# Multivariable Fractional Polynomials

## Introduction

The mfp package is a collection of R (R Core Team 2022) functions targeted at the use of fractional polynomials (FP) for modelling the influence of continuous covariates on the outcome in regression models, as introduced by Royston and Altman (1994) and modified by Sauerbrei and Royston (1999). The model may include binary, categorical or further continuous covariates which are included in the variable selection process but without need of FP transformation. It combines backward elimination with a systematic search for a ‘suitable’ transformation to represent the influence of each continuous covariate on the outcome. An application of multivariable fractional polynomials (MFP) in modelling prognostic and diagnostic factors in breast cancer is given by Sauerbrei and Royston (1999). The stability of the models selected is investigated in Royston and Sauerbrei (2003). Briefly, fractional polynomials models are useful when one wishes to preserve the continuous nature of the covariates in a regression model, but suspects that some or all of the relationships may be non-linear. At each step of a `backfitting’ algorithm MFP constructs a fractional polynomial transformation for each continuous covariate while fixing the current functional forms of the other covariates. The algorithm terminates when no more covariate is excluded and the functional forms of the continuous covariates do not change anymore.

## Inventory of functions

mfp.object is an object representing a fitted mfp model. Class mfp inherits from either glm or coxph depending on the type of model fitted. In addition to the standard glm/coxph components the following components are included in an mfp object:

- x: the final FP transformations that are contained in the design matrix x. The covariate “z” with 4 df (second-degree FP) has corresponding columns “z.1” and “z.2” in x. A first-degree FP covariate “z” would have one column “z.1”.

- powers: a matrix containing the best FP powers for each covariate. If a covariate has less than two powers NAs will fill the appropriate cell of the matrix.

- pvalues: a matrix containing the P-values from the ‘’closed test procedure’’ together with the best powers chosen. Briefly p.null is the P-value for the test of inclusion (see mfp), p.lin corresponds to the test of nonlinearity and p.FP the test of simplification by comparing first degree (FP1) and second degree (FP2) transformations. The best first degree FP power (power2) and best second degree FP powers (power4.1 and power4.2) are also given. The numbers 2 and 4 describe the corresponding degrees of freedom.

- scale: all covariates are shifted and rescaled before being power transformed if nonpositive values are encountered or the range of values of the covariates is reasonably large. If x’ would be used instead of x where x’ = (x+a)/b the parameters a (shift) and b (scale) are contained in the matrix scale.

- df.initial: a vector containing the degrees of freedom allocated to each covariate corresponding to the degree of FP (m=4 for second degree FP, m=2 for first degree FP).

- df.final: a vector containing the degrees of freedom of each covariate at convergence of the backfitting algorithm (m=4 for second degree FP, m=2 for first degree FP, m=1 for untransformed variable, m=0 if covariate was excluded).

- dev: the deviance of the final model.

- dev.lin: the deviance of the model that uses the linear predictor of untransformed covariates.

- dev.null: the deviance of the null model.

- fptable: the table of the final fp transformations.

- fit: a call of the corresponding glm or cox model using the selected and (possibly) FP transformed variables of the final model.

## Usage in R

An mfp.object will be created by application of function mfp.

A typical call of an mfp model has the form response \(\sim\) terms where response is the (numeric) response vector and terms is a series of terms, separated by \(+\) operators, which specifies a linear predictor for response provided by the formula argument of the function call.

    library(mfp)
    #> Lade nötiges Paket: survival

    str(mfp)
    #> function (formula, data, family = gaussian, method = c("efron", "breslow"),
    #>     subset = NULL, na.action = na.omit, init = NULL, alpha = 0.05, select = 1,
    #>     maxits = 20, keep = NULL, rescale = FALSE, verbose = FALSE, x = TRUE,
    #>     y = TRUE)

Fractional polynomial terms are indicated by fp.

For binomial models the response can also be specified as a factor. If a Cox proportional hazards model is required then the outcome need to be specified using the Surv() notation.

The argument family describes the error distribution and link function to be used in the model. This can be a character string naming a family function, a family function or the result of a call to a family function.

Argument alpha sets the global FP selection level for all covariates. Different selection levels for individual covariates can be chosen by using the fp function. The variable selection level for all covariates is set by select. Values for individual fractional polynomials may be set using the fp function.

The function fp defines a fractional polynomial object for a single input variable.

    str(fp)
    #> function (x, df = 4, select = NA, alpha = NA, scale = TRUE)

In addition to alpha and select the scale argument of the fp function denotes the use of pre-transformation scaling to avoid possible numerical problems or for variables with non-positive values.

### Model selection

The original Stata implementation of mfp uses two different selection procedures for a single continuous covariate \(x\), a sequential selection procedure and a closed testing selection procedure (ra2, Ambler and Royston (2001)). In the R implementation only the ra2 algorithm is used which is also the default in the Stata and SAS implementations of mfp.

The ra2 algorithm is described in Ambler and Royston (2001) and Sauerbrei and Royston (2002). It uses a closed test procedure Marcus, Peritz, and Gabriel (1976) which maintains approximately the correct Type I error rate for each component test. The procedure allows the complexity of candidate models to increase progressively from a prespecified minimum (a null model) to a prespecified maximum (an FP) according to an ordered sequence of test results.

The algorithm works as follows:

1. Perform a 4 df test at the \(\alpha\) level of the best-fitting second-degree FP against the null model. If the test is not significant, drop \(x\) and stop, otherwise continue.

2. Perform a 3 df test at the \(\alpha\) level of the best-fitting second-degree FP against a straight line. If the test is not significant, stop (the final model is a straight line), otherwise continue.

3. Perform a 2 df test at the \(\alpha\) level of the best-fitting second-degree FP against the best-fitting first-degree FP. If the test is significant, the final model is the FP with \(m=2\), otherwise the FP with \(m=1\).

The tests in step 1, 2 and 3 are of overall association, non-linearity and between a simpler or more complex FP model, respectively.

## Example

### Cox proportional hazards model

We use the dataset GBSG which contains data from a study of the German Breast Cancer Study Group for patients with node-positive breast cancer.

    data(GBSG)
    str(GBSG)
    #> 'data.frame':    686 obs. of  11 variables:
    #>  $ id      : int  1 2 3 4 5 6 7 8 9 10 ...
    #>  $ htreat  : Factor w/ 2 levels "0","1": 1 2 2 2 1 1 2 1 1 1 ...
    #>  $ age     : int  70 56 58 59 73 32 59 65 80 66 ...
    #>  $ menostat: Factor w/ 2 levels "1","2": 2 2 2 2 2 1 2 2 2 2 ...
    #>  $ tumsize : int  21 12 35 17 35 57 8 16 39 18 ...
    #>  $ tumgrad : Factor w/ 3 levels "1","2","3": 2 2 2 2 2 3 2 2 2 2 ...
    #>  $ posnodal: int  3 7 9 4 1 24 2 1 30 7 ...
    #>  $ prm     : int  48 61 52 60 26 0 181 192 0 0 ...
    #>  $ esm     : int  66 77 271 29 65 13 0 25 59 3 ...
    #>  $ rfst    : int  1814 2018 712 1807 772 448 2172 2161 471 2014 ...
    #>  $ cens    : int  1 1 1 1 1 1 0 0 1 0 ...

The response variable is recurrence free survival time (Surv(rfst, cens)). Complete data on 7 prognostic factors is available for 686 patients. The median follow-up was about 5 years, 299 events were observed for recurrence free survival time. We use a Cox proportional hazards regression to model the hazard of recurrence by the 7 prognostic factors of which 5 are continuous, age of the patients in years (age), tumor size in mm (tumsize), number of positive lymphnodes (posnodal), progesterone receptor in fmol (prm), estrogen receptor in fmol (esm); one is binary, menopausal status (menostat); and one is ordered categorical with three levels, tumor grade (tumgrad). The additional variable htreat describes if a hormonal therapy was applied and is used as stratification variable.

We use mfp to build a model from the initial set of 7 covariates using the backfitting model selection algorithm. We set the global variable selection level to 0.05 and use the default FP selection level.

By using fp() in the model formula a fractional polynomial transformation with possibly pre-transformation scaling is estimated. This is done here for tumsize, posnodal, prm, and esm. Otherwise a linear form of the unscaled variable is used, as for age. Categorical factors are included without transformation. Hormonal therapy (htreat) was used as stratification variable.

By verbose=TRUE the process of FP and variable selection is printed.

    f <- mfp(Surv(rfst, cens) ~ strata(htreat)+age+fp(tumsize)+fp(posnodal)+fp(prm)+fp(esm)
            +menostat+tumgrad, family = cox, data = GBSG, select=0.05, verbose=TRUE)
    #>
    #>  Variable    Deviance    Power(s)
    #> ------------------------------------------------
    #> Cycle 1
    #>  posnodal
    #>              3135.218
    #>              3103.245    1
    #>              3081.123    0
    #>              3074.213    0.5 3
    #>
    #>  prm
    #>              3095.43
    #>              3074.213    1
    #>              3067.746    0.5
    #>              3066.502    -2 0.5
    #>
    #>  tumgrad2
    #>              3081.253
    #>              3074.213    1
    #>
    #>
    #>
    #>  tumgrad3
    #>              3082.613
    #>              3074.213    1
    #>
    #>
    #>
    #>  tumsize
    #>              3075.813
    #>              3074.213    1
    #>              3072.091    -1
    #>              3071.882    -1 3
    #>
    #>  menostat2
    #>              3076.922
    #>              3075.813    1
    #>
    #>
    #>
    #>  age
    #>              3076.922
    #>              3076.922    1
    #>
    #>
    #>
    #>  esm
    #>              3077.795
    #>              3076.922    1
    #>              3073.627    3
    #>              3071.028    -0.5 3
    #>
    #> Cycle 2
    #>  posnodal
    #>              3152.737
    #>              3108.965    1
    #>              3085.051    0
    #>              3077.795    0.5 3
    #>
    #>  prm
    #>              3099.562
    #>              3077.795    1
    #>              3071.74     0.5
    #>              3070.548    0 0.5
    #>
    #>  tumgrad2
    #>              3085.024
    #>              3077.795    1
    #>
    #>
    #>
    #>  tumgrad3
    #>              3086.686
    #>              3077.795    1
    #>
    #>
    #>
    #>  tumsize
    #>              3077.795
    #>              3076.471    1
    #>              3074.077    -1
    #>              3073.759    -0.5 0
    #>
    #>  menostat2
    #>              3077.795
    #>              3076.973    1
    #>
    #>
    #>
    #>  age
    #>              3077.795
    #>              3077.737    1
    #>
    #>
    #>
    #>
    #> Tansformation
    #>           shift scale
    #> posnodal      0    10
    #> prm           1   100
    #> tumgrad2      0     1
    #> tumgrad3      0     1
    #> tumsize       0    10
    #> menostat2     0     1
    #> age           0     1
    #> esm           1   100
    #>
    #> Fractional polynomials
    #>           df.initial select alpha df.final power1 power2
    #> posnodal           4   0.05  0.05        4    0.5      3
    #> prm                4   0.05  0.05        1      1      .
    #> tumgrad2           1   0.05  0.05        1      1      .
    #> tumgrad3           1   0.05  0.05        1      1      .
    #> tumsize            4   0.05  0.05        0      .      .
    #> menostat2          1   0.05  0.05        0      .      .
    #> age                1   0.05  0.05        0      .      .
    #> esm                4   0.05  0.05        0      .      .
    #>
    #>
    #> Transformations of covariates:
    #>                                          formula
    #> age                                         <NA>
    #> tumsize                                     <NA>
    #> posnodal I((posnodal/10)^0.5)+I((posnodal/10)^3)
    #> prm                           I(((prm+1)/100)^1)
    #> esm                                         <NA>
    #> menostat                                    <NA>
    #> tumgrad                                  tumgrad
    #>
    #>
    #> Deviance table:
    #>           Resid. Dev
    #> Null model    3198.026
    #> Linear model  3103.245
    #> Final model   3077.795

After two cycles the final model is selected. Of the possible input variables tumor size (tumsize), menopausal status (menostat), age and estrogen receptor (esm) were excluded from the model. Only for variable posnodal a nonlinear transformation was chosen. Prescaling was used for esm, prm and tumsize.

Details of the model fit are given by summary.

    summary(f)
    #> Call:
    #> coxph(formula = Surv(rfst, cens) ~ strata(htreat) + I((posnodal/10)^0.5) +
    #>     I((posnodal/10)^3) + I(((prm + 1)/100)^1) + tumgrad, data = GBSG)
    #>
    #>   n= 686, number of events= 299
    #>
    #>                          coef exp(coef) se(coef)      z Pr(>|z|)
    #> I((posnodal/10)^0.5)  1.79086   5.99461  0.21339  8.392  < 2e-16 ***
    #> I((posnodal/10)^3)   -0.03251   0.96801  0.01334 -2.438  0.01478 *
    #> I(((prm + 1)/100)^1) -0.21321   0.80799  0.05381 -3.963 7.41e-05 ***
    #> tumgrad2              0.61624   1.85195  0.24877  2.477  0.01324 *
    #> tumgrad3              0.74897   2.11482  0.26798  2.795  0.00519 **
    #> ---
    #> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
    #>
    #>                      exp(coef) exp(-coef) lower .95 upper .95
    #> I((posnodal/10)^0.5)     5.995     0.1668    3.9457    9.1075
    #> I((posnodal/10)^3)       0.968     1.0330    0.9430    0.9936
    #> I(((prm + 1)/100)^1)     0.808     1.2376    0.7271    0.8978
    #> tumgrad2                 1.852     0.5400    1.1373    3.0157
    #> tumgrad3                 2.115     0.4729    1.2507    3.5758
    #>
    #> Concordance= 0.679  (se = 0.016 )
    #> Likelihood ratio test= 120.2  on 5 df,   p=<2e-16
    #> Wald test            = 116.4  on 5 df,   p=<2e-16
    #> Score (logrank) test = 122.9  on 5 df,   p=<2e-16

Details of the FP transformations are given in the fptable value of the resulting mfp.object.

    f$fptable
    #>           df.initial select alpha df.final power1 power2
    #> posnodal           4   0.05  0.05        4    0.5      3
    #> prm                4   0.05  0.05        1      1      .
    #> tumgrad2           1   0.05  0.05        1      1      .
    #> tumgrad3           1   0.05  0.05        1      1      .
    #> tumsize            4   0.05  0.05        0      .      .
    #> menostat2          1   0.05  0.05        0      .      .
    #> age                1   0.05  0.05        0      .      .
    #> esm                4   0.05  0.05        0      .      .

The final model uses a second degree fractional polynomial for the number of positive lymphnodes with powers 0.5 and 3. The estimated functional from can be visualized using predict.mfp.

    vizmfp <- predict(f, type = "terms", terms = "posnodal", seq = list(1:50), ref = list(5))

    plot(vizmfp$posnodal$variable, exp(vizmfp$posnodal$contrast), type = "n", log = "y",
         xlab = "posnodal", ylab = "Hazard Ratio", ylim = c(0.1, 5))
    polygon(x = c(vizmfp$posnodal$variable, rev(vizmfp$posnodal$variable)),
            y = exp(c(vizmfp$posnodal$contrast - 1.96 * vizmfp$posnodal$stderr,
                  rev(vizmfp$posnodal$contrast + 1.96 * vizmfp$posnodal$stderr))),
            col = "grey", border = NA)
    grid()
    lines(vizmfp$posnodal$variable, exp(vizmfp$posnodal$contrast), type = "l", col = 4, lwd = 2)

The value fit of the resulting mfp object can be used for survival curve estimation of the final model fit.

    pf <- survfit(f$fit)
    plot(pf, col=c("red","green"), xlab="Time (years)", ylab="Recurrence free survival rate", xscale=365.25)

## References

Ambler, G., and P. Royston. 2001. “Fractional Polynomial Model Selection Procedures: Investigation of Type I Error Rate.” Journal of Statistical Simulation and Computation 69: 89–108.
Marcus, R., E. Peritz, and K. Gabriel. 1976. “On Closed Test Procedures with Special Reference to Ordered Analysis of Variance.” Biometrika 76: 655–60.
R Core Team. 2022. R: A Language and Environment for Statistical Computing. Vienna, Austria: R Foundation for Statistical Computing. https://www.R-project.org/.
Royston, P., and D. G. Altman. 1994. “Regression Using Fractional Polynomials of Continuous Covariates: Parsimonious Parametric Modelling (with Discussion).” Applied Statistics 43 (3): 429–67.
Royston, P., and W. Sauerbrei. 2003. “Stability of Multivariable Fractional Polynomial Models with Selection of Variables and Transformations: A Bootstrap Investigation.” Statistics in Medicine 22: 639–59.
Sauerbrei, W., and P. Royston. 1999. “Building Multivariable Prognostic and Diagnostic Models: Transformation of the Predictors by Using Fractional Polynomials.” Journal of the Royal Statistical Society (Series A) 162: 71–94.
———. 2002. “Corrigendum: Building Multivariable Prognostic and Diagnostic Models: Transformation of the Predictors by Using Fractional Polynomials.” Journal of the Royal Statistical Society (Series A) 165: 399–400.