🔎 What's the problem

Political scientists often report quantities of interest derived from model coefficients—predicted probabilities, first differences, and marginal effects—rather than the coefficients themselves. These derived quantities do not automatically inherit the small-sample properties of coefficient estimates. An important consequence is that unbiased coefficient estimates are neither necessary nor sufficient for unbiased estimates of quantities of interest. This phenomenon is labeled transformation-induced bias.

📐 How the bias is characterized and approximated

The article (1) defines transformation-induced bias formally, (2) derives an analytic approximation to quantify its magnitude, and (3) explains why transformations from coefficients to target quantities can produce bias even when coefficient estimators are unbiased.

Key conceptual points:

• Quantities of interest considered include predicted probabilities, first differences, and marginal effects
• Bias can arise from nonlinear transformations and the sampling distribution of coefficients
• An approximation is provided to estimate the transformation-induced bias in practice

🧪 What the simulations demonstrate

Two simulation studies illustrate the practical importance of transformation-induced bias. The simulations show how often and how large the bias can be under realistic small-sample conditions, highlighting cases where practitioners relying on unbiased coefficients would nonetheless report biased quantities.

💡 Why this matters for methodology and practice

Transformation-induced bias has direct implications for methodological research and applied work. It suggests that evaluations of estimator performance must consider the small-sample behavior of the final quantities researchers report, not just coefficient estimators. The findings motivate greater attention to bias assessment for derived quantities and to methods that target those quantities directly.