Multicollinearity among predictor traits is a major challenge in path coefficient analysis, distorting the estimation of direct and indirect effects on grain yield in rice. This study addresses this issue by applying ridge regression to an M2 population of rice (Oryza sativa L.), evaluating its effectiveness in stabilizing coefficients and improving model interpretability. Severe multicollinearity was identified in the conventional path analysis model, with traits like plant height (PH), Culm height (CH), days to maturity (DM), panicle weight (PW), and grain number per panicle (GNP) exhibiting inflated Variance Inflation Factor (VIF) values (>10). Ridge regression significantly reduced multicollinearity, stabilizing path coefficients while retaining key traits. In the modified Model 6, ridge regression combined with the exclusion of highly collinear traits (e.g., plant height (PH) and days to flowering (DF)) achieved superior model fit (R2=0.870R2 = 0.870) and reduced residual effects. Total tillers (TT) and biomass yield (BY) consistently emerged as primary yield determinants, with realistic direct effects (0.340 and 0.428, respectively). Additionally, ridge regression preserved essential interdependencies among traits like PW and GNP, which were distorted in the conventional model. This study provides a robust framework for handling multicollinearity in path coefficient analysis, emphasizing the practical utility of ridge regression in breeding programs.

Rice (Oryza sativa L.) is one of the most widely consumed staple crops globally and plays a vital role in ensuring food security, particularly in many developing countries where it is a primary source of nutrition for millions of people. Improving rice yields is crucial to meet the increasing global demand, and understanding relationships between morphological traits can assist plant breeders in identifying key yield-contributing traits to optimize selection criteria, and gaining a deeper understanding of the genetic architecture of yield to improve breeding efficiency. To get insights into these complex relationships between yield and its component traits, path coefficient analysis plays a crucial role. An appropriate utilising path coefficient analysis as an aid to the selection of genotype can make significant strides in developing high-yielding, resilient rice varieties to meet the growing global demand for food.  The technique was proposed by Dewey & Lu (1959) to separate the simple correlation coefficient between the seed yield (as the dependent variable) and its components (as predictor variables) into direct and indirect effects (that affect grain yield through the other variables). However, the path coefficient is a standardized partial regression coefficient, therefore, no or weak associations among predictor variables are a necessary assumption needed to satisfy the goodness of fit for the path analysis model (Neter et al., 1996). Correlation and path analysis in rice was done by various scientists before like Singh et al. (2018); Zahid et al. (2006); Patil et al. (2005); Mahto et al. (2003); Sarawgi et al. (1997), etc.

Multicollinearity, a condition where two or more predictor variables are highly correlated, poses a significant challenge in path analysis, as it can lead to inaccurate estimates of the relationships between yield component traits, thereby hindering the identification of the most critical yield component traits contributing to grain yield. In fact, independence among yield components is rarely found under field conditions. This case is usually called a multicollinearity problem (EL-Taweel et al., 2012). Multicollinearity can lead to an overestimation of direct effects, producing effect values greater than 1, which are not meaningful for interpreting path analysis (Satyanarayana et al., 2023; Muthuramu et al. 2023). Similarly, Gravois & Helms (1992) mentioned that when the ordinary path analysis model is used in the presence of multicollinearity, some path coefficients may exceed one, which is considered a negative effect of multicollinearity. Williams et al. (1979) stated that the negative effects of multicollinearity may be enough to reject the ordinary path analysis model. Various statistical methods and techniques have been employed to handle multicollinearity in path analysis, including ridge regression, principal component regression, and partial least squares regression. These methods can be effective in addressing multicollinearity, but the choice of method depends on the specific characteristics of the data and the research question at hand. Path coefficient analysis relies on the accurate estimation of regression weights of the predictor variables (yield attributing traits) to interpret causal relationships among them. Ridge regression improves the reliability of these coefficients when predictors are highly correlated, avoiding exaggerated effects of multicollinearity (Xu et al., 2014; Pelzer et al., 2009). It retains all original predictors in the model while principal component regression and partial least squares regression methods reduce predictors to latent components and may lose some granular details (Zou & Hastie 2005). Thus, no information is completely discarded in the ridge regression method. The ridge regression method adds a penalty term proportional to the squared magnitude of coefficients that effectively shrinks coefficients towards zero without fully eliminating variables. This reduces the variance of coefficient estimates in the presence of multicollinearity, improving their stability (Hoerl & Kennard 1970). Thus, ridge regression preferred method for resolving multicollinearity in path coefficient analysis which improves model interpretability and predictive accuracy. In this study, we have made a framework for dealing with multicollinearity in path coefficient analysis on the mutant M2 population of rice using the strategies suggested by Olivoto et al. (2017).

The experimental material used for this study was an M2 population of the rice cultivar Gobindabhog. The 910 mutant seedlings were transplanted at the spacing of 20 × 20 cm as one seedling per hill and each individual was phenotyped for 14 traits as mentioned in Table 1.

Table 1: List of the traits evaluated.

Statistical analysis. The simple correlation matrix was computed among traits as outlined by Snedecor & Cochran (1989). The conventional and modified models of path analysis were formulated by considering the grain yield per plant (GY) as a dependent variable and the other 13 traits as independent variables. The pairs. panels() and path_coeff() functions of the “psych” and “metan” R packages were used for the analysis. A common measure called the Variance Inflation Factor (VIF) was used to test the presence of multicollinearity among the predictor variables (yield attributing traits) in different models (Hair et al., 1992). If the VIF value is above 10 for any trait, this indicates the presence of multicollinearity, then such models could not estimate the correct contribution of predictor variables. Here we have applied two important strategies for the ideal model construction, (i) exclude the traits from the model if VIF>10 and (ii) apply the modified path analysis model (ridge regression-based model). 

The modified path analysis model was proposed by Carvalho et al. (1999) to correct the negative effects of the multicollinearity problem. The proposed model adds a small bias constant value (k) to the diagonal elements (unity value) of the correlation matrix of yield components (predictor variables). The k value would range from 0 to 1. When k=0, the modified model would behave like the normal model. Two methods have been used to determine the optimum value of k, (a) ridge trace exam (Hoerl & Kennard 1970 a and b) uses a 2D chart showing how the path coefficients vary as a function of k (0 < k < 1) to select the smallest value of k which is capable of stabilising most path coefficients (<1) and (b) VIF based approach select the smallest value of k at which VIF values become less than 10.

A total of 5 different models of path coefficients were estimated as mentioned in Table 2. The impact of different models was evaluated by analysing the direct effects of traits with multicollinearity, residual effect square and coefficient of determinants (R2).

Table 2: Models of Path Coefficient Analysis.

Correlations among traits. The strongest correlation was observed between culm height (CH) and plant height (PH) (r = 0.98), followed by days to flowering (DF) and days to maturity (DM) (r = 0.96), and panicle weight (PW) and grain number per panicle (GNP) (r = 0.94). High correlations were also noted for PW and spikelet fertility (SF) (r = 0.85), total tillers (TT) and biomass yield (BY) (r = 0.77), and GNP and SF (r = 0.75). These inter-correlations indicated significant multicollinearity among predictor traits.

Fig. 1. Correlation Plot.

Fig. 2. Ridge trace examination plot 1 (left) and plot 2 (right).

Table 3: The VIF of different traits in the Model.

The optimum k value for the ridge regression in models 5 and 6 has been fixed at 0.04 by analysing the ridge trace examination plot 1. Model 5 has been evaluated with all traits except PH. The ridge regression was found to deal with multicollinearity without excluding the correlated traits. Still, the direct effects of DF(0.038) and DM(-0.027) were observed in opposite directions. Ridge regression combined with the exclusion of PH and DF in model 6 achieved the highest stability and interpretability, with a coefficient of determination (R2=0.870) and residual effect square of 0.130. Ridge regression effectively resolved multicollinearity, stabilizing direct and indirect effects. For instance, CH (0.002) and DM (0.008) showed realistic and interpretable direct effects and TT (0.340) retained its strong direct effect on grain yield, emphasizing its role as a key yield determinant. Ridge regression significantly stabilizes VIF values, reducing them to below the critical threshold of 10, as seen for GNP (Model 1: 16.4; Model 6: 7.879). In Model 6, traits like TT and BY emerged as the most consistent predictors of grain yield, with direct effects of 0.340 and 0.428, respectively. Indirect effects between traits like PW and GNP were more consistent, highlighting their interdependence without overestimation.

Table 4: Direct effects of predictor variables.

Table 5: Path Coefficient Matrix (Model 6) of direct and indirect effects of yield components.

This study demonstrates that ridge regression is an effective method for addressing multicollinearity in path coefficient analysis. By stabilizing coefficients and retaining critical predictors, ridge regression provides robust insights into the relationships among yield-related traits. Traits such as TT and BY consistently emerged as key contributors to grain yield, highlighting their potential as selection criteria in rice breeding. The proposed approach offers a valuable framework for addressing multicollinearity in other crop improvement studies. The stability of coefficients in Models 5 and 6 underscores the importance of integrating ridge regression into path coefficient analysis. This approach allows for the inclusion of interrelated traits like GNP and PW, which are vital for understanding complex yield dynamics in rice (Hoerl & Kennard 1970; Xu et al., 2014).