Partial Likelihood as Profile Likelihood
Consider generally with time-dependent covariates:
i(t) = 0(t) expcfw_ 0Zi(t),
where i = 1, ., n denotes the subjects.
The likelihood is
L=
Y
i
i(Xi)i Si(Xi)/Si(Qi),
where Xi is the possibly censored event time, i

Lecture 3
COMPARISON OF SURVIVAL
CURVES
We talked about some nonparametric approaches for esti
mating the survival function, S(t),
over time for a group of
individuals.
Now we want to compare the survival estimates between two
or more groups.
Nonparametri

Lecture 4
PARAMETRIC SURVIVAL MODELS
Some Parametric Survival Distributions (defined on t 0):
The Exponential distribution (1 parameter)
f (t) = et ( > 0)
Z
S(t) =
f (u)du
= e
t
t
f (t)
S(t)
=
constant hazard!
Z t
(u) du
(t) =
(t) =
0
t
Z
=
du
0
= t
C

Lecture 2
ESTIMATING THE SURVIVAL
FUNCTION
One-sample nonparametric methods
There are commonly three methods for estimating a survivorship function
S(t) = P (T > t)
without resorting to parametric models:
(1) Kaplan-Meier
(2) Nelson-Aalen or Fleming-Harr

Biometrika (1981), 68, 2, pp. 373-79
373
Printed in Great Britain
On the regression analysis of multivariate failure time data
BY R. L. PRENTICE
Fred Hitichinson Cancer Research Center, Seattle
B. J. WILLIAMS
Oonzaga University, Spokane, Washington
AND A.

Rong Huang
Zeyun Lu
Yanxing Zhang
Weiwei Li
May 25th, 2016
Outline
Introduction to multivariate survival data
Regression analysis by modeling marginal distributions
Example 1
Example 2
Multivariate Failure Time
Univariate survival data: independent ev

Multivariate Survival Analysis
Previously we have assumed that either (Xi, i) or (Xi, i, Zi),
i = 1, ., n, are i.i.d. This may not always be the case.
Multivariate survival data can arise in practice in difference
ways:
Clustered survival data. This happ

Lecture 8
Stratified Cox Model
So far, weve been considering the following Cox model (with
possibly time-dependent covariates):
(t|Z(t) = 0(t) expcfw_ 0Z(t)
Here the baseline hazard 0(t) is common to all the individuals in a study.
But there are cases whe

Lecture 9
Assessing the Fit of the Cox Model
The Cox (PH) model:
(t|Z(t) = 0(t) expcfw_ 0Z(t)
Assumptions of this model:
(1) the regression effect is constant over time (PH assumption)
(2) linear combination of the covariates (including possibly
higher or

Lecture 5
THE PROPORTIONAL HAZARDS
REGRESSION MODEL
Now we will explore the relationship between survival and
explanatory variables by mostly semiparametric regression
modeling. We will first consider a major class of semiparametric regression models (Cox

Competing Risks in Survival Analysis
So far, weve assumed that there is only one survival endpoint
of interest, and that censoring is independent of the event of
interest.
However, in many contexts it is likely that we can have several different types of

REVIEW
Likelihood Inference
Likelihood functions
Setting: Let Y1, ., Yn be independent random variables,
with Yi having probability (or density) function
f (yi; ),
where is some unknown parameter.
For example, in the Bernoulli distribution, all the Yis

Lecture 7
Time-dependent Covariates in Cox
Regression
So far, weve been considering the following Cox PH model:
(t|Z) = 0(t) exp( 0Z)
X
= 0(t) exp( j Zj )
where j is the parameter for the the j-th covariate (Zj ).
Important features of this model:
(1) the

Journal of the American Statistical Association
ISSN: 0162-1459 (Print) 1537-274X (Online) Journal homepage: http:/www.tandfonline.com/loi/uasa20
Regression Analysis of Multivariate Incomplete
Failure Time Data by Modeling Marginal
Distributions
L. J. Wei

Model Selection in Cox regression
Suppose we have a possibly censored survival outcome that
we want to model as a function of a (possibly large) set of
covariates. How do we decide which covariates to use?
An illustration example:
Survival of Atlantic Hal

Lecture 1
INTRODUCTION TO SURVIVAL
ANALYSIS
Survival Analysis typically focuses on time to event data.
In the most general sense, it consists of techniques for positivevalued random variables, such as
time to death
time to onset (or relapse) of a diseas

The Design of a Survival Study
The design of survival studies are usually based on the logrank test, and sometimes assumes the exponential distribution.
As in standard designs, the power depends on
The Type I error (significance level)
The difference o

Lecture 6
PREDICTING SURVIVAL UNDER
THE PH MODEL
The Cox PH model: (t|Z) = 0(t) exp( 0Z).
How do we estimate the survival probability, Sz (t) = S(t|Z) =
P (T > t|Z), for an individual with covariates Z?
For the baseline (reference) group, we have:
S0(t) =