the hazards of hazard ratios - paul dickman · time varying hazard ratio for a vs c.25.5 1 2 5 20...

The hazards of hazard ratios

Paul W Dickman

Sodersjukhuset16 May 2016

Abstract

I will discuss the ”2 problems with hazard ratios” described byMiguel Hernan [1].

1 the HR may change over time2 the HR has a built-in selection bias

First, I will define some of the basic measures used in survivalanalysis (survivor function, hazard function) by looking at twosurvival curves and asking ‘Which treatment is associated withthe best survival?’.

We will begin by assuming that these curves arise from ahypothetical perfect randomised trial where everyone adheres tothe allocated treatment/intervention.

Paul Dickman Hazards of Hazard Ratios Sodersjukhuset16 May 2016 2

Which treatment (A or C) has the best survival?

0.0

0.2

0.4

0.6

0.8

1.0

Sur

viva

l Fun

ctio

n

0 .2 .4 .6 .8 1Time since treatment (years)

Treatment ATreatment C

Which treatment is associated with the best survival?


Which treatment (A or C) has the best survival?

0.0

0.2

0.4

0.6

0.8

1.0

Sur

viva

l Fun

ctio

n

0 1 2 3 4 5Time since treatment (years)




The two hazard functions

0

.1

.25

.5

1

2

4

6

Haz

ard

Rat

e

0 1 2 3 4 5Years since diagnosis


Hazard function for each treatment group


What about if we further extend the follow-up?

0.0

0.2

0.4

0.6

0.8

1.0

Sur

viva

l Fun

ctio

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0 5 10 15Time since treatment (years)




Time varying hazard ratio for A vs C

.25

.5

1

2

5

20

Haz

ard

Rat

io (

A v

s C

)

0 5 10 15Years since diagnosis

True Hazard RatioHR from PH model

Hazard Ratio (A vs C)


Relation between the survivor and hazard functions

h(t) = lim∆t→0

Pr(event in (t, t + ∆t] | alive at t)

∆t

= lim∆t→0

F (t + ∆t)− F (t)

S(t)×∆twhere F (t) = 1− S(t)

= lim∆t→0

S(t + ∆t)− S(t)

∆t× −1

S(t)

=dS(t)

dt× −1

S(t)by definition of a derivative

= − d ln S(t)

dtsince d/dx ln(f (x)) = f ′(x)/f (x)


What does this mean in practice?

h(t) = − ddt

ln S(t)

In practical terms, this means that the event rate is proportionalto the rate at which the survival function decreases.

That is, if the survival function is decreasing sharply with timethen the mortality rate is high (and vice versa).

If the survival function is flat then the hazard is zero (and viceversa).

The derivative of a function at a point is the slope of the[tangent to the] curve at that point. A curve that is decreasing(like the survival function) has a negative slope, hence thenegative sign in the formula above.

We can think of the hazard as being proportional to the rate ofchange of S(t).


Other relationships (for completeness)

− d log S(t)

dt= h(t)

m

S(t) = exp

(−∫ t

0

h(u) du

)= exp (−H(t))

H(t) =∫ t

0h(u) du is called the integrated hazard or cumulative

hazard.

h(t) = − d log(S(t))

dt= −S ′(t)

S(t)=

F ′(t)

1− F (t)=

f (t)

S(t)


Women’s Health Initiative

Findings from a Women’s Health Initiative randomizedexperiment that compared the risk of coronary heart disease ofwomen assigned to combined hormone therapy with that ofwomen assigned to placebo. [2]

By using a randomized experiment as an example, discussion canfocus on the HR, setting aside issues of confounding and otherproblems that arise in observational studies.

The Women’s Health Initiative followed over 16,000 women foran average of 5.2 years before the study was halted due to safetyconcerns. The primary result from the trial was a HR.

As stated in the abstract and shown in Table 1 of the article,’Combined hormone therapy was associated with a hazard ratioof 1.24.’


Women’s Health Initiative 2

In addition, Table 2 provided the HRs during each year offollow-up: 1.81, 1.34, 1.27, 1.25, 1.45, and 0.70 for years 1, 2,3, 4, 5, and 6+, respectively.

Thus, the HR reported in the abstract and Table 1 can beviewed as some sort of weighted average of the period-specificHRs reported in Table 2.

The average HR in the WHI would have been 1.8 if the studyhad been halted after 1 year of follow-up, 1.7 after 2 years, 1.2after 5 years, and – who knows – perhaps 1.0 after 10 years.

The 24% increase in the rate of coronary heart disease thatmany researchers and journalists consider as the effect ofcombined hormone therapy is the result of the arbitrary choice ofan average follow-up period of 5.2 years.


Phase III Trial Assessing Bevacizumab in Stages II and IIICarcinoma of the Colon: Results of NSABP Protocol C-08Carmen J. Allegra, Greg Yothers, Michael J. O’Connell, Saima Sharif, Nicholas J. Petrelli, Linda H. Colangelo,James N. Atkins, Thomas E. Seay, Louis Fehrenbacher, Richard M. Goldberg, Seamus O’Reilly, Luis Chu,Catherine A. Azar, Samia Lopa, and Norman Wolmark

See accompanying editorial on page 1

From the National Surgical AdjuvantBreast and Bowel Project; GraduateSchool of Public Health, University ofPittsburgh; and Allegheny GeneralHospital, Pittsburgh, PA; University ofFlorida, Gainesville; and Florida CancerSpecialists–Sarasota, FL; Helen F.Graham Cancer Center at ChristianaCare, Newark, DE; Southeast CancerControl Consortium–Community ClinicalOncology Program (CCOP), Goldsboro,NC; Atlanta Cancer Care RegionalCCOP, Atlanta, GA; Kaiser PermanenteNorthern California, Vallejo, CA; Univer-sity of North Carolina at Chapel Hill,Chapel Hill, NC; All-Ireland CooperativeOncology Research Group, Dublin,Ireland; and Kaiser PermanenteMidtown I, Denver, CO.

Submitted May 17, 2010; acceptedAugust 25, 2010; published onlineahead of print at www.jco.org onOctober 12, 2010.

Supported by Public Health ServiceGrants No. U10-CA-12027, U10-CA-69651, U10-CA-37377, and U10-CA-69974 from the National CancerInstitute, Department of Health andHuman Services.

Authors’ disclosures of potential con-flicts of interest and author contribu-tions are found at the end of thisarticle.

Clinical Trials repository link available onJCO.org.

Corresponding author: Carmen J.Allegra, MD, University of Florida, Divi-sion of Hematology and Oncology,1600 SW Archer Ave, Rm N-503,Gainesville, FL 32610; e-mail:[email protected].

© 2010 by American Society of ClinicalOncology

0732-183X/11/2901-11/$20.00

DOI: 10.1200/JCO.2010.30.0855

A B S T R A C T

PurposeThe National Surgical Adjuvant Breast and Bowel Project C-08 trial was designed to investigate the safetyand efficacy of adding bevacizumab to modified FOLFOX6 (mFOLFOX6; ie, infusional/bolus fluorouracil,leucovorin, and oxaliplatin) for the adjuvant treatment of patients with stages II to III colon cancer.

MethodsPatients received mFOLFOX6 every 2 weeks for 26 weeks alone or modified as FOLFOX6 �bevacizumab (5 mg/kg every 2 weeks for 52 weeks [ie, experimental group]). The primary endpoint was disease-free survival (DFS).

ResultsAmong 2,672 analyzed patients, demographic factors were well balanced by treatment. With amedian follow-up of 35.6 months, the addition of bevacizumab to mFOLFOX6 did not result in anoverall significant increase in DFS (hazard ratio [HR], 0.89; 95% CI, 0.76 to 1.04; P � .15). The pointestimates for 3-year DFS for the overall population were 77.4% and 75.5% for the experimentaland control arms, respectively. For patients with stages II and III diseases, these same estimateswere 87.4% and 84.7%, respectively, for stage II and 74.2% and 72.4%, respectively, for stageIII. Exploratory analyses found that the effect of bevacizumab on DFS was different before andafter a 15-month landmark (time-by-treatment interaction P value � .0001). Bevacizumab had astrong effect before the landmark (HR, 0.61; 95% CI, 0.48 to 0.78; P � .001) but no significanteffect after (HR, 1.22; 95% CI, 0.98 to 1.52; P � .076).

ConclusionBevacizumab for 1 year with mFOLFOX6 does not significantly prolong DFS in stages II and IIIcolon cancer. However, a significant but transient effect during bevacizumab exposure wasobserved in the experimental arm. We postulate that this observation reflects a biologic effectduring bevacizumab exposure. Given the lack of improvement in DFS, the use of bevacizumabcannot be recommended for use in the adjuvant treatment of patients with colon cancer.

J Clin Oncol 29:11-16. © 2010 by American Society of Clinical Oncology

INTRODUCTION

The state-of-the-art treatment for the manage-ment of patients with stage III colon cancer is tooffer 6 months of adjuvant fluorouracil (FU)-and-oxaliplatin–containing chemotherapy. Twolarge, multinational, randomized studies demon-strated that the addition of oxaliplatin to the combi-nation of FU and leucovorin resulted in significantimprovement in disease-free survival (DFS) whencompared with FU plus leucovorin alone for theadjuvant treatment of patients with stages II and IIIcolon cancer.1,2 The Multicenter InternationalStudy of Oxaliplatin/Fluorouracil/Leucovorin in the

Adjuvant Treatment of Colon Cancer (MOSAIC)trial compared FU, leucovorin, and oxaliplatin(FOLFOX4) with infusion/bolus FU plus leucov-orin, and the National Surgical Adjuvant Breastand Bowel Project (NSABP) C-07 trial evaluatedthe FLOX regimen, which uses oxaliplatin plusbolus FU with leucovorin, in patients with stagesII and III colon cancer. Both trials demonstrated asignificant increase in DFS for patients treatedwith the oxaliplatin-containing regimen.

Bevacizumab, a humanized monoclonal anti-body with a high binding affinity for circulating vas-cular endothelial growth factor A (VEGF-A), hasbeen demonstrated to enhance the response rate,

JOURNAL OF CLINICAL ONCOLOGY O R I G I N A L R E P O R T

VOLUME 29 � NUMBER 1 � JANUARY 1 2011

© 2010 by American Society of Clinical Oncology 11

BLACKWELL INC on May 16, 2016 from 130.229.54.182Information downloaded from jco.ascopubs.org and provided by at Karolinska Institutet University Library / SWETS

Copyright © 2011 American Society of Clinical Oncology. All rights reserved.


Phase III Trial Assessing Bevacizumab in Stages II and IIICarcinoma of the Colon: Results of NSABP Protocol C-08Carmen J. Allegra, Greg Yothers, Michael J. O’Connell, Saima Sharif, Nicholas J. Petrelli, Linda H. Colangelo,James N. Atkins, Thomas E. Seay, Louis Fehrenbacher, Richard M. Goldberg, Seamus O’Reilly, Luis Chu,Catherine A. Azar, Samia Lopa, and Norman Wolmark

See accompanying editorial on page 1

From the National Surgical AdjuvantBreast and Bowel Project; GraduateSchool of Public Health, University ofPittsburgh; and Allegheny GeneralHospital, Pittsburgh, PA; University ofFlorida, Gainesville; and Florida CancerSpecialists–Sarasota, FL; Helen F.Graham Cancer Center at ChristianaCare, Newark, DE; Southeast CancerControl Consortium–Community ClinicalOncology Program (CCOP), Goldsboro,NC; Atlanta Cancer Care RegionalCCOP, Atlanta, GA; Kaiser PermanenteNorthern California, Vallejo, CA; Univer-sity of North Carolina at Chapel Hill,Chapel Hill, NC; All-Ireland CooperativeOncology Research Group, Dublin,Ireland; and Kaiser PermanenteMidtown I, Denver, CO.

Submitted May 17, 2010; acceptedAugust 25, 2010; published onlineahead of print at www.jco.org onOctober 12, 2010.

Supported by Public Health ServiceGrants No. U10-CA-12027, U10-CA-69651, U10-CA-37377, and U10-CA-69974 from the National CancerInstitute, Department of Health andHuman Services.

Authors’ disclosures of potential con-flicts of interest and author contribu-tions are found at the end of thisarticle.

Clinical Trials repository link available onJCO.org.

Corresponding author: Carmen J.Allegra, MD, University of Florida, Divi-sion of Hematology and Oncology,1600 SW Archer Ave, Rm N-503,Gainesville, FL 32610; e-mail:[email protected].

© 2010 by American Society of ClinicalOncology

0732-183X/11/2901-11/$20.00

DOI: 10.1200/JCO.2010.30.0855

A B S T R A C T

PurposeThe National Surgical Adjuvant Breast and Bowel Project C-08 trial was designed to investigate the safetyand efficacy of adding bevacizumab to modified FOLFOX6 (mFOLFOX6; ie, infusional/bolus fluorouracil,leucovorin, and oxaliplatin) for the adjuvant treatment of patients with stages II to III colon cancer.

MethodsPatients received mFOLFOX6 every 2 weeks for 26 weeks alone or modified as FOLFOX6 �bevacizumab (5 mg/kg every 2 weeks for 52 weeks [ie, experimental group]). The primary endpoint was disease-free survival (DFS).

ResultsAmong 2,672 analyzed patients, demographic factors were well balanced by treatment. With amedian follow-up of 35.6 months, the addition of bevacizumab to mFOLFOX6 did not result in anoverall significant increase in DFS (hazard ratio [HR], 0.89; 95% CI, 0.76 to 1.04; P � .15). The pointestimates for 3-year DFS for the overall population were 77.4% and 75.5% for the experimentaland control arms, respectively. For patients with stages II and III diseases, these same estimateswere 87.4% and 84.7%, respectively, for stage II and 74.2% and 72.4%, respectively, for stageIII. Exploratory analyses found that the effect of bevacizumab on DFS was different before andafter a 15-month landmark (time-by-treatment interaction P value � .0001). Bevacizumab had astrong effect before the landmark (HR, 0.61; 95% CI, 0.48 to 0.78; P � .001) but no significanteffect after (HR, 1.22; 95% CI, 0.98 to 1.52; P � .076).

ConclusionBevacizumab for 1 year with mFOLFOX6 does not significantly prolong DFS in stages II and IIIcolon cancer. However, a significant but transient effect during bevacizumab exposure wasobserved in the experimental arm. We postulate that this observation reflects a biologic effectduring bevacizumab exposure. Given the lack of improvement in DFS, the use of bevacizumabcannot be recommended for use in the adjuvant treatment of patients with colon cancer.

J Clin Oncol 29:11-16. © 2010 by American Society of Clinical Oncology

INTRODUCTION

The state-of-the-art treatment for the manage-ment of patients with stage III colon cancer is tooffer 6 months of adjuvant fluorouracil (FU)-and-oxaliplatin–containing chemotherapy. Twolarge, multinational, randomized studies demon-strated that the addition of oxaliplatin to the combi-nation of FU and leucovorin resulted in significantimprovement in disease-free survival (DFS) whencompared with FU plus leucovorin alone for theadjuvant treatment of patients with stages II and IIIcolon cancer.1,2 The Multicenter InternationalStudy of Oxaliplatin/Fluorouracil/Leucovorin in the

Adjuvant Treatment of Colon Cancer (MOSAIC)trial compared FU, leucovorin, and oxaliplatin(FOLFOX4) with infusion/bolus FU plus leucov-orin, and the National Surgical Adjuvant Breastand Bowel Project (NSABP) C-07 trial evaluatedthe FLOX regimen, which uses oxaliplatin plusbolus FU with leucovorin, in patients with stagesII and III colon cancer. Both trials demonstrated asignificant increase in DFS for patients treatedwith the oxaliplatin-containing regimen.

Bevacizumab, a humanized monoclonal anti-body with a high binding affinity for circulating vas-cular endothelial growth factor A (VEGF-A), hasbeen demonstrated to enhance the response rate,

JOURNAL OF CLINICAL ONCOLOGY O R I G I N A L R E P O R T

VOLUME 29 � NUMBER 1 � JANUARY 1 2011

© 2010 by American Society of Clinical Oncology 11




Treatment

For patients randomly assigned to the control and experimentalarms, 73% versus 77% were able to receive at least 10 of 12 planneddoses of oxaliplatin (P � .028). Similarly, 80% versus 84% (P � .007)were able to receive at least 10 of 12 doses of FU. Fifty-eight percent ofpatients were able to receive at least 21 of 26 planned doses of bevaci-zumab, and the median duration of bevacizumab administration was11.5 months.

DFS

As shown in Figure 2, the addition of bevacizumab tomFOLFOX6 did not result in an overall significant increase in DFS(HR, 0.89; CI, 0.76 to 1.04; P � .15). The point estimates for 3-yearDFS were 77.4% and 75.5% for the experimental and control arms,respectively. The effect of bevacizumab treatment did not vary by stage(interaction P � .68). For patients with stage II disease, the 3-year DFSrates were 87.4% and 84.7% for the experimental and control arms,respectively (HR, 0.82; CI, 0.54 to 1.25; P� .35); for patients with stageIII disease, the rates were 74.2% and 72.4%, respectively (HR, 0.90; CI,0.76 to 1.08; P � .25). Tests for a potential interaction of the effect ofbevacizumab with sex, age, and nodal status were not statisticallysignificant, which suggests that no group benefits more than anotherwith the addition of bevacizumab.

Figure 3 gives the DFS hazard rates over time for each arm. Thesecurves clearly illustrate a lower event rate in the bevacizumab-containing arm during the first 15 months. When the HR over time isexamined, as shown in Figure 4, the ratio favors the bevacizumab arm

0

Treatment Patients Events HR (95% CI) P mFF6 1,338 312 ReferencemFF6 + Bev 1,334 291 0.89 (0.76 to 1.04) .15

No. at riskmFF6 1,338 1,282 1,175 1,096 1,009 783 465 204mFF6 + Bev 1,334 1,295 1,236 1,154 1,055 812 501 219

Aliv

e an

d Di

seas

e Fr

ee (%

)

Time Since Random Allocation (months)

100

80

60

40

20

6 1812 24 30 4236

Fig 2. Disease-free survival (DFS). Overall DFS for patients treated with mFF6(mFOLFOX6) alone for 6 months or mFF6 for 6 months plus bevacizumab for 12months (mFF6 � Bev). HR, hazard ratio.

0

DFS

Haza

rd R

ate

(eve

nts

per p

atie

nt-m

onth

)


0.009

0.006

0.003

6 1812 24 30 4236

mFF6mFF6 + Bev

Fig 3. Disease-free survival (DFS) hazard rates. The DFS hazard rates over timeare illustrated for patients treated with either mFF6 (mFOLFOX6) alone or mFF6plus bevacizumab (mFF6 � Bev).

Table 1. Demographic and Clinical Characteristics of Analyzed Patients

Characteristic

mFF6(n � 1,338)

mFF6 � Bev(n � 1,334)

Total(N � 2,672)

No. % No. % No. %

Age, years� 50 332 24.8 342 25.6 674 25.250-59 447 33.4 435 32.6 882 33.060-69 347 25.9 367 27.5 714 26.7� 70 212 15.8 190 14.2 402 15.0

SexFemale 673 50.3 668 50.1 1,341 50.2Male 665 49.7 666 49.9 1,331 49.8

EthnicityWhite 1,172 87.6 1,160 87.0 2,332 87.3Black 101 7.6 114 8.6 215 8.0Other 50 3.7 43 3.2 93 3.5Multiracial 2 0.2 1 0.1 3 0.1Unknown 13 1.0 16 1.2 29 1.1

ECOG performance0 � fully active 1,089 81.4 1,075 80.6 2,164 81.01 � no strenuous activity 249 18.6 259 19.4 508 19.0

Nodal stageN0 � node negative 332 24.8 334 25.0 666 24.9N1 � 1-3 positive nodes 611 45.7 607 45.5 1,218 45.6N2 � � 4 positive nodes 395 29.5 393 29.5 788 29.5

Abbreviations: mFF6, modified FOLFOX6 (ie, infusional/bolus fluorouracil,leucovorin, and oxaliplatin); Bev, bevacizumab; ECOG, Eastern CooperativeOncology Group.

DFS log (hazard ratio)95% simultaneous conf. band

DFS

Log

(haz

ard

ratio

)


1.2

0.8

0.4

0

-0.4

-0.8

-1.260 12 18 24 30 36

Fig 4. Disease-free survival (DFS) log (hazard ratio, HR). The smooth estimatesof the logarithm of the DFS HR over time along with the 95% simultaneousconfidence intervals. Values less than zero favor the bevacizumab (Bev)-containing arm.

Efficacy Report of NSABP C-08

www.jco.org © 2010 by American Society of Clinical Oncology 13




Treatment


DFS



0



Aliv

e an

d Di

seas

e Fr

ee (%

)


100

80

60

40

20

6 1812 24 30 4236


0

DFS

Haza

rd R

ate

(eve

nts

per p

atie

nt-m

onth

)


0.009

0.006

0.003

6 1812 24 30 4236

mFF6mFF6 + Bev



Characteristic

mFF6(n � 1,338)

mFF6 � Bev(n � 1,334)

Total(N � 2,672)

No. % No. % No. %

Age, years� 50 332 24.8 342 25.6 674 25.250-59 447 33.4 435 32.6 882 33.060-69 347 25.9 367 27.5 714 26.7� 70 212 15.8 190 14.2 402 15.0

SexFemale 673 50.3 668 50.1 1,341 50.2Male 665 49.7 666 49.9 1,331 49.8






DFS

Log

(haz

ard

ratio

)


1.2

0.8

0.4

0

-0.4

-0.8

-1.260 12 18 24 30 36







during therapy and at up to 3 months after completion of bevaci-zumab. Importantly, the simultaneous confidence band for the log ofthe HR excludes zero (no difference) from 5 to 11.5 months, whichindicates a statistically significant improvement in DFS for bevaci-zumab during this interval.

The apparent time-varying treatment effect observed in Figures 2through 4 prompted an exploratory analysis to examine this apparentinitial effect of bevacizumab. We used the 15-month landmark todichotomize time, because the maximum separation of the DFSKaplan-Meier estimates is near 15 months (Fig 2); the hazard rateestimates cross near 15 months (Fig 3); and the log HR estimatecrosses zero near 15 months (Fig 4). We identified a highly significantinteraction between time and treatment on DFS (P � .0001), whichsuggests the effect of bevacizumab on DFS is different before the15-month landmark (ie, period of bevacizumab therapy plus wash-out time) than after the landmark. The HR before the landmarkstrongly favored bevacizumab (HR, 0.61; CI 0.48 to 0.78; P � .001),whereas this benefit was entirely lost subsequently (HR, 1.22; CI, 0.98to 1.52; P � .076; Figs 5A and 5B).

Potential Rebound Effect

Two hundred forty-eight patients experienced recurrence in thecontrol arm versus 227 in the experimental arm; 146 died in thecontrol arm versus 132 in the experimental arm; and 46 in the controlarm versus 47 in the experimental arm had a second malignancy.None of these outcomes differed significantly between the arms. Toadditionally evaluate the possibility that anti-VEGF exposure resultedin the development of a more aggressive phenotype, as suggested byseveral preclinical studies,16-20 we compared survival after recurrencefor patients with recurrent disease. We found that 41% of patients inthe control arm and 37% of patients in the experimental arm werealive 2 years after recurrence; this difference was not statistically signif-icant. The proportion of patients with more than a single site ofrecurrence and the organ distribution of recurrence were also similarbetween the arms.

Imaging Follow-Up

Because differences in the timing of follow-up imaging couldimpact the observed differences in DFS, we investigated the timing ofimaging in each arm of the study. Because the timing of imagingstudieswasnotspecifiedinthestudyprotocol,weretrospectivelycollectedthe date of first diagnostic imaging for 2,219 (81.9%) of 2,710 patients;1,795ofthesepatientsdidnothavearecurrenceorsecondprimarycancerdiagnosed during the first 2.5 years on study and thus are useful forestimating time to first imaging. These 1,795 patients have similar patientcharacteristics to the corresponding subset of the overall trial.

The overall time to imaging did not differ significantly betweenthe two arms by the log-rank test (P � .15), and an HR from a Coxmodel did not exclude 1.0 (HR, 0.93; CI, 0.83 to 1.03; Appendix FigA1, online only). However, the median time to first imaging was 10.75months (CI, 10.06 to 11.54 months) in the control arm, and 12.03months (CI, 11.64 to 12.43 months) in the experimental arm—asignificant difference of 1.28 months (P � .05). Many of the 52% ofpatients with first imaging in the interval of 6 to 22 months may havehad delayed imaging in the experimental arm. The difference in areasunder the Kaplan-Meier curves (shaded region) gives an estimate of1.09 months for the average delay in imaging in the experimental armfor these patients and, therefore, serves as an estimate of the possible biasintroduced into the DFS end point by differential timing of imaging.Although some proportion of patients would have been diagnosed in anunbiased fashion, we assume all patients in the experimental arm withrecurrence or second primary cancer have event times biased by 1.09months, an assumption that may overcorrect for the true bias.

Appendix Figure A2 (online only) presents the overall bias-adjusted results for DFS. The HR (0.90; CI, 0.77 to 1.06) and P value(.19) are similar to the unadjusted results given in Figure 2. Thetime-by-treatment interaction on DFS at the 15-month landmark,illustrated in Appendix Figures A3A and A3B (online only), remainssignificant after bias adjustment (P � .0066), as does the result beforethe landmark strongly favoring bevacizumab (HR, 0.71; CI, 0.56 to0.89; P � .0033).

A B

0

Treatment Patients EventsmFF6 1,339 170mFF6 + Bev 1,334 108

HR (95% CI), 0.61 (0.48 to 0.78)P < .0001

Treatment Patients EventsmFF6 1,124 142mFF6 + Bev 1,199 183

HR (95% CI), 1.22 (0.98 to 1.52)P = .076

Aliv

e an

d Di

seas

e Fr

ee (%

)


100

80

60

40

20

6 12 15 18 24 30 36 42

Time-treatment interaction P < .0001

mFF6mFF6 + Bev

mFF6mFF6 + Bev Fig 5. Disease-free survival (DFS) time

by treatment interaction. (A) DFS beforethe 15-month landmark. The DFS duringthis initial period strongly and significantlyfavored the bevacizumab (Bev) arm (haz-ard ratio [HR], 0.61). (B) DFS subsequentto the 15-month landmark. The differencebetween the curves during this time wasnot significant (HR, 1.22). The time-by-treatment interaction using the 15-monthlandmark is highly significant (P � .0001).

Allegra et al

14 © 2010 by American Society of Clinical Oncology JOURNAL OF CLINICAL ONCOLOGY




Limited (D1) vs. extended (D2) lymph node

dissection for gastric cancer

STATISTICS IN MEDICINEStatist. Med. 2005; 24:2807–2821Published online in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/sim.2143

Long-term survival with non-proportional hazards: results fromthe Dutch Gastric Cancer Trial

H. Putter1;∗;†, M. Sasako2, H. H. Hartgrink3, C. J. H. van de Velde3

and J. C. van Houwelingen1

1Department of Medical Statistics and Bioinformatics; Leiden University Medical Centre;University of Leiden; The Netherlands

2Department of Surgery; National Cancer Centre Hospital; Tokyo; Japan3Department of Surgery; Leiden University Medical Centre; University of Leiden; The Netherlands

SUMMARY

Randomized clinical trials with long-term survival data comparing two treatments often showKaplan–Meier plots with crossing survival curves. Such behaviour implies a violation of the proportionalhazards assumption for treatment. The Cox proportional hazards regression model with treatment as a�xed e�ect can therefore not be used to assess the in�uence of treatment of survival. In this paper weanalyse long-term follow-up data from the Dutch Gastric Cancer Trial, a randomized study comparinglimited (D1) lymph node dissection with extended (D2) lymph node dissection. We illustrate a numberof ways of dealing with survival data that do not obey the proportional hazards assumption, each ofwhich can be easily implemented in standard statistical packages. Copyright ? 2005 John Wiley &Sons, Ltd.

KEY WORDS: long-term survival; non-proportional hazards; time-dependent covariate e�ects

1. INTRODUCTION

Many randomized clinical trials in oncology concern long-term survival data, comparing anexperimental treatment with a standard treatment or control. To test for equality of the survivalrates of the treatments, the log-rank test is used [1]. Often in these trials, characteristics ofthe patient and of the tumour that are known before treatment are also recorded. The Coxproportional hazards regression model is the most popular choice to study the e�ect of thoseprognostic factors on survival [2]. One of the assumptions underlying the Cox regressionmodel is the assumption of proportional hazards, meaning that the ratio of the hazard ratesfor di�erent levels of the prognostic factor or for treatment versus control is constant over

∗Correspondence to: Hein Putter, Department of Medical Statistics and Bioinformatics, Leiden University MedicalCentre, University of Leiden, P.O. Box 9604, Leiden, 2300 RC, The Netherlands.

†E-mail: [email protected]

Received 3 June 2004Copyright ? 2005 John Wiley & Sons, Ltd. Accepted 9 December 2004


2810 H. PUTTER ET AL.

Months since surgery

Sur

viva

l

0.0

0 24 48 72 96 120

0.2

0.4

0.6

0.8

1.0D1D2

Figure 1. Kaplan–Meier plots of the survival curves for D1- and D2-dissection. Thesurvival curves cross after 53 months.

The Cox regression with only randomization as a time-�xed e�ect gives an estimated hazardratio of 0.97 of D2 dissection compared to D1-dissection, with a p-value of 0.73. The survivalcurves resulting from this univariate Cox regression are depicted in Figure 2. The higherpost-operative mortality in the D2 group is not visible from this plot, nor is the crossingof the survival curves, so clearly Figure 2 does not give a realistic picture of the e�ect oftreatment.One way of studying how the e�ect of treatment changes over time is by using the life-

table method. This method was used by epidemiologists long before the Cox regression modelbecame popular. Divide time into a number of disjoint intervals I1; : : : ; Im. The hazard hk ofdying in interval Ik is then given by the number of deaths in that interval (dk) divided bythe number of person years in that interval (yk). The number of person years is the sum overall patients still alive at the beginning of the interval (at risk) of the number of years aliveduring that interval. The standard error of hk , based on a Poisson approximation, is

√dk=yk .

If hk1 and hk2 denote the estimated hazards at Ik for D1 and D2, respectively, and dk1 anddk2 the number of deaths at Ik for D1 and D2, respectively, then the delta-method impliesthat

se2 log(hk1hk2

)≈ se

2(hk1)h2k1

+se2(hk2)h2k2

=1dk1

+1dk2

The left plot of Figure 3 shows the estimated hazards on a yearly basis using the life-tablemethod for each of the treatment groups separately. The plot on the right shows the resultinghazard ratio and associated error bars. The initial advantage and subsequent disadvantage of

Copyright ? 2005 John Wiley & Sons, Ltd. Statist. Med. 2005; 24:2807–2821


LONG-TERM SURVIVAL WITH NON-PROPORTIONAL HAZARDS 2813

Months since surgery

0 24 48 72 96 120

Haz

ard

ratio

0.5

1.0

1.5

2.0

2.5

3.0

Figure 4. The estimated hazard ratio with 95 per cent con�dence intervals basedon Cox regression with treatment as time-dependent e�ect. A hazard ratio of one

indicates equality of the hazard rates of D1 and D2.

Standard statistical packages like SPSS, SAS and S-plus are able to perform Cox regressionwith time-dependent covariates (although for S-plus and R the original data needs to beexpanded), but most of them do not return the baseline hazard functions automatically in thepresence of time-dependent covariates. The survival library in S-plus and R contains a functionbasehaz() to obtain an estimate of the baseline hazard. To show how this is done, we focuson the situation of a single covariate Z given by two values, 0 and 1. The time-dependenttreatment e�ect is modelled by a function f(t). The Cox proportional hazards model statesthat the hazard rate of an individual with covariate Z is given by

h(t)= h0(t) exp(�FZ + �TZf(t)) (1)

where �F and �T denote the �xed and time-dependent regression coe�cients, respectively.Here h0 is the baseline hazard corresponding to Z =0, and if we denote the hazard functioncorresponding to Z =1 by h1, then this means that h1(t)= h0(t) exp(�F+�Tf(t)) and exp(�F+�Tf(t)) is the hazard ratio varying over time. The regression coe�cients are estimated by anextension of the well known partial likelihood (see e.g. Section 9.2 of Klein and Moeschberger[3]). With estimated regression coe�cients �F and �T obtained in this way, the baselinecumulative hazard rate H0(t) is estimated by Breslow’s estimator, given by

H 0(t)=∑

ti6t; ti∈D

1∑

j∈R(ti) exp(�FZj + �TZjf(tj))(2)

Copyright ? 2005 John Wiley & Sons, Ltd. Statist. Med. 2005; 24:2807–2821


Causal graphs (for a generic problem)

ZX Y

U

Assume our knowledge about the associations of interest in ourgeneric problem can be summarised in this causal graph.

If we wish to estimate the direct effect of X on Y then we shouldnot condition on Z (since it opens a backdoor path between Xand Y via U).


Illustration of built-in selection bias in the HR

Y1X Y2

U

Y1 is the outcome at one point in the follow-up and Y2 is theoutcome at a later point in the follow-up. U represents theunmeasured susceptibility.

The estimated HR at time 2 is conditional on being alive attime 1 so does estimate the direct effect of X on Y2.

See Appendix Figure 1 in Hernan et al. (2004) [3].


Hazard plots are informative (Johansson 2011)

More difficult to see this pattern on survival curves.

Hazards are what we model.Paul Dickman Hazards of Hazard Ratios Sodersjukhuset16 May 2016 24

Hazard plots are potentially informative

Breast Cancer Adjuvant Therapy: Time to ConsiderIts Time-Dependent EffectsIsmail Jatoi, University of Texas Health Science Center, San Antonio, TXWilliam F. Anderson, National Cancer Institute, Bethesda, MDJong-Hyeon Jeong and Carol K. Redmond, University of Pittsburgh, Pittsburgh, PA

Breast cancer is a chronic and heterogeneous disease that mayrecur many years after initial diagnosis and treatment.1 This has im-portant implications for the practicing oncologist. For instance, anearly effect of adjuvant treatment may diminish over time after cessa-tion of therapy, or, alternatively, there may exist a lag time before sometreatment effects become pronounced. Indeed, the risk of breast can-cer recurrence and death (hazard rate) varies over time (ie, is nonpro-portional) according to prognostic and predictive factors (Figs 1 and 2;Table 1).6,13 The hazard curve for breast cancer death peaks between 2and 3 years after initial diagnosis and then declines sharply, suggestingthat the biologic mechanisms responsible for early and late cancer-specific events are fundamentally different. Thus the early and lateeffects of adjuvant therapy may vary accordingly.

For example, Figure 1 shows the annual hazard rates for breastcancer deaths (percent per year) after initial diagnosis among womenin the National Cancer Institute’s Surveillance, Epidemiology, andEnd Results 13 Registries database.2 The average annual rate of breastcancer deaths is nonproportional overall and by estrogen receptor(ER) expression.14 Thus the annual hazard rate for all cases peaks near3% per year between the second and third years after diagnosis andthen declines to 1% to 2% per year by the sixth through eighth years.The hazard rates for ER-negative and ER-positive tumors peak atapproximately 6.5% and 2% per year, respectively, between the firstand third years (ie, � three-fold difference). Notably, ER-negative toER-positive hazard rates cross between the seventh and eighth years,after which women with ER-negative tumors have a lower rate ofbreast cancer death than those with ER-positive tumors. Table 1 fur-ther shows the fold difference for ER-negative compared with ER-positive tumors over time. ER-negative to ER-positive hazard ratios(HR) were more than 1.0 before the eighth year, after which HRs wereless than 1.0.

Similar nonproportional hazard rates are evident for largeversus small tumors, positive versus negative lymph nodes, highversus low tumor grade,13 the intrinsic molecular breast cancersubtypes,6,8 and the molecular prognostic signatures OncotypeDX12 and Mammaprint9-11 (Fig 2). Thus hazard rates for relapseamong high-risk tumors (eg, nonluminal A, Mammaprint poor sig-nature, and Oncotype high-risk score) show a sharp peak soon afterinitial diagnosis, similar to ER-negative cancers (Fig 1). Conversely,hazard rates for low-risk tumors (eg, luminal A, Mammaprint goodsignature, and Oncotype low- and intermediate-risk score) lack a

sharp peak, similar to ER-positive tumors. These hazard curves sug-gest that the biologic mechanisms responsible for early and late breastcancer events differ and may therefore respond differently to thesame treatment.

0

Annu

al H

azar

d Ra

te fo

r Bre

ast

Canc

er D

eath

(%)

Time After Initial Breast CancerDiagnosis (years)

7

6

4

5

2

3

1

2 4 6 8 10 12

All cases

ER negative

ER positive

Fig 1. Annual hazard rates for breast cancer death and ER-negative to ER-positive hazard ratios (Table 1) using the National Cancer Institute’s Surveillance,Epidemiology, and End Results 13 Registries Databases (1992 to 2007) forinvasive female breast cancer.2 Annual hazard rates for breast cancer deathoverall (all cases combined, n � 401,693), estrogen receptor (ER) –negative(n � 74,567), and ER-positive (n � 257,426) breast cancers. The annual hazardrate for cancer-specific death describes the instantaneous rate of dying fromcancer in a specified time interval after initial cancer diagnosis. Hazard rate curveswere modeled using cubic splines with join-points selected by Akaike’s informa-tion criteria3,4; 95% CIs were applied with bootstrap resampling techniques.5

Under the null hypothesis of no interaction over time, annual hazard rates forER-positive and ER-negative breast cancers would be proportional (or similar)with follow-up after initial breast cancer diagnosis. The overall rate of breastcancer death for all cases peaks near 3% per year between the second and thirdyears after initial breast cancer diagnosis and then declines to 1% to 2% per yearby the sixth through eighth years. The annual hazard rates for women withER-negative and ER-positive tumors demonstrate peaks of approximately 6.5%and 2% near the first through third years after initial breast cancer diagnosis,respectively (� three-fold difference). An ER-negative to ER-positive hazard ratecross-over occurs between the seventh and eighth years after breast cancerdiagnosis, and then women with ER-negative tumors had a somewhatparadoxically lower rate of breast cancer death than those with ER-positivebreast cancers.

JOURNAL OF CLINICAL ONCOLOGY COMMENTS AND CONTROVERSIES

VOLUME 29 � NUMBER 17 � JUNE 10 2011

© 2011 by American Society of Clinical Oncology 2301Journal of Clinical Oncology, Vol 29, No 17 (June 10), 2011: pp 2301-2304

Downloaded from jco.ascopubs.org on June 21, 2011. For personal use only. No other uses without permission.Copyright © 2011 American Society of Clinical Oncology. All rights reserved.


Adjusted survival curves

Hernan (2010) [1] suggests estimating survival curves adjustedfor baseline covariates.

These are extremely easy easy to obtain after fitting a flexibleparametric model in Stata using stpm2. I’ll show an exampleshortly but first a general comment.

There is not a single definition of ‘average survival curve’.

Approaches include using the mean value of all covariates [4],the mean of all predicted survival curves [5] and inverseprobability weighting [6].

The first is more common in standard software; Hernan isreferring to the ‘the mean of all predicted survival curves’.


Adjusted survival curves: mean covariate method

Most software (e.g., stcurve) uses the mean covariate method.

This gives the survival for an individual who happens to have themean value of all covariates. For example, for a Cox model themean survival is,

Sind(t) = exp (−H0(t) exp (β1x1 + β2x2))

This is the survival of an ‘average’ individual, who happens tohave the average values of all covariates.

Problem with categorical covariates. May be for someone with aproportion of each stage and who is 50% male.


Adjusted survival curves:

average of individual predictions

The predicted survival for individual i is

Si(t) = exp (−H0(t) exp (β1x1i + β2x2i))

We then average over all predicted survival curves

SP(t) =1

N

N∑

i=1

Si(t)

The model can be as complex as required (continuous covariates,interactions, non-linear functions, non-proportional hazards).

SP(t) will be smaller than Sind(t).

Note that we are predicting a curve, not S(t) evaluated at asingle time point.


Standardized survival curves

When interest lies in comparing the survival of (two) exposuregroups we need to standardize to the same covariate distribution.

Let X be the exposure of interest.

Let Z denote the set of measured covariates.

SP(t|X = x ,Z ) =1

N

N∑

i=1

Si (t|X = x ,Z )

Note that the average is over the marginal distribution of Z , notover the conditional distribution of Z among those with X = x .

This is needed to compare like-with-like.

We are forcing the same covariate distribution on both exposuregroups.


Average and adjusted survival in stpm2

In stpm2, the meansurv option to predict produces an averageof predicted survival curves for each observation.

Works for continuous covariates, time-dependent effects, etc.

For adjusted survival curves we force the distribution of one ormore covariates (e.g., age) to be the same when comparinggroups of interest.

When comparing calendar periods we can predict survival in onecalendar period assuming it has the age distribution of another(reference) calendar period.

R users can use the stdReg package (Arvid Sjolander).


Renal dialysis example

252 patients entering a renal dialysis program in Leicestershire,England 1982-1991 with follow-up to the end of 1994.

Interest in difference in survival by ethnicity (Non-South Asian vsSouth Asian).

At the time of the study approximately 25% of population inLeicester of South Asian origin (currently around 36%)


Kaplan-Meier Curves - Renal Replacement Therapy

Unadjusted HR = 0.62 (0.41, 0.94)Age adjusted HR = 1.14 (0.73, 1.79)

Mean Age = 62.9Mean Age = 55.5

0.0

0.2

0.4

0.6

0.8

1.0S

urvi

val F

unct

ion

0 2 4 6 8Survival Time (years)

Non−AsianAsian


Predictions for Standardised Survival Curves

The meansurv optionstpm2 asian age, df(3) scale(hazard)

/* Age distribution for study population as a whole */

predict meansurv pop0, meansurv at(asian 0)

predict meansurv pop1, meansurv at(asian 1)

/* Age distribution for non-asians */

predict meansurv pop0b if asian == 0, meansurv at(asian 0)

predict meansurv pop1b if asian == 0, meansurv at(asian 1)

/* Age distribution for asians */

predict meansurv pop0c if asian == 1, meansurv at(asian 0)

predict meansurv pop1c if asian == 1, meansurv at(asian 1)

Survival curve calculated for each subject in the studypopulation and then averaged.The adjusted curves show the survival we would expect to see inboth groups if each had the age distribution of the studypopulation as a whole.In large studies use the timevar() option to predict survival foreach individual at fewer time points.


Adjusted Survival Curve 1

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1.0

Sur

viva

l Fun

ctio

n


Non−AsianAsian

Age Distribution in Whole Study Population



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Sur

viva

l Fun

ctio

n


Non−AsianAsian

Age Distribution in Non−Asians



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1.0

Sur

viva

l Fun

ctio

n


Non−AsianAsian

Age distribution in Asians


Difference in adjusted survival curves

We can quantify the difference between survival curves.

To obtain confidence intervals we make use of the delta method.Stata has a very powerful inbuilt command, predictnl, thatimplements the delta method on functions of model parameters.

To obtain a difference in adjusted survival curves use thefollowing.

Using predictnl

predictnl sdiff = predict(meansurv at(asian 1)) - ///predict(meansurv at(asian 0)), ci(sdiff lci sdiff uci)


References

[1] Hernan MA. The hazards of hazard ratios. Epidemiology 2010;21:13–15.

[2] Manson JE, Hsia J, Johnson KC, Rossouw JE, Assaf AR, Lasser NL, et al.. Estrogen plusprogestin and the risk of coronary heart disease. N Engl J Med 2003;349:523–534.

[3] Hernan MA, Hernandez-Dıaz S, Robins JM. A structural approach to selection bias.Epidemiology 2004;15:615–625.

[4] Cupples LA, Gagnon DR, Ramaswamy R, D’Agostino RB. Age-adjusted survival curveswith application in the Framingham study. Statistics in Medicine 1995;14:1731–1744.

[5] Nieto FJ, Coresh J. Adjusting survival curves for confounders: a review and a new method.American Journal of Epidemiology 1996;143:1059–1068.

[6] Cole SR, Hernan MA. Adjusted survival curves with inverse probability weights. ComputMethods Programs Biomed 2004;75:45–49.


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