DIY fractional polynomials
Patrick Royston
MRC Clinical Trials Unit , London
10 September 2010
Overview
• Introduction to fractional polynomials
• Going off-piste: DIY fractional polynomials
• Examples
3
Fractional polynomial models
• A fractional polynomial of degree 1 with power p1 is defined
as FP1 = β1 X p1
• A fractional polynomial of degree 2 with powers (p1,p2) is
defined as FP2 = β1 X p1 + β2 X p2
• Powers (p1,p2) are taken from a predefined set
S = {2,  1,  0.5, 0, 0.5, 1, 2, 3}
where 0 means log X
 Also, there are ‘repeated’ powers FP2 models
Example: FP1 [power 0.5] = β1 X0.5
Example: FP2 [powers (0.5, 3)] = β1 X0.5 + β2 X3
Example: FP2 [powers (3, 3)] = β1 X3 + β2 X3lnX
Some examples of fractional
polynomial (FP2) curves
(-2, 1)
(-2, 2)
(-2, -2)
(-2, -1)
Royston P, Altman DG (1994) Applied Statistics 43: 429-467.
FP analysis for the prognostic
effect of age in breast cancer
Degree 1
Power Model
chisquare
-2
6.41
-1
3.39
-0.5
2.32
0
1.53
0.5
0.97
1
0.58
2
0.17
3
0.03
7
Powers
-2
-2
-2
-2
-2
-2
-2
-2
-1
-1
-1
-1
-2
-1
-0.5
0
0.5
1
2
3
-1
-0.5
0
0.5
Model
chisquare
17.09
17.57
17.61
17.52
17.30
16.97
16.04
14.91
17.58
17.30
16.85
16.25
Degree 2
Powers
Model Powers
chisquare
-1
1
15.56
0
2
-1
2
13.99
0
3
-1
3
12.37 0.5 0.5
-0.5 -0.5
16.82 0.5 1
-0.5
0
16.18 0.5 2
-0.5
0.5
15.41 0.5 3
-0.5
1
14.55
1
1
-0.5
2
12.74
1
2
-0.5
3
10.98
1
3
0
0
15.36
2
2
0
0.5
14.43
2
3
0
1
13.44
3
3
Model
chisquare
11.45
9.61
13.37
12.29
10.19
8.32
11.14
8.99
7.15
6.87
5.17
3.67
FP function selection procedure
Simple functions are preferred. More complicated functions
are accepted only if the fit is much better
Effect of age significant at 5% level?
χ2
df
P-value
Any effect?
Best FP2 versus null
17.61
4
0.0015
Linear function suitable?
Best FP2 versus linear
17.03
3
0.0007
FP1 sufficient?
Best FP2 vs. best FP1
11.20
2
0.0037
Fractional polynomials in Stata
• fracpoly command
• Basic syntax:
. fracpoly [, fp_options]: regn_cmd [yvar] xvar1 [xvars] …
• xvar1 is a continuous predictor which may have a curved
relationship with yvar
• xvars are other predictors, all modelled as linear
• Can use the fp_option compare to compare the fit of
different FP models
• uses the FP function selection procedure
Example (auto data)
• fracpoly, compare: regress mpg displacement
Fractional polynomial model comparisons:
-------------------------------------------------------------------------displacement
df
Deviance
Res. SD
Dev. dif. P (*) Powers
-------------------------------------------------------------------------Not in model
0
468.789
5.7855
70.818
0.000
Linear
1
417.801
4.12779
19.830
0.000 1
m = 1
2
400.592
3.67467
2.621
0.284 -2
m = 2
4
397.971
3.6355
--- -2 3
-------------------------------------------------------------------------(*) P-value from deviance difference comparing reported model with m = 2
model
• Show FP1 and FP2 models in Stata (+ fracplot)
But what if fracpoly can’t fit my
model … ?
• fracpoly supports only some of Stata’s rich set of
regression-type commands
• Provided we know what the command we want to fit looks
like with a transformed covariate, we can fit an FP model to
the data
• We just create the necessary transformed covariate values,
fit the model using them, and assess the fit
• A new, simple command fracpoly_powers helps by
generating strings (local macros) with the required powers:
. fracpoly_powers [, degree(#) s(list_of_powers) ]
Fitting an FP2 model in the auto
example
// Store FP2 powers in local macros
fracpoly_powers, degree(2)
local np = r(np)
forvalues j = 1 / `np' {
local p`j' `r(p`j')'
}
// Compute deviance for each model with covariate displacement
local x displacement
local y mpg
local devmin 1e30
quietly forvalues j = 1 / `np' {
fracgen `x' `p`j'', replace
regress `y' `r(names)'
local dev = -2 * e(ll)
if `dev' < `devmin' {
local pbest `p`j''
local devmin `dev'
}
}
di "Best model has powers `pbest', deviance = " `devmin'
A real example: modelling fetal
growth
• Prospective longitudinal study of n = 50 pregnant women
• There are about 6 repeated measurements on each fetus at
different gestational ages (gawks)
• gawks = gestational age in weeks
• Wish to model how y = log fetal abdominal circumference
changes with gestational age
• There is considerable curvature!
4
4.5
5
Log AC
5.5
6
The raw data
10
20
30
Gestational age, wk
40
A mixed model for fetal growth
Multilevel (mixed) model to fit this relationship:
. xtmixed y FP(gawks) || id: FP(gawks),
covariance(unstructured)
But how do we implement “FP(gawks)” here?
We want the best-fitting FP function of gawks, with random
effects for the parameters (β’s) of the FP model
Fitting an FP2 mixed model to the
fetal AC data
[First run fracpoly_powers to create local macros with powers]
// Compute deviance for each FP model with covariate gawks
gen x = gawks
gen y = ln(ac)
local devmin 1e30
forvalues j = 1 / `np' {
qui fracgen x `p`j'', replace adjust(mean)
qui xtmixed y `r(names)' || id: `r(names)', ///
nostderr covariance(unstructured)
local dev = -2 * e(ll)
if `dev' < `devmin' {
local p `p`j''
local devmin `dev'
}
di "powers = `p`j''" _col(20) " deviance = " %9.3f `dev'
}
di _n "Best model has powers `p', deviance = " `devmin'
Plots of some results
.1
4
-.2
-.1
0
Residuals
5.5
4.5
5
Log AC
10
20
30
Gestational age, wk
40
-.2
-.1
0
.1
.2
Residuals and fitted residuals
Residuals
Residuals at the individual level
.2
6
Fitted curves at the individual level
10
20
30
Gestational age, wk
40
10
20
30
Gestational age, wk
40
An “ignorant” example!
• I know almost nothing about “seemingly unrelated
regression” (Stata’s sureg command)
• It fits a set of linear regression models which have
correlated error terms
• The syntax therefore has a set of “equations”
. sureg (depvar1 varlist1) (depvar2 varlist2) ... (depvarN
varlistN)
• There may be non-linearities lurking in these “equations”
• How can we fit FP models to varlist1, varlist2, … ?
Example: modelling learning
scores
Stata FAQ from UCLA
(http://www.ats.ucla.edu/stat/stata/faq/sureg.htm):
What is seemingly unrelated regression and how can I
perform it in Stata?
Example: High School and Beyond study
Example: modelling learning
scores
Contains data from hsb2.dta
obs:
200
highschool and beyond (200 cases)
vars:
11
5 Jul 2010 13:23
size:
9,600 (99.9% of memory free)
------------------------------------------------------------------------------storage display
value
variable name
type
format
label
variable label
------------------------------------------------------------------------------id
float %9.0g
female
float %9.0g
fl
race
float %12.0g
rl
ses
float %9.0g
sl
schtyp
float %9.0g
scl
type of school
prog
float %9.0g
sel
type of program
read
float %9.0g
reading score
write
float %9.0g
writing score
math
float %9.0g
math score
science
float %9.0g
science score
socst
float %9.0g
social studies score
-------------------------------------------------------------------------------
• [It is unclear to me what “ses” (low, middle, high) is]
Example (ctd.)
• As an example, suppose we wish to model 2 outcomes
(read, math) as predicted by “socst female ses” and
“science female ses” using sureg as follows:
. sureg (read socst female ses) (math science female
ses)
• Are there non-linearities in read as a function of socst?
In math as a function of science?
• For simplicity here, will restrict ourselves to FP1 functions
of socst and science
• not necessary in principle
• We fit the 8 × 8 = 64 FP1 models and look for the bestfitting combination
Stata
gen x1 = socst
gen x2 = science
gen y1 = read
gen y2 = math
local devmin 1e30
forvalues j = 1 / `np' {
qui fracgen x1 `p`j'', replace adjust(mean)
local x1vars `r(names)'
forvalues k = 1 / `np' {
qui fracgen x2 `p`k'', replace adjust(mean)
local x2vars `r(names)'
qui sureg (y1 `x1vars' female ses) (y2 `x2vars' female ses)
local dev = -2 * e(ll)
if `dev' < `devmin' {
local px1 `p`j''
local px2 `p`k''
local devmin `dev'
}
}
}
[Run fpexample3.do in Stata]
Comments
• The results suggest that there is indeed curvature in both
relationships
• Can reject the null hypothesis of linearity at the 1%
significance level
• FP1 vs linear: χ2 = 10.08 (2 d.f.), P = 0.0065
• Shows the importance of considering non-linearity
read as a function of socst
(adjusted female ses)
30
40
50
60
70
80
Fractional Polynomial (3),
adjusted for covariates
30
40
50
social studies score
60
70
math as a function of science
(adjusted female ses)
30
40
50
60
70
80
Fractional Polynomial (2),
adjusted for covariates
20
40
60
science score
80
Conclusions
• Fractional polynomial models are a simple yet very useful
extension of linear functions and ordinary polynomials
• If you are willing to do some straightforward do-file
programming, you can apply them in a bespoke manner to
a wide range of Stata regression-type commands and get
useful results
• For (much) more, see Royston & Sauerbrei (2008) book