Patrick Royston MRC Clinical Trials Unit, London, UK

Interactions With Continuous Variables – Extensions of the Multivariable Fractional

Polynomial Approach

Willi SauerbreiInstitut of Medical Biometry and Informatics University Medical Center Freiburg, Germany

Patrick RoystonMRC Clinical Trials Unit, London, UK

2

Overview

• Issues in regression models• (Multivariable) fractional polynomials

(MFP)• Interactions of continuous variable with

– Binary variable– Continuous variable– Time

• Summary

3

Observational StudiesSeveral variables, mix of continuous and (ordered) categorical

variables

Different situations:– prediction– explanation

Explanation is the main interest here:• Identify variables with (strong) influence on the outcome• Determine functional form (roughly) for continuous variables

The issues are very similar in different types of regression models (linear regression model, GLM, survival models ...)

Use subject-matter knowledge for modelling ...... but for some variables, data-driven choice inevitable

4

Regression modelsX=(X1, ...,Xp) covariate, prognostic factorsg(x) = ß1 X1 + ß2 X2 +...+ ßp Xp (assuming effects are linear)

normal errors (linear) regression model

 Y normally distributedE (Y|X) = ß0 + g(X)Var (Y|X) = σ2I

logistic regression model

Y binary

 Logit P (Y|X) = ln

survival times T survival time (partly censored) Incorporation of covariates

0β)X0P(Y)X1P(Y g(X)

(t)expλ)Xλ(t 0 (g(X))

5

Central issue

To select or not to select (full model)?

Which variables to include?

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Continuous variables – The problem

“Quantifying epidemiologic risk factors using non-parametric regression: model selection remains the greatest challenge”Rosenberg PS et al, Statistics in Medicine 2003; 22:3369-3381

Discussion of issues in (univariate) modelling with splines

Trivial nowadays to fit almost any modelTo choose a good model is much harder

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Rosenberg et al, StatMed 2003

Alcohol consumption as risk factor for oral cancer

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Building multivariable regression models

Before dealing with the functional form, the ‚easier‘ problem of model selection:

variable selection assuming that the effect of each continuous variable is linear

9

Multivariable models - methods for variable selection

Full model– variance inflation in the case of multicollinearity

Stepwise procedures prespecified (in, out) and actual significance level?

• forward selection (FS)• stepwise selection (StS)• backward elimination (BE)

All subset selection which criteria?• Cp Mallows = (SSE / ) - n + p 2 • AIC Akaike Information Criterion = n ln (SSE / n) + p 2• BIC Bayes Information Criterion = n ln (SSE / n) + p ln(n)

fit penalty

Combining selection with ShrinkageBayes variable selection Recommendations???

Central issue: MORE OR LESS COMPLEX MODELS?

2σ̂

10

Backward elimination is a sensible approach

- Significance level can be chosen- Reduces overfitting

Of course required• Checks• Sensitivity analysis• Stability analysis

11

 Traditional approaches a) Linear function

- may be inadequate functional form- misspecification of functional form may lead to

wrong conclusions

b) ‘best‘ ‘standard‘ transformation

c) Step function (categorial data)- Loss of information- How many cutpoints?- Which cutpoints?- Bias introduced by outcome-dependent choice

Continuous variables – what functional form?

12

StatMed 2006, 25:127-141

13

Continuous variables – newer approaches

• ‘Non-parametric’ (local-influence) models– Locally weighted (kernel) fits (e.g. lowess)– Regression splines– Smoothing splines

• Parametric (non-local influence) models– Polynomials– Non-linear curves– Fractional polynomials

• Intermediate between polynomials and non-linear curves

14

Fractional polynomial models

• Describe for one covariate, X• Fractional polynomial of degree m for X with powers p1, … ,

pm is given byFPm(X) = 1 X p1 + … + m X pm

• Powers p1,…, pm are taken from a special set {2, 1, 0.5, 0, 0.5, 1, 2, 3}

• Usually m = 1 or m = 2 is sufficient for a good fit• Repeated powers (p1=p2)

1 X p1 + 2 X p1log X• 8 FP1, 36 FP2 models

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(-2, 1) (-2, 2)

(-2, -2) (-2, -1)

Examples of FP2 curves- varying powers

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Examples of FP2 curves- single power, different coefficients

(-2, 2)

Y

x10 20 30 40 50

-4

-2

0

2

4

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Our philosophy of function selection• Prefer simple (linear) model• Use more complex (non-linear) FP1 or FP2

model if indicated by the data• Contrasts to more local regression modelling

– Already starts with a complex model

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299 events for recurrence-free survival time (RFS) in 686 patients with complete data

7 prognostic factors, of which 5 are continuous

Example: Prognostic factors

GBSG-study in node-positive breast cancer

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FP analysis for the effect of age

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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

Function selection procedure (FSP)

Effect of age at 5% level?

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Many predictors – MFPWith many continuous predictors selection of best FP for each becomes more difficult MFP algorithm as a standardized way to variable and function selection

(usually binary and categorical variables are also available)

MFP algorithm combinesbackward elimination withFP function selection procedures

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P-value 0.9 0.2 0.001

Continuous factors Different results with different analysesAge as prognostic factor in breast cancer (adjusted)

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Results similar? Nodes as prognostic factor in breast cancer (adjusted)

P-value 0.001 0.001 0.001

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Multivariable FP

f X X X X Xa12

10 5

4 5 60 50 1 2

, ; ; ex p . ;. .

Model choosen out ofmore than a million possible models, one model selected

Model - Sensible?- Interpretable?- Stable?

Bootstrap stability analysis (see R & S 2003)

Final Model in breast cancer example

age grade nodes progesterone

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Example: Risk factors

• Whitehall 1– 17,370 male Civil Servants aged 40-64 years,

1670 (9.7%) died

– Measurements include: age, cigarette smoking, BP, cholesterol, height, weight, job grade

– Outcomes of interest: all-cause mortality at 10 years logistic regression

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Fractional polynomials First degree Second degree Power p

Deviance Difference

Powers p q

Deviance Difference

Powers p q

Deviance Difference

Powers p q

Deviance difference

-2 -74.19 -2 -2 26.22* -1 1 12.97 0 2 7.05 -1 -43.15 -2 -1 24.43 -1 2 7.80 0 3 3.74 -0.5 -29.40 -2 -0.5 22.80 -1 3 2.53 0.5 0.5 10.94 0 -17.37 -2 0 20.72 -0.5 -0.5 17.97 0.5 1 9.51 0.5 -7.45 -2 0.5 18.23 -0.5 0 16.00 0.5 2 6.80 1 0.00 -2 1 15.38 -0.5 0.5 13.93 0.5 3 4.41 2 6.43* -2 2 8.85 -0.5 1 11.77 1 1 8.46 3 0.98 -2 3 1.63 -0.5 2 7.39 1 2 6.61 -1 -1 21.62 -0.5 3 3.10 1 3 5.11 -1 -0.5 19.78 0 0 14.24 2 2 6.44 -1 0 17.69 0 0.5 12.43 2 3 6.45 -1 0.5 15.41 0 1 10.61 3 3 7.59

Whitehall 1Systolic blood pressureDeviance difference in comparison to a straight line for FP(1) and FP(2) models

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Similar fit of several functions

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Presentation of models for continuous covariates

• The function + 95% CI gives the whole story• Functions for important covariates should always be

plotted• In epidemiology, sometimes useful to give a more

conventional table of results in categories• This can be done from the fitted function

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Whitehall 1Systolic blood pressure  Odds ratio from final FP(2) model LogOR= 2.92 – 5.43X-2 –14.30* X –2 log XPresented in categories Systolic blood pressure (mm Hg) Range ref. point

Number of men at risk dying

OR (model-based) Estimate 95%CI

90 88 27 3 2.47 1.75, 3.49 91-100 95 283 22 1.42 1.21, 1.67 101-110 105 1079 84 1.00 - 111-120 115 2668 164 0.94 0.86, 1.03 121-130 125 3456 289 1.04 0.91, 1.19 131-140 135 4197 470 1.25 1.07, 1.46 141-160 150 2775 344 1.77 1.50, 2.08 161-180 170 1437 252 2.87 2.42, 3.41 181-200 190 438 108 4.54 3.78, 5.46 201-240 220 154 41 8.24 6.60, 10.28 241-280 250 5 4 15.42 11.64, 20.43

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Whitehall 1MFP analysis

Covariate FP etc. Age Linear Cigarettes 0.5 Systolic BP -1, -0.5 Total cholesterol Linear Height Linear Weight -2, 3 Job grade In

No variables were eliminated by the MFP algorithm

Assuming a linear function weight is eliminated by backward elimination

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Detecting predictive factors (interaction with treatment)

• Don’t investigate effects in separate subgroups!• Investigation of treatment/covariate interaction requires

statistical tests• Care is needed to avoid over-interpretation• Distinguish two cases:

- Hypothesis generation: searching several interactions- Specific predefined hypothesis

• For current bad practise - see Assmann et al (Lancet 2000)

InteractionsMotivation – I

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Continuous by continuous interactions

• usually linear by linear product term

• not sensible if main effect (prognostic effect) is non-linear

• mismodelling the main effect may introduce spurious interactions

Motivation - II

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Detecting predictive factors(treatment – covariate interaction)

• Most popular approach- Treatment effect in separate subgroups- Has several problems (Assman et al 2000)

• Test of treatment/covariate interaction required - For `binary`covariate standard test for interaction

available• Continuous covariate

- Often categorized into two groups

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Categorizing a continuous covariate

• How many cutpoints?• Position of the cutpoint(s)• Loss of information loss of power

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Standard approach• Based on binary predictor• Need cut-point for continuous predictor• Illustration - problem with cut-point approach

TAM*ER interaction in breast cancer (GBSG-study)

P-v

alue

(log

sca

le)

ER cutpoint, fmol/l0 5 10 15 20

.01

.02

.05

.1

.2

.5

36

Treatment effect by subgroup

Log

haza

rd ra

tio fo

r tre

atm

ent

ER cutpoint, fmol/l

Low ER High ER

0 5 10 15 20-.5

0

.5

1

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New approaches for continuous covariates

• STEPPSubpopulation treatment effect pattern plotsBonetti & Gelber 2000

• MFPIMultivariable fractional polynomialinteraction approach Royston & Sauerbrei 2004

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STEPPSequences of overlapping subpopulations

Sliding window Tail oriented

Con

tin. c

ovar

iate

2g-1 subpopulations(here g=8)

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STEPPEstimates in subpopulations

• No interaction treatment effects ‚similar‘ in all subpopulations

• Plot effects in subpopulations

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STEPP

Overlapping populations, therefore correlation between treatment effects in subpopulations

Simultaneous confidence band and tests proposed

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MFPI• Have one continuous factor X of interest • Use other prognostic factors to build an adjustment

model, e.g. by MFP

Interaction part – with or without adjustment

• Find best FP2 transformation of X with same powers in each treatment group

• LRT of equality of reg coefficients• Test against main effects model(no interaction) based on

2 with 2df• Distinguish

predefined hypothesis - hypothesis searching

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At risk 1: 175 55 22 11 3 2 1

At risk 2: 172 73 36 20 8 5 1

0.00

0.25

0.50

0.75

1.00

Pro

porti

on a

live

0 12 24 36 48 60 72Follow-up (months)

(1) MPA(2) Interferon

RCT: Metastatic renal carcinomaComparison of MPA with interferon N = 347, 322 Death

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• Is the treatment effect similar in all patients?Sensible questions?- Yes, from our point of view

• Ten factors available for the investigation of treatment – covariate interactions

Overall: Interferon is better (p<0.01)

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-4-2

02

Trea

tmen

t effe

ct, l

og re

lativ

e ha

zard

5 10 15 20White cell count

Original data

 Treatment effect function for WCC

Only a result of complex (mis-)modelling?

MFPI

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0.00

0.25

0.50

0.75

1.00

Pro

porti

on a

live

0 12 24 36 48 60 72

Group I

0.00

0.25

0.50

0.75

1.00

0 12 24 36 48 60 72

Group II

0.00

0.25

0.50

0.75

1.00

Pro

porti

on a

live

0 12 24 36 48 60 72Follow-up (months)

Group III

0.00

0.25

0.50

0.75

1.00

0 12 24 36 48 60 72Follow-up (months)

Group IV

Treatment effect in subgroups defined by WCC

HR (Interferon to MPA; adjusted values similar) overall: 0.75 (0.60 – 0.93)I : 0.53 (0.34 – 0.83) II : 0.69 (0.44 – 1.07)III : 0.89 (0.57 – 1.37) IV : 1.32 (0.85 –2.05)

Does the MFPI model agree with the data?Check proposed trend

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STEPP – Interaction with WCCSLIDING WINDOW (n1 = 25, n2 = 40)

TAIL ORIENTED (g = 8)

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STEPP as check of MFPI

.1.2

.51

2R

elat

ive

haza

rd

4 8 12 16 20White cell count

STEPP – tail-oriented, g = 6

48

0.5

11.

5D

ensi

ty

0 .2 .4 .6 .8 1p

MFPI – Type I error Random permutation of a continuous covariate (haemoglobin) no interaction

Distribution of P-value from test of interaction1000 runs, Type I error: 0.054

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Continuous by continuous interactionsMFPIgen

• Have Z1 , Z2 continuous and X confounders

• Apply MFP to X, Z1 and Z2, forcing Z1 and Z2 into the model. FP functions f1(Z1) and f2(Z2) will be selected for Z1 and Z2

• Add term f1(Z1)* f2(Z2) to the model chosen and use LRT for test of interaction

• Often f1(Z1) and/or f2(Z2) are linear

• Check all pairs of continuous variables for an interaction

• Check (graphically) interactions for artefacts

• Use forward stepwise if more than one interaction remains

• Low significance level for interactions

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InteractionsWhitehall 1

Consider only age and weight

Main effects:age – linearweight – FP2 (-1,3)

Interaction?

Include age*weight-1 + age*weight3

into the model

LRT: χ2 = 5.27 (2df, p = 0.07)

no (strong) interaction

51

Erroneously assume that the effect of weight is linear

Interaction?

Include age*weight into the model

LRT: χ2 = 8.74 (1df, p = 0.003)

hightly significant interaction

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Model check:categorize age in 4 equal sized groups

Compute running line smooth of the binary outcome on weight in each group

53

Whitehall 1: check of age x weight interaction-4

-3-2

-10

Logi

t(pr(

deat

h))

40 60 80 100 120 140weight

1st quartile 2nd quartile3rd quartile 4th quartile

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Running line smooth are about parallel across age groups no (strong) interactions

smoothed probabilities are about equally spaced effect of age is linear

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Erroneously assume that the effect ofweight is linear

Estimated slopes of weight in age-groups indicates strong qualitative interaction between age und weight

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Whitehall 1:P-values for two-way interactions from MFPIgen

*FP transformations

chol*age highly significant

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Presentation of interactionsWhitehall 1: age*chol interaction

Effect (adjusted) for 10th, 35th, 65th and 90th centile

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Age*Chol interaction

Chol ‚low‘ : age has an effectChol ‚high‘ : age has no effect

Age ‚low‘ : chol has an effectAge ‚high‘ : chol has no effect

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Age*Chol interactionDoes the model fit? Check in 4 subgroups

Linearity of chol: okBut: Slopes are not monotonically ordered Lack of fit of linear*linear interaction

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More complicated model?Interaction ‚real‘?Validation in new data!

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Survival data

effect of a covariate may vary in time time by covariate interaction

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Extending the Cox model

• Cox model(t | X) = 0(t) exp (X)

• Relax PH-assumption dynamic Cox model

(t | X) = 0(t) exp ((t) X)

HR(x,t) – function of X and time t

• Relax linearity assumption(t | X) = 0(t) exp ( f (X))

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Causes of non-proportionality

• Effect gets weaker with time• Incorrect modelling

– omission of an important covariate– incorrect functional form of a covariate– different survival model is appropriate

64

Non-PH – What can be done?

Non-PH - Does it matter ? - Is it real ?

Non-PH is large and real– stratify by the factor

(t|X, V=j) = j (t) exp (X )• effect of V not estimated, not tested • for continuous variables grouping necessary

– Partition time axis– Model non-proportionality by time-dependent

covariate

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Example: Time-varying effectsRotterdam breast cancer data

2982 patients1 to 231 months follow-up time1518 events for RFI (recurrence free interval)Adjuvant treatment with chemo- or hormonal therapy according to clinic guidelines70% without adjuvant treatment

Covariatescontinuous age, number of positive nodes, estrogen, progesteronecategorical menopausal status, tumor size, grade

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• Treatment variables ( chemo , hormon) will be analysed as usual covariates

• 9 covariates , partly strong correlation (age-meno; estrogen-progesterone; chemo, hormon – nodes )

variable selection • Use multivariable fractional polynomial approach

for model selection in the Cox proportional hazards model

67

Assessing PH-assumption• Plots

– Plots of log(-log(S(t))) vs log t should be parallel for groups– Plotting Schoenfeld residuals against time to identify patterns in

regression coefficients– Many other plots proposed

• Testsmany proposed, often based on Schoenfeld residuals,most differ only in choice of time transformation

• Partition the time axis and fit models seperatly to each time interval

• Including time-by-covariate interaction terms in the model and estimate the log hazard ratio function

68

Smoothed Schoenfeld residuals –multivariable MFP model assuming PH

69

Factor

SE

p-value t rank(t) Log(t) Sqrt(t)

X1 – age -0.01 0.002 0.082 0.243 0.329 0.149 X3a – size 0.29 0.057 0.000 0.000 0.001 0.000 X4b – grade 0.39 0.064 0.189 0.198 0.129 0.164 X5e – nodes -1.71 0.081 0.002 0.000 0.000 0.000 X8 - chemo-T -0.39 0.085 0.091 0.008 0.023 0.034 X9 – horm-T -0.45 0.073 0.014 0.001 0.000 0.002 Index 1.00 0.039 0.000 0.000 0.000 0.000

estimates test of time-varying effect for different time transformations

Selected model with MFP

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Including time – by covariate interaction(Semi-) parametric models for β(t)

• model (t) x = x + x g(t)calculate time-varying covariate x g(t) fit time-varying Cox model and test for 0plot (t) against t

• g(t) – which form?– ‘usual‘ function, eg t, log(t)– piecewise– splines– fractional polynomials

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MFPTime algorithmMotivation

• Multivariable strategy required to select– Variables which have influence on outcome– For continuous variables determine functional form of

the influence (‚usual‘ linearity assumption sensible?)– Proportional hazards assumption sensible or does a

time-varying function fit the data better?

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MFPTime algorithm (1)

• Determine (time-fixed) MFP model M0

possible problems variable included, but effect is not constant in time variable not included because of short term effect only

• Consider short term period only Additional to M0 significant variables?

This gives M1

73

MFPTime algorithm (2)For all variables (with transformations) selected from full time-period and short time-period

• Investigate time function for each covariate in forward stepwise fashion - may use small P value

• Adjust for covariates from selected model• To determine time function for a variable

compare deviance of models ( 2) fromFPT2 to null (time fixed effect) 4 DFFPT2 to log 3 DFFPT2 to FPT1 2 DF

• Use strategy analogous to stepwise to add time-varying functions to MFP model M1

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Development of the modelVariable Model M0 Model M1 Model M2

β SE β SE β SE

X1 -0.013 0.002 -0.013 0.002 -0.013 0.002X3b - - 0.171 0.080 0.150 0.081X4 0.39 0.064 0.354 0.065 0.375 0.065X5e(2) -1.71 0.081 -1.681 0.083 -1.696 0.084X8 -0.39 0.085 -0.389 0.085 -0.411 0.085X9 -0.45 0.073 -0.443 0.073 -0.446 0.073X3a 0.29 0.057 0.249 0.059 - 0.112 0.107logX6 - - -0.032 0.012 - 0.137 0.024X3a(log(t)) - - - - - 0.298 0.073logX6(log(t)) - - - - 0.128 0.016Index 1.000 0.039 1.000 0.038 0.504 0.082Index(log(t)) - - - - -0.361 0.052

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Time-varying effects in final model

76

Final model includes time-varying functions for

progesterone ( log(t) ) and tumor size ( log(t) )

Prognostic ability of the Index vanishes in time

77

Software sources MFP

• Most comprehensive implementation is in Stata– Command mfp is part since Stata 8 (now Stata 10)

• Versions for SAS and R are available– SASwww.imbi.uni-freiburg.de/biom/mfp

– R version available on CRAN archive• mfp package

• Extensions to investigate interactions• So far only in Stata

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Concluding comments – MFP

• FPs use full information - in contrast to a priori categorisation

• FPs search within flexible class of functions (FP1 and FP(2)-44 models)

• MFP is a well-defined multivariate model-building strategy – combines search for transformations with BE

• Important that model reflects medical knowledge, e.g. monotonic / asymptotic functional forms

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Towards recommendations for model-building by selection of variables and functional forms for continuous predictors under

several assumptionsIssue Recommendation

Variable selection procedure Backward elimination; significance level as key tuning parameter, choice depends on the aim of the study

Functional form for continuous covariates

Linear function as the 'default', check improvement in model fit by fractional polynomials. Check derived function for undetected local features

Extreme values or influential points Check at least univariately for outliers and influential points in continuous variables. A preliminary transformation may improve the model selected. For a proposal see R & S 2007

Sensitivity analysis Important assumptions should be checked by a sensitivity analysis. Highly context dependent

Check of model stability The bootstrap is a suitable approach to check for model stability

Complexity of a predictor A predictor should be 'as parsimonious as possible'

Sauerbrei et al. SiM 2007

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Interactions

• Interactions are often ignored by analysts• Continuous categorical has been studied in

FP context because clinically very important• Continuous continuous is more complex• Interaction with time important for long-term FU

survival data

81

MFP extensions

• MFPI – treatment/covariate interactions In contrast to STEPP it avoids categorisation

• MFPIgen – interaction between two continuous variables

• MFPT – time-varying effects in survival data

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Summary

Getting the big picture right is more important than optimising aspects and ignoring others

• strong predictors

• strong non-linearity

• strong interactions• strong non-PH in survival model

83

Harrell FE jr. (2001): Regression Modeling Strategies. Springer.

Royston P, Altman DG. (1994): Regression using fractional polynomials of continuous covariates: parsimonious parametric modelling (with discussion). Applied Statistics, 43, 429-467.

Royston P, Altman DG, Sauerbrei W (2006): Dichotomizing continuous predictors in multiple regression: a bad idea. Statistics in Medicine, 25, 127-141.

Royston P, Sauerbrei W. (2004): A new approach to modelling interactions between treatment and continuous covariates in clinical trials by using fractional polynomials. Statistics in Medicine, 23, 2509-2525.

Royston P, Sauerbrei W. (2005): Building multivariable regression models with continuous covariates, with a practical emphasis on fractional polynomials and applications in clinical epidemiology. Methods of Information in Medicine, 44, 561-571.

Royston P, Sauerbrei W. (2007): Improving the robustness of fractional polynomial models by preliminary covariate transformation: a pragmatic approach. Computational Statistics and Data Analysis, 51: 4240-4253.

Royston P, Sauerbrei W (2008): Multivariable Model-Building - A pragmatic approach to regression analysis based on fractional polynomials for continuous variables. Wiley.

Sauerbrei W. (1999): The use of resampling methods to simplify regression models in medical statistics. Applied Statistics, 48, 313-329.

Sauerbrei W, Meier-Hirmer C, Benner A, Royston P. (2006): Multivariable regression model building by using fractional polynomials: Description of SAS, STATA and R programs. Computational Statistics & Data Analysis, 50, 3464-3485.

Sauerbrei W, Royston P. (1999): Building multivariable prognostic and diagnostic models: transformation of the predictors by using fractional polynomials. Journal of the Royal Statistical Society A, 162, 71-94.

Sauerbrei W, Royston P, Binder H (2007): Selection of important variables and determination of functional form for continuous predictors in multivariable model building. Statistics in Medicine, 26:5512-28.

Sauerbrei W, Royston P, Look M. (2007): A new proposal for multivariable modelling of time-varying effects in survival data based on fractional polynomial time-transformation. Biometrical Journal, 49: 453-473.

Sauerbrei W, Royston P, Zapien K. (2007): Detecting an interaction between treatment and a continuous covariate: a comparison of two approaches. Computational Statistics and Data Analysis, 51: 4054-4063.

Schumacher M, Holländer N, Schwarzer G, Sauerbrei W. (2006): Prognostic Factor Studies. In Crowley J, Ankerst DP (ed.), Handbook of Statistics in Clinical Oncology, Chapman&Hall/CRC, 289-333.

References