fractional polynomial vs multivariable fractional polynomial

[Pakeezah Saadat]

Hello,

I am a little confused between the two concepts listed in the subject line. FP in stata can be used with the command fpfit to draw a curve whereas the other concept is regression. Stata goes in depth on MFP (regression) but not FP. I could be wrong. Can someone clarify the two or point me to sources pertaining FP only?

Thanks.

[FernandoRios]

In my view, the difference between both concepts is if you are interest in estimating the nonlinear effects of only one variable, or all variables in your model.
For example,
Say that you know y=f(x1,x2,x3)
Standard OLS approach says that you can estimate this model under the linear assumption. y=a0+a1*x1+a2*x2+a3*x3+error
This is just a simple approximation for f(x1,x2,x3).
Now, lets relax the linearity assumption, somewhat, by saying that ONE of your variables has an unknown nonlinear effect on the outcome.
y=a0+a1*x1+a2*x2+g(x3)+error
In this case, im assuming that X3 has a non linear effect, here represented with g(x3). How do you estimate g(.)?
Fractional polynomial will allow you to obtain an approximation for g(.) using, as the name says, a Fractional Polynomial. This is a type of global fitting, that compares a large set of possible models (see the details on -fp- to check what are the models used) based on polynomial and fractional polynomials using only the variable x3.

Now, lets relax the assumption further, and say that all variables have a nonlinear effect on y
y=g1(x1)+g2(x2)+g3(x3)+error
Even though this is more flexible than the linear model mode, it is still an approximation for f(x1,x2,x3), as Im assuming that the effects are additive.
To estimate a model like this, you would need to use -mfp-. This command works similarly to -fp-, but search for the best model using more than one explanatory variables.

in terms of your specific question, fpfit would only allow for fractional polynomial, because in 2 dimensions, your "model" is simply y=g(x). So only one variable has a non linear effect on y.
there is no multivariable fractional polynomial because that would require 3 dimensions, y=g1(x1)+g2(x2)+error. And beyond that it would not be possible to obtain any graphical representation of the effect.

HTH
Fernando