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Survival Analysis
Survival Analysis
Semiparametric Proportional
Hazards Regression (Part I)
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Survival Analysis
Abbreviated Outline

The proportional hazards regression model
provides a method for
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Exploring the association of covariates with
failure rates,
Studying the effect of a primary covariate
while adjusting for other variables.
The model is neither fully parametric nor
fully nonparametric.
Inference for the model is based on a
likelihood type function that, in censored
data, only approximates the probability
function of any observed set of values.
Survival Analysis
Identification of Prognostic
Factors
Prognosis: the prediction of the future
of a patient with respect to duration,
course, and outcome of a disease
 Prognosis plays an important role in
medical practice.
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Survival Analysis
Identification of Prognostic
Factors
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Many medical charts contain a large
number of patient characteristics:
Demographic variables,
 Physiological variables,
 Factors associated with lifestyle.
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A compact summary of the data is
useful in revealing the relationship of
these characteristics to prognosis.
Survival Analysis
Example: PBC
Primary biliary cirrhosis (PBC) is a
rare but fatal chronic liver disease.
The bile ducts within the liver become
inflamed and are destroyed.
 A randomized trials conducted by
Mayo Clinic during 1974 to 1984 to
compare the drug D-penicillamine
(DPCA) with a placebo; 312 patients
per group.
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Survival Analysis
Example: PBC
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Data:
Survival Analysis
Regression Modeling for
Survival Data
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As in the linear regression modeling,
regression models for survival data
can be used to examine the
dependence of survival time on other
covariates
Survival Analysis
Regression Modeling for
Survival Data
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We model the hazard function directly
instead of the survival time since our
interest centers on the risk of failure.
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Data:
( zi ,  i , xi ),i  1,...,n.
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Survival Analysis
Cox Regression Model
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h(y|x): the hazard rate at time y for an
individual with risk vector x.
Survival Analysis
Cox Regression Model
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Survival Analysis
Baseline Hazard Function
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The baseline hazard function, h0,
characterizes the dependence of the
hazard rate on time.
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It is the hazard function for an
individual with x = (0,…,0)’.
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Survival Analysis
Prognostic Index
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Survival Analysis
Cox Regression Model
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Semiparametric
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Proportional hazard
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Survival Analysis
Interpretation of Regression
Coefficients
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Single covariate:
 Categorical
 Continuous
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Multiple covariates
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Survival Analysis
Single Categorical Coefficient
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Consider a single dichotomous covariate
x, coded 0 and 1
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The hazard ratio, also known as relative
risk, is
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h( y | 1) / h( y | 0)  e .
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Survival Analysis
Example: PBC
In the DPCA group:
 125 out of 312 patients had died.
 (right-) censored observations:
11 deaths were not attributable to PBC
 8 were lost to follow up
 19 had undergone liver transplantation
 187 were survived at the end of study
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Survival Analysis
Example: PBC
The covariate X is coded as 0 for
DPCA and as 1 for placebo.
 Therefore, the baseline hazard
function h0 is the hazard function for
DPCA
 Interpretation:
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Estimate of e^ is 0.944
 Its 95% C.I. is (0.665, 1.342)
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Survival Analysis
Single Continuous Covariate

The interpretation of the coefficient for
a continuous covariate is similar.
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For a continuous covariate, the
hazard ratio should be reported for a
clinically interesting unit of change.
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Survival Analysis
Example: PBC
Consider X = age (in days)
 is estimated as 1.095x10^-4
 The hazard ratio for a 1-day change in
age
 The hazard ratio for a 5-year change
in age
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Survival Analysis
Multiple Covariates
Similar to the linear regression model,
each coefficient represents the effect
of the corresponding covariate after
adjusting for other covariate effects.
 Such adjustment is called

Analysis of covariance in statistical
applications
 Control of confounding in clinical and
epidemiological investigations
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Survival Analysis
Example: PBC
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Survival Analysis
Example: PBC
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Survival Analysis
Example: PBC
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Survival Analysis