Analyzing Time-to-Event Analyzing Time-to-Event Data Data

Survival Analysis and Survival Analysis and Cox Proportional Hazards Cox Proportional Hazards

RegressionRegression

Robert Boudreau, PhDRobert Boudreau, PhDCo-Director of Methodology CoreCo-Director of Methodology Core

PITT-Multidisciplinary Clinical Research Center PITT-Multidisciplinary Clinical Research Center for Rheumatic and Musculoskeletal Diseasesfor Rheumatic and Musculoskeletal Diseases

Effect of acyclovir on time to Effect of acyclovir on time to resolution of postherpetic resolution of postherpetic

neuralgianeuralgia

Spruance SL, Reid JE, Grace M, Samore M. Hazard Ratio in Clinical Trials. Antimicrob Agents and Chemotherapy Aug 2004:2787-2792.

Flow chart for regression Flow chart for regression modelsmodels

Outcome variable continuous or dichotomous?

dichotomouscontinuous

Time-to-event available ?

No Yes

Multiple logistic regression

Cox proportionalhazards regression

Predictor variable categorical?

No Yes(e.g. groups)

Multiple linear regression

ANCOVA

Effect of acyclovir on time to Effect of acyclovir on time to resolution of postherpetic resolution of postherpetic

neuralgianeuralgia

Event: Resolution of Herpes Zoster Pain Time-to-event also available

Statistical Modeling Approaches

Logistic Regression: Would do separate rate comparisons at distinct timepoints % with Pain Resolution by 60days, by 120 days …

Cox Proportional Hazards Regression:

Comparison of survival curves across all timepoints

> Uses more information: Event (Yes/No), Time-to-event

> More powerful in identifying systematic differences

ExamplesExamplesCompare Compare MTX+Enbrel vs MTX+HumiraMTX+Enbrel vs MTX+Humira Time until RemissionTime until Remission Time until ACR 20/50/70Time until ACR 20/50/70 Time until DAS drop > 1.2Time until DAS drop > 1.2

Longitudinal cohort study (on aging)Longitudinal cohort study (on aging) Time until participant develops mobility Time until participant develops mobility

limitationlimitation Time until participant has CVD eventTime until participant has CVD event Time until mortality eventTime until mortality event

CensoringCensoring

Generally, three reasons why censoring Generally, three reasons why censoring might occur:might occur:

A subject does not experience the event A subject does not experience the event before the study endsbefore the study ends

A person is lost to follow-up during the A person is lost to follow-up during the study periodstudy period

A person withdraws from the studyA person withdraws from the study

These are all examples of right-censoringThese are all examples of right-censoring

CensoringCensoring

Censored

Non-Events

o

o

Most typical to consider start of time-to-event “clock” as t=0

Life TablesLife Tables

Life TablesLife Tables

Life TablesLife Tables

Censored observations are counted in the denominator of those “at risk” until they are censored

146-30

Life TablesLife Tables

Censored observations are counted in the denominator of those “at risk” until they are censored

146-30

Survival CurveSurvival Curve

Kaplan-Meier Survival Kaplan-Meier Survival Curve Curve

Generalization of Life Table methodGeneralization of Life Table method Assumes (i.e. can handle) continuous Assumes (i.e. can handle) continuous

event timesevent times Updates “at risk” denominator at Updates “at risk” denominator at

each event or censor timepointeach event or censor timepoint

400 meter walk time in 400 meter walk time in elderlyelderly

predicts mobility predicts mobility limitationlimitation Newman AB, Simonsick EM, Naydeck BL, Newman AB, Simonsick EM, Naydeck BL,

Boudreau RM, Kritchevsky SB, Nevitt MC, Pahor Boudreau RM, Kritchevsky SB, Nevitt MC, Pahor M, Satterfield S, Brach JS, Studenski SA, Harris M, Satterfield S, Brach JS, Studenski SA, Harris TB. TB. Association of Long Distance Corridor Association of Long Distance Corridor Walk Performance with Mortality, Walk Performance with Mortality, Cardiovascular Disease, Mobility Limitation, Cardiovascular Disease, Mobility Limitation, and Disabilityand Disability. JAMA 2006;295:2018-2026.. JAMA 2006;295:2018-2026.

EventEvent: Persistent Mobility Limitation: : Persistent Mobility Limitation: 2 consecutive reports (6 month contacts) 2 consecutive reports (6 month contacts)

of having any self-reported difficulty of having any self-reported difficulty walking a quarter of a mile or climbing walking a quarter of a mile or climbing stairsstairs

% of Women With Mobility % of Women With Mobility LimitationLimitation

by Quartile of Baseline 400m by Quartile of Baseline 400m Walk TimeWalk Time

Quartile 1Lowest times(Fastest Pace)

Proportional Hazards Proportional Hazards ModelModel

Example:Example: Compare Treatment to Control Compare Treatment to Control GroupGroup

Dummy variable for group (random) Dummy variable for group (random) assignment:assignment:

Z= 0 if control groupZ= 0 if control group

= 1 if treatment group= 1 if treatment group

Survival Curves Survival Curves (group specific)(group specific) Control

Treatment

Effect of acyclovir on time to Effect of acyclovir on time to resolution of postherpetic resolution of postherpetic

neuralgianeuralgia

Hazard Ratio (HR)Hazard Ratio (HR)

Example:Example: Compare Treatment to Compare Treatment to Control GroupControl Group

Survival Curves (group specific)Survival Curves (group specific) Control

Treatment

HR = (same relationship to regression coeff “beta” as OR in logistic regression)

Cox Proportional Cox Proportional Hazards RegressionHazards Regression

proc phreg data=acyclovir;proc phreg data=acyclovir;

model time*event(0)=drug;model time*event(0)=drug;

run;run;

* event=0 if censored (non-event)* event=0 if censored (non-event)

* =1 if event (resolution of pain)* =1 if event (resolution of pain)

HR = exp( 0.77919) = 2.180 (acyclovir vs placebo)

Cox PH RegressionCox PH RegressionAdjusted for AgeAdjusted for Age

proc phreg data=acyclovir;proc phreg data=acyclovir;

model time*pain_resolved(0)=drug age;model time*pain_resolved(0)=drug age;

run;run;

Adjusted HR = exp( 0.94108) = 2.563 (acyclovir vs placebo)

Age HR=1.096 => Increasing “pain resolve” response with age

400 meter walk time in 400 meter walk time in elderlyelderly

predicts mobility predicts mobility limitationlimitation

Note: “Completed the 400m walk” is the referent group here

400 meter walk time 400 meter walk time (continuous) (continuous) predicts predicts

mortality, CVD and mortality, CVD and mobility limitationmobility limitation Of those who completed 400 meters, Of those who completed 400 meters, each each

additional minute of performance timeadditional minute of performance time was was associated with an adjusted HR of associated with an adjusted HR of

HR= 1.29 (95% CI, 1.12-1.48) for mortality HR= 1.29 (95% CI, 1.12-1.48) for mortality

HR= 1.20 (95% CI, 1.01-1.42) for incident HR= 1.20 (95% CI, 1.01-1.42) for incident

cardiovascular diseasecardiovascular disease

HR= 1.52 (95% CI, 1.41-1.63) for mobility HR= 1.52 (95% CI, 1.41-1.63) for mobility limitationlimitation

400 meter walk time vs 400 meter walk time vs mortalitymortality

(best vs worst quartile) (best vs worst quartile) After adjusting for potential confounders, After adjusting for potential confounders,

those in the poorest quartile of functional those in the poorest quartile of functional capacity capacity

(walk time > 362 seconds) had a higher risk (walk time > 362 seconds) had a higher risk of death of death

over 6 years than those in the best quartile over 6 years than those in the best quartile

(walk time < 290 seconds). (walk time < 290 seconds).

Adjusted HR = 3.23; 95% CI, 2.11-4.94; Adjusted HR = 3.23; 95% CI, 2.11-4.94; P .001).P .001).

Thank you !Thank you !

Any Questions?Any Questions?

Robert Boudreau, PhDRobert Boudreau, PhDCo-Director of Methodology CoreCo-Director of Methodology Core

PITT-Multidisciplinary Clinical Research Center PITT-Multidisciplinary Clinical Research Center for Rheumatic and Musculoskeletal Diseasesfor Rheumatic and Musculoskeletal Diseases