Introduction

Regression analysis is a fundamental statistical technique used to understand the relationship between a dependent variable (often denoted as y) and one or more independent variables (often denoted as x1, x2, x3, ..., xn). In many real-world applications, the choice of the dependent variable can significantly influence the outcome of the analysis. This article explores how to determine which dependent variable is better than the others, especially when the independent variables are the same across different scenarios.

Cases and Scenarios in Regression Analysis

Case I: Multiple Dependent Variables with Different Independent Variables

In real-world scenarios, it is common to have multiple dependent variables (y) and independent variables (x1, x2, x3, ..., xn), where the independent variables are different across different cases. This scenario is represented as 'Case I.' For instance, if you are studying the effect of temperature, humidity, and wind speed on crop yield, each dependent variable (e.g., yield, protein content, sugar concentration) would represent a different crop attribute.

Case II: Single Dependent Variable with Multiple Identical Independent Variables

On the other hand, if all independent variables are the same across different scenarios, they collapse into a single independent variable (x), and the analysis simplifies to a univariate regression. This scenario is represented as 'Case II.' For example, if you are measuring the impact of sunlight on plant growth, and all other conditions remain constant, your independent variable (sunlight) is uniform.

Evaluation of Dependent Variables

In Case I, where multiple dependent variables are involved, it is crucial to determine which dependent variable is better suited for the analysis. This involves evaluating the contribution of each dependent variable to the total variance in the data. In contrast, in Case II, since all independent variables are the same, the evaluation is straightforward and typically involves simpler statistical methods.

Recursive Feature Elimination (RFE) and Recursive Feature Selection (RFS)

For Case I, one effective method to determine which dependent variable is better is through Recursive Feature Elimination (RFE) or Recursive Feature Selection (RFS). These techniques involve iteratively removing the least important features (in this case, dependent variables) and re-evaluating the model's performance. The process is repeated until a satisfactory subset of features is obtained.

Evaluation Process

The evaluation process involves several steps:

Model Training: Train a regression model using all available dependent variables. Feature Relevance: Rank the dependent variables based on their contribution to the model's performance. Feature Iteration: Remove the least contributing dependent variable and retrain the model. Performance Comparison: Compare the performance of the model before and after the removal of the feature. Iteration: Repeat the process until a satisfactory subset of dependent variables is obtained.

The key metric for evaluating the dependent variables is the variance explained by each variable. The variable that explains the maximum variance is considered the best.

Case Studies and Examples

Consider a study where a researcher is evaluating the impact of altitude, temperature, and precipitation on a plant's growth. The dependent variables could be height, number of leaves, and root length. Here, the researcher might use RFE to determine which of these dependent variables is the most significant in explaining the plant's growth.

Another example could be a study on the impact of various dietary factors on health outcomes. The independent variable (dietary factors) remains the same across different health outcomes (e.g., cholesterol levels, weight, blood pressure). RFS can help identify the health outcome most influenced by the dietary factors.

Conclusion

Choosing the right dependent variable is crucial in regression analysis. In scenarios where multiple dependent variables are involved, methods such as recursive feature elimination can help identify the best variable. For cases where independent variables are identical, simpler models can be applied.

By understanding the appropriate methods and techniques, researchers can better interpret their data and make informed decisions about the dependent variables they choose to analyze.