive points on sympodia (-0.0048), sympodial length at 50 % plant height (-0.0091), inter branch distance (-0.0051) and seed index (-0.0012).
Number of sympodia per plant
The direct influence of number of sympodia per plant towards seed cotton yield was positive (0.0103). Tomar and Singh (1992), Bhatade (1982) found the same result.Although number of sympodia per plant recorded non significant positive correlation with seed cotton yield. The contribution of other characters indirectly on seed cotton yield through number of sympodia per plant showed to be positive value in respect ofplant height (0.0053), number of bolls per plant (0.0027), mean boll weight (0.0007), reproductive points on sympodia (0.0019), sympodial length at 50% plant height (0.0032), inter branch distance (0.0022), seed index (0.0010), ginning outturn (0.0011) and lint index (0.0016). The contribution showed negative indirect effect via number of monopodia per plant (-0.0004).
Number of bolls per plant
The importance of number of bolls per plant towards seed yield has also been reported by Dedaniy and Pethani (1994), Tyagi et al (1988), Muthu et al (2004) and Verma et al (2006).Therefore this trait appears to be most important in influencing seed cotton yield in cotton and selection should be oriented towards high boll number. The direct effect of number of bolls per plant on seed cotton yield was negative (-0.0024), although number of bolls per plant exhibited non significant positive association with seed cotton yield. The contribution of other characters indirectly on seed cotton yield through number of bolls per plant appeared to be positive value in respect of inter branch distance (0.0002). The contribution exhibited negative indirect effects via plant height (-0.0001), number of monopodia per plant (-0.0003), number of sympodia per plant (-0.0006), mean boll weight (-0.0002), reproductive points on sympodia (-0.0002), sympodial length at 50 % plant height (-0.0002), seed index (-0.0005), ginning outturn (-0.0003) and lint index (-0.0005).
Mean boll weight (g)
The direct contribution of mean boll weight on seed cotton yield was positive (0.1241). The highest contribution of mean boll weight has been reported by Balakotaiah (1973), Gill and Singh (1981), Vijendaradradas (1981), Salimath (1975) and Krishnarao and Mary (1990). Although mean boll weight showed significant positive association with seed cotton yield. The contribution of other characters indirectly on seed cotton yield through mean boll weight appeared to be positive value in respect of number of sympodia per plant (0.0078), number of bolls per plant (0.0105), seed index (0.0149) and lint index (0.0029). The association exhibited negative indirect effects via plant height (-0.0043), number of monopodia per plant (-0.0047), reproductive points on sympodia (-0.0094), sympodial length at 50 % plant height (-0.010), inter branch distance (-0.0046) and ginning outturn (-0.0079).
Reproductive points on sympodia
It was observed that this character appeared to influence seed cotton yield directly as negative value (-0.0687), reproductive points on sympodia recorded non significant negative correlation with seed cotton yield. The contribution of other characters indirectly on seed cotton yield through reproductive points on sympodia to be positive value in respect of number of monopodia per plant (0.0069), mean boll weight (0.0052), seed index (0.0024), ginning outturn (0.0033) and lint index (0.0055). The association showed negative indirect effects via plant height (-0.0153), number of sympodia per plant (-0.0129), number of bolls per plant (-0.0058), sympodial length at 50 % plant height (-0.0441) and inter branch distance (-0.0099).
Sympodial length at 50 %plant height (cm)
The direct effect of sympodial length at 50% plant height on seed cotton yield was positive (0.0601), although sympodial length at 50% plant height recorded non significant negative correlation with seed cotton yield. The indirect contribution to seed cotton yield through sympodial length at 50% plant height was positive value in respect of plant height (0.0313), number of sympodia per plant (0.0189), number of bolls per plant (0.0057), reproductive points on sympodia (0.0386) and inter branch distance (0.0206). The association exhibited negative indirect effects via number of monopodia per plant (-0.0114), mean boll weight (-0.0048), seed index (-0.0024), ginning outturn (-0.0034) and lint index (-0.0057).
Inter branch distance (cm)
It expressed a considerably negative direct effect of inter branch distance on seed cotton yield (-0.0571) and had non significant negative correlation with seed cotton yield. The indirect effect of inter branch distance was positive through number of monopodia per plant (0.0060), number of bolls per plant (0.0055), mean boll weight (0.0021), seed index (0.0108) and lint index (0.0030). The association recorded negative indirect effects via plant height (-0.0273), number of sympodia per plant (-0.0124), reproductive points on sympodia (-0.0082), sympodial length at 50 % plant height (-0.0195) and ginning outturn (-0.0053).
Seed index (g)
It was observed that this character appeared to influence seed cotton yield directly as negative value (-0.4620), seed index recorded non significant negative correlation with seed cotton yield. The contribution of other characters indirectly on seed cotton yield through seed index to be positive value in respect of number of monopodia per plant (0.0114), reproductive points on sympodia (0.0161), sympodial length at 50 % plant height (0.0184), inter branch distance (0.0876) and ginning outturn (0.0263). The association recorded negative indirect effects via plant height (-0.0247), number of sympodia per plant (-0.0448), number of bolls per plant (-0.0896), mean boll weight (-0.0554) and lint index (-0.2545).
Ginning outturn (%)
The direct effect of ginning outturn on seed cotton yield was negative (-0.4280), ginning outturn recorded significant positive correlation with seed cotton yield. The contribution of other characters indirectly on seed cotton yield through ginning outturn to be positive value in respect of mean boll weight (0.0271), reproductive points on sympodia (0.0204), sympodial length at 50% plant height (0.0245) and seed index (0.0244). The association exhibited negative indirect effects via plant height (-0.0007), number of monopodia per plant (-0.0116), number of sympodia per plant (-0.0451), number of bolls per plant (-0.0572), inter branch distance (-0.0394) and lint index (-0.3400).
Lint index (g)
It expressed a considerably positive direct effect of lint index on seed cotton yield (0.6884) and had significant positive correlation with seed cotton yield. These findings are in line with the earlier findings of Dedaniya and Pethani (1994) and Tomar and Singh (1991). The indirect effect of lint index was positive through plant height (0.0153), number of monopodia per plant (0.0104), number of sympodia per plant (0.1039), number of bolls per plant (0.1594), mean boll weight (0.0163), seed index (0.3792) and ginning outturn (0.5468). The association recorded negative indirect effects via reproductive points on sympodia (-0.0549), sympodial length at 50 % plant height (-0.0656) and inter branch distance (-0.0360).
2 Conclusion
The measure of association and path co-efficient analysis was done in the study of Line×Tester inter specific crosses studies. It is necessary to work out path co-efficient analysis which partitions the observed correlation into direct and indirect effects and also reveals the cause and effect relationship between yield and their related traits. At phenotypic level, seed cotton yield exhibited significant positive correlation with mean boll weight, ginning outturn and lint index. Among these, seed cotton yield recorded significant positive strong correlation with ginning outturn.
Seed cotton yield being a complex polygenic character, direct selection based on this trait would not yield fruitful results without giving due importance to its genetic background. The association of yield and its component traits reflects the nature and degree of relationship between them. The correlation analysis helps in examining the possibility of improving yield through indirect selection of its component traits which are highly correlated.
3 Materials and Methods
The main objective of the Line×Tester study was to determine the combining ability status of the barbadense and hirsutumlines included in the heterotic box. These lines were compared with other barbadense and hirsutum lines. For the pattern of combining ability and the potentiality of the inter specific hybrids developed based on them.
In this experiment, 92 (YHB trial) inter specific crosses (G.hirsutum×G.barbadense) along with three checks (MRC 6918 Bt check, RAHB 87 and DCH 32 non Bt checks) were subjected to Line×Tester analysis (23 hirsutum testers and 4 barbadense lines). This experiment was laid out in Randomized Block Design (RBD) with two replications. Each entry was sown in 2 row plots spaced at 90 cm×60 cm with recommended dose of fertilizer and seeds were sown on 28-6-2010, 2~3 seeds were dibbled per spot in each row and thinning was attended to retain one healthy plant per hill at 25 days after sowing. All the recommended package of practices were followed to raise healthy crop.
Three plants in each entry were selected randomly and tagged. These tagged plants were used for recording observations on the following characters.
Plant height (cm): Plant height was measured in centimeters from the base of the plant to apex of the plant at maturity; Number of monopodia per plant: The total number of vegetative branches on the main stem were counted and recorded at the time of harvest; Number of sympodia per plant: The total number of sympodial branches i.e., number of fruiting branches present in the plant at maturity was counted; Number of bolls per plant: The number of bolls on a plant which contributed to seed cotton yield was counted and recorded at the time of harvest; Mean boll weight (g): Seed cotton obtained from a random sample of 20 bolls per plant was used to determine boll weight in grams; Reproductive points on sympodia: The total numbers of reproductive points on the sympodial branches were counted; Sympodial length at 50% plant height (cm): Sympodial branch length at 50 per cent plant height was chosen for measurement and expressed in centimeter; Inter branch distance (cm): The distance between two branches at 50% plant height was taken and expressed in centimeters; Seed index (g): Hundred good and bold seeds were weighed to determine the seed index in grams;
Ginning outturn (%)
A random sample of 300 g seed cotton from each entry was ginned and the lint yield obtained from it was utilized for working out the ginning outturn in the following manner.
 
 
 
Lint index (g)
It is the weight of the lint obtained from hundred seeds and expressed in grams. This was calculated by using the formula.
 
 
 
Seed cotton yield (kg ha-1)
It is the mean seed cotton yield harvested till final picking from the net plot area and expressed in kg/ha.
Analysis of covariance was computed in a fashion similar to that of analysis of variance formula and these statistics were made use of in calculating, phenotypic correlation co-efficient.
Phenotypic correlation co-efficient (rp)
The degree of phenotypic association amongst eleven characters was computed as per the formula given by Weber and Moorthy (1952).
 
 
Where, r p=Phenotypic correlation co-efficient; Cov P1.2=Phenotypic covariance between two traits (1 and 2); σ P1=Phenotypic standard deviation of first trait (1); σ P2=Phenotypic standard deviation of second trait (2)
Path co-efficient analysis
Path co-efficient analysis was carried out using phenotypic correlation values of yield components on kapas yield as suggested by Wright (1921) and illustrated by Dewey and Lu (1959). Standard path co-efficients which are the standardized partial regression co-efficients were obtained using statistical software package called GENRES. These values were obtained by solving the following set of ‘P’ simultaneous equations by using the above package.
P01 + P02 r12 + ----------- + P0P r1P = r 01
P01 + P12 r02 + ----------- + P0P r2P = r 02
 
                     
 
P01 + r1P+ P02 r2 P ----------- + P0P = r 0P
Where P01, P02, --------------P0P are the direct effects of variables 1,2, --------------p on the dependent variable 0 and r12, r13 ,-------r1P------ rP(P-1 ) are the possible correlation co-efficients between various independent variables and r01, r02, r03 ------- r0P are the correlations between dependent and independent variables.
The indirect effect of the ith variable via jth variable is attained as (Poj×rij). The contribution of remaining unknown factor is measured as the residual factor, which is calculated as given below.
P2ox=1- [P2 01 + 2P01 P02 r12 + 2 P01 P03 r13 + ---------- + P202 + 2P02 P03 r13 + -------- +P20P]
Residual factor = (P2ox)½
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