# Analyzing Time-to-Event Data Cox Proportional Hazards Regression

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Analyzing Time-to-Event Data Cox Proportional Hazards Regression
Survival Analysis and Cox Proportional Hazards Regression Robert Boudreau, PhD Co-Director of Methodology Core PITT-Multidisciplinary Clinical Research Center for Rheumatic and Musculoskeletal Diseases

Effect of acyclovir on time to resolution of postherpetic neuralgia
Spruance SL, Reid JE, Grace M, Samore M. Hazard Ratio in Clinical Trials. Antimicrob Agents and Chemotherapy Aug 2004:

Flow chart for regression models
Outcome variable continuous or dichotomous? continuous dichotomous Predictor variable categorical? Time-to-event available ? No Yes (e.g. groups) No Yes Multiple linear regression ANCOVA Multiple logistic regression Cox proportional hazards regression

Effect of acyclovir on time to resolution of postherpetic neuralgia
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

Examples Compare MTX+Enbrel vs MTX+Humira Time until Remission
Time until ACR 20/50/70 Time until DAS drop > 1.2 Longitudinal cohort study (on aging) Time until participant develops mobility limitation Time until participant has CVD event Time until mortality event

Censoring Generally, three reasons why censoring might occur:
A subject does not experience the event before the study ends A person is lost to follow-up during the study period A person withdraws from the study These are all examples of right-censoring

Censoring  Most typical to consider start of time-to-event “clock” as t=0 Censored Non-Events o o

Life Tables

Life Tables

Life Tables 146-30 Censored observations are counted in the denominator of those “at risk” until they are censored

Life Tables 146-30 Censored observations are counted in the denominator of those “at risk” until they are censored

Survival Curve

Kaplan-Meier Survival Curve
Generalization of Life Table method Assumes (i.e. can handle) continuous event times Updates “at risk” denominator at each event or censor timepoint

400 meter walk time in elderly predicts mobility limitation
Newman AB, Simonsick EM, Naydeck BL, Boudreau RM, Kritchevsky SB, Nevitt MC, Pahor M, Satterfield S, Brach JS, Studenski SA, Harris TB. Association of Long Distance Corridor Walk Performance with Mortality, Cardiovascular Disease, Mobility Limitation, and Disability. JAMA 2006;295: Event: Persistent Mobility Limitation: 2 consecutive reports (6 month contacts) of having any self-reported difficulty walking a quarter of a mile or climbing stairs

% of Women With Mobility Limitation by Quartile of Baseline 400m Walk Time
Lowest times (Fastest Pace)

Proportional Hazards Model
Example: Compare Treatment to Control Group Dummy variable for group (random) assignment: Z= 0 if control group = 1 if treatment group Survival Curves (group specific) Control Treatment

Effect of acyclovir on time to resolution of postherpetic neuralgia

Hazard Ratio (HR) Example: Compare Treatment to Control Group Survival Curves (group specific) Control Treatment HR = (same relationship to regression coeff “beta” as OR in logistic regression)

Cox Proportional Hazards Regression
proc phreg data=acyclovir; model time*event(0)=drug; run; * event=0 if censored (non-event) * =1 if event (resolution of pain) HR = exp( ) = (acyclovir vs placebo)

Cox PH Regression Adjusted for Age
proc phreg data=acyclovir; model time*pain_resolved(0)=drug age; run; Adjusted HR = exp( ) = (acyclovir vs placebo) Age HR= => Increasing “pain resolve” response with age

400 meter walk time in elderly predicts mobility limitation
Note: “Completed the 400m walk” is the referent group here

400 meter walk time (continuous) predicts mortality, CVD and mobility limitation
Of those who completed 400 meters, each additional minute of performance time was associated with an adjusted HR of HR= 1.29 (95% CI, ) for mortality HR= 1.20 (95% CI, ) for incident cardiovascular disease HR= 1.52 (95% CI, ) for mobility limitation

400 meter walk time vs mortality (best vs worst quartile)
After adjusting for potential confounders, those in the poorest quartile of functional capacity (walk time > 362 seconds) had a higher risk of death over 6 years than those in the best quartile (walk time < 290 seconds).  Adjusted HR = 3.23; 95% CI, ; P .001).

Thank you ! Any Questions? Robert Boudreau, PhD
Co-Director of Methodology Core PITT-Multidisciplinary Clinical Research Center for Rheumatic and Musculoskeletal Diseases