Introduction to Survival Analysis October 19, 2004

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Introduction to Survival AnalysisOctober 19,
2004

• Brian F. Gage, MD, MSc
• with thanks to Bing Ho, MD, MPH
• Division of General Medical Sciences

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Presentation goals

• Survival analysis compared w/ other regression
  techniques
• What is survival analysis
• When to use survival analysis
• Univariate method Kaplan-Meier curves
• Multivariate methods
• Cox-proportional hazards model
• Parametric models
• Assessment of adequacy of analysis
• Examples

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Regression vs. Survival Analysis
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Regression vs. Survival Analysis
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What is survival analysis?

• Model time to failure or time to event
• Unlike linear regression, survival analysis has a
  dichotomous (binary) outcome
• Unlike logistic regression, survival analysis
  analyzes the time to an event
• Why is that important?
• Able to account for censoring
• Can compare survival between 2 groups
• Assess relationship between covariates and
  survival time

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Importance of censored data

• Why is censored data important?
• What is the key assumption of censoring?

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Types of censoring

• Subject does not experience event of interest
• Incomplete follow-up
• Lost to follow-up
• Withdraws from study
• Dies (if not being studied)
• Left or right censored

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When to use survival analysis

• Examples
• Time to death or clinical endpoint
• Time in remission after treatment of disease
• Recidivism rate after addiction treatment
• When one believes that 1 explanatory variable(s)
  explains the differences in time to an event
• Especially when follow-up is incomplete or
  variable

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Relationship between survivor function and hazard
function

• Survivor function, S(t) defines the probability
  of surviving longer than time t
• this is what the Kaplan-Meier curves show.
• Hazard function is the derivative of the survivor
  function over time h(t)dS(t)/dt
• instantaneous risk of event at time t
  (conditional failure rate)
• Survivor and hazard functions can be converted
  into each other

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Approach to survival analysis

• Like other statistics we have studied we can do
  any of the following w/ survival analysis
• Descriptive statistics
• Univariate statistics
• Multivariate statistics

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Descriptive statistics

• Average survival
• When can this be calculated?
• What test would you use to compare average
  survival between 2 cohorts?
• Average hazard rate
• Total of failures divided by observed survival
  time (units are therefore 1/t or 1/pt-yrs)
• An incidence rate, with a higher values
  indicating more events per time

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Univariate method Kaplan-Meier survival curves

• Also known as product-limit formula
• Accounts for censoring
• Generates the characteristic stair step
  survival curves
• Does not account for confounding or effect
  modification by other covariates
• When is that a problem?
• When is that OK?

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Time to Cardiovascular Adverse Event in VIGOR
Trial
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Comparing Kaplan-Meier curves

• Log-rank test can be used to compare survival
  curves
• Less-commonly used test Wilcoxon, which places
  greater weights on events near time 0.
• Hypothesis test (test of significance)
• H0 the curves are statistically the same
• H1 the curves are statistically different
• Compares observed to expected cell counts
• Test statistic which is compared to ?2
  distribution

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Comparing multiple Kaplan-Meier curves

• Multiple pair-wise comparisons produce cumulative
  Type I error multiple comparison problem
• Instead, compare all curves at once
• analogous to using ANOVA to compare gt 2 cohorts
• Then use judicious pair-wise testing

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Limit of Kaplan-Meier curves

• What happens when you have several covariates
  that you believe contribute to survival?
• Example
• Smoking, hyperlipidemia, diabetes, hypertension,
  contribute to time to myocardial infarct
• Can use stratified K-M curves for 2 or maybe 3
  covariates
• Need another approach multivariate Cox
  proportional hazards model is most common -- for
  many covariates
• (think multivariate regression or logistic
  regression rather than a Students t-test or the
  odds ratio from a 2 x 2 table)

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Multivariate method Cox proportional hazards

• Needed to assess effect of multiple covariates on
  survival
• Cox-proportional hazards is the most commonly
  used multivariate survival method
• Easy to implement in SPSS, Stata, or SAS
• Parametric approaches are an alternative, but
  they require stronger assumptions about h(t).

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Cox proportional hazard model

• Works with hazard model
• Conveniently separates baseline hazard function
  from covariates
• Baseline hazard function over time
• h(t) ho(t)exp(B1XBo)
• Covariates are time independent
• B1 is used to calculate the hazard ratio, which
  is similar to the relative risk
• Nonparametric
• Quasi-likelihood function

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Cox proportional hazards model, continued

• Can handle both continuous and categorical
  predictor variables (think logistic, linear
  regression)
• Without knowing baseline hazard ho(t), can still
  calculate coefficients for each covariate, and
  therefore hazard ratio
• Assumes multiplicative riskthis is the
  proportional hazard assumption
• Can be compensated in part with interaction terms

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Limitations of Cox PH model

• Does not accommodate variables that change over
  time
• Luckily most variables (e.g. gender, ethnicity,
  or congenital condition) are constant
• If necessary, one can program time-dependent
  variables
• When might you want this?
• Baseline hazard function, ho(t), is never
  specified
• You can estimate ho(t) accurately if you need to
  estimate S(t).

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Hazard ratio

• What is the hazard ratio and how to you calculate
  it from your parameters, ß
• How do we estimate the relative risk from the
  hazard ratio (HR)?
• How do you determine significance of the hazard
  ratios (HRs).
• Confidence intervals
• Chi square test

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Assessing model adequacy

• Multiplicative assumption
• Proportional assumption covariates are
  independent with respect to time and their
  hazards are constant over time
• Three general ways to examine model adequacy
• Graphically
• Mathematically
• Computationally Time-dependent variables
  (extended model)

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Model adequacy graphical approaches

• Several graphical approaches
• Do the survival curves intersect?
• Log-minus-log plots
• Observed vs. expected plots

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Testing model adequacy mathematically with a
goodness-of-fit test

• Uses a test of significance (hypothesis test)
• One-degree of freedom chi-square distribution
• p value for each coefficient
• Does not discriminate how a coefficient might
  deviate from the PH assumption

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Example Tumor Extent

• 3000 patients derived from SEER cancer registry
  and Medicare billing information
• Exploring the relationship between tumor extent
  and survival
• Hypothesis is that more extensive tumor
  involvement is related to poorer survival

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Log-Rank ?2 269.0973 p lt.0001
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Example Tumor Extent

• Tumor extent may not be the only covariate that
  affects survival
• Multiple medical comorbidities may be associated
  with poorer outcome
• Ethnic and gender differences may contribute
• Cox proportional hazards model can quantify these
  relationships

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Example Tumor Extent

• Test proportional hazards assumption with
  log-minus-log plot
• Perform Cox PH regression
• Examine significant coefficients and
  corresponding hazard ratios

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Example Tumor Extent 5

• The PHREG
  Procedure
• 
  
• Analysis of Maximum
  Likelihood Estimates
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• Parameter Standard
  Hazard 95 Hazard Ratio Variable
• Variable DF Estimate Error Chi-Square Pr
  gt ChiSq Ratio Confidence Limits Label
• 
  
• age2 1 0.15690 0.05079 9.5430
  0.0020 1.170 1.059 1.292 70ltagelt80
• age3 1 0.58385 0.06746 74.9127
  lt.0001 1.793 1.571 2.046 agegt80
• race2 1 0.16088 0.07953 4.0921
  0.0431 1.175 1.005 1.373 black
• race3 1 0.05060 0.09590 0.2784
  0.5977 1.052 0.872 1.269 other
• comorb1 1 0.27087 0.05678 22.7549
  lt.0001 1.311 1.173 1.465
• comorb2 1 0.32271 0.06341 25.9046
  lt.0001 1.381 1.219 1.564
• comorb3 1 0.61752 0.06768 83.2558
  lt.0001 1.854 1.624 2.117
• DISTANT 1 0.86213 0.07300 139.4874
  lt.0001 2.368 2.052 2.732
• REGIONAL 1 0.51143 0.05016 103.9513
  lt.0001 1.668 1.512 1.840
• LIPORAL 1 0.28228 0.05575 25.6366
  lt.0001 1.326 1.189 1.479
• PHARYNX 1 0.43196 0.05787 55.7206
  lt.0001 1.540 1.375 1.725
• treat3 1 0.07890 0.06423 1.5090
  0.2193 1.082 0.954 1.227 both

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Summary

• Survival analyses quantifies time to a single,
  dichotomous event
• Handles censored data well
• Survival and hazard can be mathematically
  converted to each other
• Kaplan-Meier survival curves can be compared
  statistically and graphically
• Cox proportional hazards models help distinguish
  individual contributions of covariates on
  survival, provided certain assumptions are met.