CHAPTER 18
SURVIVAL ANALYSIS
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SURVIVAL ANALYSIS (SA)
 The primary goals of survival analysis are to:
 (1) Estimate and interpret survivor or hazard functions from
survival data
 (2) Assess the impact of explanatory variables on survival time
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Survival analysis goes by various names, such as:
Duration analysis
Event history analysis
Reliability or failure time analysis
Transition analysis
Hazard rate analysis
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TERMINOLOGY OF SURVIVAL ANALYSIS
 Event: “An event consists of some qualitative change that
occurs at a specific point in time….The change must consist
of a relatively sharp disjunction between what precedes and
what follows.”
 Duration Spell: The length of time before an event occurs.
 Discrete Time Analysis: Some events occur only at discrete
times.
 Continuous Time Analysis: Continuous time SA analysis
treats time as continuous.
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CUMULATIVE DISTRIBUTION FUNCTION OF TIME
 If we treat T, the time until an event occurs, as a continuous
variable, the distribution of the T is given by the CDF:
F (t )  Pr(T  t )
which gives the probability that the event has occurred by
duration t.
 If F(t) is differentiable, its density function can be expressed
as:
dF (t )
f (t ) 
 F '(t )
dt
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SURVIVAL AND HAZARD FUNCTIONS
 The Survivor Function S(t): is the probability of surviving
past time t and is defined as:
S (t )  1  F (t )  Pr(T  t )
 The Hazard Function h(t): Consider the following function:
h(t )  lim
Pr(t  T  t  h) T  t
h 0
h
where the numerator is the conditional probability of leaving the initial
state in the (time) interval {t, t+h}, given survival up to time t.
 The hazard function is the ratio of the density function to the
survivor function for a random variable:
h(t ) 
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f (t )
f (t )

1  F (t ) S (t )
SOME PROBLEMS ASSOCIATED WITH SA
 1. Censoring: A frequently encountered problem in SA is
that the data are often censored.
 2. Hazard Function With or Without Covariates: We
have to determine if covariates are time-variant or timeinvariant.
 3. Duration Dependence: If the hazard function is not
constant, there is duration dependence.
 4. Unobserved Heterogeneity: No matter how many
covariates we consider, there may be intrinsic heterogeneity
among individuals.
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SOME PROBLEMS ASSOCIATED WITH SA
 There are several parametric models that are used in
duration analysis.
 Each depends on the assumed probability distribution, such
as:
Exponential Distribution
Weibull Distribution
Lognormal Distribution
Loglogistic Distribution
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EXPONENTIAL DISTRIBUTION
 Suppose the hazard rate is constant and is equal to h.
 A constant hazard implies the following CDF and PDF:
F (t )  1  e
 ht
f (t )  F '(t )  he
 ht
 The hazard rate function is a constant, equal to h:
 ht
f (t ) he
h(t ) 
  ht  h
S (t ) e
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WEIBULL DISTRIBUTION
 If h(t) is not constant, we have the situation of duration
dependence—a positive duration dependence if the hazard
rate increases with duration, and a negative duration
dependence if this rate decreases with duration.
 For this distribution, we have:
and
h(t )  t
 1
;  0,   0
S (t )  e
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( ht )
PROPORTIONAL HAZARD MODEL
 Originally proposed by Cox
 The PH model assumes that the hazard rate for the ith individual
can be expressed as:
h(t | X i )  h0 (t )e BXi
where h0(t) is the baseline hazard
 In PH, the ratio of the hazards for any two individuals depends
only on the covariates or regressors but does not depend on t, the
time.
 The hazard rate is proportional to the baseline hazard rate for all
individuals:
h(t | X )
i
h0 (t )
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 e BXi
SALIENT FEATURES OF SOME DURATION MODELS
Probability Distribution
Hazard Function
Survival Function
Exponential
h (t) = h
S (t )  e ht
Weibull
h(t )   t  1
S (t )  e( ht )

  0,   0
Lognormal
f (t )  ( p / t )[ p ln(ht )]
S (t )  [ p ln(ht )]
Loglogistic
 (ht ) 1
h(t ) 
1   t
S (t ) 
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1
1  ( t )