Patrick Royston
MRC Clinical Trials Unit,
London, UK
Willi Sauerbrei
Institut of Medical Biometry and Informatics
University Medical Center Freiburg, Germany
Multivariable regression modelling –
a pragmatic approach based on fractional
polynomials for continuous variables
Outline
• Prognostic factor studies
• Continuous variables
– categorizing data
– fractional polynomials
– interactions
• Reporting
• Conclusions
2
Mc Guire 1991
Guidelines for evaluating new prognostic factors
1.
2.
3.
4.
5.
6.
7.
Begin with a biological hypothesis for the new factor
Differentiate between a pilot study and a definitve study
Perform sample size calculations prior to initiating the study
Identify possible patient selection biases
Validate the methodologies used to measure the new factor
Include optimal representations of the factor in the analyses
Perform multivariate analyses that also include standard
factors
8. Validate the reproducibility of the results in internal and
external validation sets
3
Observational Studies
•
one spezific variable of interest,
necessity to control for confounders
•
many variables measured, pairwiseand multicollinearity present
• model should fit the data
• identification of important variables
• model and single effects
• sensible
• interpretable
•
Use subject-matter knowledge for modelling...
... But for some variables, data-driven choice inevitable
Modelling in the framework of
• Regression models
• Trees
• Neutral Net
Selection of important variables
4
Methods for variable selection
full model
- variance inflation in the case of multi-collinearity
* Wald-statistic
stepwise procedures
- prespecified (αin, αout) and actual selection level?
* forward selection (FS)
* stepwise selection (StS)
* backward elimination (BE)
all subset selection
- which criteria?
*
Cp
Mallows
*
AIC
Akaike
*
SBC
Schwarz
Bayes variable selection
MORE OR LESS COMPLEX MODELS?
5
Evaluation of prognostic factors is often based on
historical data
• Advantage
Patient data with long-term follow-up
information available in a database
•
Disadvantages
Insufficient quality of data
Important variables not availabe
Study population heterogeneous with
respect to prognostic factors and therapy
6
Assessment of a ‘new‘ factor
Population
•
•
•
ideally from a clinical trial
most often registry data from a clinic
often too small
Analysis
•
•
•
•
Often only univariate analysis
cutpoint for division into two groups
cutpoint derived data-dependently
multivariate analysis required
7
Example to demonstrate issues
Freiburg DNA study (Pfisterer et al 1995)
N= 266, Median follow-up 82 months
115 events for recurrence free survival time
Prognostic value of SPF
SPF missing: 2.5% of diploid tumours (N=122)
38.9% of aneuploid tumours (N=144)
8
´Optimal´ cutpoint analysis – serious problem
SPF-cutpoints used in the literature (Altman et al 1994)
Cutpoint
Reference
Method
Cutpoint
Reference
Method
2.6
Dressler et al 1988
median
8.0
Kute et al 1990
median
3.0
Fisher et al 1991
median
9.0
Witzig et al 1993
median
4.0
Hatschek et al 1990
1)
10.0
O'Reilly et al 1990a
'optimal'
5.0
Arnerlöv et al 1990
not given
10.3
Dressler et al 1988
median
6.0
Hatschek et al 1989
median
12.0
Sigurdsson et al 1990 'optimal'
6.7
Clark et al 1989
'optimal'
12.3
Witzig et al 1993
2)
7.0
Baak et al 1991
not given
12.5
Muss et al 1989
median
7.1
O'Reilly et al 1990b
median
14.0
Joensuu et al 1990
'optimal'
7.3
Ewers et al 1992
median
15.0
Joensuu et al 1991
'optimal'
7.5
Sigurdsson et al 1990
median
1) Three Groups with approx. equal size 2)Upper third of SPF-distribution
9
Searching for optimal cutpoint
minimal p-value approach
SPF in Freiburg DNA study
Problem
multiple testing => inflated type I error
10
Searching for optimal cutpoint
% significant
Inflation of type I errors
(wrongly declaring a variable as important)
Cutpoint selection in inner interval (here 10% - 90%) of
distribution of factor
Sample size
Simulation study
• Type I error about 40% istead of 5%
• Increased type I error does not disappear with increased
sample size (in contrast to type II error)
11
Freiburg DNA study
Study and 5 subpopulations (defined by nodal and
ploidy status
Optimal cutpoints with P-value
Pop 1
Pop 2
Pop 3
Pop 4
Pop 5
Pop 6
All
Diploid
N0
N+
N+, Dipl.
N+, Aneupl
n = 207
n = 119
n = 98
n = 109
n = 59
n = 50
optimal cutpoint
5.4
5.4
9.0 - 9.1
10.7 - 10.9
3.7
10.7 - 11.2
P-value
0.037
0.051
0.084
0.007
0.068
0.003
corrected p-value
0.406
>0.5
>0.5
0.123
>0.5
0.063
relative risk
1.58
1.87
0.28
2.37
1.94
3.30
12
Continuous factor
Categorisation or determination of
functional form ?
a) Step function (categorical analysis)
–
–
–
–
Loss of information
How many cutpoints?
Which cutpoints?
Bias introduced by outcome-dependent choice
b) Linear function
–
–
–
May be wrong functional form
Misspecification of functional form leads to wrong
conclusions
c) Non-linear function
–
Fractional polynominals
13
StatMed 2006, 25:127-141
14
Fractional polynomial models
• Fractional polynomial of degree m with powers
(p1,…, pm) is defined as
FPm  1 X
p1
 2 X
p2
   m X
p=
pm
( conventional polynomial p1 = 1, p2 = 2, ... )
•
•
•
•
•
Notation: FP1 means FP with one term (one power),
FP2 is FP with two terms, etc.
Powers p are taken from a predefined set S
We use S = { 2, 1, 0.5, 0, 0.5, 1, 2, 3}
Power 0 means log X here
15
Fractional polynomial models
• Describe for one covariate, X
– multiple regression later
• 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
• 8 FP1, 36 FP2 models
16
Examples of FP2 curves
- varying powers
(-2, 1)
(-2, 2)
(-2, -2)
(-2, -1)
17
Examples of FP2 curves
- single power, different coefficients
(-2, 2)
4
Y
2
0
-2
-4
10
20
30
x
40
50
18
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
19
GBSG-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
20
FP analysis for the effect of age
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
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
7
21
Effect of age at 5% level?
χ2
df
p-value
Any effect?
Best FP2 versus null
17.61
4
0.0015
Effect linear?
Best FP2 versus linear
17.03
3
0.0007
FP1 sufficient?
Best FP2 vs. best FP1
11.20
2
0.0037
22
Multivariable Fractional Polynomials (MFP)
With multiple continuous predictors selection
of best FP for each becomes more difficult 
MFP algorithm as a standardized way to
variable and function selection
MFP algorithm combines
• backward elimination with
• FP function selection procedures
23
Continuous factors
Different results with different analyses
Age as prognostic factor in breast cancer (adjusted)
P-value
0.9
0.2
0.001
24
Results similar?
Nodes as prognostic factor in breast cancer(adjusted)
P-value
0.001
0.001
0.001
25
Multivariable FP
Final Model in breast cancer
 2

 0.5
0.5
f  X 1 , X 1 ; X 4 a ; exp  012
. X 5 ; X 6 


Model choosen out of
5760 possible models,
one model selected
Model
– Sensible?
– Interpretable?
– Stable?
Bootstrap stability analysis
26
Main interest of clinicians:
Individualized treatment
This requires knowledge about several
predictive factors
27
Detecting predictive factors
• 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
28
Categorizing a continuous covariate
• How many cutpoints?
• Position of the cutpoint(s)
• Loss of information  loss of power
29
FP approach can also be used
to investigate predictive factors
30
MRC RE01 trial
RCT in metastatic renal carcinoma
1.00
N = 347; 322 deaths
0.00
0.25
0.50
0.75
(1) MPA
(2) Interferon
At risk 1:
175
55
22
11
3
2
1
At risk 2:
172
73
36
20
8
5
1
0
12
24
36
48
60
72
Follow-up (months)
31
Renal Carcinoma
Overall conclusion:
Interferon is better (p<0.01)
MRCRCC, Lancet 1999
Is the treatment effect
similar in all patients?
32
Predictive factors
Treatment – covariate interaction
Treatment effect function for WCC
-4
-2
0
2
Original data
5
10
White cell count
15
20
Only a result of complex (mis-)modelling?
33
Check result of MFPI modelling
Treatment effect in subgroups defined by WCC
0
12
24
36
48
60
72
12
24
36
48
0
12
60
Follow-up (months)
72
24
36
48
60
72
Group IV
0.00 0.25 0.50 0.75 1.00
0.00 0.25 0.50 0.75 1.00
Group III
0
Group II
0.00 0.25 0.50 0.75 1.00
0.00 0.25 0.50 0.75 1.00
Group I
0
12
24
36
48
60
72
Follow-up (months)
HR (Interferon to MPA) 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)
34
Assessment of WCC as a predictive factor
• Retrospective, searching for hypothesis
• 10 factors investigated, for one an interaction
was identified
• ‚Dose-response‘ effect in RE01 trial
• Validation in independent data
Worldwide collaboration:
Don‘t we have other trials to check this result?
35
REPORTING –
Can we believe in the published literature?
• Selection of published studies
• Insufficient reporting for assessment of quality of
– planing
– conducting
– analysis
•
•
too early publications
Usefullness for systematic review (meta-analysis)
Begg et al (1996) Improving the Quality of Reporting of
Randomized Controlled Trials – The CONSORT
Statement, JAMA,276:637-639
Moher et al JAMA (2001), Revised Recommendations
36
Reporting of prognostic markers
Riley et al BJC (2003)
Systematic review of tumor markers for neuroblastoma
260 studies identified, 130 different markers
The reporting of these studies was often inadequate,
in terms of both statistical analysis and presentation,
and there was considerable heterogeneity for many
important clinical/statistical factors. These problems
restricted both the extraction of data and the
meta-analysis of results from the primary studies,
limiting feasibility of the evidence-based approach.
37
Papers useful for overview ?
Prognostic markers for neuroblastoma
Marker
OS & Total
Different Different Different
name Papers DFS successful cutoff
stage
age
U/A
reports estimates groups groups groups
MYC-N
151
194
94
9
9
4
77/17
CD44
8
8
3
1
1
2
3/0
MDR
16
30
16
8
3
3
13/3
38
EJC 2007, 43:2559-79
Database 1: 340 articles included in meta-analysis
Database 2: 1575 articles published in 2005
39
• examined whether the abstract reported any statistically significant
prognostic effect for any marker and any outcome (‘positive’
articles).
• ‘Positive’ prognostic articles comprised 90.6% and 95.8% in
Databases 1 and 2, respectively.
• ‘Negative’ articles were further examined for statements made by
the investigators to overcome the absence of prognostic statistical
significance.
• Most of the ‘negative’ prognostic articles claimed significance for
other analyses,expanded on non-significant trends or offered
apologies that were occasionally remote from the original study
aims.
• Only five articles in Database 1 (1.5%) and 21 in Database 2 (1.3%)
were fully ‘negative’ for all presented results in the abstract and
without efforts to expand on non-significant trends or to defend the
importance of the marker with other arguments.
• Of the statistically non-significant relative risks in the meta-analyses,
25% had been presented as statistically significant in the primary
papers using different analyses compared with the respective metaanalysis.
• Under strong reporting bias, statistical significance loses its
discriminating ability for the importance of prognostic markers.
40
We expect some improvements by the REMARK guidelines
published simultaneously in 5 journals, August 2005
41
Prognostic markers – current situation
number of cancer prognostic markers validated as clinically useful is
pitifully small
Evidence based assessment is required, but
collection of studies difficult to interpret due to
inconsistencies in conclusions or a lack of comparability
Small underpowered studies, poor study design, varying and
sometimes inappropriate statistical analyses, and differences in
assay methods or endpoint definitions
More complete and transparent reporting
distinguish carefully designed and analyzed studies from
haphazardly designed and over-analyzed studies
Identification of clinically useful cancer prognostic factors: What are we missing?
McShane LM, Altman DG, Sauerbrei W; Editorial JNCI July 2005
42
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
• MFP extensions
• Interactions
• Time-varying effects
Investigation of properties required
Comparison to splines required
43
References
McShane LM, Altman DG, Sauerbrei W, Taube SE, Gion M, Clark GM for the Statistics Subcommittee of the NCI-EORTC
Working on Cancer Diagnostics (2005): REporting recommendations for tumor MARKer prognostic studies (REMARK).
Journal of the National Cancer Institute, 97: 1180-1184.
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. (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, 561571.
Royston P, Sauerbrei W. (2008): Multivariable Model-Building - A pragmatic approach to regression analysis based on
fractional polynomials for continuous variables.Wiley.
Sauerbrei W, Meier-Hirmer C, Benner A, Royston P. (2006): Multivariable regression model building by using fractional
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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, to appear
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.
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