# Sequential path model for determining interrelationships among grain yield and related characters in maize.

PATH COEFFICIENT ANALYSIS has been widely used in crop breeding to determine the nature of relationships between grain yield and its contributing components, and to identify those components with significant effects on yield for potential use as selection criteria (Puri et al., 1982; Kang et al., 1983; Milligan et al., 1990; Williams et al., 1990; Board et al., 1997; Moghaddam et al., 1998; Samonte et al., 1998). Wright (1921) first used this approach to organize and graphically portray the relationships between predictor variables and the response variable through a path diagram based on experimental results or a priori grounds. A path coefficient is a standardized partial regression coefficient, and measures the direct influence of a predictor variable on the response variable. This permits the separation of the correlation coefficient into direct effect (path coefficient) and indirect effects (effects exerted through other independent variables) (Li, 1975). The proportion of variance in the response variable explained by variance in the predictor variable (partial coefficient of determination) is the square of the path coefficient.

In most studies involving path analysis, researchers considered the predictor characters as first-order variables to analyze their effects over a dependent or response variable such as yield (Xu, 1986; Han et al., 1991; Simon, 1993; Agrama, 1996; Board et al., 1997; Kumar et al., 1999). This approach might result in multicollinearity for variables, particularly when correlations among some of the characters are high (Hair et al., 1995; Samonte et al., 1998). There may also be difficulties in (i) interpretation of the actual contribution of each variable, since the effects are mixed or confounded because of collinearity (Hair et al., 1995), and (ii) supplementation of unique explanatory predictions from additional variables. Samonte et al. (1998) adopted a sequential path analysis for determining the relationships among yield and related characters in rice (Oryza sativa L.) by organizing and analyzing various predictor variables in first-, second-, and third-order paths. However, the collinearity of predictor variables was not tested before organization of variables in different path orders. Also, validation of such a path model on datasets other than those used for designing the path model is important for further possible application of the model.

Our objectives were to determine the usefulness of a sequential path model relative to the conventional path model, and to analyze the associations between grain yield and related characters in maize by applying the model to different datasets, with special attention on the analysis of collinearity of various predictor variables and analyzing the predictive value of the model.

MATERIALS AND METHODS

Our study utilized (i) 56 cross combinations generated from an 8 x 8 diallel, involving the Indian maize inbred lines CM115, CM116, CM117, CM128, CM132, CM138, CM139, and NAI147; and (ii) 34 experimental crosses generated by a line x tester mating design (involving CM104, CM119, CM123, CM136, CM137, and CM207 as lines and CM111, CM128, CM129, CM209, CM211, and CM209 as testers). All the inbred lines used as parental lines, except NAI147, are elite maize inbred lines released under the All India Coordinated Maize Improvement Project, and are parental lines of some popular maize hybrids in India. The cross combinations were evaluated for their agronomic performance during the monsoon season (Kharif) of 1999 at the IARI Experimental Farm, New Delhi, and during winter/dry season (Rabi) of 1999-2000 at the Maize Winter Nursery, Hyderabad (referred subsequently as DM-1999 and HW-1999, respectively).

The field tests were performed in an [alpha]-lattice design with two replications, two rows per replication. Row length and width were 5 and 0.75 m, respectively. Plots were overplanted and thinned to a uniform plant stand of approximately 25 plants per row. All plots were hand weeded as necessary to maintain proper weed control. Data were collected on the following 12 characters in all replications on five competitive plants per plot. Plant height (PH) and ear height (EH) were measured as the distance (cm) from the soil surface to the node of the flag leaf and to the highest ear-bearing node, respectively, at the harvest stage. Other plant characters recorded were: total number of leaves per plant (NL); number of leaves above the primary ear (NLE); and number of tassel branches (NT). Number of ears per plant (NE) is the total number of ears harvested from the plot divided by the number of plants in the plot. The following characters were recorded on ears from five competitive plants from each plot: ear length (EL) (cm); ear diameter (ED) (cm); number of kernel rows (NR); total number of kernels per ear (TNK); kernel length (KL) (cm); 100-grain weight (100GW) (g); and total grain weight per ear (TGW), measured as an average weight (g) of shelled kernels from ears, adjusted to 15 g [kg.sub.-1] moisture content. In addition, number of kernels per row (NKR), kernel thickness (KT), and kernel width (KW), were computed on the basis of the data recorded on ED and CD, TGW and 100GW, TNK/NR ratio, EL/NKR ratio, and ED/NR ratio, respectively. Abbreviations shall be used in the proceeding text and tables.

The datasets were first tested for skewness and kurtosis by MSTATC statistical software. Appropriate transformation (angular, logarithm and square root transformations) was applied for specific characters that showed nonnormal distributions. Data from each trial were subjected to analysis of variance (ANOVA) using appropriate models. Test for homogeneity of error variance between various datasets obtained from the diallel and line x tester evaluations was performed using Hartley's [F.sub.max] test (Ott, 1988). Homogeneity of error variances facilitated pooling of datasets obtained from the diallel and line x tester evaluations. Correlation coefficients between various pairs of characters were computed. A preliminary analysis was performed by means of the conventional path model in which all yield-related characters were considered as first-order predictor variables with TGW as the response variable.

Sequential stepwise multiple regression was performed to organize the predictor variables into first-, second- and third-order paths (Fig. 1-3) on the basis of their respective contributions to the total variation of grain yield per plant (TGW) and minimal collinearity. The sequential path model consisted of six component paths, each component with its respective predictor and response variables. The level of multicollinearity in each component path was measured from two common measures, namely the "tolerance" value and its inverse, the "variance inflation factor," as suggested by Hair et al. (1995), by means of SPSS 9.0 statistical software. Tolerance value is the amount of variability of the selected independent variable not explained by other independent variables (1 - [R.sup.2.sub.i], where [R.sup.2.sub.i] is the coefficient of determination for the prediction of variable i by the predictor variables). Variance inflation factor (VIF) indicates the extent of effects of other independent variables on the variance of the selected independent variable [VIF = 1/(1- [R.sup.2.sub.i])]. Thus, very small tolerance values (much below 0.1) or large variance inflation factor values (above 10) indicate high collinearity (Hair et al., 1995). On the basis of tolerance and variance inflation factor values, besides the magnitude of direct effects, TNK and 100GW were considered as first-order variables among various yield characters under study. This procedure was again performed separately taking TNK and 100GW as dependent variables to find out first-order variables for these two response variables, which shall be, consequently, second-order variables for TGW. Similar procedure was followed to determine the third-order variables for TGW. This analysis resulted in identification of ED, EL, NKR, and NR as second-order variables and the rest of characters as third-order variables (Fig. 1-3). EH, NLE, and NT were not considered in the path model because of their high multicollinearity.

[FIGURES 1-3 OMITTED]

Direct effects of yield characters in different order paths were estimated by the procedure described by Williams et al. (1990). Partial coefficients of determination (analogous to [R.sup.2] of linear regression) were calculated from the path coefficients for all predictor variables. To estimate the standard error of path coefficients, bootstrap analysis (Efron and Tibshirani, 1993) was performed by Amos-4.0 (SPSS Inc.) statistical package, followed by the standard t test to analyze the significance of path coefficients.

A goodness-of-fit test was performed by comparing the observed correlation matrix among the variables in the model to that predicted through the estimated path coefficients. The test statistics are distributed as chi-square. The validity of the path model was further analyzed by two approaches. First, estimates of path coefficients from the DM-1999 dataset were used to predict the values of response variables for the HW-1999 dataset. Second, the estimated path coefficients from the individual (DM-1999 and HW-1999) and combined datasets (which were employed to generate the model) were used for predicting the values of respective response variables in two additional datasets obtained by evaluation of 92 testcross combinations. The testcrosses were generated by crossing 46 Indian maize inbred lines with two heterotic testers, CM111 and CM202) and were evaluated during Kharif (monsoon season)-2000 at Delhi and Hyderabad (subsequently referred to as DM-2000 and HM-2000, respectively). Observations on various yield characters were recorded for this set of experimental material following the same procedure mentioned earlier.

RESULTS

At DM-1999, all characters except KT and KW showed significant correlations with TGW. A similar pattern of correlations was observed for the HW-1999 dataset and for the combined dataset. Correlation coefficients computed between different pairs of characters from the DM-1999 and HW-1999 datasets are presented in Table 1. The highest correlation was between EL and TGW [DM-1999 (0.779); HW-1999 (0.882)]. Estimation of direct effects by conventional path analysis (Table 2), where the yield-related characters were considered as first-order variables with TGW as the response variable, and analysis of multicollinearity (Table 3) indicated inconsistent patterns of relationships among the variables. For example, at DM-1999, TNK, 100GW, and EL showed positive and high direct effects, whereas at HW-1999 the direct effect of TNK on TGW was considerably smaller. Number of kernels per row had a negative direct effect on TGW (-0.29) for DM-1999, but a positive direct effect on TGW (0.27) for HW-1999. Besides such inconsistent patterns in direct effects, high multicollinearity was observed for some characters, particularly for those showing high direct effects such as TNK (VIF = 92.03 and 241.90, for DM-1999 and HW1999, respectively). High correlations were observed between some of the predictor variables, namely PH and EH (r = 0.86), TNK and NKR (r = 0.86), and ED and KD (r = 0.71)], leading to high multicollinearity and consequent inability to ascertain actual contribution of each independent variable to the total variance of TGW due to mixed or confounded effects.

In contrast to the above results, sequential path analyses (illustrated in Fig. 1, 2, and 3 for DM-1999, HW1999, and combined datasets, respectively) provided a better understanding of the interrelationships among various variables and their relative contribution to TGW. The mean direct effects estimated from a set of 200 bootstrap samples were in close agreement with the observed direct effects of various characters (Table 4). The low standard error for all the direct effects and low bias also indicated the robustness of the sequential path model. The t test of significance, using standard error values obtained through bootstrap resampling, revealed that all the direct effects were significant. One-hundred grain weight and TNK as first-order variables accounted for nearly 97% of the variation in TGW in both seasons (Table 4); both the predictor variables also displayed high and positive direct effects on TGW. Although the magnitude of effects differed between the two seasons, direction of the effects remained unchanged. The direct effect of TNK on TGW (0.81) was found to be relatively higher than that of 100GW (0.71) during DM-1999. Because of a low and nonsignificant correlation between 100GW and TNK (0.14), their indirect effects on TGW were found to be low and negligible. The path analysis of second-order variables over the first-order variable showed that 76% (DM-1999) and 66% (HW-1999) of the total variation for 100GW were explained by four characters, namely ED, NR, EL, and NKR (Table 4). Among these characters, ED and EL showed positive direct effects, while NR and NKR recorded negative direct effects on 100GW. The direct effects were significant in both seasons and the highest effect was recorded for NKR at DM-1999 (p = -0.81) and EL at HW-1999 (p = 0.90). In the same order path, only NR and NKR had significant effects on TNK, and together these characters accounted for about 90% of variation in TNK. Both the characters had high positive effects on TNK, but their indirect effects were small because of a low and nonsignificant correlation (-0.08) between the two characters (Table 3).

Results of path analyses when the third-order variables were used as predictors and second-order variables as response variables indicated that CD and KL positively influenced ED and accounted for more than 98% of observed variation in ED (Table 4). Since the correlation between CD and KL was also significant (0.25), these two characters exerted considerable indirect effects on ED through each other. Kernel width along with CD and KL explained 81% (DM-1999) and 82 % (HW-1999) of the total variation for NR. However, the direct effect of KW on NR was negative and high (p = -0.79). In the same order path when NL, NE, and KT were regressed on EL, the direct effect of KT on EL was smaller than those of NL and NE. Together, the contribution of NE, NL, and KT to EL was relatively small, as indicated by the adjusted [R.sup.2] of the model (0.295 for DM-1999, and 0.239 for HW-1999). The situation was similar when NKR was considered as the response variable with NE, KT, and PH as predictor variables. These three variables together accounted for 53% (DM-1999) and 39% (HW-1999) of variation in NKR (Table 4). Unlike the direct effect of NE and PH, the effect of KT on NKR was found to be negative and high at DM-1999.

The validity of the path model was verified by three methods. First, the path-predicted correlation coefficients were compared with the observed correlation coefficients (Table 5). The analysis revealed very low and nonsignificant chi-square values for all the three datasets (0.0148, 0.00001, and 0.0068, respectively), indicating the usefulness of this model in the datasets employed for constructing the path. Second, the path coefficients revealed by analysis of DM-1999 dataset were used to predict the corresponding values for response variables m the HW-1999 dataset. The results, presented in Table 6, clearly showed high, positive correlations between predicted and observed values for all response variables, although EL showed smaller lower value than other characters. Third, the general applicability of this path model on datasets obtained from subsequent experiments (DM-2000 and HM-2000) was verified (Table 6). Positive and high correlations obtained between the predicted and observed values of response variables (DM-1999 and DM-2000; HW-1999 and HM-2000 etc.) indicated the utility of the model in analysis of datasets other than those used for the formulation of the sequential path model.

DISCUSSION

Several researchers have explored earlier the utility of path-coefficient analysis for estimating the direct and indirect effects of various yield-related characters on grain yield in an array of crop plants, including maize. The characters often highlighted in maize in this regard were number of ears per plant (Agrama, 1996), ear length (Shalygina, 1990; Tyagi et al., 1988), number of kernels per ear (Singh and Singh, 1993; Agrama 1996; Wang et al., 1999), kernel weight (Debnath and Khan, 1991; Simon, 1993), ear height (Farhatullah, 1990), and kernel abortion (Fisher and Palmer, 1983; Saha and Mukherjee, 1985).

The basic assumption while carrying out multiple regression is that the characters used as predictor variables are independent of each other. In reality, yield-related characters are intricately interrelated, often leading to high multicollinearity. This, in turn, confounds the detection and interpretation of actual contribution of a specific character. Path analyses performed in earlier studies on maize considered different yield components as first-order variables and yield as the response variable, and did not take into account the multicollinearity factor. A novel approach of organizing the variables into different order paths, based on character relationships indicated by earlier studies, was first adopted in crop plants by Samonte et al. (1998). In the present study, we have attempted to improve this sequential path model further by carefully considering minimal multicollinearity of the variables and maximum direct effects in each component path and by testing the validity of the model on additional datasets.

Our study demonstrates the utility of the sequential path model over conventional path analysis in discerning direct and indirect effects of various yield-related characters. When all the characters were used as firstorder variables as per conventional path model, we detected the occurrence of moderate to severe multicollinearity for salient yield components. This highlighted the inadequacy of the conventional path model in determining the actual effect of each predictor variable on the response variable. For instance, when the conventional path model was applied, ear diameter (ED) had a negligible effect on TGW in all three datasets (Table 2), indicating that this character had no significant contribution to grain yield. However, the sequential path model clearly showed that in all three datasets, ED had a positive and significant effect on TGW through 100GW (Fig. 1-3). Such effects could not be detected through the conventional path model because of high multicollinearity of characters, such as ED, TKR, and NKR.

Stepwise regression, in which characters with nonsignificant influence on the response variable are removed from analysis, could probably reduce the amount of collinearity for remaining characters in the model. However, in this process some important information might be lost. Therefore, a better strategy would be to carry out a sequential stepwise regression in which characters removed after the first-order path analysis are reanalyzed as possible predictor variables in the next order path. This strategy, when adopted in the present study, minimized the collinearity measures of all characters, thereby facilitating detection of actual contribution of each predictor variable in different path components with negligible confounding effects and interference.

In an analysis of 11 traits in high-lysine maize, Hart et al. (1991) found that yield components that develop earlier physiologically had a negative effect on those that developed later, that is, NR on NKR, NR on 100GW, and NKR on 100GW. The effects of PH and EH on yield components were small. Our results are largely in agreement with the above observations, as reflected by negative and significant direct effects of NR on 100GW (p = -0.38), and NKR on 100GW (p = -0.82), whereas the correlation between NR and NKR was found to be nonsignificant (Fig. 3). Plant height fits only as a third-order predictor variable, indicating its relatively lower influence on grain yield in comparison with characters such as 100-grain weight and number of kernels per ear. We have also found that despite a positive and significant correlation with grain yield [r = 0.40** (DM-1999) and r = 0.46** (HW-1999)], the direct effect of plant height on TGW was nonsignificant in the two datasets (p = 0.039 in DM-1999; p = 0.006 in HW-1999).

Path analysis of correlation coefficients between TGW and first-order variables, namely 100GW and TNK, into direct and indirect effects revealed high direct effects of both characters in all datasets. The importance of these two characters in influencing TGW is indicated by the observation that nearly 97% of observed variation for TGW was explained only by these two first-order variables. The utility of these two characters as predictor variables for TGW was reconfirmed in DM2000 and HM-2000 datasets (Table 6). Earlier studies have also demonstrated strong direct effects of TNK and 100GW on grain yield in maize (Xu, 1986; Han et al., 1991; Kumar et al., 1999).

The sequential path model not only indicated that other ear characters, namely ear length, ear diameter, number of kernel rows, and number of kernels per row, exercise their influence as second-order variables, but also provided a better understanding of their relative contributions to the first-order variables. For instance, the analysis revealed that ear length and ear diameter have strong positive influence on 100-grain weight, with apparently no significant direct effect on total number of kernels per ear. This might be due to the very low overall variation for these characters in the germplasm analyzed; standard deviation values for EL were 1.78 (DM-1999) and 1.52 (HW-1999), and 0.22 (DM-1999) and 0.12 (HW-1999). It is possible that EL and ED could have a relatively greater influence on 100-grain weight than total number of kernels per ear, as was indicated by the results (Fig. 1-3).

The negative direct effects of NR and NKR, in combination with their positive indirect effects through other characters, resulted in no significant correlation between these characters and 100GW, indicating that selection based on correlations alone may not be efficient. The high positive direct effects of these characters on 100GW were largely neutralized by their negative indirect effects through NR and NKR, resulting in reduced correlation coefficients. In the same order path, about 99% of variation in TNK was accounted for by two second-order variables (NR and NKR) having significant and positive direct effects on TNK. Since the model was organized mainly to minimize multicollinearity, correlations between predictor variables in most cases were small and nonsignificant.

In summary, the sequential path model presented in this study is unique with respect to (i) ordering of various predictor variables in first-, second- and third-order paths on the basis of minimal multicollinearity using sequential stepwise regression analyses; (ii) use of bootstrap analyses to determine the standard error of path coefficients for the consequent test of significance; and (iii) testing the validity of the path model by means of datasets other than those used for generation of the model. Through this approach, we have attempted to use path analysis as a predictive tool for analysis of interrelationships among yield-related characters rather than only as a descriptive tool. The study indicated that character associations revealed by the sequential path model have high congruity between datasets obtained from evaluation of hybrid combinations using a specific set of Indian maize inbred lines. The stability of these associations is based on evaluation of the two sets of hybrid combinations in two seasons (monsoon and winter seasons during 1999) and two locations (Delhi and Hyderabad in India). Character associations revealed by path analysis could be influenced by different factors, including (i) the germplasm used; (ii) the traits considered for analysis; (iii) the environment(s) used for evaluation; and (iv) the statistical methodologies employed to resolve the associations. Therefore, the general applicability of the present sequential path model can be ascertained by analysis of data from different sets of germplasm and in different production conditions.
```Table 1. Correlation coefficients between 14 characters measured in
the DM-1999 environment (above diagonal) and the HW-1999 environment
(below diagonal).

PH         NL           NE          EL          NR

PH          --        0.65 **     0.12        0.46 **     0.04
NL        0.26 *        --        0.06        0.42 **     0.19
NE        0.18        0.39 **       --        0.31 **     0.13
EL        0.51 **     0.36 **     0.28 **       --        0.05
NR       -0.03        0.11       -0.01       -0.10         --
ED        0.29 **     0.33 **     0.23 *      0.33 **     0.50 **
CD        0.29 **     0.27 *      0.11        0.42 **     0.40 **
TGW       0.54 **     0.33 **     0.28 **     0.82 **     0.23 *
100GW     0.37 **     0.13        0.15        0.62 **    -0.13
KL        0.16        0.26 *      0.25 *      0.08        0.38 **
TNK       0.42 **     0.42 **     0.33 **     0.63 **     0.46 **
NKR       0.49 **     0.42 **     0.39 **     0.76 **    -0.08
KT       -0.06       -0.02       -0.02        0.30 **    -0.07
KW        0.26 *      0.10        0.11        0.33 **    -0.68 **

ED          CD         TGW        100GW         KL

PH        0.25 *      0.24 *      0.45 **     0.28 *      0.17
NL        0.11        0.10        0.46 **     0.18        0.06
NE        0.21 *      0.09        0.35 **     0.11        0.25 *
EL        0.18        0.22 *      0.78 **     0.34 **     0.11
NR        0.45 **     0.43 **     0.26 *     -0.04        0.29 **
ED          --        0.78 **     0.64 **     0.51 **     0.79 **
CD        0.79 **       --        0.49 **     0.39 **     0.25 *
TGW       0.65 **     0.56 **       --        0.58 **     0.54 **
100GW     0.44 **     0.35 **     0.76 **       --        0.43 **
KL        0.76 **     0.22 *      0.43 **     0.31          --
TNK       0.54 **     0.45 **     0.73 **     0.14        0.37 **
NKR       0.29 **     0.25 *      0.68 **     0.23 *      0.19
KT       -0.01        0.20        0.12        0.48 **    -0.22 *
KW        0.15        0.14        0.19        0.44 **     0.08

TNK         NKR          KT          KW

PH        0.33 **     0.34 **     0.02        0.10
NL        0.41 **     0.37 **    -0.07       -0.08
NE        0.33 **     0.42 **    -0.22 *      0.19
EL        0.64 **     0.67 **     0.15        0.10
NR        0.38 **    -0.04        0.09       -0.69 **
ED        0.34 **     0.17       -0.09        0.18
CD        0.27 *      0.11        0.04        0.09
TGW       0.71 **     0.64 **    -0.09        0.18
100GW    -0.16       -0.14        0.48 **     0.40 **
KL        0.29 **     0.18       -0.17        0.19
TNK         --        0.90 **    -0.51 **    -0.18
NKR       0.88 **       --       -0.59 **     0.12
KT       -0.34 **    -0.34 **       --       -0.09
KW       -0.13        0.25 *      0.12          --

* Significant at the 0.05 probability level.

** Significant at the 0.01 probability level.

([dagger]) PH: plant height; NL: total number of leaves per plant; NE:
number of ears per plant; EL: ear length; NR: number of kernel rows;
ED: ear diameter; CD: cob diameter; TGW: total grain weight per ear;
100GW: 100-grain weight; KL: kernel length; TNK: total number of kernels
per ear, NKR: number of kernels per row: KT: kernel thickness; KW:
kernel width.

Table 2. Direct effects of first-order predictor variables on total
grain weight per ear and measures of collinearity in Model 1 (all
predictor variables as first-order variables).

DM-1999

Character                                         VIF
([dagger])    Direct effect    Tolerance    ([double dagger])

PH                -0.03          0.43              2.32
NI                 0.02          0.36              2.77
NE                 0.01          0.72              1.38
EL                 0.36          0.06             16.33
NR                 0.09          0.04             25.23
ED                -0.08          0.02             62.80
CD                 0.04          0.04             26.23
100GW              0.60          0.18              5.53
KL                 0.08          0.04             26.16
TNK                0.67          0.01             92.03
NKR               -0.29          0.01             93.91
KT                -0.25          0.09             10.89
KW                 0.09          0.20              4.96

HW-1999
Character
([dagger])    Direct effect    Tolerance       VIF

PH                0.03           0.62          1.61
NL               -0.02           0.65          1.54
NE               -0.02           0.74          1.35
EL                0.11           0.04         28.41
NR                0.22           0.01         72.02
ED                0.08           0.01        100.28
CD               -0.002          0.02         41.84
100GW             0.65           0.26          3.92
KL               -0.06           0.03         36.45
TNK               0.21           0.00        241.90
NKR               0.27           0.01        203.54
KT               -0.05           0.09         10.74
KW               -0.02           0.21          4.69

Combined data
Character
([dagger])    Direct effect       Tolerance       VIF

PH               -0.01              0.48           2.08
NL                0.001             0.53           2.08
NE                0.004             0.63           1.58
EL                0.41              0.03          29.85
NR                0.20              0.02          48.29
ED                0.18              0.01          90.66
CD               -0.14              0.03          36.94
100GW             0.66              0.20           4.98
KL               -0.09              0.03          37.55
TNK               0.43              0.01         175.24
NKR              -0.07              0.01         153.98
KT               -0.24              0.07           4.11
KW                0.06              0.16           6.46

([dagger]) PH: plant height; NIL: total number of leaves per plane NE:
number of ears per plant; EL: ear length; NR: number of kernel rows;
ED: ear diameter; CD: cob diameter, TGW: total grain weight per ear,
100GW: 100-grain weight; KL kernel length; TNK: total number of kernels
per ear; NKR: number of kernels per row; KT: kernel thickness; KW:
kernel width.

([double dagger]) Variance Inflation Factor.

Table 3. Tolerance and variance inflation factor (VIF) values for the
predictor variables in Model 1 (all predictor variables as first-order
variables) and Model 2 (predictors grouped into first-, second-, and
third-order variables).

DM-1999

Tolerance
Predictor    Response
variables    variables    M1 ([double dagger])    M2 ([double dagger])

100GW        TGW                  0.18                    0.97
TNK                               0.01                    0.97
ED           100GW                0.02                    0.75
EL                                0.06                    0.53
NR                                0.04                    0.77
NKR                               0.01                    0.53
NR           TNK                  0.04                    0.99
NKR                               0.01                    0.99
CD           ED                   0.04                    0.93
KL                                0.04                    0.93
CD           NR                   0.04                    0.93
KL                                0.04                    0.93
KW                                0.20                    0.96
NL           EL                   0.36                    0.99
NE                                0.72                    0.95
KT                                0.09                    0.95
NE           KR                   0.72                    0.93
KT                                0.09                    0.95
PH                                0.43                    0.98

DM-1999                  HW-1999

VIF          Tolerance            VIF
Predictor
variables     M1       M2      M1      M2       M1       M2

100GW         5.53    1.03    0.26    0.98      3.92     1.02
TNK          92.03    1.03    0.00    0.98    241.90    10.20
ED           62.80    1.32    0.01    0.60    100.30     1.66
EL           16.33    1.86    0.04    0.39     28.41     2.57
NR           25.23    1.29    0.01    0.67     72.02     1.49
NKR          93.91    1.89    0.01    0.41    203.50     2.43
NR           25.23    1.00    0.01    0.99     72.02     1.01
NKR          93.91    1.00    0.01    0.99    203.50     1.01
CD           26.23    1.07    0.02    0.95     41.48     1.05
KL           26.16    1.07    0.03    0.95     36.45     1.05
CD           26.23    1.07    0.02    0.93     41.47     1.07
KL           26.16    1.07    0.03    0.94     36.45     1.06
KW            4.96    1.04    0.21    0.97      4.69     1.02
NL            2.77    1.01    0.65    0.84      1.54     1.18
NE            1.38    1.05    0.74    0.83      1.35     1.20
KT           10.89    1.05    0.09    0.98     10.74     1.02
NE            1.38    1.07    0.74    0.95      1.35     1.05
KT           10.89    1.05    0.09    0.98     10.74     1.02
PH            2.32    1.02    0.62    0.96      1.61     1.03

Combined data

Tolerance            VIF
Predictor
variables     M1      M2       M1       M2

100GW        0.20    0.98      4.98     1.01
TNK          0.01    0.98    175.20     1.02
ED           0.01    0.83     90.66     1.20
EL           0.03    0.48     29.85     2.07
NR           0.02    0.83     48.29     1.21
NKR          0.01    0.48    154.00     2.08
NR           0.02    0.98     48.29     1.03
NKR          0.01    0.98    154.00     1.03
CD           0.03    0.97     36.94     1.03
KL           0.03    0.97     37.55     1.03
CD           0.03    0.96     36.94     1.04
KL           0.03    0.95     37.55     1.05
KW           0.16    0.97      6.46     1.03
NL           0.53    0.97      1.89     1.04
NE           0.63    0.88      1.88     1.04
KT           0.07    0.91     14.11     1.10
NE           0.63    0.86      1.88     1.16
KT           0.07    0.87     14.11     1.15
PH           0.48    0.93      2.08     1.07

([dagger]) PH: plant height; NL: total number of leaves per plant; NE:
number of ears per plant; EL: ear length; NR: number of kernel rows;
ED: ear diameter; CD: cob diameter; TGW: total grain weight per ear;
100GW: 100-grain weight; KL: kernel length; TNK: total number of
kernels per ear; NKR: number of kernels per row; KT: kernel thickness;
KW: kernel width.

([double dagger]) M1-Model 1; M2-Model 2.

Table 4. Estimation of standard error values of path coefficients using
bootstrap analysis.

DM-1999

Predictor    Response                                Bootstrap
variables    variables    [R.sup.2]   Direct
([dagger])   ([dagger])     Adj.      effect    SE      Mean     Bias

100GW        TGW            0.97        0.71   0.59     0.713    0.007
TNK                                     0.81   0.61     0.813    0.005
ED           100GW          0.76        0.71   0.61     0.709    0.003
NR                                     -0.44   0.074   -0.439    0.000
EL                                      0.78   0.093    0.787    0.003
NKR                                    -0.81   0.096   -0.814   -0.001
NR           TNK            0.99        0.42   0.040    0.421    0.005
NKR                                     0.92   0.045    0.922    0.004
CD           ED             0.98        0.62   0.041    0.627    0.005
KL                                      0.63   0.049    0.633    0.004
CD           NR             0.81        0.42   0.060    0.418   -0.001
KL                                      0.34   0.048    0.339    0.002
KW                                     -0.79   0.065   -0.790   -0.001
NL           EL             0.29        0.42   0.092    0.417   -0.005
NE                                      0.34   0.092    0.331   -0.005
KT                                      0.25   0.072    0.252    0.003
NE           NKR            0.53        0.26   0.075    0.260   -0.004
KT                                     -0.54   0.068   -0.543   -0.001
PH                                      0.33   0.070    0.332    0.000

HW-1999

Predictor                               Bootstrap
variables    [R.sup.2]   Direct
([dagger])     Adj.      effect    SE      Mean     Bias

100GW          0.98        0.67   0.056    0.682    0.007
TNK                        0.64   0.044    0.639    0.001
ED             0.66        0.50   0.071    0.502   -0.001
NR                        -0.35   0.083   -0.351   -0.003
EL                         0.90   0.103    0.900   -0.003
NKR                       -0.63   0.145   -0.625   -0.007
NR             0.99        0.53   0.065    0.537    0.003
NKR                        0.89   0.047    0.882   -0.004
CD             0.99        0.65   0.037    0.656    0.003
KL                         0.62   0.041    0.625    0.008
CD             0.82        0.43   0.048    0.439    0.007
KL                         0.35   0.059    0.350    0.001
KW                        -0.77   0.065   -0.772    0.000
NL             0.24        0.29   0.077    0.291    0.005
NE                         0.21   0.104    0.197   -0.015
KT                         0.33   0.130    0.339    0.005
NE             0.39        0.28   0.086    0.270   -0.010
KT                        -0.28   0.076   -0.291   -0.008
PH                         0.42   0.087    0.405   -0.017

Combined data

Predictor                              Bootstrap
variables    [R.sup.2]   Direct
([dagger])     Adj.      effect   SE      Mean     Bias

100GW          0.98       0.74   0.060    0.745    0.009
TNK                       0.78   0.069    0.780    0.003
ED             0.76       0.57   0.052    0.565   -0.004
NR                       -0.38   0.075   -0.379    0.002
EL                        0.89   0.097    0.899    0.011
NKR                      -0.82   0.100   -0.819    0.001
NR             0.99       0.50   0.052    0.502    0.001
NKR                       0.94   0.044    0.941   -0.002
CD             0.98       0.64   0.039    0.634   -0.005
KL                        0.66   0.045    0.658    0.004
CD             0.83       0.37   0.053    0.381    0.007
KL                        0.33   0.047    0.332    0.005
KW                       -0.84   0.060   -0.837   -0.002
NL             0.36       0.40   0.086    0.410    0.006
NE                        0.33   0.080    0.326   -0.009
KT                        0.38   0.080    0.390    0.014
NE             0.44       0.30   0.083    0.300   -0.001
KT                       -0.41   0.085   -0.411    0.003
PH                        0.36   0.087    0.356    0.000

([dagger]) PH: plant height; NL: total number of leaves per plant; NE:
number of ears per plant; EL: ear length; NR: number of kernel rows;
ED: ear diameter; CD: cob diameter; TGW: total grain weight per ear;
100GW: 100-grain weight; KL: kernel length; TNK: total number of
kernels per ear; NKR: number of kernels per row; KT: kernel thickness;
KW: kernel width.

Table 5. Goodness-of-fit between observed and predicted correlation
coefficients in different path orders.

DM-1999                  HW-1999

Character pairs    Observed    Estimated    Observed    Estimated

TGW-100GW              0.58         0.58        0.76         0.76
TGW-TNK                0.70         0.70        0.73         0.73
100GW-ED               0.51         0.51        0.44         0.44
100GW-NR              -0.04        -0.04       -0.13        -0.13
100GW-EL               0.32         0.34        0.62         0.62
100GW-NKR             -0.01        -0.14        0.23         0.23
TNK-NKR                0.90         0.90        0.84         0.84
TNK-NR                 0.38         0.38        0.46         0.46
ED-CD                  0.78         0.78        0.79         0.79
ED-KL                  0.79         0.79        0.76         0.76
NR-CD                  0.43         0.43        0.40         0.40
NR-KL                  0.29         0.29        0.38         0.38
NR-KW                 -0.69        -0.69       -0.68        -0.68
EL-NL                  0.42         0.43        0.36         0.36
EL-NE                  0.31         0.31        0.28         0.28
EL-KT                  0.15         0.15        0.30         0.30
NKR-NE                 0.42         0.42        0.39         0.39
NKR-KT                -0.59        -0.59       -0.34        -0.34
NKR-PH                 0.34         0.42        0.49         0.49

Combined

Character pairs    Observed    Estimated

TGW-100GW              0.62         0.62
TGW-TNK                0.69         0.67
100GW-ED               0.51         0.51
100GW-NR              -0.09        -0.09
100GW-EL               0.40         0.40
100GW-NKR             -0.11        -0.11
TNK-NKR                0.86         0.86
TNK-NR                 0.35         0.35
ED-CD                  0.76         0.76
ED-KL                  0.77         0.77
NR-CD                  0.34         0.34
NR-KL                  0.27         0.27
NR-KW                 -0.75        -0.75
EL-NL                  0.46         0.46
EL-NE                  0.29         0.29
EL-KT                  0.27         0.27
NKR-NE                 0.49         0.49
NKR-KT                -0.46        -0.46
NKR-PH                 0.35         0.35

([dagger]) PH: plant height; NL: total number of leaves per plant; NE:
number of ears per plant; EL: ear length; NR: number of kernel rows;
ED: ear diameter; CD: cob diameter; TGW: total grain weight per ear;
100GW: 100-grain weight; KL: kernel length; TNK: total number of
kernels per ear; NKR: number of kernels per row; KT: kernel thickness;
KW: kernel width.

Table 6. Correlations between observed and predicted values of response
variables in different paths based on sequential path model.

Response
variables     HW-1999 based on    DM-2000 based on    HW-1999 based on
([dagger])     DM-1999 model       combined model      combined model

TGW                 0.99                0.99                0.98
100GW               0.78                0.79                0.87
TNK                 0.99                0.98                0.99
NKR                 0.60                0.71                0.86
EL                  0.49                0.13                0.21
NR                  0.91                0.82                0.86
ED                  0.99                0.99                0.99

Response
variables     DM-2000 based on    HM-2000 based on
([dagger])     DM-1999 model       HW-1999 model

TGW                 0.99                0.99
100GW               0.83                0.90
TNK                 0.97                0.99
NKR                 0.75                0.61
EL                  0.11                0.18
NR                  0.83                0.86
ED                  0.76                0.99

([dagger]) PH: plant height; NL: total number of leaves per plant; NE:
number of ears per plant; EL: ear length; NR: number of kernel rows;
ED: ear diameter; CD: cob diameter; TGW: total grain weight per ear;
100GW: 100-grain weight; KL: kernel length; TNK: total number of
kernels per ear, NKR: number of kernels per row; KT: kernel thickness;
KW: kernel width.
```

ACKNOWLEDGMENTS

We thank the authorities of the Indian Agricultural Research Institute, New Delhi, and Maize Research Station, Acharya N.G. Ranga Agricultural University, Hyderabad, for providing necessary support for field experimentation. Assistance received from Mr. Prabhu B. Patil, Drs. Rajesh Kumar, J.C. Sekbar and S. Venkatesh in recording observations from field experiments is gratefully acknowledged. S.A.M. was supported by a Ph.D. Research Fellowship at IARI from the Government of Islamic Republic of Iran.

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S. A. Mohammadi, B. M. Prasanna, * and N. N. Singh

S.A. Mohammadi, Department of Crop Production and Breeding, University of Tabriz, Tabriz 51664, Islamic Republic of Iran; B.M. Prasanna, Division of Genetics, Indian Agricultural Research Institute, New Delhi 110012, India; N.N. Singh, Directorate of Maize Research, New Delhi 110012, India. Received 6 Sept. 2001. * Corresponding author (prasanna@ndf.vsnl.net.in)
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