Choosing the Right Statistical Test: When One Independent Variable Meets Three Continuous Dependent Variables

When faced with a dataset that has one continuous independent variable and three continuous dependent variables, selecting the appropriate statistical test can be challenging. Many might turn to multiple multivariate regression, but it is often misunderstood to require multiple independent variables. In reality, you can run a simple multivariate regression that will help you understand the relationship between your one independent variable and the three dependent variables.

Understanding Simple Multivariate Regression

Simple multivariate regression, rather than requiring two or more independent variables, is a method that can handle a single independent variable and multiple dependent variables. The term "simple" here refers to the fact that only one independent variable is involved, not the number of dependent variables.

How Simple Multivariate Regression Works

Conceptually, you are running three simple univariate regressions, each with one independent variable (your income of parents) and one dependent variable (income at age 40, wealth at age 40, and age at death). Each of these regressions will provide you with a coefficient estimate, also known as a beta (β) value, and a constant term (intercept).

Example Scenario

Consider an example where your independent variable is the income of parents in the year of a person’s birth. The dependent variables might include the person’s income at age 40, their wealth at age 40, and their age at death. This would give you three separate regressions:

Regression 1: Income at age 40 ~ Income of parents Regression 2: Wealth at age 40 ~ Income of parents Regression 3: Age at death ~ Income of parents

Each regression provides you with a coefficient estimate and a constant term, giving you a comprehensive understanding of how each dependent variable relates to the independent variable.

The Multivariate Aspect

The "multivariate" aspect comes into play when the residuals from these three regressions are not independent. In most cases, this is likely to be the case, as there are likely correlations among the residuals due to the nature of the data. For example, a person's income at age 40 might influence their wealth at age 40. This interdependence can affect your confidence intervals, hypothesis tests, and predictions.

Joint Testing

If you want to test the hypothesis that all three coefficient estimates are positive, you would typically perform a joint test rather than testing each regression individually. This is because the residuals from each regression are likely to be correlated, and testing each regression separately would not adequately capture this.

When to Use Simple Multivariate Regression

Simple multivariate regression is particularly useful when you have a single independent variable and multiple dependent variables, and you wish to understand the relationships between them. The primary advantages are:

It can be used for hypothesis testing of the relationship between the independent variable and each dependent variable. It provides a structured approach to understanding the influence of the independent variable on each dependent variable. It helps in identifying significant relationships and coefficients that may not be apparent in univariate analysis.

However, it is essential to consider the following:

Check the assumptions of the OLS (Ordinary Least Squares) method, such as linearity, normality of errors, and no omitted variables. Consider the nature of your data to ensure that the assumptions are met. Use diagnostic tests to check for issues like heteroscedasticity, multicollinearity, and outliers.

Conclusion

When you have one continuous independent variable and three continuous dependent variables, simple multivariate regression is a powerful tool that can help you understand the relationships between them. It is not about the number of independent variables but the complexity of the model and the relationships it seeks to uncover. If the assumptions are met, you can use this method to gain insights into how each dependent variable is influenced by the independent variable.

Keywords

Simple Multivariate Regression, Statistical Test, Independent Variable