Fractional Polynomial Models
Suppose that we have an outcome variable, a single continuous covariate X, and a suitable regression model relating them. Our starting point is the straight line model, β 1 X (for simplicity, we suppress the constant term, β 0). Often a straight line is an adequate description of the relationship, but other models must be investigated for possible improvements in fit. A simple extension of the straight line is a power transformation model, β 1 X p. The latter model has often been used by practitioners in an ad hoc way, utilising different choices of P. Royston and Altman (1994) formalize the model slightly by calling it a first-degree fractional polynomial or FP1 function. The power p is chosen from a pragmatically chosen restricted set \(S =\{ -2,-1,-0.5,0,0.5,1,2,3\}\), where X 0 denotes logX.
As with polynomial regression, extension from one-term FP1 functions to the more complex and flexible two-term FP2 functions follows immediately. Instead of \({\beta...
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References and Further Reading
Ambler G, Royston P (2001) Fractional polynomial model selection procedures: investigation of Type I error rate. J Stat Comput Simul 69:89–108
Binder H, Sauerbrei W (2010) Adding local components to global functions for continuous covariates in multivariable regression modeling. Stat Med 29:808–817
Faes C, Aerts M, Geys H, Molenberghs G (2007) Model averaging using fractional polynomials to estimate a safe level of exposure. Risk Anal 27:111–123
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Royston P, Sauerbrei W (2007) Improving the robustness of fractional polynomial models by preliminary covariate transformation. Comput Stat Data Anal 51:4240–4253
Royston P, Sauerbrei W (2008) Multivariable model-building – a pragmatic approach to regression analysis based on fractional polynomials for modelling continuous variables. Wiley, Chichester
Sauerbrei W, Royston P (1999) Building multivariable prognostic and diagnostic models: transformation of the predictors using fractional polynomials. J R Stat Soc A 162:71–94
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Sauerbrei W, Royston P, Binder H (2007b) Selection of important variables and determination of functional form for continuous predictors in multivariable model-building. Stat Med 26:5512–5528
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Sauerbrei, W., Royston, P. (2011). Multivariable Fractional Polynomial Models. In: Lovric, M. (eds) International Encyclopedia of Statistical Science. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-04898-2_393
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