Fractional polynomials

Fractional polynomials are an alternative to regular polynomials that provide flexible parameterization for continuous variables.

For example, say we have an outcome \(y\), a regressor \(x\), and our research interest is in the effect of \(x\) on \(y\). We know that \(y\) is also affected by \(age\). One solution to this problem would be to fit a linear regression of the form

\(y_1 = b_0 + b_1x_i + b_2age_i + b_3age_i\!^2 + u_i\)

An alternative would be to control for age using fractional polynomials:

\(y_i = b_0 + b_1x_i + b_2age_i\!^{(p2)} + u_i\)

Fractional powers are different from regular powers, and we emphasize this difference by enclosing the fractional power in parentheses. For instance, \(age^{(0)}\) is \(ln(age)\). You can see the full definition, but one example will demonstrate the power of fractional polynomials.

To fit the fractional polynomial model, we type

. fp <age>, scale: reg y x <age>
(fitting 44 models)
(....10%....20%....30%....40%....50%....60%....70%....80%....90%....100%)

Fractional polynomial comparisons:



            
 
 Test              Residual   Deviance                  

        age 
 
   df   Deviance   std. dev.      diff.       P   Powers

 
 
 

    omitted 
 
    4    -83.145      0.237    625.517    0.000               

     linear 
 
    3   -179.287      0.232    529.376    0.000   1           

      m = 1 
 
    2   -295.392      0.225    413.270    0.000   3           

      m = 2 
 
    0   -708.663      0.203      0.000       --   -.5 -.5     



Note: Test df is degrees of freedom, and P = P > F is sig. level for tests
      comparing models vs. model with m = 2 based on deviance difference,
      F(df, 1994).



      Source 
 
       SS           df       MS 
   Number of obs   =     2,000

 
 
 
   F(3, 1996)      =    347.37

       Model 
 
  42.8971855         3  14.2990618
   Prob > F        =    0.0000

    Residual 
 
  82.1620986     1,996  .041163376
   R-squared       =    0.3430

 
 
 
   Adj R-squared   =    0.3420

       Total 
 
  125.059284     1,999  .062560923
   Root MSE        =    .20289




           y 
 
 Coefficient  Std. err.      t    P>|t|     [95% conf. interval]

 
 
 

           x 
 
   .7954347   .0466225    17.06   0.000     .7040008    .8868686

       age_1 
 
   11.04709   .4085977    27.04   0.000     10.24577    11.84841

       age_2 
 
   13.17616   .4991682    26.40   0.000     12.19722    14.15511

       _cons 
 
  -8.059393   .5469551   -14.74   0.000    -9.132056    -6.98673

We find that \(b_1\), the effect of \(x\) on \(y\), is 0.80, but before we take that result seriously, we must ask ourselves whether we have adequately controlled for age.

. fp plot, residuals(none)

Notice the shape of the \(y\) versus \(age\) curve. We could not have obtained that shape using a standard quadratic. Fractional polynomials provide a wide range of shapes that include all the shapes provided by ordinary polynomials and more. The fractional polynomial parameterization did not predetermine that the shape we obtained was skewed right. Fractional polynomials can just as easily produce skewed left shapes.

We still need to do more to convince ourselves that the curve above is adequate, but we will not do so here