Standardized beta coefficients in nonlinear regression

Question

In linear models $Y=X\beta+\epsilon$, where the errors $\epsilon_i\sim\text{Normal}(0,\sigma^2)$ are independent, the standardized beta coefficients are given by $$ \beta_i^*=\beta_i\frac{\sigma_{x_i}}{\sigma_y}, $$ where $\sigma_{x_i}$ is the standard deviation of the $i$-th column of $X$ and $\sigma_y$ is the standard deviation of $Y$.

For nonlinear models $Y=f(\beta)$, one linearizes $f$ and works with $X=Jf(\beta_{\text{ls}})$, where $Jf$ is the Jacobian matrix of $f$ and $\beta_{\text{ls}}$ is the least-squares fitting of $y=f(\beta)$, where $y$ is real data. The common formulas for linear regression are valid for the nonlinear case using the corresponding matrix $X$.

My question is whether the formulas for the standardized beta coefficients also hold in the nonlinear setting, namely taking as $\sigma_{x_i}$ the standard deviation of the $i$-th column of $Jf(\beta_{\text{ls}})$. I am asking this question because I have a nonlinear model and I would like to assess which variables have the greatest association with the response variable individually.

Answer 1

In nonlinear models, the relationship between the response variable and the predictors is not directly linear, making the interpretation of standardized beta coefficients less straightforward than in linear models. Although you can linearize the nonlinear function and use the Jacobian matrix to approximate a linear relationship, the standardized beta coefficients from this linear approximation may not accurately represent the true effect of each predictor in the original nonlinear model.

The idea of standardizing beta coefficients in the linearized version of a nonlinear model using the standard deviation of the i-th column of the Jacobian matrix might provide some insights into the relative importance of the predictors. However, this approach has limitations:

1. The linearized version of the model is only an approximation, and the standardized beta coefficients may not represent the true effect of the predictors in the original nonlinear model.
2. Standardized beta coefficients are more interpretable in linear models, where the effect of each predictor on the response variable is constant. In nonlinear models, the effect of predictors can change depending on the values of the other predictors.

Instead of relying on standardized beta coefficients, you may want to consider alternative methods to assess the importance of predictors in a nonlinear model, such as:

1. Sensitivity analysis: Examine how changes in each predictor affect the model's predictions. This approach can help you understand the relative importance of each predictor and how their effects might change depending on the values of other predictors.
2. Variable importance measures: Use techniques like Random Forests, which can handle nonlinear relationships and provide an importance score for each predictor.

In summary, the formulas for standardized beta coefficients in linear models do not directly translate to nonlinear models. Although you can use the linearized version of the model and the Jacobian matrix to approximate the relative importance of predictors, this approach has limitations. You may want to explore alternative methods, such as sensitivity analysis or variable importance measures, to assess the association between predictors and the response variable in a nonlinear model.