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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
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
We model the hazard function directly
instead of the survival time since our
interest centers on the risk of failure.
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
The baseline hazard function, h0,
characterizes the dependence of the
hazard rate on time.
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
Semiparametric
Proportional hazard
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Survival Analysis
Interpretation of Regression
Coefficients
Single covariate:
Categorical
Continuous
Multiple covariates
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Survival Analysis
Single Categorical Coefficient
Consider a single dichotomous covariate
x, coded 0 and 1
The hazard ratio, also known as relative
risk, is
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:
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.
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