Interactions With Continuous Variables Extensions of the Multivariable Fractional Polynomial Approac

1
Interactions With Continuous Variables
Extensions of the Multivariable Fractional
Polynomial Approach
Willi SauerbreiInstitut of Medical Biometry and
Informatics University Medical Center Freiburg,
Germany
Patrick Royston MRC 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 Studies

• Several 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 models

• X(X1, ...,Xp) covariate, prognostic factors
• g(x) ß1 X1 ß2 X2 ... ßp Xp (assuming
  effects are linear)
• normal errors (linear) regression model
•  Y normally distributed
• E (YX) ß0 g(X)
• Var (YX) s2I
• logistic regression model
• Y binary
•  Logit P (YX) ln
• survival times
•  T survival time (partly censored)
• Incorporation of covariates

g(X)
(g(X))

5
Central issue

• To select or not to select (full model)?
• Which variables to include?

6
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
223369-3381 Discussion of issues in (univariate)
modelling with splines Trivial nowadays to fit
almost any model To choose a good model is much
harder
7
Alcohol consumption as risk factor for oral cancer
Rosenberg et al, StatMed 2003
8
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 Shrinkage
• Bayes variable selection
• Recommendations???

Central issue MORE OR LESS COMPLEX MODELS?
10
Backward elimination is a sensible approach

• Significance level can be chosen
• Reduces overfitting
• Of course required
• Checks
• Sensitivity analysis
• Stability analysis

11
Continuous variables what functional form?
 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
12
StatMed 2006, 25127-141
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 by FPm(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 (p1p2)
• ?1 X p1 ?2 X p1log X
• 8 FP1, 36 FP2 models

15
Examples of FP2 curves- varying powers
16
Examples of FP2 curves- single power, different
coefficients
17
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

18
Example Prognostic factorsGBSG-study in
node-positive breast cancer
299 events for recurrence-free survival time
(RFS) in 686 patients with complete data 7
prognostic factors, of which 5 are continuous
19
FP analysis for the effect of age
20
Function selection procedure (FSP)Effect of age
at 5 level?
?2 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
21
Many predictors MFP

• With 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 combines
• backward elimination with
• FP function selection procedures

22
Continuous factors Different results with
different analysesAge as prognostic factor in
breast cancer (adjusted)
P-value 0.9 0.2
0.001
23
Results similar? Nodes as prognostic factor in
breast cancer (adjusted)
P-value 0.001 0.001 0.001
24
Multivariable FP
Final Model in breast cancer example
age grade nodes
progesterone

• Model choosen out of
• more than a million possible models,
• one model selected
• Model - Sensible?
• - Interpretable?
• - Stable?
• Bootstrap stability analysis (see R S 2003)

25
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

26
Whitehall 1 Systolic blood pressure Deviance
difference in comparison to a straight line for
FP(1) and FP(2) models
27
Similar fit of several functions
28
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

29
Whitehall 1 Systolic blood pressure  Odds ratio
from final FP(2) model LogOR 2.92 5.43X-2
14.30 X 2 log X Presented in categories
30
Whitehall 1MFP analysis
No variables were eliminated by the MFP
algorithm Assuming a linear function weight is
eliminated by backward elimination
31
InteractionsMotivation I

• Detecting predictive factors (interaction with
  treatment)
• Dont 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)

32
Motivation - II

• 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

33
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 binarycovariate standard test for
  interaction available
• Continuous covariate
• Often categorized into two groups

34
Categorizing a continuous covariate

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

35
Standard approach

• Based on binary predictor
• Need cut-point for continuous predictor
• Illustration - problem with cut-point approach
• TAMER interaction in breast cancer (GBSG-study)

36
Treatment effect by subgroup
37
New approaches for continuous covariates

• STEPP
• Subpopulation treatment effect pattern plots
• Bonetti Gelber 2000
• MFPI
• Multivariable fractional polynomial
• interaction approach
• Royston Sauerbrei 2004

38
STEPP

• Sequences of overlapping subpopulations
• Sliding window Tail oriented

Contin. covariate
2g-1 subpopulations (here g8)
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STEPP
Estimates in subpopulations

• No interaction ?
• treatment effects similar in all
  subpopulations
• Plot effects in subpopulations

40
STEPP

• Overlapping populations, therefore correlation
  between treatment effects in subpopulations
• Simultaneous confidence band and tests proposed

41
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

42
RCT Metastatic renal carcinoma
Comparison of MPA with interferon N
347, 322 Death
43
Overall Interferon is better (plt0.01)

• 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

44
MFPI

•  Treatment effect function for WCC
• Only a result of complex (mis-)modelling?

45
Does the MFPI model agree with the data?Check
proposed trend
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)
46
STEPP Interaction with WCC
SLIDING WINDOW (n1 25, n2 40)
TAIL ORIENTED (g 8)
47
STEPP as check of MFPI
STEPP tail-oriented, g 6
48
MFPI Type I error

• Random permutation of a continuous covariate
  
• (haemoglobin)
• ? no interaction
• Distribution of P-value from test of interaction
• 1000 runs, Type I error 0.054

49
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

50
InteractionsWhitehall 1

• Consider only age and weight
• Main effects
• age linear
• weight FP2 (-1,3)
• Interaction?
• Include ageweight-1 ageweight3
• into the model
• LRT ?2 5.27 (2df, p 0.07)
• ? no (strong) interaction

51

• Erroneously assume that the effect of weight is
  linear
• Interaction?
• Include ageweight into the model
• LRT ?2 8.74 (1df, p 0.003)
• ? hightly significant interaction

52

• 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
54

• Running line smooth are about
• parallel across age groups ?
• no (strong) interactions
• smoothed probabilities are about equally spaced ?
  effect of age is linear

55

• Erroneously assume that the effect of
• weight is linear
• Estimated slopes of weight in age-groups
  indicates strong qualitative interaction between
  age und weight

56
Whitehall 1

• P-values for two-way interactions from MFPIgen

FP transformations
?? cholage highly significant
57
Presentation of interactionsWhitehall 1
agechol interaction
Effect (adjusted) for 10th, 35th, 65th and 90th
centile
58
AgeChol interaction

• Chol low age has an effect
• Chol high age has no effect
• Age low chol has an effect
• Age high chol has no effect

59
AgeChol interaction

• Does the model fit? Check in 4 subgroups

Linearity of chol ok But Slopes are not
monotonically ordered Lack of fit of
linearlinear interaction
60

• More complicated model?
• Interaction real?
• Validation in new data!

61
Survival data

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

62
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))

63
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
• ?(tX, Vj) ?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

65
Example Time-varying effectsRotterdam breast
cancer data

• 2982 patients
• 1 to 231 months follow-up time
• 1518 events for RFI (recurrence free interval)
• Adjuvant treatment with chemo- or hormonal
  therapy according to clinic guidelines
• 70 without adjuvant treatment
• Covariates
• continuous
• age, number of positive nodes, estrogen,
  progesterone
• categorical
• menopausal status, tumor size, grade

66

• 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
• Tests
• many 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
Selected model with MFP
test of time-varying effect for different time
transformations
estimates
70
Including time by covariate interaction(Semi-)
parametric models for ß(t)

• model ?(t) x ? x ? x g(t)
• calculate time-varying covariate x g(t)
• fit time-varying Cox model and test for ? 0
• plot ?(t) against t
• g(t) which form?
• usual function, eg t, log(t)
• piecewise
• splines
• fractional polynomials

71
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?

72
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) from
• FPT2 to null (time fixed effect) 4 DF
• FPT2 to log 3 DF
• FPT2 to FPT1 2 DF
• Use strategy analogous to stepwise to add
• time-varying functions to MFP model M1

74
Development of the model
75
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
• SAS
• www.imbi.uni-freiburg.de/biom/mfp
• R version available on CRAN archive
• mfp package
• Extensions to investigate interactions
• So far only in Stata

78
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

79
Towards recommendations for model-building by
selection of variables and functional forms for
continuous predictors under several assumptions
Sauerbrei et al. SiM 2007
80
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

82
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
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