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Introduction to Survival Analysis October 19, 2004PowerPoint Presentation

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

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

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

Importance of censored data

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

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

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

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

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

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

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?

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

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 > 2 cohorts
- Then use judicious pair-wise testing

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 Student’s t-test or the odds ratio from a 2 x 2 table)

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).

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(B1X+Bo)

- Covariates are time independent
- B1 is used to calculate the hazard ratio, which is similar to the relative risk

- Baseline hazard function over time
- Nonparametric
- Quasi-likelihood function

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 risk—this is the proportional hazard assumption
- Can be compensated in part with interaction terms

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?

- Luckily most variables (e.g. gender, ethnicity, or congenital condition) are constant
- Baseline hazard function, ho(t), is never specified
- You can estimate ho(t) accurately if you need to estimate S(t).

Hazard ratio

- What is the hazard ratio and how to you calculate it from your parameters, β
- 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

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)

Model adequacy: graphical approaches

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

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

Example: Tumor Extent test

- 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

Log-Rank test2 = 269.0973 p <.0001

Example: Tumor Extent test

- 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

Example: Tumor Extent test

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

Example: Tumor Extent 5 test

The PHREG Procedure

Analysis of Maximum Likelihood Estimates

Parameter Standard Hazard 95% Hazard Ratio Variable

Variable DF Estimate Error Chi-Square Pr > ChiSq Ratio Confidence Limits Label

age2 1 0.15690 0.05079 9.5430 0.0020 1.170 1.059 1.292 70<age<=80

age3 1 0.58385 0.06746 74.9127 <.0001 1.793 1.571 2.046 age>80

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 <.0001 1.311 1.173 1.465

comorb2 1 0.32271 0.06341 25.9046 <.0001 1.381 1.219 1.564

comorb3 1 0.61752 0.06768 83.2558 <.0001 1.854 1.624 2.117

DISTANT 1 0.86213 0.07300 139.4874 <.0001 2.368 2.052 2.732

REGIONAL 1 0.51143 0.05016 103.9513 <.0001 1.668 1.512 1.840

LIPORAL 1 0.28228 0.05575 25.6366 <.0001 1.326 1.189 1.479

PHARYNX 1 0.43196 0.05787 55.7206 <.0001 1.540 1.375 1.725

treat3 1 0.07890 0.06423 1.5090 0.2193 1.082 0.954 1.227 both

treat2 1 0.47215 0.06074 60.4215 <.0001 1.603 1.423 1.806 rad

treat0 1 1.52773 0.08031 361.8522 <.0001 4.608 3.937 5.393 none

Summary test

- 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.

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