8.4: Conclusion

In this chapter, we systematically addressed the challenge of applying ordinary least squares (OLS) regression to dependent variables with bounds. The core strategy centered on variable transformation, a process designed to remap bounded data into an unbounded space where the assumptions of OLS — particularly linearity and unbounded error — are more tenable. We began by classifying continuous variables into three fundamental types: unbounded, once-bounded (e.g., variables with a minimum of zero), and twice-bounded (e.g., proportions between 0 and 1). For unbounded variables, transformation is often unnecessary for compliance with OLS, though it may still be employed to improve residual normality and homoskedasticity. For once-bounded variables, the approach required a two-step procedure: first, an algebraic shift to ensure a lower bound of zero, followed by a log transformation to stretch the scale to negative infinity. For twice-bounded variables, the procedure similarly involved an algebraic transformation to standardize the bounds to 0 and 1, succeeded by a logit transformation, which maps the unit interval to the entire real line.

Crucially, the interpretative power of any model lies in its connection to the original, substantive scale of the data. Therefore, this chapter emphasized that the transformation process is not complete without the correct back-transformation. After fitting the model and generating predictions on the transformed scale, one must first apply the appropriate inverse function — the exponential for log-transformed variables or the logistic for logit-transformed variables. This step returns values to the intermediate, algebraically shifted scale. The final, essential step is to apply the inverse of the initial algebraic transformation to restore the predictions to the original units of the dependent variable. This sequence — inverse function first, then inverse algebraic shift — is paramount; reversing the order produces nonsensical results. Through this disciplined framework, we can harness the simplicity and interpretability of linear regression while responsibly modeling the complex, bounded realities present in much social, biological, and economic data.