Beyond Linearity: Mastering Fractional Polynomial Regression in Stata

Описание: Fractional Polynomial (FP) regression is a flexible parametric approach designed to model nonlinear relationships between a continuous predictor and an outcome variable. Unlike conventional polynomial models that are restricted to positive integer powers (squares, cubes), FP regression selects powers from a predefined set S={−2,−1,−0.5,0,0.5,1,2,3}, where 0 represents the natural logarithm. By allowing combinations of these powers—including repeated powers which introduce logarithmic terms—FP models can capture a diverse array of curve shapes, such as sigmoids or asymptotes. This method is particularly valuable because it avoids the artifacts, such as edge effects and "waves," that often plague high-degree standard polynomials.

Implementing FP regression in Stata is highly automated using the fp prefix command. The basic syntax is fp term: estimation_command, where term is the variable to be transformed. For example, to model the nonlinear effect of vehicle weight on mileage, you would type fp weight: regress mpg weight foreign. Stata automatically iterates through possible power combinations (typically 44 for a second-degree polynomial) and selects the best-fitting model based on the lowest deviance. Essential options include scale, which normalizes large variable values to prevent numerical errors, and center, which simplifies the interpretation of the intercept. Post-estimation, users can utilize fp generate to store the transformed variables for further analysis or prediction.

Comparing Fractional Polynomials and Splines for Curve Fitting

Fractional Polynomial (FP) regression and Splines represent two distinct philosophies for modeling nonlinear relationships: global versus local fitting. FP regression is a parametric approach that extends conventional polynomials by selecting powers from a predefined set that includes negative integers, fractions, and logarithms. A major advantage of FP is parsimony; complex curve shapes can often be modeled effectively using only one or two power terms, thereby avoiding the unstable "waves" and artifacts often produced by high-degree standard polynomials. This makes FP highly effective for capturing smooth, global trends with a concise functional form.

Conversely, regression Splines rely on a piecewise approach, dividing the data range into regions separated by knots and fitting low-degree polynomials within each interval. This piecewise nature grants Splines superior flexibility to model local data features that a global function might miss. However, this flexibility introduces the challenge of selecting the number and placement of knots, which directly influences the model's degrees of freedom. While Splines can adapt to sharp local changes, they risk overfitting or becoming "wiggly" if too many knots are used, whereas FP generally produces smoother, stable curves over the entire range of data.

Comparing Efficiency: Fractional Polynomials vs. High-Degree Polynomials

To compare the effectiveness of fractional polynomial (FP) regression against high-degree conventional polynomials, researchers must evaluate the balance between curve flexibility and model stability. Standard polynomials are restricted to positive integer powers. While increasing the polynomial degree (e.g., to cubic, quartic, or higher) allows for more complex curves, it notoriously introduces undesirable artifacts, such as artificial "waves" or "wiggles" in the fitted function, and results in poor stability at the boundaries of the data, known as edge effects.

In contrast, FP regression addresses these limitations by expanding the candidate powers to include negative integers, fractions, and logarithms. This extension allows FPs to capture a diverse range of functional forms—such as asymptotes or sharp changes in slope—using only one or two terms, whereas a standard polynomial might require a much higher degree to approximate the same shape. Consequently, FP models are generally more parsimonious, offering a superior fit with fewer parameters while avoiding the erratic tail behavior and overfitting problems common in high-degree global polynomials. Ultimately, FPs provide a practical compromise by maintaining the global nature of polynomials while offering the flexibility required for complex nonlinear relationships.