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Modelling time dependent hazard ratios in relative survival: application to

colon cancer.

BOLARD P, QUANTIN C, ABRAHAMOWICZ M, ESTEVE J, GIORGI R, CHADHA-BOREHAM H,

BINQUET C, FAIVRE J.

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INTRODUCTION Previous results of the Flexible generalisation of the Cox model

- The PH hypothesis does not hold for most prognostic factors for all-causes mortality in colon cancer

-Some of these effects may reflect the inability of the method to separate cancer-related mortality from all-causes mortality

- Analyses of our BRDC colon cancer data require simultaneous modelling of both relative survival and possibly non-proportional hazards.

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METHODS

PH Relative survival model of Esteve et al.

Non PH Relative survival models

piecewise PH modelparametric time-by-covariate interactionnon-parametric time-by-covariate (spline)

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PH Relative survival model of Esteve

1,,,, zxtZtxZt ec

c = disease-related mortality hazard for the k-th interval

p

j

jjkc ZZt1

exp, for tk-1<t<tk

e = general population mortality hazard, depends on age (x)and gender (z1)

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For the k-th time-segment, k = 1 … r

p

jjjkjkc ZZt

1

exp, for tk-1<t<tk

PIECEWISE PH MODEL

Test of the PH: j2 = j3 = …… jr = 0

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PARAMETRIC TIME-BY-COVARIATE INTERACTION

For the k-th time-segment

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1

exp, ztfZZtp

jjjkc

fo r t k -1 < t< t k

w h e re f( t ) is a p re -s p e c if ie d fu n c t io n o f t im e ( lin e a r , lo g a r ith m ic … )

T e s t o f th e P H : 2 = 0 .

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L i n e a r i t y c o n s t r a i n t s b e f o r e t h e 1 s t k n o t a n d a f t e r t h e l a s t k n o t

a r e s t r i c t e d c u b i c s p l i n e f u n c t i o n w i t h v k n o t s ( v p a r a m e t e r s ) :

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1

131

1

310

v

ir

rr

irr

rr

irii tt

tt

tttt

tt

ttttttS

I f w e u s e v = 3 k n o t s :

3

323

1232

23

1331210 tt

tt

tttt

tt

ttttttS

t 1 , t 2 a n d t 3 a r e t h e t i m e p o i n t s w h e r e t h e k n o t s a r e p l a c e d

CUBIC SPLINE FUNCTIONS FOR MODELLING TIME-BY-COVARIATE INTERACTIONS

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RESTRICTED CUBIC SPLINE FUNCTIONS FOR MODELLING TIME-BY-

COVARIATE INTERACTION F o r t h e k - t h t i m e - s e g m e n t ,

p

jjjc ZkkaZt

1

exp,

kk jjj 10

33

23

1232

23

13312 kk

kkkkkk

kkkkkkj

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TESTS

Any type of dependence with time

j1 = j2 = 0

Non linear dependencej2 = 0

Effect of covariate Zjj0 = j1 = j2 = 0

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NUMBER OF KNOTS AND THEIR LOCATION

Number: can be restricted between 3 and 5 knots in most cases [Stone ] 3 knots.

Location: * both

- quantiles of the distribution function of deaths.

- percentiles of the distribution function of the follow-up times.

* In our restricted cubic spline model, we cannot fix the knots too near the extremes because of the linearity constraints.

5th, 50th and 95th quantiles

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APPLICATION: PATIENTS

2075 cases of colon cancer diagnosed between 76 and 90 (Burgundy Registry of Digestive Cancers)

end of follow-up: December 31, 1994.

1334 deaths at 5 yearsMedian survival time of 12 months

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Prognostic factors:

* gender

* age (< 65, 65-74, 75)

* periods of diagnosis (76-78, 79-81, 82-84, 85-87, 88-90)

* cancer TNM stage

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RESULTS

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Comparison of crude (Cox model) and relative survival (Esteve model) Proportional Hazard model in multivariate analyses

Cox Model Esteve ModelAGE HR 95% CI HR 95% CI< 65 years 1 165-74 years 1.44 [1.25 - 1.66] 1.19 [0.98-1.44]75 years or older 2.29 [2.03 - 2.65] 1.49 [1.23-1.74]

GENDERwoman 1 1man 1.20 [1.09-1.33] 1.06 [0.95-1.21]

STAGE AT DIAGNOSISstage I 1 1stage II 1.36 [1.13 - 1.63] 1.62 [1.11-2.38]stage III 2.33 [1.93 - 2.82] 4.09 [2.82-5.95]stage IIIb 5.23 [4.06 -6.73] 11.56 [7.66-17.43]stage IV 7.63 [6.31 - 9.23] 18.10 [12.63-25.93]

PERIOD OF DIAGNOSIS76-78 1 179-81 1.05 [0.90 - 1.24] 0.96 [0.76-1.22]82-84 1.02 [0.86 - 1.19] 0.87 [0.70-1.09]85-87 0.91 [0.77 - 1.07] 0.73 [0.58 - 0.92]88-90 0.81 [0.68 - 0.95] 0.62 [0.49 - 0.79]

Click for larger picture

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Testing the Proportional Hazard assumption in multivariate Relative Survival analysis

Time by covariate interaction Piecewise PHrelative survival

t log (t)

AgeGlobal < 0.001 0.008 <0.001

65-74 years 0.003 0.143 0.01275 years or older 0.004 0.004 <0.001

Gender 0.256 0.337 0.183

Stage at diagnosisGlobal <0.001 0.002 <0.001Stage II 0.172 0.008 0.287Stage III 0.024 0.043 0.004

Stage IIIb 0.256 0.214 0.024Stage IV 0.007 0.092 0.002

Period of diagnosisGlobal 0.164 0.005 0.00179-81 0.229 0.037 0.01682-84 0.181 0.350 0.01385-87 0.264 0.194 0.00688-90 0.368 0.226 0.002

Click for larger picture

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Change of the Hazard Ratio associated to age (reference category: < 65 years) using piecewise Proportional Hazard models in crude and relative survival

HR estimated by piecewise PH model in crude survival (Moreau's model)Time-segments (months)

[0;1[ [1;6[ [6;12[ [12;24[ [24;60[ pa

AGE

65-74 years 1.83* 1.62* 1.48* 0.94 1.08 0.02975 years or older 3.40* 2.25* 1.60* 1.47* 1.93* <0.001

HR estimated by piecewise PH model in relative survivalTime-segments (months)

[0;1[ [1;6[ [6;12[ [12;24[ [24;60[ pa

AGE

65-74 years 1.77* 1.52* 1.27 0.73 0.74 0.01275 years or older 3.54* 2.06* 1.13 0.96 0.70 <0.001

* : Hazard Ratio significantly different from 1

Click for larger picture

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0

1

2

3

4

5

6

0 20 40 60 80 100 120

Months

Ha

zard

Ra

tio

Piecewise PH crudesurvival (model 2)

Piecewise PH relativesurvival (model 5)

Age

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Test of proportional hazard assumption obtained with model 3 using restricted cubic spline functions for modelling different time-by-covariate interactions.

Age Site Stage Period ofdiagnosis

Gender

df -2Log L df -2Log L df -2Log L df -2Log L df -2Log L

Null 0 4947 0 4955 0 3955 0 4913 0 4970

Linear 2 4914 1 4949 4 3931 4 4886 1 4968

Cubic spline (3knots)

4 4903 2 4944 8 3900 8 4875 2 4964

Any type ofdependence(Betaj1 = Betaj2 = 0)

< 0.0001 < 0.0001 < 0.0001 < 0.0001 < 0.0001

Non lineardependence(Betaj2 = 0)

< 0.0001 < 0.0001 < 0.0001 <0.004 < 0.0001

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Age > 74 years versus age < 65 years

-1,5

-1

-0,5

0

0,5

1

1,5

0 10 20 30 40 50 60

Months

Lo

g H

azar

d R

atio

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Stage 2 versus Stage 1

-2,00

-1,00

0,00

1,00

2,00

3,00

4,00

0 10 20 30 40 50 60

Months

Lo

g H

azar

d R

atio

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Period of diagnosis 85-87 versus 76-78

-1,50

-1,00

-0,50

0,00

0,50

1,00

1,50

0 10 20 30 40 50 60

Months

Lo

g H

aza

rd R

ati

o

1,50

-1,50

0,00

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CONCLUSION

Both flexible modelling of non-proportional hazards and the relative survival approach are important:

differences between relative survival and the conventional Cox model.

Restricted cubic spline model * better fit than a linear time-by-covariate interaction* more parsimonious than a piecewise PH relative survival model * allows to represent both simple and complex patterns of changes

Number and the location of knots