NormData (version 1.1)

Fract.Poly: Fit fractional polynomials

Description

Fit a fractional polynomial model with \(m\) terms of the form \(X^{p}\), where the exponents \(p\) are selected from a small predefined set \(S\) of both integer and non-integer values. This function can be useful to model the mean or variance prediction function in a more flexible way than by using linear, quadratic or cubic polynomials.

Usage

Fract.Poly(IV, Outcome, 
S=c(-3, -2.5, -2.0, -1.5, -1, -0.5, 0.5, 1, 1.5, 2, 2.5, 3), 
Max.M=3)

Value

All.Results

The results (powers and AIC values) of the fractional polynomials.

Lowest.AIC

Table with the fractional polynomial model that has the lowest AIC.

Best.Model

The best fitted model (lm object).

IV

The IV tha was considered in the model.

Outcome

The outcome that was considered in the model.

Arguments

IV

The Independent Variable to be considered in the model.

Outcome

The outcome to be considered in the model.

S

The set \(S\) from which each power \(p^{m}\) is selected. Default S={-3, -2.5, -2.0, -1.5, -1, -0.5, 0.5, 1, 1.5, 2, 2.5, 3}.

Max.M

The maximum order \(M\) to be considered for the fractional polynomial. This value can be \(5\) at most. When \(M=5\), then fractional polynomials of order \(1\) to \(5\) are considered. Default Max.M=3.

Author

Wim Van der Elst

References

Van der Elst, W. (2024). Regression-based normative data for psychological assessment: A hands-on approach using R. Springer Nature.

Examples

data(VLT)
# Fit fractional polynomials of orders 1 to 2
FP <- Fract.Poly(IV = VLT$Age, Outcome = VLT$Total.Recall, 
  Max.M=2)
FP$Lowest.AIC
FP$Best.Model  
# Model with lowest AIC: 127.689 + (-190.731 * (Age**(-0.5))) +
#  (-7.586 * (Age**(0.5)))

# Make plot
plot(x=VLT$Age, y=VLT$Total.Recall, col="grey")
  # add best fitted fractional polynomial
Age.Vals.Plot <- 20:80
Pred.Vals <- 127.689 + (-190.731 * (Age.Vals.Plot**(-0.5))) +
   (-7.586 * (Age.Vals.Plot**(0.5)))
lines(x=Age.Vals.Plot, y=Pred.Vals, lwd=2, col="red", lty=2)
legend("topright", lwd=2, col="red", lty=2, 
  legend="Mean Prediction Function, Fractional Polynomial")